Letter to the Editors
The Mathematical Intelligencer
Counting Ambiguous Meanings
encourages comments about the
---, .
material in this issue. Letters to the editor should be sent to either of the editors-in-chief, Chandler Davis or Marjorie Senechal.
here is a story that a certain pro fessor said A and wrote B, when he meant to say C and should have said D. Let me apologize: I have nearly provided an example of the fable. In my article "Pregroups and natural language processing" (The Mathemati cal Intelligencer 28 (2006), no. 2 , 4 1--48), I pointed out that police police police police police has two interpreta tions as a grammatical sentence:
I suggested that police repeated 2 n + 3 times can be parsed as a gram matical sentence in n! different ways. What I had meant to write was that po lice repeated 2 n + 1 times (n 2: 1 ) can he parsed in (n + 1)! different ways. When I posed this problem to a group of bright students, Telyn Kusalik con vinced me that I had counted the same interpretation more than once and that 1 the correct answer is -the (n + 1 )st Catalan number.
police control police [whom] police control
Department of Mathematics and Statistics
J
(1)
(2n) n+l n ,
J. Lambek
McGill University
and
Montreal, H3C 2K6
police [whom] police control control police
Canada
(2)
e-mail:
[email protected]
A Department Head Departs Ranjy Bore has ailed away. His like we'll see no more; For he could add up two and two And always come to four.
You And And And
may well ask if he can add, do it out of ight? can he really multiply get the answer right?
And he could multiply by three, While others merely double, And seem to get the answer right With very little trouble.
My god, he raises things to power· And exponentiates; And he will work for hours and hours With hi •her potentates.
Ranjy Bore has sailed away; He's clone just as he should. We saw him little it is true, But now he's gone for good.
And even though a smiling face Is not his stock in trade, The question is, it really is, The stuff of which he's made.
He used to smile and smile again To everyone he saw, But d1en he'd go away for days, Or weeks, or sometimes more.
For we all know what matter Is whether he will do The job he's paid for, and we do Believe he knows that too.
Ranjy Bore has sailed away, And let the headship go; And someone else has risen From those of us below.
Ranjy Bore has sailed away. A breeze has blown him off, And we're so happy now because We'd really had enough. Wino Weritas
4
THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+Business Media, Inc.
Cou nti ng G rou ps: G n us/ Moas1 and othe r Exotica JOHN H. CONWAY, HEIKO DIETRICH, AND E. A. O'BRIEN
1._
l
ow many distinct abstract groups have a given finite order n? We shall call this number the group number of n, and denote it by gnu(n) . Given the long history of group constructions, a study initiated by Cayley [4] in 1854, it is perhaps surprising that only in this decade has a sizeable table of group numbers become available. The table in the appendix, adapted and slightly extended from that which appeared in [2], tabulates gnu(n) for 0 < n < 2048. The next value, gnu(2048), is still not pre cisely known, but it strictly exceeds 1 7742741 16992170, which is the exact number of groups of order 2048 that have exponent-2 class 2, and can confidently be expected to agree with that number in its first 3 digits. In this article we study some properties of the gnu func tion. We also introduce and study a new and related func tion: moa(n) is the smallest of the numbers m for which gnu(m) = n, provided any exist. (The name abbreviates minimal order attaining a given group number, and the name also honours the country in which this paper was written.) We refer the reader to the survey [2] for a detailed ac count of the history of the problem. Relying on this, we do not provide extensive references to the various contribu tions, usually citing only those that are immediately rele vant or recent. After this article was written, we learned that the recent book [3, Chapter 2 1 ] provides a more scholarly discussion of the group number function.
to groups; so for example a square-free group is one of square-free order. We now display gnu(n) for n at most 1 00 according to the multiprimality m of n, together with what we call the estimate, which is the mth Bell number, defined as the num ber of equivalence relations on a set of m objects.
The Gnu Function and Multiprimality
6. Sextiprimes (estimate 203):
I
The first thing that influences gnu(n) is the number of primes (counting repetitions) of which n is the product. This well-known function, O(n), which does not seem hith erto to have received a standard name, we call the multi primality of n, and we describe n as prime, biprime, triprime, etc . , according as its multiprimality is 1 , 2 , 3, etc. We let adjectives that usually apply to numbers also apply
6
THE MATHEMATICAL INTELLIGENCER © 2008 Splinger Science+Business Media. Inc.
1 . Primes (estimate 1 ) : gnu
=
1 for
2, 3, 5, 7, 1 1 , 1 3 , 1 7 , 1 9 , 2 3 , 29, 3 1 , 37, 4 1 , 43, 47, 53, 59, 6 1 , 67, 7 1 , 73, 79, 83, 89, 97. 2. Biprimes (estimate 2): gnu = 2 for 4 , 6, 9, 1 0, 1 4 , 2 1 , 2 2 , 2 5 , 26, 34, 38, 39, 46, 49, 55, 57, 58, 62, 74, 82, 86, 93, 94 but gnu
=
1 for
1 5 , 33, 35, 5 1 , 65, 69, 77, 85, 87, 9 1 , 95. 3. Triprimes (estimate 5): 8 12 18 20 2 7
5
5
5
5
5
2830 42 44 45 50 52 63 66 6 4
46
4
2
5
5
4
4
8 70 7 576 5
43
7 8 92 98 99
46
4
5
2
4 . Quadruprimes (estimate 1 5) : 16 14
24 15
36 14
40 14
54 15
56 13
60 13
81 15
84 15
88 12
90 10
1 00 16
5 . Quinqueprimes (estimate 52): 32 51
48 52
64 267
72 50
80 52
96 231
These values show that when n and its multiprimality m are both small, gnu(n) does not differ much from the esti mate. We think this remarkable approximation deserves an explanation, even though for larger numbers it ceases to hold.
Many other oddities will be noticed among the values in the table in the Appendix. For example, is it merely a coincidence that there are three numbers n with gnu(n) 1 387, and a fourth with gnu(n) 1 388?
=
=
If n is a power up to the fourth of some prime, then in deed gnu( n) equals its estimate, except that gnu(l6) 14 rather than 1 5 . B u t from the fifth power onward, the situa tion is different. We summarize the known results. In addi tion to those cited in [2] , the new sources are [ 1 4] and [151.
=
THEOREM 3.1 1. There are 51
groups of order 67 of order 35,
25'
and 61 + 2p+ 2 gcdCp-1 , 3) + gcdCp - 1 , 4)
of order p5 for prime p 2: There are 267
5.
groups of order of order 36 ,
26 ,
n
2:8,
=
gnu(256) 56092, gnu( 5 1 2 ) = 1 0494 2 1 3 , gnu(1024) = 49487365422.
and 3J} + 39p + 344 + 24 gcdCp - 1 , 3) + 1 1 gcdCp - 1 , 4) + 2 gcdCp - 1, 5)
of order p6 for prime p 2: There are
The greatest values of gnu are those just mentioned, namely its values at powers of 2, which dominate the others in a surprising way. For example, if a group is selected at ran dom from all the groups of order <2048, the odds are more than 1 00 to 1 that it will have order 1 024. The 423 1 7 1 1 9 1 groups of all the other orders < 2048 are swamped by the 49487365422 of order 1024. The vast majority, 48803495722, of the latter have exponent-2 class 2 , and it is the fact that the number of groups of this special type can be counted without explicit construction that has enabled gnu(1024) to be precisely calculated; an algorithm to perform this cal culation is described in (7]. The asymptotic estimates of Higman [ 1 0] and Sims [ 1 7] rfl13 show that the number of groups of order p» is p2rz3!27+ O( l. M. F. Newman ( private communication) and C. Seeley have shown that the exponent 8/3 can be reduced to 5/2. Py ber [ 1 6] has shown that 7 gnu( n):::::; n<212 +o0llp..(nl2,
504
3.
For
Great Gnus
Powerful Gnus
2.
4.
as p,(n), the largest exponent in the prime-power factor ization of n, tends to infinity.
Powerless Gnus
5. 7
2328 groups of order 2 , 7 9310 groups of order 3 , 7 34297 groups of order 5 ,
and 3P"' + 1 2 p4 + 44p + 170p + 707p + 2455 + (4j.l + 44 p + 291 ) gcdCp - 1 , 3) + Cf} + 19p + 1 35) gcdCp - 1 , 4) + (3p + 31) gcdCp - 1, 5) + 4 gcdCp - 1, 7) + 5 gcdCp - 1 , 8) + gcdCp - 1 , 9)
of order p7 for prime p 2: 7.
This title refers to the group numbers for square-free or ders. The results for prime and biprime numbers are pre sented in every beginning course on group theory. There is a unique group of each prime order (agreeing with the estimate of 1 in that case); and either one or two groups of order pq, the number being two (which is the estimate) if and only if either p = q, or one of p and q is congruent to 1 modulo the other. The first group is cyclic, and the second, supposing that q = 1 mod p, has presentation
lA,B
II= AP
=
Efl,
A-1BA = Bkl
where k is some number having order p modulo q. In such a case, we say that A acts on B, and call A the
ac-
JOHN H. CONWAY's interests range through
HEIKO DIETRICH is a doctoral student under
group theory, number theory, and geometry.
the supervision of Bettina Eick He works primar
Nonmathematicians know him best as the in
ily on algorithmic aspects of group theory, and he
ventor ofthe Game of Life. Among many other
is the author of a GAP package to construct
prizes, he has received the Berwick and P61ya Prizes of the London Mathematical Society and the Leroy P. Steele Prize for exposrtion from the American Mathematical Society.
of cube-free order. L-....::.o::�..mL.....J lnstrtute of Computational Mathematics
Technische Universrt:at Braunschweig 380 I 6 Braunschweig
Department of Mathematics
Germany
Princeton, NJ 08544
e-mail:
[email protected]
USA e-mail:
[email protected]
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
7
tor, and B the reactor. Since replacing A by Ai (0 < j < p) re places k by ki, which is another number having order p (mod ulo q), all the possible choices for k yield the same group. Holder [ 1 1 ] generalized these results to all groups of square-free order. Namely, every such group has a pre sentation with a generator Ap for each prime divisor p of n, whereas the generators for distinct primes p and q ei ther commute or one (say Ap) acts on the other (Aq) by re placing it by its kth power for some k not congruent to 1 mod q. We shall say that "P (or Ap) acts as k on q (or Aq)." There are some restrictions on these presentations. The same generator cannot be both an actor and a reactor. More over, p can only act on q if q = 1 mod p, which condition we therefore call an opportunity (for action). Moreover, if p does act as k on q, then k must be a number that has order p (modulo q), and as before we can replace k by k-i (0 < j < p) by replacing Aq by A�. This entails that if Ap acts on several generators, then the value of k for one of them may be freely chosen, but then the rest are deter mined (by the given group). The groups of square-free order n pqr . . . may there fore be specified by graphs having a node for each of the primes p, q, r, . . . , there being an arrow (directed edge) marked k from node p to node q just when Ap acts as k on Aq. If we replace Ap by its jth power, then all the marks k on the arrows from node p are replaced by their jth pow ers (modulo p), and so if there is only one arrow from node p, the mark on it is unimportant and may be omitted. The number of possibilities for such graphs, taking the previously described restrictions and equivalences into ac count, is therefore the number of groups of the given square-free order. It is completely determined by specify ing the opportunities for action among the primes p, q, r, . . . , namely which of them are congruent to 1 modulo which others. Holder [ 1 1 ] summarized the results in an elegant (if some what opaque) formula. ==
THEOREM 5.1 gnu(n)
( ) � II epo - pp p,e == d n pjd
_
1)
Cp - 1)
where de== n and pld and opp(p, e) is the number of op portunities for p to act on the primes dividing e. Murty and Murty [ 1 2] generalized this to count all the groups of any prescribed order whose Sylow subgroups are all cyclic. Their result is the following.
THEOREM 5. 2
The number of groups �forder n, all of whose Sylow subgroups are cyclic, is: '\' II � (J!'pp(pi, e) [f'PP
(
where
gcd(d,
)
_
_
a is the largest power �f p dividing d, de e)== 1 , and p l d, and now
p>pp(pi,el
==
==
n,
II gcdC p� q - 1 ) q\m
where p and q denote primes and j is a positive integer. All Triprime Groups
It is well-known that gnu C p3) 5, the estimate. For square free triprimes pqr, where we may assume p < q < r, the re sults are: ==
No opportunity Just one opportunity Two consecutive opportunities Two inward opportunities Two outward opportunities Three opportunities
p p
1 2 3 4 + 2 + 4
group groups groups groups groups groups
To interpret and explain these results, it suffices to draw the possible opportunity graphs, having an arrow from p to q when p has the opportunity to act on q, and then to in dicate the different ways for some of these opportunities to be seized. Such graphs also appear in [ 1 3] . The oppor tunity graphs for groups of order pqr appear in Figure 1 . For the remaining triprimes n == pcf, Holder found that gnu(n) is: if we have none of plq- 1 , Plq + 1 , qlp- 1 ; if both p, q> 2 and p q + 1 ; i f p> 3 , qlp- 1 , but not q2IP- 1 ; 2 o r q2IP- 1 ; 3, q if p 2 < q or p
2 3 4 5
l
==
==
==
and finally
EAMONN A. O'BRIEN received a BSc from the National Univers� of Ireland, Galway, in 1 98 3, and a PhD from the Australian National Univer s� in 1 988. After appointments in the USA Australia, and Germany, he joined the Univers� of Auckland in 1 997. His research is primarily in algorithmic and computational aspects of group theory. Department of Mathematics Univers� of Auckland Auckland New Zealand e-mail:
[email protected]
8
THE MATHEMATICAL INTELLIGENCER
(p +
9)/2
if
p
is odd and divides
q-
1.
The Baby Gnus It is not hard to prove the following theorems, which to gether find all the numbers n for which gnu(n) ::5 4. We list first the form of n, where letters represent dis tinct primes and any repeated prime is explicitly shown. Thus ''form pqr . . . " means square-free. Recall that if primes p and q both divide n, we call the condition j),(q- 1) an opportunity, in which p is the actor and q the reactor, if p2 also divides n, then p2ICq - 1) is a double-opportunity and qiCp2 - 1 ) a half-opportunity.
THEOREM 7.1 gnu(n)
==
1 if and
number with no opportunity.
only ifn is a squarefree
(i) No opportunities:
•
•
•
•
1 •
Table 1. Occurrences of small values j of gnu(n) for n :s 2048
group
656
•
•
•
•
•
(iii) Two consecutive opportunities:
•
•
3
groups
/.
•
. \
•
•
•
•
•
. \
L
•
•
•
•
/.
•
L
•
p �l
Figure I.
•
•
•
232
5
102
6
62
7
4
8
14
9
12
10
39
L
. \
/.
p�l
Opportunity graphs for groups of order
= 3
((and only if n has:
=
((and only if n bas:
form pqr , , , and just two consecutive opportunities; or form p2qr . with one ha!fopportunity hut no opportu ni�y. 4
form pqr ... with just two opportunities, that are either inward, disjoint, or both have 2 as the actor; or form p2qr . with no double or half-opportunity, and just one opportunity, whose reactor is distinct from p; or form p2q2r . . with no opportunity or half-opportunity.
These results make it easy to compute the number of n below any reasonable bound for which gnu( n) is at most 4. The densities of 1, 2, 3, 4 for 1 :s n :s 10H are, respec tively, 0.285, 0.132, 0.003, and 0.093. Table 1 records the number of occurrences of 1 :5 j :5 10 for n :5 2048.
(vi) Three opportunities: p + 4 groups
•
11
4
THEOREM 7.4 gnu( n)
(v) Two outward opportunities: p + 2 groups •
3
THEOREM 7. 3 gnu( n)
('iv) Two inward opportunities: 4 groups
•
393
2
( ii) One opportunity: 2 groups •
Number
j
Hunting Gnus and Moas
pqr,
We now formulate the
gnu-bunting conjecture.
CONJECTURE 8.1 Every positive integer This condition is equivalent to the coprimality of its Euler function r:f>( n),
THEOREM 7.2 gnu(n) • •
=
2
n with
is
a group number.
For the history of this conjecture see [3, §21.6]. It seems likely that every number is a value of gnu(n) for some square-free n, as has been conjectured by R. Keith Dennis [6]. He has established this for all numbers up to 10000000. What is the next term in the sequence 1, 4, 75, 28, 8, 42? This is, of course, the moa sequence: There is 1 group
if and only ((n has,'
form pqr , , with just one opportuni�V,' or form p2qr , , , with no opportunity or ha{fopportunity, Table 2. Values and guesses for the moa function +0
+1
0
+2
+3
+4
+5
+6
+7
+8
+9
4
75
28
8
42
375
510
308
10
90
140
88
56
16
24
100
675
156
1029
20
820
1875
6321
294
546
2450
2550
1210
2156
1380
30
270
?11774
630
?163293
450
616
612
180
1372
264
40
280
420
176
112
392
108
252
120
2730
300
50
72
32
48
656
272
162
500
168
4650
6875
60
378
312
702
3630
1596
?59150
588
243
882
1215
?26010
2420
2964
1092
?51772
3612
70
4100
?9139263
3660
1638
?16807
2394
?35322
?18620
?34914
2028
4140
?23460
?28308
?85484
6930
6498
4950
1188
80
6050
6820
?126945
90
?13300
?12324
?24990
100
3822
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
9
of order 1, 2 of order 4, 3 of order 75, 4 of order 28, and so on. The required answer is therefore 375, since that is the smallest order for which there are exactly 7 groups. It is distinctly harder to hunt moas than gnus, since to show that moa(n) = m one must not only verify that there are exactly m groups of order n, but show that no order smaller than n has exactly this number of groups. Table 2 lists the moa values we know up to moa(lOO) = 3822, to gether with our guesses about the rest. The known answers, say moa(g) = m, when not less than 2048, were found by computing, for each number n < m, either the exact value of gnu(n) or showing that gnu(n) exceeds g. We used four methods to find (lower bounds for) gnu(n).
d of some number n, then, by the inclusion-exclusion method, the number
1. If gnu(d) is known for each divisor
of indecomposable groups for each such divisor can also be computed, and one lower bound for gnu(n) is ob tained by multiplying such numbers over various fac torizations of n. (Recall that a decomposable group is one that can be expressed as a nontrivial direct product.) 2. The formula of [ 1 2] for the number of groups of a given order that have cyclic Sylow subgroups can be supple mented by estimates for those that don't; thence we ob tain another useful bound. 3. If the number is cube-free, we use the group construc tion package CUBEFREE of Dietrich and Eick [5] to count explicitly the number of groups of this order. 4. Otherwise, the groups of order n can be explicitly enu merated by the GRPCONST package of Besche and Eick [ 1 ] until all groups (or enough to establish a sufficient lower bound) are found. The last two options can be expensive, and they are only practical for limited ranges: for those cube-free num bers n E (2048, . . . , 163293} that gave lower bounds at most 30, we successfully calculated gnu(n) using the CuBE FREE package in GAP [8]. The guesses in Table 2 are prefixed by ? For 83, the guess is 75; for 73, it is the smallest square-free integer hav ing this value of gnu; all our other guesses are the small est cube-free possibilities. "
".
Good Gnus and Bad Gnus Is there any hope of proving the gnu-hunting conjecture? We address Dennis's stronger form that every number arises as gnu(n) for some square-free n. Holder's formula for the group number of a square-free number pqr . . . is a sum of products of powers of the primes p, q, r, . . . , of which the formula is composed. For example, as we saw in the "All Triprime Groups" sec tion, when p < q < r, the value of gnu(pqr) is 1, 2 , 3, 4 , p + 2 , o r p + 4 , according t o the number and nature o f the opportunities among p, q, r. A reason one can still hope to prove the conjecture is that, as well as the bad forms (such as p + 2 and p + 4 as previously mentioned) that involve unknown primes, there are good ones (such as 1, 2 , 3, 4 above) that don't. It seems likely that the latter type of forms already suffices to prove the conjecture.
10
THE MATHEMATICAL INTELLIGENCER
Let n = pqr . . . be the prime factorization of a square free number, for which we are given only the opportuni ties for action among the primes p, q, r, . . . . We say that gnu is good at pqr . . . , or loosely that the expression gnu(pqr . . . ) is a good gnu, if its value is a constant that does not depend on the particular primes in volved. The bad gnus are those expressions gnu(pqr . . . ) for which some prime, p say, has a number k > 1 of op portunities to act, since then the number of cases in which it seizes those opportunities involves the factor (p- 1)k-l. The opportunity graph for a good gnu, having at most one arrow leading from any node, must therefore be a for est based on the primes involved-we call it the primeval forest-consisting of rooted trees (the roots corresponding to the primes that have no opportunities for action). We define the uprooted forest to be the smaller forest obtained from the primeval one by removing the roots of all these trees, along with the edges that led to them, and then also removing the arrowheads on any edges that remain. The following surprising result is remarkably easy to prove.
THEOREM
9.1 The value of a good gnu expression . .. ) depends only on the shape of the uprooted for est. In particular, it is independent of the number and arrange ment of the roots, and of the directions of the arrows on any of the remaining edges.
gnu(pqr
PROOF. Since each prime has at most one opportunity for action, the groups correspond in a one-to-one manner to the possible sets of acting primes, and these sets are just the in dependent subsets of the uprooted forest, whose definition doesn't need the directions of its edges. (An independent sub set of the vertices of a graph is a set that does not contain both endpoints of any edge.) 0 For gnu-hunters, this theorem is an unfortunate one, since it forces many different numbers to yield just one value of gnu. It implies, for instance, that the four primeval trees of Figure 2 all yield the same gnu value of 1 4 . The theorem has an interesting corollary.
COROLLARY 9. 2 The set of values of good gnus (of square free numbers) is closed under multiplication. PROOF. The number of independent sets in a forest is the product of those numbers for its component trees. Any forest can be realized since Dirichlet's Theorem [9, p. 13] implies that any shape of tree can be realized by a square-free num ber with arbitrarily large prime divisors. 0
Figure 2.
Primeval forests.
Now the number of independent sets for a tree with e edges varies in the range from . fe+3 (the (e + 3 ) rd Fibonacci number) to 2e + 1. The numbers in the first few cases are the following:
-
0 edges:
edge: edges: edges: edges: edges: 6 edges: 7 edges:
1 2 3 4 5
8 edges:
720
,.....
840 1640
384
20169
5 8, 9 13, 14, 17
128
2328
960
11394
864
4725
21, 22, 23, 24, 26, 33 34, 35, 36, 37, 38, 40, 41, 43, 44, 50, 65
1344
55, 57-62, 64-66, 68-70, 76, 77, 80, 83, 84, 98, 129 89, 92-102, 104-110, 112-114, 116, 118, 120-122, 124, 126, 128, 133, 134, 145, 148,
1248
Galloping Gnus Recall that a number n is traditionally called pe�fect, abun dant, or deficient accordingly, as the sum of its proper di visors equals, exceeds, or falls short of n. We mirror this by calling a number n group-pe�fect, group-abundant, or group-deficient according to gnu( n) = n, gnu( n) > n, or gnu( n) < n. Murty and Murty [12] prove that gnu( n) ::S r:f;( n) for square-free n, and so all square-free numbers greater than 1 are group-deficient. We do not know if there is any group perfect number other than 1, but there are plenty of group abundant ones, for instance, all numbers of the form 1024n below 49487365422. However, it seems that the proportion of group-abundant numbers gradually falls to zero. Again, we do not know if there is any group-amicable pair of numbers (that is, m > n with gnu( m) = n and gnu( n) = m). A negative answer would follow from the p,al loping gnus conjecture which we now formulate:
CONJECTURE 10.1 For every positive integer n, the se quence gnu(n)
49487365422
2 3
These results imply that any product of the displayed numbers is the value of gnu( n) for infinitely many square free numbers n. However, the observant reader will notice that the numbers 7, 11, 19, 29, 31, . . . are missing. These do not arise as "good gnus" of any square-free numbers. This might not matter. Dennis (see [3, §21.6]) lists 508 numbers that he conjectures are the only ones missing from the continued form of the previous table, and he has ver ified that each of these is the (bad) gnu of some square free number. All that remains is to prove that this list is complete! We have already mentioned that the first 10 million in tegers are values of gnu; it is not hard to prove that every number whose prime factors are all smaller than 140 is of the form gnu( n) for infinitely many numbers n.
�
1280
1024
320
149, 152, 163, 164, 194, 257
n
Table 3. Verifying the galloping gnu conjecture 672
�
gnu 2 (n)
consists ultimately �( 1s.
=
gnu(gnu(n))
�
u3( n) �
gn
To check this for all starting numbers n < 2048, it suf fices to follow the 47 group-abundant numbers among them. We do this in Table 3 in descending order of gnu2,
1440
,.....
,..... ,..... ,..... ,.....
,..... ,.....
11720 5958 1460
256
56092
1728
47937
512
10494213
1536
408641062
1664
21507
1280
1116461
1116461
1
240
208
186
6
68 67 64
,..... ,.....
5
,.....
267
,.....
13
60
51 2
51 49 16 15 11
,.....
6
,.....
4
5
,..... ,..... ,.....
2 14
2
2
,.....
2
2
except that we have omitted two more numbers 48, 448 with gnu2 = 5; also 160, 432, 832, 1408, 1458, 1920, 2016 with gnu2 4; also 96, 288, 1088, 1296 with gnu 2 2; and finally 17 further numbers with gnu2 = 1. In fact, every num ber less than 2048 reaches 1 after at most 5 steps.
=
=
ACKNOWLEDGMENTS
John H. Conway was supported by a Maclaurin Fellowship from the New Zealand Institute of Mathematics and its Ap plications. All three authors were partially supported by the Marsden Fund of New Zealand via grant UOA412. REFERENCES
[ 1 ] H ans Ulrich Besche and Bettina Eick, "Construction of finite groups", J. Symbolic Comput., 27 (1 999), 387-404.
[2] H ans Ulrich Besche, Bettina Eick, and E. A O'Brien, "A millennium project: constructing small groups , " lnternat. J. Algebra Comput. , 1 2 (2002), 623-644. [3] Simon R. Blackburn, Peter M. Neumann, and Geetha Venkatara m an, Enumeration of Finite Groups, Cambridge University Press, 2007. [4] A. Cayley, "On the theory of groups, as depending on the sym bolic equation en
=
1 ," Phi/as. Mag. (4), 7 (1 854), 40-47.
[5] Heiko Dietrich and Bettina Eick, " Groups of cubefree order, " J. Al gebra, 292 (2005), 1 22-1 37.
[6] R. Keith Dennis, "The number of groups of order n," in prepara tion.
[7] Bettina Eick and E. A O'Brien, "Enumerating p-groups , " J. Aus tral. Math. Soc. Ser. A, 67 (1 999), 1 91 -205.
[8] The GAP Group, GAP-Groups, Algorithms, and Programming, Version 4. 4. 1 0; 2007. (http://www. gap-system.org). [9] G. H. H ardy and E. M. Wright, An Introduction to the Theory of Numbers. Fourth edition. Oxford University Press, Oxford, 1 963.
[1 0 ] Graham Higman, "Enumerating p-groups. 1: inequalities , " Proc. London Math. Soc. (3), 1 0 (1960), 2 4-30. [1 1] Otto H older, "Die Gruppen der Ordnungen p3, pq2, pqr, p4 , " Math. Ann., 43 (1 893), 301 -41 2 .
..
© 2008 Springer Science+ Business Media, Inc
Volume 30, Number 2 , 2008
11
[1 2] M. Ram Murty and V. Kumar Murty, "On groups of square-free or
[15] E. A. O'Brien and M. R. Vaughan-Lee, "The groups of order p7 for odd prime p," J. Algebra 292 (2005), 243-258.
der," Math. Ann. 267 (1984), 299-309.
[1 3] P. Erdos, M. Ram Murty, and M. V. Murty, "On the enumeration
[16] L. Pyber, " Enumerating finite groups of given order." Ann. of Math.
[1 4] M. F. Newman, E. A. O'Brien, and M. R. Vaughan-Lee, " Groups
[17] Charles C. Sims, " Enumerating p-groups," Proc. London Math.
of finite groups," J. Number Theory 25 (1987), 360-378.
(2) 1 37 (1993), 203-220.
and nilpotent Lie rings whose order is the sixth power of a prime," J.
Soc. (3), 15 (1965), 151-1 66.
Algebra, 278 (2004), 383-401 .
Appendix The number of groups for each order <2048 +0
+1
+8
+9
2
2
5
2
5
2
14
5
2
15
+2
+3
0 10
2
20
5
2
+4
+5
2
+6
2
4
51
40
14
6
4
2
2
50
5
5
15
2
13
60
13
2
4
50
80
52
90 100
2
4
15
2
1
10
4
2
16
4
14
15
3
4 2
2
1
52
2
2
2 5
4
267 2
5
14
30
70
+7
4
6
2
12
231
5
2
2
45
2
5
4
2
110
6
2
43
6
120
47
2
2
4
5
16
2328
130
4
10
2
5
15
4
140
11
2
197
2
6
150
13
12
4
2
18
2
238
55
5
2
2
57
170
4
4
4
2
42
2
180
37
4
6
4
190
4
1543
2
2
12
10
200
52
12
2
210
12
5
2
220
15
6
197
230
4
240
208
250
15
260
15
270 280
5
2
2
2
2
2
12
16
14
2
2
2
15 2
4
2
56092
6
39
4
4
4
10
2
4
2
4
1045
2
46
2
2
1
30
54
5
2
40
4
4
5
2
14
5
2
290
4
2
5
23
300
49
2
2
42
2
10
g
310
6
61
2
4
4
4
320
1640
4
176
2
330
12
4
5
2
18
5
12
340
15 10
14
195
360
162
2
2
370
4
4 3
4
7 2
11
2
20169
12
44
2
400
221
6
5
410
6
4
420
41
2
430
4
775
440
51
4
13
16
14
5
5 12 2
2 42
12
60
2
4
2
6
46 2
2
5
30
5
4
4
4
4
5
6
18
2
1396
2
2
15
6
235
10 2
2
2 2
15
390
2 228
11
380
2
2
12
67
350
13
51
177
4
2
4
5
2
6
2
2
5
160
2
2
2
(continued)
12
THE MATHEMATICAL INTELLIGENCER
Appendix The number of groups for each order <2048 (continued) +0
+1
+2
+3
+4
450
34
5
2
2
460
11
12
------
51
4
6
2
470
4
480
1213
2
2
12
490
10
12
1
4
500
56
2
202
510
8
10494213
520
49
10
4
530
4
9
540
119
2
2
12
2
15
2
+5
+6
+8
+9
54
+7
2
5
2
55
11
2
2
261
1
14
4
42
2
4
2
6
6
4
15
4
2
170
4
12
2
2
246
24
5
4
2 6
550
16
39
2
560
180
2
10
570
12
11
4
2
8681
5
580
15
6
15
4
2
66
4
7
4
2
49
590
4
51
205
6
610
6
36
2
2
35
620
15
2
260
15
2
2
630
32
12
12
2
640
21541
4
650
10
660
40
2
670
4
1280
680
53
4
690
8
5
700
36
710
6
720
840
5
730
4
740
15
750
39
760
39
6
770
12
5
780
53
4
790
4
2
800
1211
2
810
113
16
820
20
4
830
4
840
186
850
10
860
11
870
8
880
221
890
4
900
150
5
5
4
5
30
2
4 2
2 9
2
4
6
2
53
12 2
4
5
17
4
5
4 4 4
51
15
2
44
2
2
10
2
42
12
2
4
2
5
2
13
40
18
2
6
195
2
10
4
54
2
2
11
42
4
2
189
2
4
2
2
1090235
16
4
15
2 2
5
2
4
24
15
4
14
4
205
2
5
4
30
9
2
4
2
10
10
4
5
5
18
4725
2 14
6
4
4
51
12
1
12
2
3
15
4
4
1
19349
15
2 4
7
61
2
1 2
5
30
222
2
2
34
2
2
930
18
2
5
2
2
222
4
940
11
6
42
13
4
15
10
2
970
4
980
34
990
30
42 4
2
900 5
2
2 196
5
2
6
2
4
2
2
2
6
8
10
11
2
2
39
11394
2
9
920
960
504
2
910
950
2
2
2
68
4
13
2
2
4
1630
2
5
172
12
2
757
16
2
15
5
1387
137
3
6
15
5
2
4 195
18
2
2
2
12
62
2
2
16
4 2
600
4
2
4
235 2
4
2
12
42
2
51
6
2
46
4
6
10
2
11 2
15
(continued)
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
13
Appendix The number of groups for each order <2048 (continued) +0 1000
+1
199
+2
1010
6
1020
37
4
1030
4
54
2
13
1040
231
2
1050
40
4 10
1060
15
1070
4
4
42
1080
583
2
2
1090
4
1100
51 12
12
1120
1092
8
4
1140
41
1150
10
1160
49
1170
32
1180
11
1190
8
1200 1210
2
51
6
77
1110
1130
+3
4
2
2
4
+5
+6
4
2
2
5
+8
2
2
12
2
4
66
2
4
2
11
36
2
2
4
2
1028
5
35
2
4
12
5
2
12
4
2
5
10
2
6
1681
2
2
15
16
4
2
46 14
5
2
254
42
2
39
4
2 18
4
5
2
2
4
235
2
2 11
16
6
4
53
279
4
14
6
1040
2
13
2
16
12
27
12
2
69
1387
16
164
4
1220
20
1230
12
153
1240
51
30
2
4 2
2
1460 4
12
2
14
42
3
6
5
44
10
3
11
10
2
10
2
11
2
3609
4
2
2
18
2
12
55
2
1270
4
1280
1116461
1290
12
1300
50
24
12
1310
6
1320
181
2
2
4
131
5 2
2
9
18
244 51
4
4
2
5
4
15 1
1250
5
2
2
4
1260
4
2
2
99
2 11
2
39
14
2
6
5 4
2
4
19
4
4
170
+9
954
23
157877
3
+7
49487365422
5
4 2
13
2
+4
2
5
35 2
2 42
1330
8
5
61
4
12
6
1340
11
2
4
11720
1
2
5
1350
112
52
2
2
12
4
4
1360
245
4
9
5
2
211
2
1370
4
38
6
15
195
1380
29
2
14
1390
4
198
4
8
5
1400
153
2
227
2
4
2
5
2
4
40
1430
12
5
12
1440
5958
4
1450
10
1460
15
1470
46
1387
6
1480
49
24
11
4
1510
6
1520
178
1530
20
3
4
2
4
2
12
2
6
14
40
2
2
179
1798
4
61
5
2
5
2
36
2
10
2
10
5
41
2
2
4
4
15
889
2
2
4
12
2
15
4
4
12
408641062
2
4
5
4
4
222
2
2
2
4
195
2
13
19324
39
16
174
4 5
1
4
1490
2
6
15
1500
2
4
4
8
1420
2
32
6
53
1410
4
15
2
60
(continued)
14
THE MATHEMATICAL INTELLIGENCER
Appendix The number of groups for each order <2048 (continued) +0
+4
+5
+6
36
4
15
2
2
1550
16
54
24
2
1560
221
4
11
1570
4
1
1580
11
2
1590
8
12
1600
10281
10
1540
+1
+2
10
1610
8 477
2
1630
4
1213
1640
68 38
9
1660
11
6
2
13
2
2
5 4 2
2 2
5
+7
+8
2
4 928
14
4 3
4
4
54 6
5
6
11
4
2
5
72
12
1
2
2
2
137
2
21507
5
10
15
2
4
4
1680
1005
1690
10
235
6
1700
50
309
4
2
39
7
2
11
1710
36
2
42
2
2
5
40
2
1720
39
4
3
2
47937
1730
4
1740
37
1750
30
1760
1285
1770
8
1780
15
34
2
60
5
2
5
30
10
12 2
5 4
2
2
2
13
35
51
16
64
2
2
228
4
2
110
5
4
14
2
4
4
4 5
53
2
4
260
6
12
2
4
12
4
5
4
1800
749
4
2
11
3
30
54
1810
6
15
2
2
9
12
10
1820
35
1264
2
4
1830
18
14
1840
178
6
5
1850
10
4
16
2
1083553
2
1860
56
10
1870
12
1096
1880
39
1890
120
9
1900
36
4
5
12
2
15
4
2
21
6
5 2
4
162
2
2
2
4
4
2
9
6
2
18
4
2
54
4
2
2
36
186
2
1910
6
12
8
241004
5
15
4
10
15
1930
4
34
2
4
167
12
3973
4
4
2
15
1940
15
2
1950
40
235
1960
144
18
11
4
5
2
195
1920
15
13
117
1630
15 2
2
4
2
9
1790
5
5
12
4 2
5
42
1670
2
2
2
56
13
2
5
64
51
+9
46
30
4
697 5
11
1650
2
2
6
1620
2
+3
4
4 2
6
4
2
2
12
2
39
4
6
13 66
2
203
1970
4
1980
120
2
1990
4
39
2
5
4
2000
963
8
10
2
4
12
2
2010
12
4
2
4
2
6538
2
2
2020
20
6
2
46
63
2
88
10
2
2030
12
2040
175
15
15 2
42 2
2
2
1388
11
5
4
5
4
2
12
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
15
[email protected]§j:@hi£11jfii§4fii!'i•i§•ld
Expected Va l ue Road Tri p SAM VANDERVELDE
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
Michael Kleber and Ravi Vakil, Editors
s a result of a questionable de cision to combine a cross-coun try move with a four-week stay at a summer math program, I found my self driving solo from San Francisco to Salt Lake City in a single day during the summer of 2007. To combat the lethargy of the midafternoon stretch, I decided to reconstruct mentally the solution to a classic problem that dates back at least to Laplace, according to Uspensky [1] namely, to determine on average how many numbers, chosen uniformly at random from the interval [0, 1], one must select before their sum exceeds 1. The delightful answer, as we all know, is e. This question has been the subject of renewed interest lately. In [2], c':urgus and Jewett examine the function a(t), defined as the expected number of random se lections from [0, 1] that must be made before the cumulative total exceeds t. They go on to find an explicit formula for a(t) using the theory of delay func tions. It stands to reason that a(t) 2 t, since each selection is equal to _:_2 , on average. Their surprising observation is that
( ( - 2 t)
lim a t)
=
l. 3
One can independently show that the value of a(t) may be expressed in terms of powers of e for integral values of t. We are thus led to statements such as
e6 - Se'i
+ 8e1
- 9e5 2
--
+
2 e2 3
--
Mathematical Entertai nments Editor,
Department of Mathematics, Bldg. 380, Stanford , CA 94305-2125 , USA e-mail:
[email protected]
(1 - qn)- (1 - qn-!)
=
qn-! - qn,
valid for n 2: 1. The expected value we seek is 'YO
L
rz=1 =
n(qn-1- qJ
"'
00
n=l
n=l
00
L qn-! + ,Lcn-1)qn-1-nL nqn =l
The fact that all sums converge ab solutely will be clear once we obtain a closed form expression for qn. The tick lish part is finding the formula. The region R, in the n-cube [1, e]" consisting of points (x1, ... , xJ, the product of whose coordinates is at most e, is described by
e - 120
--
=
Ravi Vakil, Stanford Un iversity,
e
Readers may test their intuition by pre dicting whether this quantity is greater than or less than e. The assault proceeds in predictable fashion: For n 2: 1, let q11 be the prob ability that a product of n numbers cho sen from [ 1 , e] is not greater than e. (It will be convenient to define q0 = 1 as well.) Then the probability that the product exceeds e for the first time at the nth selection is
=
/--->00
Please send a l l submissions to the
e- 1 ee�J + --
12I3'
by considering a(6), for example. The two quantities agree to six digits past the decimal point. Our purpose here is to pursue a thought which struck me as yet another low mountain range passed by to the south somewhere in Nevada. We wish to know how many numbers, chosen uni formly at random from the interval [1, eJ, one must select before their product ex ceeds e. To cut the suspense, the answer is the somewhat unlikely expression
1 :S: Xn :S:
e
Xn - 1
-----
X1
·
·
We may then compute qn via
qn=
1
(e- 1)"
J
R,
dxn
·
·
·
dx1.
We focus on the integral by defining en= IR dxn ... dx1. The first six val · ues are 1 ' 1 ' le- 1 -le + 1 ie2 3 ' 8 1, and -_ll_e + 1, found by electronic 50 means. Evaluating these integrals di-
�
-
'
© 2008 Springer Science+ Business Med1a, Inc., Vo!ume 30, Number 2, 2008
17
n =e(-1) (bn+1- bn )
rectly becomes an increasingly arduous affair as n grows, so it comes as a pleas ant surprise to discover that their value may be ascertained with hardly any ef fort. Indeed, we find that
(
(-1)"
-
= e( - 1)"
-n!
e
tween these two quantities? In the spirit of the result mentioned previously, one might hope for a limiting ratio, such as
)
I
lim {3( t)· e e-I = C, f--->00
n.I ' as desired. This provides the sought after expression for qm namely n ( -1) (l- bne) n (e- l)
by performing the easy integration with respect to Xn+ 1. It follows that
l l
Making the change of variables Yk = In Xk so that dyk =__1_ dxk , we find that xk
On+1 + 0 = "
f
R' n
e dy1 · · · dyn.
It is clear that the region R� consists of those points (y1, . . . , Yn ) satisfying Yk :=:::: 0 and Y1 + · · · + Yn � 1, that is, R;, is a right unit n-simplex. It is well known that the volume of this region is __1_, so we have shown that
for n:::::: 1 . Clearly qn � (e- 1)-", which justifies the manipulations of the infinite series above. The final step of summing the qn presents a very satisfying exercise in volving geometric and exponential se ries. To begin, X
I
n�o
q
1 + " =
e On+1 + On=-.
n!
It is conceivable that one might stumble upon this approach while nav igating northern Nevada by car. In fact, I initially headed down another road by asking how many numbers from the interval [1, 2] would be required to ob tain a product that exceeded 2. Several mental integrations later it became clear that I had taken a wrong turn when the accumulation of ln2's became over whelming. By the time the question had been properly formulated, the Great Salt Flats beckoned, and the in teresting task of answering the ques tion was postponed until a later time. Resuming the argument, we now find an expression for On. For n :=:::: 1 let hn be the nth partial sum of the usual series for e-1, so that 1 1 hn =1--+--· 2 1
+
.. c-
n
n 1 -
--1
(n- 1)!
We claim that On =(-1)"(1- bne). The quantities agree for n = 1, and for n :=:::: 2 we compute On
18
+
n On+1 =( -1) (l- bne) + (-1)"+1(1- bn+Ie)
THE MATHEMATICAL INTELLIGENCER
IF!
_I
n�l
n!
=
e-
(-1)/7
X
I
(-1)"Chne)
1
e
(e- 1)"
(e- 1)" _
I
n (-1) (bne) (e- 1)"
n�l
.
We evaluate the remaining term by writing bn as a sum and interchanging the order of summations, obtaining X
-I
n=1
(-1)"(bne) (e-
1)" oc
=
-e
=-e =-e oc
=
I k�o
i
k�o
I
k�o 1
k!
{3(1) _ ee-l
( _1) 11
n 1 _ k - ( 1)
Ik!
I
k
c-1) k!
(-1)k
k!
c-1)"
e- 1 e
which now seems slightly mysterious, given that a difference appears, rather than a ratio. Incidentally, since lnx is concave down, a number x chosen uniformly from [1, e] will yield a value for lnx that is closer to 1, on average, than we would have obtained by simply having chosen a number uniformly from [0, 1 ] . In other words, a random selection from [1, e] contributes more to a product than a selection from [0, 1] will contribute to a sum. So we expect that fewer selec tions are required for our product to ex ceed a given value, implying that {3(t) < a(t) for all positive t. As anticipated, {3(1 )
n n�I (e - 1) k�O
I
for some constant C > 1. However, it is not yet clear that the limit exists, and if so, what an exact value for C might be. Of course, so far we have established that
=
2.42169 < e,
settling the matter raised earlier. And with this observation, we conclude our mathematical journey, or at least this leg of the trip.
n�k+1 (e - 1)" .
1 (-1)k+ e(e- 1)k
1 k (e- l)
This completes the computation. It is only natural to speculate whether the phenomenon described by Curgus and Jewett occurs in this con text. So let {3(t) be the expected num ber of random selections that must be made from the interval [1, e] before the product exceeds r!. Since the average value of In x on the interval [ 1 , e ] is -1 , e-l we anticipate that I
{3( t) =ee-l. But can a more precise statement be made regarding the relationship be-
ACKNOWLEDGMENTS
I am grateful to Michael Sheard for read ing through this piece and making sev eral helpful suggestions. I also wish to thank the editor for encouraging me to document this mathematical morsel. REFERENCES
[ 1 ] J. Uspensky, Introduction to Mathematical Probability, New York, NY, 1 937. 2] [ B. Curgus and R. I. Jewett, An unexpected
limit of expected values, Expo. Math. 25 (2007), 1 -20.
Sam Vandervelde
Department of Mathematics, CS & Statistics
St. Lawrence University 23 Ramada Drive Canton, NY 1 361 7
e-mail:
[email protected]
li.l$$@ij:i§,fihl£ili.)lh?11
O n the Equa l s S ig n Our ''Twi ns'' : A Tour throug h Orig i na l Sources SHIRLEY 8. GRAY
Does your hometown have any mathematical tourist attractions such as
statues, plaques, graves, the cap?
where the famous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels? If so, we invite you to
Dirk H u ylebrouck, Editor
"Though many stones doe beare greate price, the whetstone is for exersice . and to your self be not unkinde . " from The whetstone of witte, which is the seconde part ofArithmetike . . . Robert Recorde ( 1 5 1 9?-1 558) obert Recorde probably never dreamed he would contribute the very cor nerstone of modern symbolic notation. But his two parallel marks for a statement of equality have endured since he first proposed them in The whetstone of witte (London, 1557). Recorde's whetstone was for sharpening wits, and his explanation for his choice of this new symbol speaks for itself: ll J!n !:J UJI»,WUD"JIU"o
l(-)otubett,fo� e�tlc alteratio of r�UAti91u:j 1UUl PJ01 paunoe a fetue erap les, bicaure tbe c�trattion oftl)etr toatcs,ma:te tbe mo� aptlp bee tu�ougbte. .an!! to. 111 uoio� tl) e teoioure tepttitton of tbefe tooo�De.s : is e� qunUe ta : � tum fette ru:J 31 �oe often 1 n tuoo.;lte ure,a pal rc ofparallele.s,oJ
A transcription of this arcane English reads: Nowbeit, for each alteration of equations, I will propose a few examples, be cause the extraction of their roots, may the more aptly be wrought. And to avoid the tedious repetition of these words "is equal to" I will set as I do of ten in works use, a pair of parallels, or twin1 lines of one length, thus: ==: , because no two things can be more equaL And now mark these numbers, - - By contrast we find that Descartes, though born some 77 years after Recorde, used a, most likely because the printer could conveniently turn the ubiquitous zodiac symbol for Taurus sideways. In Jakob (James) Bernoulli's Ars Conjectandi 0 7 13), written 1 56 years after Recorde's publication, we find the a still being used in expressing what Euler later named "Bernoullian numbers."
submit an essay to this column. Be sure to include a picture, a description of its mathematical significance, and either a map or directions so that others may follow in your tracks.
Please send all submissions to
Figure 2.
Jacob Bernoulli's Numbers in his Ars Conjectandi (Basel, 1713).
Mathematical Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende , Belgium e-m a i l : dirk.h
[email protected]
1 Note: "Gemowe" is the adjective form of "gemew," an obsolete Middle English word meaning "twins," and the word suggests another noun found in the zodiac-Gemini.
© 2008 Spnnger Scrence+Business Media, Inc . Volume 30. Number 2, 2008
19
We might ask ourselves when, where, and how our most ordinary symbol-the equals sign-became universally ac cepted. Florian Cajori, a highly respected University of Cal ifornia, Berkeley, Professor of Mathematics, wrote a defini tive study on mathematical notation in the 1 920s. The extent of his scholarship in a time when travel and communica tion were far more difficult than now remains slightly amaz ing. But modern eyes beg to see, not merely read about, the symbols that have so permeated our profession. It is to Cajori that we turn for a roadmap to guide us on our quest to trace the pathway of our most universally accepted sym bol, our "twins"-the equals sign. Along the way, note the subtle transitions in its meaning. We begin our virtual tour of original sources in the Bib lioteca Apostolica Vaticana-the Vatican Library in Rome.
ibrate the value of the original books, note that a manu script illustrated by Leonardo fetched an amazing $30,800,000 in 1 994. A poster of the icosahedron is on sale in art stores and on the web.
Girolamo Cardan (1501-1576) and Niccolo Tartaglia (1500-1557) For many early writers, the expression for equality was not even a symbol, but a word. Commonly this word was the Latin aequales. Other expressions translate "it gives" or sim ply "is" and the abbreviations "ae," "aeq" and "aequ" were not uncommon. Cardan's Ars Magna ( 1 545) continued with Pacioli's use of "p : " for plus, " m: " for minus, and a rather elaborate Rx for root, but now we find the word "est" for equals where Pacioli would have left a blank.
Luca Pacioli (1445-1514) With the invention of printing and the founding of the Vat ican Library ( 1447) in the same decade, mathematics moved into previously inaccessible territory-mass communication. Among very early printed books we find the first widely distributed encyclopedia of mathematics, Summa de arith metica geometria proportioni etproportionalita ( 1 494). In his Summa, Pacioli used a dash, but his dash had several dis tinct purposes, equality being only one of its meanings. Moreover, his terms and operations were frequently ex pressed in abbreviated Latin. Often he and others simply left a large blank in the material where today we would most certainly use a symbol . Today, we might think of this as a pause-before giving the result or conclusion. For ex amples, we now turn to the original Summa. Summa
,
Subtraction
12-Xil ; 5 _
101
p'.
2'.
.
.. ,
Geometry
.
6
.!!!.!. 15
, ,
6-;-J
lS
Algebra
67 s
t.cc.i.ii pzv. t.«.iii,J6. '
I.CC.i!;:!.!Y t .«.ffi. ;6 l:ec.p..._.:-16 ' 2 .CC, ---""
Calor rd. JOl
I Of
c
20 3
•
2"" 3..
ll
6
x + ..Jx' - 36
2.< + 6
�216 2x � 2 I O
Value o f r
1 05
Length of a segment -
m(GC) � 1o 2I
Figure 3.
This 1 6th-century mathematician i s still noted in schools of business as the originator of double-entry bookkeeping. In art history classes, Pacioli is routinely mentioned for col laborating with his good friend, Leonardo da Vinci, who il lustrated Pacioli's De Divina proportione ( 1 509). A copy of their original publication has been reproduced by the Am brosiana library of Milan 0 956) and others. Some editions contain a complete set of 61 plates of mathematical solids labeled in Latin. We recommend seeing "maestro Luca" and Leonardo's craftsmanship if only in a reproduction. To cal-
20
THE MATHEMATICAL /NTELLIGENCER
7·P·f>t-·4· :49·P·�·x6.
·�·49·P·f>t..4-: I
Modem Notation
�(7+,/4)'
�·49·P·'Al·4·
�· •¢·P·*·I ¢ . �·49·P·�.x6.p. �.I¢.;E. f)t..l¢• 1)< .Sr. 8c efl.� -
·
7·P:�·:S.:
·
7.tii . �.:S· 49.m.j. p.'l)t. 14). ffi·.13.:.14)·
.J8i
and this is 9.
(7+ ;15)(7-,IS)
=
44
44·�
Figure 4.
6 .!..!.
x - ..Jx' - 36
I"
7·P·�·4·
_ _
-
Modem Notation
101
' ""
I�
Ars Mag11a
Though Cardan's rivalry with Tartaglia is legendary, they shared very similar Latin abbreviations. Ironically, Tartaglia, who stuttered, was far more given to words whereas the lo quacious Cardan used more symbols and shorter abbrevia tions. In Tartaglia, we can find "equal" as well as modern sounding phrases, for example, "ualera Ia cosa, cioe. 2 " trans lating "equals the thing, which is 2 . " Tartaglia's popular edi tions of Euclid's Elements printed over several years opened with more than one page of text merely to explain his first definition, that of only the point. Apparently Hilbert was not the first to be concerned with "undefined terms. " I n the century following the founding of the Vatican Li brary and the invention of printing, publishing spread rapidly. We now travel from Italy and the Continent to the Duke Humphrey collection in the Bodleian Library at the University of Oxford. But travel to Britain is not essential. Many libraries in North America and Europe have copies of Recorde's Whetstone ofwitte and the other highly sought af ter titles that are discussed in the following paragraphs. Iron ically, publishing spread faster than standardization of no tation. After Recorde introduced "twin" lines for equality, the no tation did not appear in print again until Edward Wright translated Napier's log tables in the Descriptio (16 18), 6 1
years later. Although printed books on mathematics were appearing with some regularity in England and Scotland, Recorde's symbol for "equals" was still not widely accepted until it was used in three influential works by highly re spected mathematicians. Wright and Napier's notation in the Descriptio reached a broader audience with the publication of Thomas Harriot's Artis analyticae, William Oughtred's Clavis mathematicae, and Richard Norwood's Trigonome tria in 163 1 . We can easily recognize Harriot's simplification of fractions and Napier's notation for logarithms applied to right triangles.
. 063 1 ) .
Harriot's Artis anaZvticae praxis, :·
tbe
�i:m:ofeqg:�Jity is (=) As fvr c:umple : , s + s c-ca. rh�t is, the Loi;a� ·
�� ��the angle B. a t the .S.f. e of a pbin ri�u-llngled. �n:mgle,incrc;lfed by the :�ddi JOo of the Lo_g:trithJil c of's C. the hypote
rnifa thereof, f! equ:tll to the Lo�a: ithme of C A- the nchctlts. .. . . -
Figure 6.
Edward Wright's translation of John Napier's A de
scription of the admirable table of logarithm��· .
. ( 1618).
With leading British mathematicians using the elongated one might assume that the symbol was stan dardized. This was not the case. In fact, confusion was com pounded by use of the same symbol with other meanings on the European Continent. In Leyden, Francis Viete's lmgoge has almost the same symbol as Recorde to note differences, while Descartes was using '-"=: to designate plus uu mains. ==: ,
minor ' & tamen fubduttio facienda eft ; n ota differenti� cfl: , id e fl: ) minus im:erto. ut propofitis A quadraro Figure 7.
Fram;ois Viete's In artem anaZyticam isagoge; seu,
Algebra nova ( 1 635 ) .
A library sleuth investigating 1 7th-century mathematical literature can find at least five different meanings among Continental writers for the symbol " = " , ranging from 102 = 857 ( for 102.857), to simply separating different terms in arithmetic. In geometry, the symbol was more logically used to denote parallel lines. Cajori writes that widespread competition from other sym bols was an even greater threat than the use of different meanings for " = " . In Germany, Wilhelm Holzmann, better known as Xylander, used a pair of vertical lines "II" instead. Scholars speculate that he may have been using the first and last letters of the Greek word " wm" meaning "equals. " This symbol appears in several letters and publications. Others adopted the use of only one vertical line. Of special inter est, Herigone wrote "3 2 " , " 2 3 " , and "2 I 2 " , meaning "3 is greater than 2 " , " 2 is less than 3", and thus 2 1 2 would
1
1
ll
Rene Descartes (1596-1650) Descartes was the first to use exponential notation as we know it today. In his writings, we find "- -" for minus as well as the aforementioned " a" for equals. We also find the birth of analytic geometry .
---
Figure 5.
1
surely mean "2 equals 2 . " But then his extension of 2 2 to other applications leads to the modern a2 + b2 = c2 he coming a2 + b22 2 c2. With all due respect to others, the greatest threat for the English use of " = " probably came from two men on the Continent admired by their contemporaries as mathemati cians of the highest caliber. If such leaders used a symbol, others were sure to follow. Here again we find a difference in the style and influence of two geniuses.
,x 4 -- 4 X J -- 1 .9 x x + x o6 x--- t io :n o
1:our vne equation en laq�elle Figure 8.
( 1637 ) .
il y -a qua1
From the first edition of Discours de Ia methode
\f
linea Q LVand<>'{uidem M primz figm:t
.. ·······...·· ,···· ,
z.
tongit circulum L 0 P,
P L M. Sunt autem bina redangula M 0 P & 0 M P z. . / .. qualia quadrato ex 0 M. .. .. .IEqualc igiturerit reebn'-. , .....L,__,.:;,_____-"":lll gulum M 0 P , una cum . quadmo ex L M , qua drato ex 0 M ; hoc dl:, erit :t.' ::0 A :t. + b•, ac per coofequens :t. ::0 f.C+Y i"' + b' : cum O N zqueruq .. , & quadratum ex N M rantun>icm vale:tt atque duo quadrat:a ex L & L M, hoc dt, i"' � b' . !d quod pri�o :'!' demo�fl:randum.
0,./
N
.,
\· :: . /p
rcchngulum 0 M quatur q,uadratocx
.
Figure 9.
From La Geometrie, probably the most famous ap pendix ever written.
Descartes was undoubtedly influenced by his years in Holland where a flourishing Dutch mathematics commu nity-Huygens, Hudde, van Schooten, dei3eaune, et al.-all adopted his " a" and continued its use throughout the cen tury. From van Schooten's 1 683 edition of La Geo m etrie, which was published 33 years after Descartes' death, we find the continued use of the 1 7th-century Continental equals sign, hut the negative was shortened to a single dash.
x• - fxl + m m x x - n l x +p• ::n o. x4 - lx1 + m m x x + nl x +p• ::n o. ' x• -fxl - m m x x - nl x + p• ::n o. x• -lxl - m m x .t + n l x +p• ::n o. ·
x• + Ix1 + m m x x - n 1 x +p• ::n o. X" + fx1 - m m x x - n1 x +p• ::n o. x4 + /x1 - m m x x + n 1 x + p• ::n o. �- lx1 + m � x x - n1 x -p4 'J::) o. Figure I 0. Frans van Schooten's illustrations for various op tions in Descartes' "Rule of Signs. "
© 2008 Springer Science+Business Media, Inc., Volume 30, Number 2 , 2008
21
Gottfried Leibniz (1646-1716) Among leaders in 1 7th-century mathematics, none surpasses Leibniz for contributions to notation. Contemporaries rec ognized him to be ever keen on using them to introduce, test, and refine the meaning of symbols before adopting a final use for himself. In fact, Cajori collected and organized nine pages of symbols from Leibniz's letters, manuscripts, and publications. Initially, Leibniz used his own version of Xylander's ver tical lines "I I" for equality, but with wider spacing and a bar over the top. On October 29, 1 675, in the very notebook where he first introduced "f" for summation of his "omn 1 " , Leibniz wrote:
f ab
l n
� b
x
J
l
1'f
a . b 1s
constant.
We recommend seeing this notebook, which has been videotaped by Jeremy Gray for the Open University in co operation with the BBC and those working on Leibniz's pa pers in Hanover, Germany. The videotape shows a closeup of the very first use of "f" as well as many examples of his equals symbol. Across the Channel in Cambridge, Newton was routinely penning equals signs of a length apparently based on his desire to keep a neat, well organized "wastebook, " that is, a notebook of ideas that he planned to publish.
Figure I I .
Newton's calculation of a value for each of the even powers in an alternating series representing area under a curve bounded by a hyperbola and the axes. When Wallis, Barrow, and Newton, all prominent lead ers in the Royal Society of London, published or wrote let ters using varying lengths of Recorde's ==: , the stage was set for international standardization. Yet, mathemati cians on the Continent continued to focus on other leaders: Huygens in Holland, the Bernoullis in Switzerland, Pascal, Fermat, and Descartes in France, and, of course, Leibniz, who was of German birth but lived for many years in Paris. "The lion is known by his claw," Johann Bernoulli is re puted to have said, identifying Newton as the author of a solution to the brachistochrone problem-a problem he had advertised as being for the "shrewdest mathematicians in all the world." Bernoulli most assuredly recognized the nota tion as being not from the Continent, but from London. In fact, as we have seen earlier, both Johann's brother, Jakob
22
THE MATHEMATICAL INTELLIGENCER
Bernoulli, and Descartes were using a for equals when Jo hann published their brachistochrone solutions in Acta Ern ditorium (1697). What did it? What led the mathematicians on the Conti nent to accept " = " as the symbol of equality? Briefly, it was the mathematics. The strength of the new Calculus was ob vious to all informed readers. When Leibniz adopted the symbol used by Newton, all others were sure to follow. In Calculus, we routinely teach that Leibniz's notation has uni versal acceptance and emphasize the integral sign represents summation. Newton's "fluents, " "fluxions," and small "o"s did not survive. Yet with both Newton and Leibniz using Recorde's equals sign, the world would follow. In conclusion, I offer a few observations. A symbol must have acceptance by leaders of a field to validate both its meaning and applications. Without the support of Leibniz and Newton, the "twin" lines for "equals" might not have survived. In addition, the topic itself must have staying power, as in our case, the Calculus, and the need to rep resent equality in an equation. Once introduced, universal acceptance is often spread in the wake of a dominant culture. A classic example would be the use of Latin and Roman numerals throughout the Ro man Empire. Over centuries, the world has known many different numbering systems, yet the Hindu-Arabic symbols are now universally accepted. One might speculate that had Fibonnaci brought another system into Western Civilization via the Liber Abaci ( 1 202), we might have adopted other symbols for our cardinal numbers. Practicality enters the arena as well. A recent example is the decline and fall of the once ubiquitous abacus in the market places of Asia. The inexpensive calculator is now found throughout the world. Notations like " = " , " + " , and " - " are now enshrined in our technology and offer a testament to their staying power. We cannot imagine keyboards and calculators with out them. Other examples from the mathematics curriculum abound. Those with memories of teaching "base sixteen" and binary arithmetic in the 1960s will agree that, although still essen tial in computer science, these topics are no longer of cen tral importance. Our vocabulary has expanded, for exam ple, fractals, wavelets. Now the community of scholars is united both by the web and the nearest international air port. Our notation will change with the demands of tech nology. But we feel comfortable in asserting that Recorde's symbol, now approaching its SOOth birthday, will survive. ACKNOWLEDGMENT
I thank the Huntington Library (Figs. 1-10) and the Cambridge University Library (Fig. 1 1) for permission to reproduce the figures from original sources. BIBLIOGRAPHY
Bernoulli, Ja. , Ars conjectandi, Basel, 1 71 3. Bernoulli, Jo. , Acta Eruditorium,
Basel,
1 697;
http://curvebank.
calstatela.edu/brach3/brach3.htm. Cajori, F. , A History of Mathematical Notations; Two Volumes Bound as One,
Dover Publications, New York, 1 993.
Cardano, G., Practica arithmetice, & menusurandi singularis, Mediolani,
Norwood, R . , Trigonornetrie, London, 1 631 . Oughtred , W . , Arithrneticae in numeris et speciesbus institutio, London,
1 539.
Descartes, R . , Oiscours de Ia methode, Leyden, 1 637; de Beaune and
1 631 . Pacioli, L. Divina proportione, Venice, 1 509.
van Schooten, eds. Amsterdam, 1 683. Descartes, R . , Geometria, Leyden, 1 649.
Fauvel, J . , and Gray J . , The History of Mathematics: A Reader, Milton Keynes, 1 987.
Gray, J . , The Birth of the Calculus [videotape Recording] Glanffrwd Thomas, producer. A production for The Open University, BBC TV,
Pacioli, L., Summa di aritmetica, geometria, proporzioni e proporzionalita,
Venice, 1 494; Tuscany, 1 523.
Recorde, R . , The whetstone of witte, London, 1 557. Tartaglia, N., Euclides, Bresica, 1 543. Viete, F., In artem analyticam isagoge; seu, Algebra nova, Leiden, 1 635.
Media Guild, 1 986.
Feingold, M . , The Newtonian Moment: Isaac Newton and the Making of Modern Culture,
New York. 2004.
Shirley B. Gray
Hariot, T., Artis analyticae praxis, London, 1 631 .
Napier, J . , A descrip tion of the admirable table of logarithms translated by Edward Wright,
London, 1 61 8.
California State University, Los Angeles Los Angeles, CA, USA e-mail:
[email protected]
------
WALMA T0 1 and the
British Societyfor the History ofMathematics a joint conference on
Robert Recorde
(151 0?-1558) - his life and times 8-1 0 July, 2008 at
Gregynog Hall, nr. Newtown, Powys h 2008 marks the 4501 anniversary of the death of Robert Recorde ( 1 5 1 0?- 1 5 58), a mathematician and physician born in Tenby, Pembrokeshire, whose work included
some of the first English texts on arithmetic, algebra and geometry as well as texts on cosmology and medicine. Recorde is principally known as the inventor of the equals sign, but his contribution to the development of mathematics and its teaching is far wider than that. The conference wil l explore Recorde's writings, in mathematics and other disciplines, drawing on recent research. Contributors include: •
June Barrow-Green
•
John Davies
•
Stephen Johnston
•
Margaret Pel.ling
•
•
•
• •
•
John Denniss Howell Lloyd Nia Powell Jackie Stedall John Tucker Jack Williams
Additional activities: •
A one-man show entitled 'Noe two !hinges can be moare equall e ' , based on Recorde's Socratic pupil-master style, presented by the actor David Ainsworth
•
A display of rare mathematical texts by The National Library of Wales, Aberystwyth
1
an organi ation of lecturers in mathematic education in the colleges and universities of Wale
Contact details: Emeritus Professor Gareth Roberts University of Wales, Bangor (0 1 248) 383249
[email protected]
© 2008 Springer Science+Business Mecta, Inc .. Volume 30, Number 2, 2008
23
A Conve rsation w ith S . R. S . Varad han RAJENDRA BHATIA
S. R. S. Varadhan was awarded the Abel Prize for the year 2007. I met him on 14 and 15 May-one week before the prize ceremony in Oslo-in his office a t the Courant Institute to interview him for the Mathematical Intelligencer. My qualifica tions to interview him were that he and I are Ph.D. 's from the same institute, my Varadhan number is 2, and his was the first research talk that I attended as a graduate student. My major disqualification was that I know little ofprobability, and I felt like someone destitute of geometry daring to enter Plato 's Academy. Though we had planned to talk for two or three hours, our conversation was spread over nearly eight hours. What follows is the record of this with very minor editing. To help the reader I have added a few "box items" that explain some of the mathematical ideas alluded to in the conversation.
Professor Varadhan, before coming here this morning I was in a Manhattan building whose designers seem to be lieve that the gods look upon the number 13 with an un favourable eye, and they can be hoodwinked if the 13th .floor is labelled as 12A . The Courant Institute building not only bas 13 jloors, your qffice here is 1313. Well, the two thirteens cancel each other.
Excellent. I am further encouraged that I saw no sign prohibiting those ignorant of Probability from entering the Academy. So we can begin right away.
Early Years Our readers would like to know the mysteries ofyour name. In South India a child is given three names. My name is Srinivasa Varadhan. To this is prefixed my father's name Ranga Iyengar, and the name of our village Sathamangalam. So my full name is Sathamangalam Ranga Iyengar Srinivasa Varadhan.
What part ofthis is abbreviated to Raghu, the name your friends use? The child is given another short name by which the fam ily calls him . Raghu is not any part of my long name.
And you were born in Madras, in 1940. Yourfather was a high-school teacher. Did he teach mathematics? He taught science and English. He had gotten a degree in physics, after which he had done teachers' training.
24
THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+Bus1ness Media, Inc.
And your mother, janaki? My mother didn't go to school after the age of 8, as in those days it was not the custom to send young girls to school. But she was a versatile woman. She learnt to read very well, was knowledgeable and smart. For example, she taught me how to play chess. I could play chess even be fore I went to school.
Was the school in your village? No, we had some land in the village but did not live there. My grandfather died when my father was 18. My fa ther became the head of the family with two younger broth ers one of whom was one year old, and he had to look for a job.
Did be teach in Madras? He was in the District School System in Chengalpat dis trict that surrounds the city of Madras on three sides. When I was born he was in Ponneri, a village 20 miles north of Madras. He moved from one place to another and I changed school thrice. I skipped some grades and was in elemen tary school for only two or three years. I spent three years in the high-school in Ponneri.
Do you remember some qfyour teachers? Yes, I remember my high-school teachers very well. My father was the science teacher. I remember my maths teacher who was very good. His name was Swaminatha Iyer. He used to call some students to his home on the
weekends and gave them problems to work on. His idea of mathematics was solving puzzles as a game. He gave us problems in geometry.
I remember that about my father too. He was a school teacher in Punjab. He would also teach on holidays and the parents of Sikh boys had to beg him to give at least one day offfor the boys to wash and dry their long hair. (Laughs) Yes, teachers those days thought it was their mission to educate. They enjoyed it. They were not very well paid hut they carried a lot of respect. Now things have changed.
Did you have any special talentfor mathematics in high school? In most exams I got everything right. I usually got 1 00 out of 1 00.
Was this so in other subjects as well? I n other subjects I was reasonably good hut I had prob lems with languages. I was not very enthusiastic about writ ing essays.
What languages did you study? English and Tamil; a little hit of Hindi hut not too much.
Were you told about Ramanujan in school? No. I learnt about him only in college.
Interesting, because in a high-school over a thousand miles away from Madras I had a teacher who worshipped Ramanujan and told us a few stories about him, including the one about the taxi number 1 729. Where did you go a:fter high-school? In those days one went to an Intermediate College. So, I went to Madras Christian College in Tamharam, and then to the Presidency College for a bachelor's degree.
At the Presidency College you studied for an honours de gree in statistics. Why did you choose that over mathematics? My school teacher Swaminatha Iyer told me that statis tics was an important subject, and that Statistics Honours was the most difficult course to get into. In the entire state of Madras there were only 14 seats for the course. Statis tics seemed to offer a possible profession in industry. My teacher had aroused my curiosity about it. So I d i d not ap ply for admission in mathematics, but in statistics, physics and chemistry.
Did you get admission in these other subjects also? I think I did in physics hut not in chemistry. I had ap-
plied for physics in the Madras Christian College, Tam baram, and for chemistry in Loyola College. You know ad missions are a nerve-racking process. They do not put up all the lists at the same time. They want you to join the course immediately, and take away all your certificates and then you cannot switch your course. The Presidency Col lege is different, being a government college. They put up all the lists on one day. My name was there in the statis tics list.
You mean it is somewhat ofa coincidence thatyou joined Statistics. If the other colleges had put up their lists earlier, you might have chosen another subject. Yes.
Did you read any special books on mathematics in Col lege? I never learnt anything more than what was taught. But I found that I was not really challenged. I could u nderstand whatever was taught. I did not have to work for the ex aminations, I could just walk in without any preparation and take the exams.
The newspapers in India have been writing that in the honours examination you scored the highest marks in the history qf Madras University. I think I scored 1 258 out of 1 400. The earlier highest score had perhaps been 1237, and one year after I passed out this course was stopped. So there was not any chance for any one to do better than me . V: S. Varadarajan was also in the same college. Did you know him there?
He was three years ahead of me . I met him for the first time in Calcutta.
I was struck by the fact that the two persons from India who won the physics Nobel Prize-C. V: Raman and S. Chandrasekhar-and now the one to win the Abel Prize, all studied at the same undergraduate college. Was there any thing special in the Presidency College? I think at one time the Presidency Colleges in Madras, Calcutta and Bombay were the only colleges offering ad vanced courses. So, it is not surprising that the earlier No bel Prize winners studied there. If you wanted to learn sci ence, these might have been the only colleges. They were showpieces of that time. In my time the Presidency Col lege was the only college in Madras that offered honours programs in all science subjects, and these were very good.
RAJENDRA BHATIA suggests that his exposure in the course of in
terviewing Professor Varadhan has been quite sufficient and a bio
graphical note about the interviewer would be overdoing rt. He quotes A••IIIIf'l'j the character lnsarov in On the Eve by Ivan Turgenev: "We are speak ing of other people: why bring in yourself?"
I...L"--:... --' IL.II ..:.:O __.
Indian Statistical Institute Delh1
New Delhi, I I 00 1 6 India
e-mail:
[email protected]
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Figure I . The Guru and his disciples: A. N. Kolmogorov, dressed i n a dhoti and kurta i n Calcutta 1962. Standing behind him are L to R, K. R. Parthasarathy, B. P. Adhikary, S . R. S. Varadhan, ]. Sethuraman, C. R. Rao, and P. K. Pathak.
Indian Statistical Institute Now that you had chosen Statistics it was but natural that on graduating in 1959 you came to the Indian Statis tical Institute (IS!) in Calcutta. Was the Institute well-known in Madras? In Delhi we had not heard about it. We knew about it because C. R. Rao's book (Advanced Statistical Methods in Biometric Research) was one of the books we used. There were not too many books available at that time. Feller's book had just come out. Before that there was a book by Uspensky. These were the only books on Probability. In Statistics there was Yule and Kendall which is unreadable. C. R. Rao's was a good book.
Did you join the Ph.D. program? Yes. My goal was to do a Ph.D. in Statistical Quality Control and work for the Industry. I did not know much mathematics at that time except some classical analysis.
26
THE MATHEMATICAL INTELLIGENCER
Then I ran into [K. R.] Parthasarathy, Ranga Rao and Varadarajan who started telling me that mathematics was much more interesting (Laughs) . . . and slowly I learnt more things.
What are your memories of the Institute? Do you recall anything about (P. C.} Mahalanobis? Yes, Mahalanobis would come and say he would like to give lectures to us.
Were they good? No! (Laughs) . . . . He wanted to teach mathematics but somehow he also made it clear that he did not think much of mathematics. It is difficult to explain . . . C. R. Rao was, of course, always there. He was very helpful to students. But he didn't give us any courses. There were lots of vis itors. For example, [R. A.] Fisher used to come often. But
his lectures on Fiducial Inference were ununderstandable. (Laughs)
Did R. R. Bahadur teach you? Yes, in my first year two courses were organised. One on Measure Theory by Bahadur and the other on Topol ogy by Varadarajan. I went through these courses but did not understand why one was doing these things. I was not enthused by what I was learning and by January was feel ing dissatisfied. By then Parathasarathy, Ranga Rao and I decided to start working on some problem in probability theory. In order to do the problem we had to learn some mathematics-and that is how I learnt and found that the things I had studied were useful.
So your getting into probability or mathematics, was he cause of the influence of your fellow students. Yes, it was because of Parthasarathy and Ranga Rao. We studied a lot of things. I was interested in Markov processes, stochastic processes, etc. We used to run our own seminar at 7 : 30 AM. ] . Sethuraman also joined us.
What did you study at this time? We went through Prohorov's work on limit theorems and weak convergence, Dynkin's work on Markov processes; mostly the work of the Russian school. At that time they were the most active in probability.
Were their papers easily available? Yes, some of them had been translated into English, and we had a biochemist Ratan Lal Brahmachary who was also an expert in languages. He translated Russian papers for us. We also learnt some languages from him. I learnt enough Russian and German to read mathematics papers.
What hooks did you read? We read Kolmogorov's book on limit theorems. Dynkin's book on Markov processes had not yet come out. We read his papers, some in English translation published by SIAM, some in Russian.
Was mathematics encouraged in the Institute, orjust tol erated? It was encouraged. C . R. Rao definitely knew what we were doing and encouraged us to do it. There was never any pressure to do anything else. Mahalanobis was too busy in other things. But he also knew what we were doing.
How did the idea of doing probability theory on groups arise? Before I came to the Institute, Ranga Rao and Varadara jan had studied group theory. So Ranga Rao knew a fair amount of groups. When we read Gnedenko and Kol mogorov's book on limit theorems it was clear that though they do everything on the real line there is no problem ex tending the results to finite-dimensional vector spaces. So there were two directions to go: infinite dimensions or groups. The main tools used by Gnedenko and Kolmogorov were characteristic functions. I did not know it at that time, but Ranga Rao knew that for locally compact groups char acteristic functions worked well, though they did not work so well for infinite dimensional spaces. So our first idea was to try it for locally compact groups. Then I did some work for Hilbert spaces.
Your first paper is joint work with Parthasarathy and Ranga Rao. The main result is that in the space ofproba bility measures on a complete separable metric abelian group indecomposable measures form a dense G& set. Why was this surprising? At that time we were learning about Banach spaces, Baire category, etc. To show that a distribution on the real line is indecomposable was hard. You can easily construct dis crete indecomposable distributions. The question (raised by H . Cramer) was whether there exist continuous indecom posable distributions. We proved that continuous distribu tions and indecomposable distributions both are dense G8 sets. So their intersection is non-empty, in fact very large.
I read a comment (by Varadarajan) that this work was sent to S. Bochner and he was very swprised by it. No . . . , I don't think so. Certain things are appearing in print [after the Abel Prize] about which I do not seem to know.
After this you studied infinitely divisible distributions on groups. We studied limit theorems on groups. The first paper was just really an exercise in soft functional analysis. The second problem was much harder. In proving limit theo rems you have to centre your distributions by removing their means before adding them. The mean is an expecta tion of something. In the group context this is clear for some groups and not for others. To figure this out for gen eral groups we had to use a fair amount of structure the ory. The main problem was defining the logarithm of a character in a consistent way.
Your Ph.D. thesis was about the central limit theorem for random variables with values in a Hilbert �pace. Yes, then we thought of extending our ideas to Hilbert spaces, and there characteristic functions are not sufficient. You need to control some other things.
Is that the Levy concentration function? Yes.
Was that the first work on infinite-dimensional analysis of this kind? Had the Russian probabilists done similar things? They had tried but not succeeded.
So this is the first work on measure theory without local compactness. Yes.
What happened after this? The work on Hilbert space suggests similar problems for Banach spaces. Here it is much harder and depends on the geometry of the Banach space. There has been a lot of work relating the validity of limit theorems of probability to the geometry of the Banach space.
Was Kolmogorov your thesis examiner? Yes, one of the three.
Some new�papers have written that C. R. Rao wanted to impress Kolmogorov with his prize student and brought him to your Ph .D. oral exam without telling you who he was. ( Laughs) Yes, the story is pure nonsense. We knew Kol-
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Figure 2. Varadhan and Kiyosi Ito at the Tanigushi Sympo sium in 1990.
Figure 3. A rare photograph of Monroe Donsker with his wife and Varadhan's son Ashok.
mogorov was going to visit and were prepared for it. He attended my talk on my work and I knew he was going to be one of my thesis examiners. My talk was supposed to be for one hour but I dragged it on for an hour and a half and the audience got restless. Then Kolmogorov got up to make some comments and some people who had been restless left the room. He got very angry, threw the chalk on the floor, and marched out. And I was worried that this would be the end of my thesis. (Laughs) So we all went after him and apologised. He said he was not an gry with us but with people who had left and wanted to tell them that when someone like Kolmogorov makes a re mark, they should wait and listen.
Varadarajan was not in Calcutta all this time. He re turned in 1962 and pulled you towards complex semi simple Lie groups.
Do you remember any of his lectures? Sure, I attended all of them. In one of them he talked about testing for randomness and what is meant by a ran dom sequence. If you do too many tests, then nothing will be random. If you do too few, you can include many sys tematic objects. He introduced the idea of tests whose al gorithmic complexity was limited and if you did all these your sequence would still be random. He insisted on giv ing his first lecture in Russian and Parthasarathy was the translator.
I learnt that Kolmogorov travelled by train to otherplaces in India. Did you accompany him? Yes, Parthasarathy and I , and perhaps some others, trav elled with him. We went to Waltair, Madras, and then to Mahabalipuram where Parthasarathy fell from one of the temple sculptures and fractured his leg. Then he did not travel further and I accompanied Kolmogorov to Bangalore and finally to Cochin, from where he caught a ship to re turn to Russia.
28
THE MATHEMATICAL INTELLIGENCER
Yes, he returned during my last year at lSI. He had met Barish-Chandra and wanted to work in that area.
Tbis was a different area, and considered son of diffi cult. Was it difficult for you? Not really. We were just learning, it was hard learning because it was different.
Veryfew people, even among those working on the topic, understood Harish-Chandra 's work at that time. What is the wall you had to climb to enter into it? I wouldn't say we understood all of it. We just made a beginning. Varadarajan, of course, knew a lot more and guided me. We had a specific goal, a specific problem. When you have a specific problem you learn what you need and expand your knowledge base. I find that more attractive than saying I want to learn this subject or that and face the whole thing at once.
Was this work completely different from what you had been doing with groups? It was completely different. So far we had been work ing on abelian groups and not on Lie groups.
Was there a feeling that Lie groups and not probability was real mathematics? No, I don't think so. Varadarajan was interested in math ematical physics, and he thought Lie groups were impor tant there.
In the preface to his book on Lie groups he says his first introduction to serious mathematics was from the works of
Harish-Chandra. That would suggest that what he had been doing earlier was not serious mathematics. Perhaps what he meant by serious mathematics is diffi cult mathematics. I think probability came easy to him. On the other hand, Harish-Chandra's work is certainly hard be cause it requires synthesizing many things . In probability theory, especially limit theorems, if you know some amount of functional analysis and have some intuition, you can get away with it.
So, he thought it was much more difficult. It was much more inaccessible. One gets much more pleasure out of going to a place that is inaccessible.
And you never had that feeling. No, for me I was quite happy doing whatever I had been doing.
Is there any other work from IS! at that time that influ enced your later work? For example, the paper by Bahadur and Ranga Rao related to large deviations? Yes, very much so. Cramer had a way of computing large deviations for sums of independent random variables and it led to certain expansions. Bahadur and Ranga Rao worked out the expansions. So I knew at that time about the Cramer transform and how large deviation probabilities are controlled by that.
Would it he correct to say that at IS! you got the best pos sible exposure to weak convergence and to limit theorems? Varadarajan was one of the early pioneers in weak con vergence. Prohorov's paper came in 1956 and he studied weak convergence in metric spaces. Varadarajan knew that and took it further to all topological spaces. Ranga Rao in his Ph.D. thesis used weak convergence ideas to prove diffi cult theorems in infinite-dimensional spaces, such as an er godic theorem for random variables with values in a Ba nach space. That was very important for me, as I saw how weak convergence can be used as a tool, and I have used that idea often.
In the preface to his hook Probability Measures on Met ric Spaces, Parthasarathy talks ofthe "Indian school ofprob abilists ". Did such a thing ever exist? Ranga Rao, Parthasarathy, Varadarajan, and I worked on a certain aspect of probability-limit theorems-where we did create a movement in the sense that our work has in t1uenced others, and we brought in new ideas and tech niques.
The "school " lasted very briefly. What makes a school? The school does not exist but the ideas exist. (Laughs).
With hindsight, do you still consider this work to he important? I think it is important. It has int1uenced others, and I have used ideas from that work again and again in other contexts.
Later generations in the Institute look at that period with a sense of reverence and of longing. The hurst of creatiui(y in Calcutta in the 1950 's and 60 's was perhaps like a comet that will not return for a long time. For the Tata Institute also that seems to have been the golden period.
One must remember that at that time if anyone wanted to do research in mathematics in India, there were only two places, the TIFR or the lSI. If you went to any uni versity, you would be attached to exactly one professor and do exactly what he did. There was no school there. But now that has changed. There are lots of places in India where a student can go. lSI is not the only place, and even lSI has other centres now.
Courant Institute You came to the Courant Institute in 1963 at the age of 23. How did you choose this place? When I learnt about Markov processes, I learnt they had links with partial-differential equations. Varadarajan had been here as a post-doctoral fellow in 1 96 1-62. When he returned to India he told me that if I wanted to learn about PDE, then this was the best place for me.
The reason for his recommending you this place was its strong tradition in differential equations, not probability. There were some probabilists here like [H. P.} McKean and [Monroe] Donsker. McKean wasn't here. Donsker had come just the year before.
And in PDE: Courant, Friedrichs, Fritz john, Nirenberg, and Lax were all here. Yes, Moser and Paul Garabedian too. Almost everybody ( important) in PDE was here.
Stroock, in one ofhis write-ups on you, says thatfew other probabilists knew statistics at that time. That was one o_fyour advantages. I think that is an exaggeration. In the United States prob abilists came either from the mathematics or the statistics departments. Those who came from the statistics depart ment surely knew statistics. Stroock himself had a mathe matics background.
How about the converse? Did statisticians know proba bility well at that time? I think they knew some probability. You cannot do sta tistics without knowing probability. Those who worked on mathematical statistics definitely knew enough probability to be proving limit theorems. That is what mathematical statistics at that time was.
What was the status o.fprobahili(y theory itself in 1960, within mathematics. For example, I have here with me an obituary of{!. L.} Dooh by Burkholder and Protter. They say that before Doob 's hook "probability had previously suffered a cloud o.fsuspicion among mathematicians, who were not sure what the subject real�y was: was it statistics? a �pecial use of measure theory? physics?" Doob's hook was the first one to put probability in a mathematical context. If you read the book, it is clear that what he is doing is mathematics; everything is proved.On the other hand people at that time were also int1uenced by Feller who came from a different background-he was a classical analyst. I don't think he cared much about Dooh's book. I think there was some friction there.
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29
Did Feller think the book was too mathematical? I think it was too . . . theoretical. It is not so much the mathematics. It is totally devoid of any intuition, it is very formal. For that reason Feller did not like the book. Doob's book is difficult to learn from. For certain topics like mar tingales it was perhaps the ideal book. I was interested in Markov processes, and Dynkin's books were the first ones that treated the subject in the way it is done today.
I return to my question about the status ofprobability in 1960. Was it indeed under a cloud of suspicion and math ematicians did not know where to place it? It is hard for me to say . . . . I think there were some like Mark Kac who knew what exactly it could do or not do. He used it very effectively to study problems in physics. Donsker knew it was a branch of mathematics and he was interested in using it to solve problems of interest in analy sis. And then there was [G. A.] Hunt who did excellent things in probability and potential theory.
I think again it was Doob who made the connection be tween probability and potentialfollowing the work ofKaku tani. Yes, Doob made the initial connections but the decisive work was done by Hunt.
Courant was 75 when you came here. Do you have any memories of him?
I met him two or three times at social dinners. I had no scientific interaction with him. He had retired and came to his office on some days.
Let me ask you a few questions about his spirit and his influence on the thinking here. In herfamous biography ofCourant, Constance Reid says he resisted the trend towards ''generality and abstraction " and tried to "shield" his studentsfrom it. She cites Friedrichs as saying Courant was "a mathematician who hates logic, who abhors abstractions, who is suspicious of 'truth ', if it is just bare truth. " Later in the book she says Courant told her he did not hate logic, he was repelled by it. At the same time he regarded himself as the "intellectual son " ofHilbert. Now Hilbert certainly solved several concrete problems. But he had a major role in promoting abstraction in mathematics, and also worked in logic itself. When Reidpointed this out to Courant he replied, "Hilbert didn 't live to see this overemphasis on abstraction and the self-emulation and self-adulation that some of these ab stractionists show. " 7bis quote in the book isfollowed by one from Friedrichs: "We at NYU recognised rather tardily the achievements of the leading members of 'Bourbaki'. We re ally objected only to the trivialities of those people whom Stoker calls 'les petits Bourbaki '. " (Laughs) I think there is a difference in the point of view. I think abstraction is good-to some extent. It tells you why certain things are valid, the reason behind it; it helps you put things in context. On the other hand the tradition in the In stitute has always been that you start with a concrete prob lem and bring the tools needed to solve it, and as you pro ceed do not create tools irrespective of their use. That is where the difference comes, with people who are so interested in the formalism that they lose track of what it is good for.
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THE MATHEMATICAL INTELLIGENCER
When you began your career, the Bourbaki style was on the rise. Did that affect your work? Not here!
Is there a clear line between "too abstract" and "too con crete "? Let me again quote from Reid's book. Lax is cited there as saying that there was ''provincialism at NYU which was somewhat G6ttingen-like. " He quotes Friedrichs to say that von Neuman n 's operator theory was considered too ab stract there. Ifind this surprising. First, I thought G6ttingen was very broad, and second, if we apply the same yardstick what do we say of Hilbert's work on the theory of invari ants? Gordan had dismissed this work ofHilbert as "too ab stract" and called it "theology", not mathematics. Now we find Friedrichs saying von Neumann was considered too ab stract. Is there some clear line here, or everyone feels com fortable with one 's own idea of abstraction? I do not attach much importance to these things. I think abstract methods are useful and one uses whatever tool is available. My philosophy always has been to start with con crete problems and bring the tools that are needed. And then you try to see if you can solve a whole class of problems that way. That is what gives you the ability to generalise.
I will persist with this question a little more. Lax says that what they felt in G6ttingen about von Neumann 's theory of operators, here at this Institute they felt the same way about Schwartz's theory of distributions. He says it is one of those theories which has no depth in it, but is extremely useful. He goes on to say they resisted it because it was different from the Hilbert space approach that Friedrichs had pio neered. Later both he and Friedrichs changed their minds because theyfound distributions useful in one of theirprob lems. One of Courant's last scientific projects was to write an appendix on distributions for Volume III of Courant Hilbert that he was planning. Is there a lesson here? The lesson is precisely that if you do not see any use for something, then it is abstract. Once you find a use for it, then it becomes concrete.
What Lax calls the ''provincialism " at NYU, did it exist at other places? Most of the elite departments in the US those days hardly had anyone in probability or combinatorics.
Yes, there has been a certain kind of snobbery. If one does algebraic geometry or algebraic topology, one believes that is the golden truth of mathematics. If you had to ac tually make an estimate of some kind, that is not high math ematics. (Laughs)
Has this changed in the last jew years? I think fashions change. Certain subjects like number theory have always been important and appeal to a lot of people. Some other subjects that had been peripheral be come mainstream as the range of their applications grows.
In this shift towards probability and combinatorics has computer science played a major role? Computer science has raised several problems for these subjects. There are whole classes of problems that cannot be solved in polynomial time in general, but for which al gorithms have been found that solve a typical problem in short time. What is 'typical' is clearly a probabilistic con-
cept. That is one way in which probabilty is useful in com puter sciences. Indirectly many of the problems of com puter science are combinatorial in nature, and probability is one way of doing combinatorics.
methods there. Some people find Hilbert spaces more con venient than (general) topological vector spaces. That is what Friedrichs did initially. When you come to a problem where one space does not work you go to another one.
I come hack to my question about admiration for and resentment against Bourbaki. Do you think this had un healthy consequences? Or, is it that mathematics is large enough to accommodate this?
Now let me ask a question to which I know your answer. But I will ask it and then put it in context. Were you dis appointed that you did not get the Fields Medal?
I think we have large enough room for different people to do different things. Even in France, those brought up on the Bourbaki tradition, if they need to learn other things, they will do it. People want to solve the problems they are working on, and they find the tools that will help them. Sometimes you have no idea where the tools come from. Ramanujan's conjectures were solved eventually by Deligne using the etale cohomology developed by Grothendieck.
What I real�y mean to ask you is whether you did not get the Fields Medal because at that time probability theory was not considered to be the kind ofmathematicsfor which Fields Medals are given.
Did you everfeel, as some others say they have felt, that some branches of mathematics have been declared to he prestigeous and very good work in others is ignored? I never felt so. At lSI there was no such thing. At the Courant Institute there was no snobbery.
Except that there was no need to do distributions' No, I don't think so. I will put it this way. Distributions are useful because they deal with objects that are hard to define otherwise . But, more or less, the same thing can be achieved in a Hilbert space context. It is true that duality in the context of topological vector spaces is much broader but a major part of it can be achieved by working in Hilbert spaces. A problem does not come with a space. You choose the space because it is convenient to use some analytical
Varadhan's Lemma There is a simple lemma due to Laplace that is useful in evaluating limits of integrals: For every continuous function h on [0, 1)
1
lim - log
n ----+X
n
(The common
fact
11 e-nh(x) dx 0
lim llf llp
P-> X
=
=
llfllx
- inf b(x). can
be used to get
a one-line proof of this lemma:
No.
I can't say. (Laughs) It is true that, historically, Fields Medals have gone much more to areas like algebraic geom etry and number theory. Analysis, even analysis, has not had as many. It is only this time that probability has got its first Fields Medal. Sure, I would have been happier if one had been given to a probabilist earlier. But after all, (at most) four medals are given every four years. Many peo ple who deserve these awards do not get them.
Let us come hack to 1963. Did you start getting involved in PDE soon after coming here?
I was still continuing my work in probability, and what ever PDE I needed I learnt as I went along. And here you do not even have to make an effort to learn PDE, you just have to breathe it.
Stroock says that the very first problem you solved after coming here was done simultaneously by the great proba bilist Kiyosi Ito, and you did not publish your work. What was the problem?
The function I(x) is now called the rate function. It is defined for spaces much more general than the unit in terval [0, 1). Let X b e any complete separable metric space (Pol ish space). A rate function I is a lower semicontinuous function from X into [0, oo] such that for every f < oo the level-set lx : I(x) :S f) is compact. A family {p,nl of prob ability measures on X is said to satisfy the large-devia tion principle ( LDP) with the rate function I if ( i) for every open set U
1
=
log sup e-hC xl
=
lim - log
n ----+x:-
n
J
o
' e- rzh( x) dp, (X). rz
In his 1966 paper Varadhan argues that if we have
( ii) for every closed set F
1
lim - log p,iF) :S - inf I(x) . Varadhan's Lemma says that if lp, rzl satisfy the LDP , then for every bounded continuous function h on X lim
- inf [ h ( x) + I(x)] .
F
n
n----+ X
then by Laplace's lemma this limit would be
u
- n
- inf h ( x) . )
Now suppose w e are given a family o f probability measures and are asked to evaluate the limit
1
lim - log p,,( U) 2:: - inf I(x),
2._ n
log
J
X
e-nh(x)
dp,11(x)
=
- inf [h(x) + I( x)] .
There is an amazing variety of situations where the LDP holds. Finding the rate function is a complex art that Varadhan has developed over the years.
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It was a question about giving a more precise meaning to Feynman integrals. In the Schrodinger equation there is a differential part (the Laplacian) and there is a potential part (some function). The measure you want to construct depends on the Laplacian. Without the i this will be the Brownian motion. The presence of i makes it the Feynman integral, and not so well-defined. If you take the Fourier transform of the Schrodinger equation, then the potential part (multiplication) becomes a convolution operator and plays the role of the differential operator, and the Lapla cian becomes multiplication by x2. The idea was that now you base your measure on (the Fourier transform of) the potential part. That is not as bad as the Feynman integral ; it may even be a legitimate integral for some nice poten tials-like functions with compact support, some functions with rapid decrease, or the function x.
Your address on this paper is given as the Courant In stitute and lSI. Were you still associated with the IS!?
Was this the first work you did after coming to the Courant Institute?
Most ofyour work has been in collaboration. You began by collaborating with a small group at lSI. 7ben in 1968 appears yourfirst paper with Stroock. Were you working by yourself between 1963 and 1967? You have single-author papers in these years, which is unusual for you.
Well, Donsker asked a very special question. There are several approximations that work for the Wiener integral. Do the same approximations work for the Feynman inte gral? If you take the Fourier transform, then they do work because the measure based on the potential is nicer. That was the context of my work.
Yourfirstpaper at the Courant Institute appeared in 1966 and has the title "Asymptotic probabilities and differential equations". Can you describe what it does? When I came here in 1 963 Donsker had a student by the name Schilder. He was interested in the solution of cer tain equations whose analysis required Laplace type of as ymptotics on the Wiener space. You have the Wiener mea sure and the Brownian motion that has very small variance, and you are interested in computing the expectation of some thing like exp (e- 1f) . So you have something with very large oscillations and you are computing its expecta tion with respect to something with very small variance. If you discretize time, then you get Gaussian densities instead of the Wiener measure and this becomes standard Laplace asymptotics. So you do it for finite dimensions and inter change limits, and that is what Schilder had done. Having been brought up in the tradition of weak convergence it was natural for me to think of splitting the problem in two parts. One was to abstract how the measures behave as ymptotically and then have a theorem linking the behav iour of the integrals to that of measures. That is not a hard theorem to prove, once you realize that is what you want to do. Then if you know a little bit of functional analysis, that Riemann integrals are limits of sums, and how to con trol errors you can work out the details. It was clear that if you have probabilities that decay exponentially and func tions that grow exponentially you can do it by formulating a variational problem that can be solved.
Is this paper the foundation for your later work with Donsker? Yes. This paper has two parts. First I prove the theorem I just mentioned and then apply it to a specific problem. Schilder had studied the case of Wiener measure with a small vari ance. I do it for all processes with independent increments.
32
THE MATHEMATICAL INTELLIGENCER
I was on leave from the lSI for three years and resigned later.
You have stayed at this Institute since 1963. What has been the major attraction? Is it New York? the Institute? some thing else? I like New York. After living in Calcutta I got used to living in big cities. In 1 964 I got married and my wife was a student here. So when the opportunity came to join the faculty here, I did so. By that time I had got used to the place, and I liked it and stayed. It is a good place and has been good for me. It is always exciting and interesting with lots of people coming here all the time.
The Martingale Problem
I was a post-doctoral fellow working mainly by myself. But I had lots of conversations with Donsker.
How did your collaboration with Stroock begin? He was a graduate student at Rockefeller and we met at joint seminars. In 1 965-66 I wrote a paper on diffusions in small time intervals and he was interested in that. He came here as a post-doc and joined the faculty after that. He was here for about six years from 1 966 to 1 972. We talked often and formulated a plan of action, a series of things we would like to accomplish together.
I have never met him but from his writings I get the im pression that he will /ike it ifI say that your coming together was a stroock of good luck. (Laughs.)
Your work with Stroock seems to have flowed like the Ganga. In three years between 1969 and 1972 you published more than 300 pages of research in a series ofpapers in the Communications on Pure and Applied Mathematics . Can we convey a .flavour of this work to the lay mathematician? Let us understand clearly what you want and what you are given. In the diffusion problem certain physical quan tities are given. These are certain diffusion coefficients {a;jx)l which form a positive-definite matrix A(x) for each x in fR d, and you are given a first-order drift, i.e., a vector field {bjx)l. We want to associate with them a stochastic process, i.e., a measure P on the space 0 consisting of continuous functions :x(t) from [O,oo) into fR d such that x(O) = .xo almost surely. When we started our work there were two ways of do ing it. One is the PDE method in which you write down the second-order PDE
a2 u au 1 "' "' - = - L aiJ(x) -- + L at
2 i.j
ax,.axi
j
au bjx) - . axi
This equation has a fundamental solution [It, x, y). You use this as the transition probability to construct a Markov process P, and the measure coming out of this process is
The Martingale Problem For the discussion that follows it might he helpful to re mind the reader about a few facts about diffusion processes. Let us begin with the prototype Brownian motion (the Wiener process) in 1ft It is a process with stationary in dependent increments that are normally distributed. The transition probability (the probability of a particle start ing at x being found at y after time t) has normal den sity p ( t, x, y) with mean x and variance at, where a is a positive constant. This is related to fundamental solu tions of the heat equation as follows. For every rapidly decreasing function rp,
u(t, X)
r
=
-%
rp(y) p(t,
X,
at
and further, lim0 u(t, y)
t-> V->X
=
- a 1
2
= rp(x).
a2 u
ax2 '
More generally, one may study a problem where the constant a is replaced by a function a (x) , and the par ticle is subjected to a dri:fi b(x). (For example, the Orn stein-Uhlenbeck process is one in which b(x) -px, an elastic force pulling the Brownian particle towards the origin.) Then we have the equation
=
---;;{ = 2 au
1
a(x)
a2 u
a >?
+
b(x)
ax
your answer. All this requires some regularity conditions on the coefficients. In the other method, due to Ito, you write down a stochastic differential equation (SDE) involving the Brownian motion {3(· ) . Let IJ he the square root of A. The associated SDE is
=
1J(x(t))d{3( t)
+
li.x( t))dt;
x(O)
=
Xo .
This equation has a unique solution under certain condi tions. This gives a map
XII from n into itself and the im age of the Wiener measure under this map is the diffusion we want. The conditions under which the two methods work overlap, but neither contains the other. The PDE method does not work very well if the coefficients are de generate (the lowest eigenvalue of [a;jx)] comes close to zero); the Ito method does not work if the coefficients are not Lipschitz. When they fail it is not clear whether it is the method or the problem that fails . We wanted to establish a direct link between P and the coefficients without any PDE or SDE coming in. This is what we formulated as the Martingale Problem: Can you find a measure P on 0 such that
Xcp( t)
=
rp(x(t) ) -
rp(x0) - r ()
rp (B( t)) -
(.ii.rp)(x(s))ds
I
t
0
1
- /).rp (B(s) ) ds 2
is a martingale. It was shown by Ito and Levy that this property characterizes the Brownian motion-any sto chastic process for which Xcp( t) defined as above is a martingale for every rp must be the Wiener process. The Martingale Problem posed by Stroock and Varad han is the following question. Let
.i1
= I; ·
az
--
a; (x) I
J
iJx;ax1·.
+
I· J
a
b (x) I
ax1·.
be a second-order differential operator on [H d_ Can one as sociate with .i1 a diffusion process with paths x(t) such that
Xcp( t)
au
In higher dimensions a(x) is replaced by a covariance matrix [aiJ (x)] whose entries are the d!ffusion coeffi cients, and b(x) is now a vector.
dx( t)
=
y)dy
--
=
1
Xcp( t)
satisfies the heat equation ( or the diffusion equation)
au -
Let (0, ?f, P) be a probability space and {X1} 1;,:0 be a family of random variables with finite expectations. Let {?Ji1}1;,:0 be an increasing family of sub-IJ-algebras of ?F. If each X1 is measurable with respect to ?F 1, and the conditional expectation E(X1 'lf 5) Xs for all s :S t, then we say {X1}1;,:0 is a martingale. (A common choice for ?F 1 is the IJ-algebra generated by the family {X, : 0 :s: s :s: t).) The Brownian motion {B(t)}1;,:0 in [H d i s a martingale. The connection goes further. Let rp be a C2-function from [H d into !H. Then
=
rp(x(t))
=
-{ 0
(.ii.rp) (x(s)) ds
is a martingale? (If .i1 � /). such a process exists and 2 is the Wiener process.)
is a martingale with respect to (0, ?F,, P), where ?F, is the IJ-field generated by {x(s) : 0 :s: s :s: t ) and
.i1
=
1
2
I i.j
a2
au . iJX;ax i
+
Ij
a
b; ax;
-
·
In this general formulation .i1 can be replaced by any operator. This method works always when the other two do, and in many other cases. (Just as integration works in more cases than differentiation.)
I believe after the completion q( your work the field q( PDE started borrowing more from probability theory, while the opposite had been happening before.
No, we too use a lot of differential equations; we do not avoid them.
Between distribution solutions of d![(erential equations and viscosizy solutions, that came later, is there another layer of solutions that one may call probability solutions? Yes, . . . , there is something to that. If you take expec tations with respect to the probability measure that you have constructed, then you get solutions to certain differential equations. Usually they will he distributions but the condi tions for the existence of a generalized solution may not be
© 2008 Springer Science+Bus1ness Media, Inc., Volume 30, Number 2, 2008
33
fulfilled. So you can call these a new class of generalized solutions, and they can be defined through martingales.
So, are we saying that for a certain class of equations there are no distribution solutions but there are solutions in this new probability sense? It is difficult to say what exactly is a distribution solu tion. It is perfectly clear what a classical solution is. Then everyone can create one's own class-nothing special about the Schwartz class-in which a unique solution exists, as long as it reduces to the classical solution when that exists.
Talking of classical solutions, what is the first instance of a major problem in PDE being solved by probabilistic methods? Is it Kakutani's paper in which be solved the Dirichlet problem using Brownian motion? Sure, that is the first connection involving probability, harmonic functions, and the Dirichlet problem.
What is it that Wiener did not know to make this con nection? Tbe relation between Brownian motion and the Laplace operator was obvious to everyone. Is it because things like the strong Markov property were not known at that time? Also, Wiener was much more of an analyst. I don't think he thought as much of Brownian paths as of the Wiener measure. Unless you think of the paths wandering around and hitting boundaries you will not get the physical intu ition needed to solve some of the problems.
of a function-space integral. Asymptotically, this integral grows like the first eigenvalue of the Schrodinger operator, and this can be seen from the usual spectral theory. Donsker asked whether the variational formulas arising in large de viations and Laplace asymptotics and the classical Rayleigh Ritz formula for the first eigenvalue have some connection through the Feynman-Kac representation. I thought about this and it turned out to be the case. This led to several questions like whether there are Sanov-type theorems for Markov chains and then for Markov processes; and if we did the associated variational analysis for the Brownian mo tion, would we recover the classical Rayleigh-Ritz formula. It took us about two years 1 973-75 to solve this problem. The German mathematician ]Urgen Gartner did very simi lar work from a little different perspective.
What are your recollections about Donsker? He had a large collection of problems, many of them a little off-beat. He had the idea that function-space integrals could be used to solve many problems in analysis, and in this he was often right. We worked together a lot for about ten years till he died, rather young, of cancer.
It is mentioned in Courant's biography that Donsker was his confidant when be worried about the direction the In stitute was taking.
Kac, for example, with the Feynman-Kac formula, surely knew the connections.
There was a special relationship between the two. I think a part of the reason was that most of the others at the In stitute were too close to Courant-they were his graduate students or sons-in-law. (Laughs) Donsker was an outsider and Courant respected the perspective of some one like him. But in the end Courant did what he wanted to do in any case.
As you were working on this, who were the other people doing similar things?
Almost all the reports say that the large-deviation prin ciple starts with Cramer.
Who were the other players in the development of this connection between probability and PDE?
In Japan: Ikeda, Watanabe, Fukushima, and many students of Ito. The brilliant Russian probabilist Girsanov. He died very young in a skiing accident. He had tremendous intuition. An other very good analyst and probabilist Nikolai Krylov, now in Minnesota. Then there were Ventcel, Freidlin, and a whole group of people coming from the Russian school. In the United States McKean who collaborated with Ito, and several people working in martingales: Burkholder, Gundy, Silver stein; and the French have their school too.
I am curious why hyperbolic equations are excludedfrom probability methods. Except one or two cases. There are some examples in the work of Reuben Hersh. But they are rare. If you want to apply probability, there has to be a maximum principle, and not all equations have that. The maximum principle forces the order to be two, and the coefficients to be pos itive-definite.
Large Deviations Your papers with Stroock seem to stop in 19 74-I guess that is because he left New York-and there begins a series ofpapers with Donsker. How did that work start? I was on sabbatical leave in 1 972-73 and on my return Donsker asked me a question about the Feynman-Kac for mula which expresses the solution of certain PDE in terms
34
THE MATHEMATICAL INTELLIGENCER
The idea comes from the Scandinavian actuarial scien tist Esscher. He studied the following problem. An insur ance company has several clients and each year they make claims which can be thought of as random variables. The company sets aside certain reserves for meeting the claims. What is the probability that the sum of the claims exceeds the reserve set aside? You can use the central limit theo rem and estimate this from the tail of the normal distribu tion. He found that is not quite accurate. To find a better estimate he introduced what is called tilting the measure (Esscher tilting). The value that you want not to be ex ceeded is not the mean, it is something far out in the tail. You have to change the measure so that this value becomes the mean and again you can use the central limit theorem. This is the basic idea which was generalized by Cramer. Now the method is called the Cramer transform.
Is Sanov's work thefirst one where entropy occurs in large deviation estimates? It is quite natural for entropy to enter here. Sanov's and Cramer's theorems are equivalent. One can be derived from the other by taking limits or by discretizing.
Tbe Shannon interpretation of entropy is that it is a mea sure of information. Is it that a rare event gives you more information than a common event and that is how entropy and large deviations are related?
. . . The occurence of a rare event gives you more in formation, but that may not be the information you were looking for. (Laughs) What happens in large deviations is something like in statistical mechanics. You want to calculate the probability of an event. That event is a combination of various micro events and you are adding their probabilities. It is often possible to split these micro events into various classes and in each of these the probability is roughly the same. It is very small, exponentially small in some parameter. So each individual event has probability = exponential of n times something. That something is called the "energy" in physics. But then the number of micro events making an event could be large-it could be the exponential of n times something. That something is the "entropy" . So the energy and entropy are fighting each other and the result gives you the correct probability. That is the picture in statistical mechanics. So, -
Coin Tossing and Large Deviations The popular description of the theory of large deviations is that it studies probabilities of rare events. Some sim ple examples may convey an idea of this. If you toss the mythical fair coin a hundred times, then the proba bility of getting 60 or more heads is less than . 1 4 . If you toss it a thousand times, then the probability of getting 600 or more heads reduces very drastically; it is less than 2 X 1 0 9 . How does one estimate such probabilities? Let us enlarge the scope of our discussion to include unfair coins. Suppose the probability for a head is p, and let Sn be the number of heads in n tosses. Then by the weak law of large numbers (which just makes formal our intuitive idea of probability) for every e > 0 -
nlim �oo P
(I PI ) 5"
n
> e = 0.
-
The elementary, but fundamental, inequality of Cheby shev gives an estimate of the rate of decay in this limit: p
(l �; l )
-p �e �
p(
��
(:
h+ ( e) = (p + e) log
p+ e
--
p
I do not know! . . . It is something like Bochner having been surprised [by our first theorem]. (Laughs)
One comment I heard about your work was that before you most people were concerned only with the sample mean, whereas you have studied many other kinds of objects and their large deviations. Let me put it this way. Large deviations is a probability estimate. In probability theory there is only one way to es timate probabilities, and that is by Chebyshev's inequality.
Bernstein's inequality is an example of a large-devi ation estimate. It is optimal in the sense that lim n�oo
1
- log P
n
(
)=
511
�p+ e
-
n
- h+ (e).
The function h+ is the rate function for this problem. The expression defining it shows that it is an entropy like quantity. Let us now go to a slightly more complicated situa tion. Let /.L be a probability distribution on IR and let . . . be independent identically distributed ran dom variables with common distribution /.L· The sample mean is the random variable
X1, X2,
-
1
Xn(w) =
-
n
n I X,( w),
i=l
and by the strong law of large numbers, a s n --'> oo this converges almost surely to the mean m = EX1 . In other words, for every e > 0 lim P
rl--'1>X
(jXn
-
m l � e)
=
Finer information about the rate of vided by Cramer's theorem. Let
0.
decay
to 0 is pro
(the cumulant function)
)
� p + e � e- "h+<sl,
where for 0 < e < 1 - p,
In the notes in their book Deutsche/ and Stroock say that Sanov 's elegant result was at first so swprising that several authors expressed doubts about its veracity. W'hy was that so?
p)
It was pointed out by Bernstein that for large n this upper hound can be greatly improved:
P
for me entropy is just a combinatorial counting. Of course you can say that if I pick a needle from a hay stack, then it gives me more information than picking a needle from a pin cushion. But then entropy is the size of the hay stack.
and
I(x) 1 -p- e --=+ ( 1 - p - e) log 1 -p -
-
As e --'> 0, h+(e) is approximately e/2p ( l - p) . For a fair coin p = 1/2 and this is 2e. So,
(In our example at the hegining we had e = . 1 , and for n we chose n = 1 00 and 1000 .)
=
sup (tx - k( t)). t
Then
,\�
1
-;;
log
p
-
c l xn - m l 2: e) =
-
:Y
i I(x). lx- � s
In other words, as n goes to oo , P m � e) goes to 0 like e- nc, where c inf! I(x) : x - m 2: el. The functions I and k are convex conjugates of each other, and I is the Fenchel-Legendre transform of k. Con vex analysis is one of the several tools used in Varad han's work on large deviations.
=
l
CIXn - l l
© 2008 Springer Science+Business Media, Inc , Volume 30, Number 2, 2008
35
Sanov's Lemma Let J.L be a probability measure on a finite set A 1 1 ,2 , . . . , m} and let X1 , X2 , . . . be A-valued i.i.d ran dom variables distributed according to J.L. Each X,. is a map from a probability space (0, �. P) into A such that P(X,. = k) = J.L(k). The "empirical distribution" associated with this sequence of random variables is defined as =
J.L,(w)
=
1
-
n
"
L
i=r
8x,(w),
w E D.
For each w and n, this is a probability measure on A. (If among the values X1 (w), . . . , X11(w) the value k is assumed r times, then J.L11(w)(k) = r/ n.) The Glivenko Cantelli lemma says that the sequence J.LrzCw) converges to J.L for almost all w. Let .M be the collection of all probability measures on A and let U be a neighbourhood of J.L in .M. Since J.Ln converges to J.L, the probability P(w : J.L 11(w) ft. U) goes
The usual Chebyshev inequality applies to second moments, Cramer's applies to exponential moments. You compute the expectation of some large function and then use a Cheby shev-type inequality to control the probability. That is you control the integral, and the probability of the set where the integrand is big cannot be very large. So people have concentrated on the expectation of the object that you want to estimate. That stems from the generating-function point of view. My attitude has been slightly different. I would start from the Esscher idea of tilting the measure. His ex ponential tilting is just one way of tilting. It works for in dependent random variables. If you have some process with some kind of a model and you are interested in some tail event, then you change the model so that this event is not in the tail but near the middle. The new model has a Radon Nikodym derivative with respect to the original model, and you can use a Jensen inequality to obtain a lower bound. This may be very small. Then you try to optimise with re spect to the choice of models. If you do this properly, then the lower bound will also be the upper bound.
What kind of optimisation theory is used here? It depends on the problem. For example for diffusion in small time for Brownian motion on a Riemannian manifold it is the geodesic problem. If you want to get Cramer's the orem by Sanov-type methods, the ideas are similar to those in equilibrium statistical mechanics. The Lagrange-multiplier method is an analogue of the Esscher tilt. If you want a Sanov-type theorem not for i.i.d. random variables but for Markov chains, then a Feynman-Kac-like term is the Es scher tilt. It is the same idea in different shapes.
It is said that whereas in the classical limit theorems the nature of individual events is immaterial, in the large deviation theory you do have to look at individual events. I guess what people mean to say is that in the large deviation theory you solve an optimisation problem. So events near the optimal solution have to be examined more carefully.
36
THE MATHEMATICAL INTELLIGENCER
to 0 as n � oo. Finer information about the rate of de cay is given by Sanov's Lemma. For every v in .M, the relative entropy of v with re spect to J.L is defined as
H (v J J.L) Let c
=
inf
v f!. U
=
{'\'m
L 00i= l
H Cvl J.L).
v( z')
l og
v( i)
lit)
if v <<J.L otherwise.
Then
1
lim - log P (w : J.L11(w) $. U) = - c.
n�oo
n
For this problem l(v) = H ( v i J.L) is the rate function, and our probability goes to 0 at the same rate as e- crz. In this case we have studied limits of measures in stead of numbers. This is what Varadhan calls the LDP at the second level. At the third, and the highest, there are LDP's at the stochastic-process level.
You and Donsker have a series ofpapers on the Wiener sausage (a tubular neighbourhood ofthe Brownian motion) where you have asymptotic estimates of its volume. What is the problem in physics that motivated this study? The Laplace operator on [R d has a continuous spectrum. If you restrict to a box of size N it has a discrete spectrum. As you let N go to infinity and count the number of eigen values in some range and normalize it properly this goes to a limit, called the density of states. If you add a poten tial, you get another density of states. A special class of po tentials of interest is where you choose random points in [R d according to a Poisson distribution, put balls of small radius around them where the potential is infinite. There are two parameters now, one is the density (of the Pois son distribution of traps) and the other is the size of the traps. Now you want to compute the density of states. This is done better if you go to the Laplace transform. Then it becomes a trace calculation, and by the Feynman-Kac for mula this can be done in terms of the Brownian motion. Entering the infinite trap means the process gets killed. So we are looking at a Brownian motion that must avoid all these traps-which are distributed at random. This prob lem was posed by Mark Kac. You are looking at the behaviour of density of states at low levels of energy. That is the same as the behaviour of the Laplace transform for large t. So you want to know what is the probability that the Brownian motion avoids these traps for a very long time. The conjecture made by the Russian physicist Lifschitz was that this probability de cays like exp( - c tdl( d+ 2l) . It is easy to calculate the probability of having a big sphere with no traps in it. Then you calculate the proba bility that a Brownian motion that is in this sphere stays there for ever, or at least up to time T This can also be easily calculated and turns out to be like e-AT, where A is the first eigenvalue of the Laplacian on this sphere with the Dirichlet boundary condition. You multiply the two prob abilities to get the answer.
Are Monte Carlo methods relevant to this kind ofproblem? No, they are not useful here. As this theory tells you, if a Brownian path has found a safe territory without traps, then it must try to stay there. A typical path will not do that. So the contribution comes from paths that are not typ ical, and Monte Carlo methods can not simulate such paths.
In another series of papers you and Donsker study the polaron problem . Can we describe this to our readers, even if loosely? Our work started with a question posed to us by E . Lieb. It comes from a problem in quantum statistical mechanics and the work of Feynman. In the usual Feynman-Kac for mula you have to calculate the expectation of integrals like
In the polaron problem you have a quantity depending on a double integral:
{ [ LL
A(t,a) = E exp a
J}·
e- I
and you have to evaluate
1
G(a) = lim
t--.x
log A(t,a).
This is complicated, and there was a conjecture about the value of the limit of G(a)/a2 as a � oo . The potential V(x
·
·
The Feynman-Kac Integral
The Lagrangian of a classical mechanical system L(x, x) .
1 �
1
B2!fi
at 2 ax2 ljf(O , x) = r,p (x) .
Is this the highest level, or can you go beyond? Level three is the highest because the output and the in put are in the same class-both are stochastic processes. (At the first level, for example, the input is random vari ables and you consider quantities like their means. ) The interesting thing i s that a t level three the rate func tion is universal. You take any two processes P and Q and the rate function is the Kolmogorov-Sinai entropy between them. So at level three there is a universal formula. They are different at the lower level because the contraction prin ciples are different.
Let me tum to lighter things now. In 1980 I attended a talk by Mark Kac. He began by saying that Gelfand, who was three months older than him, advised him that as you grow old you should talk
Kac says a large part of his scientific effort was devoted to understanding the meaning of statistical independence. For Courant, a very large part of the work is around the Dirichlet principle. Is there one major theme underlying your work? It is hard for me to say something in those words. I can talk of my attitude to probability. I don't like it if I have
where rx is the space of all paths -
Schrodinger equation
aljf
.
=
x2/2 - V(x) has its quantum mechanical counterpart - -, 2 rl.r V The wave function ljf( t, x) is a solution of the 1
measures as a stream and in addition to ox, , Ox2 , . . . , look at Ocx , , x, h Ocx,, x1 J, . . , and then at Ocx, , x2, x3), Ocxz.x,.x4J . . . , and so on with tuples of length k. Now you can first let n, and then k, go to oo. This is process level large devi ation. Here I am considering the following question: I draw a sample from one stochastic process and ask what is the probability that it looks like a sample drawn from a differ ent process? This is what I call a level-three large-deviation problem. The rate function for this can be computed and turns out to be the Kolmogorov-Sinai entropy. This is used in the solution of the polaron problem.
V(x)ljf,
Feynman's solution to this is in the form of a curious function-space integral
{x7 : 0 <
T
:S
t,
.x{) =
xl
a n d ll7 dx7 i s a "uniform measure" o n [RC O.tl . F o r a math ematician such a measure does not exist. Kac observed that if the i in Schrodinger's equation is taken out, one gets the heat equation. A solution sim ilar to Feynman's now reduces to a legitimate Wiener integral
ljf( t,x) = I.
rx
exp
{-r 0
}
V(xT)dr r,p(xT)dWx .
These function-space integrals occur very often in the work of Donsker and Varadhan.
© 2008 Springer Science+ Business Media, Inc .. Volume 30, Number 2. 2008
37
The Wiener Sausage This may give the best example of the reach, the power, and the depth of the work of Donsker and Varadhan. Let f3 be a Brownian path in [R d. The Wiener sausage is an £-tube around the trajectory of f3 till time t; i . e . ,
sfel( /3)
=
{x E [R d
:
l x - f3(s)j < £ for some s in [ 0 ,
=
I
X
exp [ - c v cs[el(f3))] W(d/3).
to do lots of calculations without knowing what the answer might turn out to be. I like it when my intuition tells me what the answer should be and I work to translate that into rigorous mathematics.
Where are these problems coming from? Usually from physics. Physicists have some intuitive feel ing for the answer and mathematics is needed to develop that.
Do you talk often to physicists? Yes, For the last few years I have been working on hy drodynamic scaling. I often talk to ]. Lebowitz at Rutgers, and to others.
What is your work connecting large deviations to statis tical mechanics, thermodynamics and fluid flow? This may loosely be described as non-equilibrium sta tistical mechanics. The ideas go back far; for example in the derivation of Euler's equations of fluid dynamics from classical Hamiltonian systems for particles. You ignore individual particles and look at macroscopic variables like pressure, density, fluid velocity. These are quantities that are locally conserved and vary slowly, and others that wiggle very fast but reach some equilibrium. There are ergodic measures indexed by values for differ ent conserved quantities. These are local equilibria or Gibbs states. If your system is not in equilibrium, it is still locally in equilibrium . So certain parameters that were constants earlier are now functions of space and time. You want to write some differential equations that describe how these evolve in time. Those are the Euler equations.
How do large deviations enter the picture? In the classical model there is no noise. I can't touch things that have no noise. (Laughs) Think of a model in which after a collision who gets to leave with what mo mentum is random.
What are the applications of these ideas in areas other than physics. I believe there are some in queuing networks. How about economics? There are some applications. I do know some people working in mathematical finance-I don't know whether that is real finance. (Laughs) But it is conceivable that these ideas are used. These days you can write "options" on anything.
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THE MATHEMATICAL INTELLIGENCER
1
lim tdl< +Z log sljel d l f->00
t]l .
Let V b e the volume (Lebesgue measure) in [R d, a n d W the Wiener measure on the space X of continuous paths from [O,oo) into fR d. For a fixed positive number c, let
:Ajel
Varadhan explains in this conversation why physicists are interested in studying the behaviour, as t � oo , of :Ajel . Donsker and Varadhan proved the marvelous for mula =
- k(c),
where
k(c)
=
d + 2 ( 2A ) d!C d+2J c21< +2 , d l -d d
and A is the first eigenvalue of the Laplace operator in the unit ball of Lz(IR").
Let us make up a problem. I write an option that if a certain stock rises to $ 1 000, then I will pay you the average closing price for the last 90 days. So what I pay does not depend just on the current value but also on the past history how it got there. But the past history counts only if it reached this high value. Therefore you want to know first the probability that the stock will reach this high value, and then if it did so what is the most likely path through which it will reach this value. For this you have to solve a large-deviation problem.
There is a joke that two economists who got the Nobel Prizefor their work on stock markets lost their money in the stock market. (Laughs) I have no idea. But whatever they lost they made up in consultancy.
It is remarkable that the equations of Brownian motion were first discovered by Bachelier in connection with the stock market, and only later by Einstein and Smoluchowski. Are there many instances of this kind where social sciences have a lead over physical sciences? I think many statistical concepts, now used in biology, were first discovered in the context of social sciences.
Another major collaborator ofyours bas been G. Papa nicolaou. Whereas your work with Stroock and Donsker was concentrated over a few years to the exclusion of other things, here it is spread over several years. Is this the begin ning ofyour interest in hydrodynamic limits? Can you sum marize it briefly? George and I were at a conference in Luminy in Mar seille. We always went for a walk after lunch. Luminy is on top of a hill and there is a steep walk down to the sea. We walked down and up for exercise after lunch, and dis cussed mathematics. George explained to me this problem about interacting Brownian motions. You have a large num ber of Brownian particles that come together and are re pelled from each other. The density of paths satisfies a non linear diffusion equation. You have to compute some scaling limits for this system. I was intrigued by the prob lem as it looked like a limit theorem, and I always thought I should be able to prove a limit theorem, especially if everyone believed it was true. But when I looked at it closely there was a serious problem-of the kind I men tioned before. How to prove certain quantities are in local
equilibrium. Pretty soon I found a way of doing it. In some sense large deviations played a role. If you are in equilib rium you can compute probabilities. If you have a small probability with respect to one measure and if there is an other measure absolutely continuous with respect to this, then the probability in this measure is small if you have control over the Radon-Nikodym derivative. This derivative is given by the relative entropy. In statistical mechanics the relative entropy of nonequilihrium with respect to the equi librium is of the order of the volume. So events that had probabilities that were super-exponentially small in the equilibrium case still have small probabilities in the non equilibrium case. My idea was that to control something in the nonequilibrium case you control it very well in the equi librium. At this time Josef Fritz gave a seminar at the In stitute where he was looking at a different problem on lat tice models. Some ideas from there could handle what we had been unable to do in our problem. That is the history of my first entry into this field.
You have worked with several collaborators. Do you have any advice on collaborations? I think you should talk to a lot of people. A part of the fun of doing mathematics is that you can talk about it. Talk ing to others is also a good source of generating problems. If you work on your own, no matter how good you are, your problems will get stale.
Have you thought ofproblems coming from areas other than physics? Problems that come from physics are better structured. In statistical mechanics one knows the laws of particle dy namics and can go from the micro level to the macro level where observations are made. There is a similar situation in economics where you want to make the transition from individual behaviour to what happens to the economy. This is fraught with difficulties. Whereas we know which parti cles are interacting, we do not know how persons interact and with whom. The challenge is to make a reasonable model. Physics is full of models.
How about biology, does it have good mathematical models?
The Ubiquitous Brownian Motion A gambler betting over the outcome of tossing a coin wins one rupee for every head and loses one for every tail. Let N( n) be the number of times that he is a net gainer in the first n tosses. For a > 0 what is lim P
n�cc
( -- ) N(n) n
The answer is that this limit is equal to the Wiener mea sure of the set of Brownian paths (in the plane) that spend less time than a in the upper half-plane. This very special example is included in a very gen eral "Invariance Principle" proved by Donsker in his Ph.D. dissertation.
It seems to me that at this moment most of their prob lems are statistical in nature, . . . like data mining.
On Probabilists I would like to talk a little about some major figures in probabili�y theory in the 20th century, and get a feel of the recent history as you see it. Does the modern theory of probability begin with Kol mogorov, as is the general view? Kolmogorov was really responsible for making it a le gitimate branch of mathematics. Before that it was always suspect as mathematics, something that was intuitively clear but was definitely not mathematics. The person who con tributed the most to probahlistic ideas of the time was Paul Levy, but he was considered an engineer by the French.
What was Wiener's role? In 1923 Wiener wrote a paper Differential space and several years before Kolmogorov he
introduced a measure on a function space. Wiener measure was just one particular measure. Kol mogorov advanced the view that, very generally, the mod els in probability or statistics have legitimate measures he hind them. After that it became easier to make new models. Kolmogorov must have known for several years what is in that book and decided to write it at some point. There is perhaps nothing there that he discovered just before he wrote it.
In the preface to Ito 's selected papers you and Stroock say Wiener (along with Paley) was the Riemann of stochastic integration. Yes, though he did it more by duality and completion arguments. The Wiener integral is very special, but it must have been the motivation for Ito's more general theory.
Do you think Cramer's Mathematical Methods of Statis tics 0945) did for statistics what Kolmogorov's book had done for probability? It is quite unreadable! When I was a student there were not any statistics books that were readable. The best I found were some lecture notes on statistical inference by Lehmann from Berkeley. Statistics has two aspects to it. One is com puting sampling distributions of various objects, and this is Let (fl, 'd', P) be any probability space and X1 , X2, i.i.d. random variables on it with mean 0 and vari ance 1 . For each n 2:: 1 associate with every point w in fl an element 'Yn of C[O, 1] as follows. Let
.
S,lw) = X1Cw) +
'Yn(O) = 0; for k = 1 ,2, Sk(w)/Vn, and define y,( t)
· · ·
+ Xn(w).
. . . , n, let y,(kl n) = for other values of t as a piecewise linear extension of this. This defines a map 'Pn from fl into C[O,l] given by cp,(w) = ')'11• Let f.L n P cp� 1 be the measure induced on C [O,l] by 'Pn· The Donsker Invariance Principle says that the se quence p,11 converges weakly to the Wiener measure on Let
=
o
C[O, l].
© 2008 Springer Science+ Business Media, Inc
.
Volume 30, Number 2, 2008
39
just an exercise in multiple integrals. This is really more analysis than statistics. The other aspect, real statistics, is inference. Very few books did that. At that time many ex positions came from Berkeley.
We already talked ofPaul Levy. Though you said be was thought of as an engineer, he was proving theorems to the effect that the set where the typical Brownian path intersects the real axis is homeomorphic to the Cantor set. How is this kind of thing useful in probability? There is an interesting way of looking at Brownian mo tion. The zero-set of a Brownian motion is a Cantor set. So there is a measure that lives on the Cantor set, and para metrised by that measure it becomes an interval. This gives a map from the Cantor set into an interval; under this map several open sets are closed up. You can try to "open" them up again. This means there are randomly distributed points on the interval where you "open up" things using Brown ian paths that wandered into the upper-half or the lower half plane. These are "Brownian loops" or "excursions" . Is it possible to reconstruct the Brownian motion by starting with the Lebesgue measure on the interval and opening random intervals with excursions? This "excursion theory" is a beautiful description of the Brownian motion. Levy saw all this, and later Ito perfected it.
Among other important names there are Khinchine, Feller, . . . Khinchine did probability, number theory, and several other things. I think he would have thought of probability as an exercise in analysis.
Did Feller like analysis? He seems to have been critical of Doob for making probability too abstract. Feller's work is all analytical . For example, the law of the iterated logarithm is hard analysis, as is his description of one-dimensional diffusion. It is not that he did not like analysis; he thought Doob's book was too technical . In his own book he cites Doob's book among "books of histori cal interest" . That made Doob very angry.
How do you assess Doob 's Stochastic Processes ( 1953)? My view of Doob's book is that it is very uneven. Some parts like martingales are very original. But if you look at a book you should compare the number of pages with what is proved in those pages. Doob's book is large, over 600 pages, but does not prove that much. In one interoiew he says he intended to minimise the use of measure theory because probabilists thought it was killing their subject. But then he found the "circumlocutions " be came so great that he bad to rewrite the whole book. Is it that even that late probabilists did not want measure the ory to intrude into their subject? I don't think so. But you see, in probability what do you do with measure theory? The only thing you need is the dominated convergence theorem, what else? It is always in the background. But to say that you were avoiding mea sure theory in an advanced book sounds strange.
No, Doob says he tried to avoid it because probabilists thought it was killing their subject. That is only because they allowed it to. Let us take
40
THE MATHEMATICAL INTELLIGENCER
Doob's own book, for example. One of the concepts in the study of stochastic processes is the notion of separability. That is where measure theory really intrudes. The problem is that sets depending on more than a countable number of operations (like those involving a supremum) are not measurable. If you change a random process on a set of measure zero nobody will notice it. But if for each t you change it on a set of measure zero, then as a function you change it on the union of these sets which is no longer of measure zero. So one has to be careful in choosing certain sets and functions from an equivalence class. Of course if you don't know measure theory this does not bother you. (Laughs) But soon you notice you can choose versions that are reasonable, and then you don't have to worry about the matter. So you should know "separability" can be a problem, learn to avoid it, and then avoid it forever. Doob, on the other hand, makes a whole theory out of it. That is because you let the measure theory intimidate you.
At another place Doob attributes the popularity of mar tingales to the catchy name. Do you believe that the name "martingale " made the theory popular? I don't think so. Maybe when Dooh started the theory, no one cared. But then it turned out to be a very useful concept. Today even people on Wall Street know of mar tingales. (Laughs)
Let us come to Ito now. In your preface to his Selecta you and Stroock say that if Wiener was the Riemann ofstochastic integration, then Ito was its Lebesgue. Is that an accurate analogy? I thought Wiener's integral is very special, while Ito 's is much more general. . . . I am sure that was written by Stroock; it is not my style. If you read Levy's work you will get some idea of what a diffusion should be like. It is locally like a Brown ian motion, but the mean and the variance depend on where you are. It is clear from Wiener's integral that you are chang ing the variances by a scale factor, but the factor depends on time and not on space. If you want it to depend on both time and space, you get an equation, and that is a stochastic differential equation. This is what Ito must have seen; and he made precise the ideas of Wiener and the in tuition of Levy, by defining this equation.
Do you have any special memories of Mark Kac? Oh, he was a lot of fun. He would often call us up and invite us to come to Rockefeller University where he would talk of many problems. He had a tremendous collection of problems.
Is there anyone else you would like to mention? Dynkin made big contributions. He started out in Lie groups and came to probability a little late, around 1 960. Then he founded a major school on Markov processes. I learnt a lot from his work, from his books, papers, and ex pository articles. Around 1 960 he wrote a beautiful paper in Uspehi on problems of Markov processes and analysis, that I remember very well.
What, in your view, is the most striking application of probabili�y in an area far away from it? Although I am not quite familiar with it, it is used in law some times. (Laughs)
I meant an application in mathematics, but in an area not traditionally associated with probability. For example, Bismut's purely probabilistic proof of the Atiyah-Singer in dex theorem. Is that unexpected? Well, McKean had already done some work on it with Singer and it was clear that probability or, at least, the fun damental solution of the heat kernel plays a role. Since the Laplacian operator is involved, the role of probability is not far fetched.
Is there an area where you would not expect probability to enter, but it does in a major way? . . . Number Theory. For example the work of Fursten berg. In PDE probability now plays a major role but that is not unexpected. If you use martingales, the maximum principle just reduces to saying that the expectation of a nonnegative function is nonnegative.
The Prize Did you anticipate your being chosen for the Abel Prize? No, not at all.
I have read that potential Nobel Prize winners are usu al(y tense in October and jump every time theirphone rings at 6 AM. I also heard a talk by a winner who told us that when they phone you about the Nobel Prize they have with them someone whom you know so that you are sure no one is pulling your leg. How was it for you?
They called me at 6 : 1 0 A.M. , gave me the news and said I should not tell anyone till 7 when they would announce it at a news conference in Oslo. They told me there would be a live interview on the Norwegian radio.
Were you allowed to tell your wife? I told my wife.
My question was whether you were allowed to. 1be Fields Medal winners have to he told in advance because the
Some Thoughts on Prizes The Nobel is awesome to most of us in the field, prob ably because of the luster of the recipients, starting with Roentgen (1901). The Prize gives a colleague who wins it a certain aura. Even when your best friend, one with whom you have peed together in the woods, wins the Prize it somehow changes him in your eyes. I had known that at various times I had been nomi nated . . . . As the years passed, October was always a nervous month, and when the Nobel names were announced, I would often be called by one or another of my loving offspring with a "How come . . . ?" In fact, there are many physicists-who will not get the Prize but whose accomplishments are equivalent to those of the people who have been recognized. Why? I don't know. It's partly luck, circumstances, the will of Allah. When the announcement finally came, in the form of a 6 A.M. phone call on October 10, 1 988, it released a hidden store of uncontrolled mirth. My wife, Ellen, and I, after very respectfully acknowledging the news,
awards are announced just before they are actually given, and they are told they can tell their spouses but no one else. 1bat must he difficult for them! But this was only between 6 : 30 and 7. You cannot call too many people anyway. I did not tell anyone except my wife till 7 .
There has been some discussion about the purpose ofsuch prize�heyond honouring an individual. Lennart Carlson said they draw public attention to the subject. When I en tered college, physics was the most prestigious subject. 1ben Hargobind Khorana got a Nobel Prize, and for a few years many top students in India wanted to study biochemistry. I see little chance of mathematics displacing management even after your Abel Prize. I think it does make the subject more visible, and may attract a few individuals who otherwise had not thought about it.
At 67 you are the baby among the Abel Prize winners. Ito got thefirst Gauss Prize last year when he was about 90. Is it good to have age limits for such prizes?
I don't think it is important. Now you have several prizes of high level. There are the Wolf Prize, the Crawfoord Prize, the Kyoto Prize, the King Faisal Prize, . . . And although they don't say it, they rarely go to the same individual.
Still most of the other prizes have not caught the public imagination in the same way as the Nobel Prize. The Nobel Prize is a century old and has got etched into people's consciousness.
One purpose the prizes could be made to serve is that a se rious attempt is made to explain the winner's work to people. It is hard to explain what a mathematician has done, compared to a new cure for cancer or diabetes.
But we don 't even explain it to mathematicians. At the ICM's there are talks on the work of the Fields Medalists.
laughed hysterically until the phone started ringing and our lives started changing. -Leon Lederman, in "The God Particle " I think it's a good thing that Fields Medals are not like the Nobel Prizes. The Nobel prizes distort science very badly, especially physics. The difference between someone getting a prize and not getting one is a toss-up-it is a very artificial dis tinction. Yet, if you get the Nobel Prize and I don't, then you get twice the salary and your university builds you a big lab; I think that is very unfortunate. But in mathematics the Fields Medals don't have any effect at all, so they don't have a negative effect. I found out that in a few countries the Medals have a lot of prestige-for example, Japan. Getting a Fields Medal in Japan is like getting a Nobel Prize. So when I go to Japan and am introduced, I feel like a Nobel Prize winner. But in this country, nobody notices at all. -Michael Atiyah 1be Intelligencer, 1 984
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
41
On Bach The Stroock-Varadhan book proceeds on its inexorable way like a massive Bach fugue.
There's nothing to it. You just have to press the right keys at the right time with the right force, and the organ will make the most beautiful music all by itself
-David Williams
Johann Sebastian Bach
(Book Review in Bull. Amer. Math. Soc.)
Some are very good and others do not convey much even to a competent mathematician from a neighbouring field.
resentation theory orprobability. Is this something you could have done, or would like to do in the future?
To explain something very clearly and very well takes a lot of effort, thought and time. It is not an easy job.
I don't know. I think the Indian psyche is different from the Chinese. The Chinese like the role of an emperor much more and Chern enjoyed that role. Indians seem to be much more individualistic, and even within India I do not see anyone with that much influence.
You have been an editor, for many years, of Communi cations on Pure and Applied Mathematics, and of the Grundlehren series. Do you make any effort to make your authors write better? In a journal it is difficult to do so. But for books in the Grundlehren series we are very meticulous. We try to have books from which people can learn.
I began our conversation with India and would like to end with it. You left India at the age of 23. Do you think you could have done something more for mathematics in India? . . . Perhaps I could have. But these things are com plicated. Since my family and my work are here, I could at best make short visits and give some lectures. Some students could then keep in contact, or come here. Some of that was done in the 1 970's when we had more schol arships. But then our funds for these things were re duced.
I have a very specific question here. Ifyou see a person like S. S. Chern, he played an enormous role in grooming mathematicians of Chinese origin, even before China opened up. Perhaps Harish-Chandra could have played a similar role for Indians but he didn 't. It could be that the two personalities were different. Several mathematicians of Chinese origin became outstanding dif.ferential geometers. Nothing like that happened to Indians in the fields of rep-
42
THE MATHEMATICAL INTELLIGENCER
What are your other interests? I like sports. I play either tennis or squash for one hour every day. I listen to music, though I do not have special knowledge of it. I like Karnatak music. I like to watch movies, I see a lot of English as well as Tamil movies.
Are these the masala movies in Tamil? Yes, a lot of them. These days you have DVD players and you can fast-forward whenever you want to. I also read Tamil books, both new and old. Nobel Prize winners are often asked a silly question: what will you do with the money? I haven't made detailed plans but I have a rough idea . I would like to put some of it for public good. My par ents' last residence was in Madras (Chennai) and they were associated with a school. There is also a hospital there which is doing good work. I would like to help such ventures. Perhaps I will use one third of the prize money for that. Then, of course, I have to pay taxes nearly one third of it. The remaining one third I will keep for my own use.
Thank you very much for giving me so much of your time, and best wishes for the Award Ceremony next week.
l)f1(9·i. (.i
Dav i d E . Rowe , E d i t o r
A Tri p to the C ircus w ith the H ydrau l is VINCENT G . HART
Send submissions to David E. Rowe,
Fachbereich 17-Mathematik,
Johannes Gutenberg University, D55099 Mainz, Germany.
l
n organ recital at a horse race seems a little bizarre nowadays, I but it was the accepted thing at the circus in Roman imperial times. At least this is implied by a reference in Petronius's Trimalchio's Feast [1], in which a person carving meat is com pared to a charioteer fighting to the sound of the hydraulis, or water organ. This ancient predecessor of the modern organ seems to have been invented be tween 270 and 250 Be by a Greek named Ctesibius. The earliest reference to this machine-by Philon [2] about 250 Be-does not give an exact de scription of it, and we have to rely on Hero (3] and Vitruvius [4], writing in about 50 AD, for accounts of later ma chines that may be similar to the orig inal hydraulis. Parts of two ancient wa ter organs have been found. Parts of the brazen pipes of a first century Be hy draulis were found in 1 992 at Diem in Greece [5]; a reconstruction can be seen and heard on the Internet at www. archaeologychannel. org/hydraulisint. html. In 1 93 1 , the remains of a Roman water organ dating from 228 AD were found in a suburb of Budapest. Again de tails can he found and a reconstruction seen online at orgona .hu/orgonaink/ tuzolto_orgona_e.html. I have not found any published technical details of either of these finds, and I am advised that the reconstructions must be con sidered largely hypothetical--especially as regards the air supply. I was asked by an organist, Dr. W. Jordan, to calculate the pressure in the hydraulis. This turned out to require only elementary calculations, as de scribed in this note. The essential feature of these ma chines appears to be that air is sup plied to the organ pipes at a uniform pressure, and this air pressure can be stabilized in part by the use of water pressure . Here I describe a conjectured hydraulis based on the somewhat vague descriptions available; the air pressure at a typical stage in the oper ation of the organ is calculated, also the efficiency of the system is evaluated for two different shapes of the de-
L
vice-called a pnigeus--that funnels air to the organ. Since we do not know the air pressure at which the hydraulis worked, I have allowed for relatively high working pressures in the first two sections of the paper, with low pres sures in the third section. For compar ison, modern church organs work at low air pressures of about three inches of water, whereas fairground organs use pressures of about eighteen inches of water. Of course, modern organs use a much more powerful supply of air, driven by electric fans, than was avail able in the hydraulis. I emphasize at the outset that the water pressure involved is just part of the control system for the air supply. (Dr. Jordan will describe the exterior control system elsewhere. )
Pressure i n the Hydraulis: A Cone as Pnigeus A conjectured typical apparatus ap pears in Figure 1 . The cone of height h, made of thin metal, is the pnigeus in this case; it is open at the bottom and is fixed at height h0 above the base of a cylindrically shaped containing vessel called a cistern. The radius of the cistern is a, and r is the radius of the base of the cone. Air can he pumped into the top of the cone by means of a pipe P, and there is a valve
1
p
1
y
I
h
L
t
ho
� a ------+i
i
Figure I . Ini ti al height
and cistern:
h+
)
of water in cone of wa
h0. Final height
ter: in cone t, in cistern y.
© 2008 Springer Science+Bus1ness Media. Inc .. Volume 30, Number 2, 2008
43
in the cone at Q that can admit air into the sounding box and pipes of the or gan above Q . With this valve closed, the cone can be filled with air. Next, water is filled into the cistern and cone to the top of the cone to a to tal height h + h0 above the cistern floor. Then, with the valve closed, air is ad mitted to the cone from P and the wa ter is driven down to a level t above the cistern floor inside the cone, and up to a level y outside the cone, as shown in Figure 1 . The air pressure in the cone is therefore in equilibrium with the pres sure caused by the head of water out side the cone. Initially y = h + h0. Al though the working state of the hydraulis involves the pnigeus being completely filled with air, with surplus air bubbling out of the bottom, that is with t = h0, the calculation with an arbitrary value of t is necessary to allow for variations of the external air pressure. Using the constancy of volume of the water, we calculate y as follows (as suming that the diameter of the air pipe leading upwards from the vertex of the pnigeus to the sounding box is negli gible). 1T
a 2 (h + h,,) = 2 y - (7T r 2/3) (h - ( t - hu)), 1T a l
where r1 is the radius of the water sur face in the cone. Thus we find
y = h + h0 + ( r/a h)2 [h - (t - hu W/3.
(1)
Then the pressure in the cone is p = w (y - t),
(2)
mit sufficient air to the pnigeus to keep the level of water in that vessel approx imately unchanged-thus stabilizing the pressure to the organ at the maximum corresponding to the cone completely filled with air. Should there be a slight excess in the pressure of the admitted air, the water pressure from the in creased height of water in the cistern op poses this pressure and tends to restore the original situation. It is interesting to speculate just what quality of sound was produced by the ancient water organs. One source (Athenaeus [6]) says it was "very sweet and joyous." That it was rea sonably loud by Nero's time seems to be confirmed by Petronius's comment [ 1 ]. A small positive increase I:J. t in the water depth will give rise to a pressure change I:J.p, given by I:J.p =
- w (I:J.t) [ 1 + ( rl a? (1 - u)Z].
(4)
Specific numbers: let h = 1 5", hu = 2", r = 7.5", a = 8.5", w = 1 . Then we find pressure p in inches of water from (3). The maximum pressure of 18.89 inches of water occurs when the cone is full of air and t = h0 (u = 0). Minimum pres sure zero occurs when the cone is full of water and t = h + h0 (u = 1 ) . (All pressures are above atmospheric.)
A Hemisphere as Pnlgeus It is interesting to compare this perfor mance when the cone is replaced by a hemisphere, as shown in Figure 2. We suppose that the base radius r of the hemisphere is the same as that of the
44
THE MATHEMATICAL INTELLIGENCER
a 2 ( r + hJ = 2 2 5 1T a y - 1T [2 r 13 - (t - hJ r 3 + ( t - ho) /3],
where we have used the result for the volume of a spherical segment. Then we find
y = r + h0 + [2 r3!3 2 - ( t - hJ r2 + (t - h0)3/3l! a .
(5)
The pressure in the hemisphere is p = w (y - t),
(6 )
or, on setting v = ( t - h0)/r, we find
p/r = w [ 1 - v + (2/3 - v + v 3/3) ( r! a)2] .
(7)
Again the function in the square brack ets gives the pressure in the hemi sphere in inches of water when w = 1 , and when multiplied by r in inches. A small increase I:J.t in t would give a de crease in pressure
+ ( r/ a)2 ( 1 - v 2)].
(8)
Taking r = 7.5", a = 8.5", h0 = 2", w = 1 , the maximum pressure of 1 1 . 39 inches of water is obtained when the hemi sphere is full of air (v = 0).
Low Air Pressures
(3)
The function in square brackets in (3) gives the pressure in inches of water when w = 1 , and h is measured in inches. If w = 0.0 361 , pressure is in pounds per square inch per inch depth of water (organists unrepentantly use Imperial units). With the cone filled with air, the valve at Q is opened and the air pressure in the cone is admitted to the sounding box of the organ. Then, when a key is opened, the air enters a pipe and a sound is heard. The exterior control mechanism immediately operates to ad-
1T
I:J.p = - w(I:J.t) [ 1
where w is the weight of a unit vol ume of water. Setting u = (t - h0 )/ h, we have, using (1),
p/h = w [1 - u + ( r/ a? (1 - u)3/3l .
cone. Then both cone and sphere have the same volume if the cone has height equal to twice its base radius. The cis tern radius is unchanged at a, and the initial water level is up to the vertex of the hemisphere. Again, by constancy of water volume, the water height in the cistern y when the water height in the hemisphere is t is given by
p
r
r 1...� ..--r ----J -
�+----- a -----+�
Il
Figure 2. Initial height of water in hemi sphere and cistern: r + h0. Final height of water: in hemisphere t, in cistern y.
So far we have supposed that the pnigeus is completely filled with water at the outset, and air pressure then dri ves the water level to the bottom of the pnigeus in working mode, in confor mity with the ancient description. How ever by partially filling the pnigeus and then driving the water level down slightly, one may achieve low working pressures. We consider only the coni cal pnigeus for illustration. Let the initial height of water in both cistern and cone be x, where, since we want low pressures, we restrict x to be less than h + h0 , and let the final heights of water inside the cone and outside in the cistern be t (less than x) and y, respectively, where y is less than
interval ( h0, h + h;J is found by use of the Maple program solve. It is useful to note a limit on x if we do not wish to drive the water level y in the cistern above the vertex of the cone. Let X0 be the initial level for which the level y just reaches the ver tex with the cone empty of water. Then y = h + h0, and t = h0 with (9) yielding 7T
a 2 x0 =
7T
a 2 (h + hJ
- 7T
il- h/3,
or
Using the previous dimensions, this gives X0 = h0 + (0.740484) h. If initially x is less than X0, then, for any t in the interval h0 < t < x, the level of water in the cistern will not reach the vertex. But if x is greater than X0, there must be a limit on t to ensure that the ver tex level is not exceeded. If at t = t0, say, the vertex is just reached with y = h + h0, then (9) gives 7T
a2x =
7T
a2 (h + h0) 3 - 7T ( r/h? ( h - Cto - ho)) /3,
thus yielding t0 as a function of x :
to = h + ho - [3 (a h/r)2 (h
+
h0 - x)J l 13 .
( 1 1)
The pressure in the cone is given by (2). We take w = 1 and quote pressure p in inches of water in Table 1 .
Discussion Regarding the calculations of the first two sections, though the total volumes are the same in cone and hemisphere, and each vessel is filled with water to the vertex at the outset, the pressures are greater in the cone when each ves sel is in the working state-that is, com pletely filled with air-because the cone has twice the height of the hemisphere, and this causes greater depth of ex pelled water in the former case, and therefore greater pressure. (Yet more pressure at the outset can of course be supplied to either system by simply top-
The Hydraulis, constructed by Mary Elisabeth Speller in 2005.
She writes, "A real hydraulis would have water chambers and pistons of cast bronze. Because bronze is expensive and difficult to work with, I used acrylic plastic instead. My goal was to create a model showing how the water and air pressure worked in the organ, not to create a replica of an ancient organ. Clear acrylic was the ideal material, because you can look inside. For simplicity (of construction), both the inner chamber Cpnigeus) and outer chamber (cistern) are rectangular boxes. " (Photo: Stan Sherer)
the vertex height. Then constancy of water volume implies 7T
a 2x =
TABLE 1
(X" - ho)lh
(t - h0)1h
(y - ho)lh 0 . 3671
2 . 506
0 . 25
0 . 15
0 . 3320
2 . 730
0 . 20
0 . 10
0 . 3003
3 . 004
0 . 20
0
0 . 4048
6 . 072
0 . 08317
0 .80
0 . 65
0 . 8093
2 . 390
0 . 27230
0 . 90
0 . 75
0 . 9038
2 . 307
(to - ho)lh
7T
a 2y 2 [(h - ( t - h0))3 - 7T ( r/ h) - (h - (y - h0))3J/3. (9)
If we prescribe both x and t, this gives a cubic equation for y. The root in the
0 . 30
0 .20
p
© 2008 Springer Science+Business Media. Inc .. Volume 30. Number 2. 2008
45
modern organs-can be achieved in the hydraulis either by just slightly fill ing the cone at the outset, or by using a higher level-that is, provided that t does not differ too much from x. Per haps the latter choice might ensure a more stable air pressure; it must have depended on the vigor and regularity of the pumpers !
ping up with water above the level of the vertex in the initial configuration.) In another direction, however, in the neighborhood of the equilibrium posi tion u = 0, the rate of change of pres sure with depth t as the water level changes in the pnigeus occurs more slowly in the hemisphere than in the cone. We can see this by comparing equations (4) and (8) and noting that v = 2 u. Thus for a given volume of air one can infer that the hemisphere af fords a more stable response to pres sure change. There seems to be no in dication in the literature whether ancient organists preferred one type of pnigeus to another. But Vitruvius [4] certainly mentions that it was an inverted fun nel-which we interpret as a cone-and Hero [3] mentions the hemisphere. With respect to the low pressure cal culations, it is clear from the table of results in the previous page that small air pressures of two to three inches of water-comparable to those used in
2. Philon. 1971 . Belopoiica , In E. W. Marsden, "Greek and Roman Artillery, Technical Trea tises," Clarendon Press, Oxford, 153. 3. Hero. 1 97 1 . The Pneumatics of Hero of Alexandria ,
M. Boas Hall, ed. , MacDonald,
London, and American Elsevier, New York, 105-D9. 4. Vitruvius. 1 934. On Architecture, F. Grainger, trans., ed. , vol I I , W. Heinemann, London, and G. P. Putnam's Sons, New York, 31 4-1 9.
ACKNOWLEDGMENTS
I am greatly indebted to Dr. Wesley Jor dan, organist at St. Stephen's Cathedral, Brisbane, for informing me about the hy draulis and for giving me basic refer ences and information about the organ and about recent finds. I am also grate ful to Emeritus Professor R. D. Milns who provided some Latin translations.
5. Clark, 0. December 2006. A recently-dis covered Hydraulis, Organ Australia,
6. Athenaeus.
1928.
The
C. B. Gulick, trans., W. Heinemann, London, and G. P. Putnam's Sons, New York, 291 .
Vincent G. Hart Mathematics, School of Physical Sciences
REFERENCES
1 . Petronius. 1 975. Petronii Arbitri, Gena Tri malchionis ,
Martin S. Smith, ed. , Clarendon
Press, Oxford, Chap. 36.
University of Queensland Queensland 4072 Australia [email protected]
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46
THE MATHEMATICAL INTELLIGENCER
4-5.
Deipnosophists ,
0 1 1 743a
I n Search of the " B i rthday" of E l l i ptic F u ncti ons ADRIAN RICE
l
n 1750, an elderly Italian nobleman and diplomat named Giulio Carlo de'Toschi di Fagnano applied for member ship of the prestigious Berlin Academy of Science. Founded in 1700 and reorganized by Frederick the Great in 1743, the Academy was one of the most distinguished sci entific centers in Europe. To be elected a member by virtue of one's academic accomplishments was one of the highest honors available to the European scientist, a fact of which Fagnano would have been well aware, since, in addition to his political and diplomatic activities, he was a highly skilled mathematician. As well as having worked in analytic geom etry and deriving a moderately well-known formula in com plex numbers, 1 his chief interest was the rectification and quadrature of curves. At the age of 68, he published a two volume collection of his mathematical papers [14], which was included as supporting material for his application to the Academy (see Fig. 1). To judge the merits of the work now before them, the Academy sent Fagnano's papers to its chief mathematical scientist, none other than Leonhard Euler, who acknowl edged receipt of the documents on 23 December 175 1 . So impressed was Euler by what he read that he was stimu lated to intensify his study of a particular area of mathe matics that today fills two volumes of his collected works [12] . The fact that this new research agenda was prompted by Euler's receipt of Fagnano's mathematical papers led no less a mathematician than Jacobi to describe 23 December 1751 as "the birthday of elliptic functions. " 2 Such a statement raises more questions than answers. First of all, in any history of elliptic functions,3 Jacobi's name
1This formula, known as Fagnano's identity, is lnY(1
-
i )/(1 + i )
=
-i7T/4.
2"den Geburtstag der elliptischen Funktionen" [ 1 5, p. 1 83], [26, p. 23]. 3See for example [7], [8], [ 1 5] , [19].
48
THE MATHEMATICAL INTELLIGENCER © 2008 Springer Science+Bus1ness Media. Inc.
P R OD U Z I O N I MATEMATICHE
DEL CONTE GIULIO CARLO DI Fl\GNANO, MARCHESE DE' TOSCHI, E DI SANT' ONORIO
NOBILE ROMANO, E PATRIZIO SENOGAGLIESE A L L A S AN. T I T A' D l N. J'.
BENEDETTO XIV. PONTEFICE MASSIMO .. r o � o P R I M.9 · , __
L' A N N O
P E S A ll O
D E L G l U B B l L E O M. D C C. I..
NELLA STAMPERIA GAVE.LUANA
CON LJCENZ4 DE S U P£R./OR./, Figure I . The title page of lished in
1750.
Fagnano's collected papers, pub
features at least as prominently as Euler's. So if the work of any one mathematician can be claimed to have served as the starting point of elliptic functions, could not that mathematician just as easily have been Jacobi himself? As we shall see, the answer to this question is yes. Secondly, if the contributions of Euler and Jacobi are two candidates for such distinction, are there not other mathematicians whose work is equally eligible? The answer to this ques tion is similarly positive. But other questions are a little more involved. For example, what do we mean by the "birthday" of a mathematical subject? Can such things be accurately described as being "born"? Finally, even given that these questions may be answered conclusively, can a satisfactory answer ever be found to the question What is
the "Birthday " of Elliptic Functions? This article will attempt to answer all of these questions. To do so, the definition of an elliptic function that will be used is as follows: Let j(z) be a single-valued meromor phic function; if f is doubly periodic and defined over C, then we say that f is an elliptic function 4 I will consider a number of possible "birthdays" from the earliest appearance of an elliptic integral in 1 694 to the publication of Jacobi's Fundamenta nova theoriae func tionum ellipticarum in 1 829 in search of the first example of a doubly periodic function defined over C. The defini tion contrasts with that historically employed, whereby an elliptic function was defined simply as the inverse of an el liptic integral. Yet, although the use of such modern-sound ing terminology may seem somewhat anachronistic when considering mathematics from this period, it should be borne in mind that this definition was made possible largely by Jacobi's work on the subject. I therefore use this defin ition to evaluate Jacobi's claim that 23 December 1 7 5 1 was "the birthday of elliptic functions."
Elliptic Integrals and Functions from Jakob Bernoulli to Carl Jacobi Problems requiring what are now known as elliptic inte grals can be traced as far back as the mid-1 7th century. For example, in his A rithmetica infinitorum of 1 655, John Wal lis investigated the arc lengths of the ellipse and the hy-
perbola. A little later, for example in the work of Christi aan Huygens, problems concerning pendulum motion yielded questions on arc lengths of the cycloid. Later still, the work of Jakob Bernoulli provided yet another example of a problem whose solution required the evaluation of an elliptic integral, that is, a nonelementary integral of the form
u
= I R( t, Vi{])) dt,
where R is a rational function and P is a third- or fourth degree polynomial with no multiple factors. In a 1 694 paper on elasticity [6], Bernoulli investi gated the rectification of the lemniscate curve, which, with half-axis 1 , would today have the Cartesian equation (x2 + y 2) 2 = x2 - y 2 , or r2 = cos 28 in polars. One of the results Bernoulli was able to prove was that, in order to find the arc length of such a curve, it was necessary to eval uate a "lemniscatic integral" of the form
j(x) =
fx o
dr ,� · v l - r·
This paper by Bernoulli marked the first explicit appearance of what would today be called an elliptic integral, and, nearly a quarter of a century later, was to serve as the basis for a work crucial to the subject's subsequent development. In 1 718, Giulio Fagnano published a paper containing perhaps his most profound discovery: a formula for dou bling or bisecting the arc length of a lemniscate [ 1 3P Tak ing as his starting point the integrand of Bernoulli's lem niscatic integral, Fagnano solved the differential equation
dt 2 dx v1=7 = � to obtain the solution 2x � ---4
t=
1 + x
What Fagnano had discovered was that, if the arc length of a lemniscate was
I
X
0
dt
vl=7 '
4A meromorphic function is analytic everywhere, except possibly at isolated poles. 5For more details on this aspect of Fagnano's work, see [4] and [27] .
ADRIAN RICE received his PhD from Middlesex Universrty, London,
in 1 997, and he also holds degrees from King's College London and
Universrty College London. His research is on the history of mathe matics, specializing in 1 9th- and early 20th-century British mathematics. In 1 998 he came to America, serving as a visiting professor at the Uni versrty of Virginia for one year, and since 1 999, he has been Associate Professor of Mathematics at Randolph-Macon. Department of Mathematics Randolph-Macon College Ashland, VA 23005-5505 USA e-mail: [email protected]
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
49
as Bernoulli had proved, then the integral
f2xvl=X' l+x4
o
u
dt
V1 - t 4
will yield double that length. It was this result that caught the attention of Euler shortly after he received Fagnano's collected works on 23 Decem ber 1 75 1 . His recognition that Fagnono's theorem was cru cial in understanding integrals of this type stimulated him to write a stream of publications on elliptic integrals. The first of these was submitted to the Berlin Academy on 27 January 1 752, just five weeks after his receipt of Fagnano's work [ 1 1]. This, together with a second paper on the sub ject [ 1 0] , read before the St. Petersburg Academy on 30 April 1753, was finally published in 1 76 1 . Clearly influenced by Fagnano's earlier research, Euler tackled the far more dif ficult problem of finding any multiple of a lemniscatic arc length. Beginning with the general differential equation m
dx
_
Vl-7 -
he showed that, when expression
m =
x
0
+ yVl-7 . V1="]J4 1 + x2y2
dt
Il(n,4J)
and
=
1"' o
o
r Y1
=
0
-
-
df)
(2)
k 2 sin2 f)
k2 sin2 f)df)
(1 + n sin2 f))
(3)
df)
Y1 -
k 2 sin2 f)
.
(4)
Legendre called the value 4> the amplitude of the function u, or am u, and the arbitrary constant k its modulus, which he usually took to be between 0 and 1 . Incidentally, the name "elliptic integral" is derived from the fact that, in or der to find the arc length of an ellipse, an integral of the second kind must be evaluated ? Of his three kinds of elliptic integrals, Legendre regarded the first as the most important. With it, he associated two important constants, determined by the modulus k and its complement k' = �:
n = 1 , it was satisfied by the
x
-vf=7
E(4J)
I"' Y1
n dy
(1)
Euler's generalization o f Fagnano's theorem was there fore that, if
J
F(4J) =
V17 '
or, in terms of the constant c,
c=
=
y J 0
dt
-vf=7
K' =
I
TT/2
df)
0
These "complete elliptic integrals of the first kind" were to feature prominently in the work of Abel and Jacobi, as would a subsequent formula, one of many derived by Le gendre. Abbreviating the function Y1 - k2 sin2 f) by tl(f)), Legendre considered the differential equation
where 4> and ljJ are variable amplitudes with a constant sum. Integrating this equation gave
are two arc lengths of a lemniscate, then their sum would be given by or the sum of two elliptic integrals of the first kind, a result of which Fagnano's theorem is obviously a special case when x = y. Clearly fascinated by the possibilities cre ated by this new subject, Euler subsequently derived a mul titude of results on elliptic integrals. 6 For example, he fur ther generalized the previously mentioned theorem, extending it to hold for such integrals featuring the square root of any fourth-degree polynomial. Euler's work effectively created a brand new mathemati cal area, namely the subject of elliptic integrals, although at the time of his death in 1 783 it plainly lacked both a name and a cohesive and unifying theory. Both were provided by the Frenchman Adrien-Marie Legendre, who for a forty-year period from 1 786 developed, refined, and systematized the subject, culminating in his three-volume Traite des Jonctions elliptiques (1825-1828) [22]. Perhaps Legendre's most funda mental contribution was his classification [2 1 , vol. 1 , p. 1 9] of elliptic integrals into three distinct types:
6For a recent study of Euler and the early history of elliptic integrals, see
F(4J) + F(ljl)
=
F(4J
+
ljl).
(5)
He was then able to prove that, providing F(4> + ljl) is con stant, sin C4> + ljl)
=
sin 4>· cos ljJ· tl(ljl) + cos 4> sin ljJ tl(4J) , (6) 1 - k2 sin2 4> sin2 ljJ ·
•
a formula that reduces to the standard addition formula for the sine function when k 0. It is ironic that, during the four decades that Legendre devoted to elliptic integrals, he was almost alone in show ing an interest in the subject. It is doubly ironic that, in 1827, just as his career was coming to an end, the subject attracted the attention of two of the brightest up-and coming mathematical minds of the time, who were to trans form it entirely, rendering much of Legendre's forty-year la bor obsolete. Interestingly, their work appears to have been produced almost simultaneously and, as far as can be de=
[g] .
71t is confusing that Legendre's "elliptic functions" are now called elliptic integrals. We shall shortly see the origin of the modern definition of elliptic functions.
50
THE MATHEMATICAL INTELLIGENCER
termined, more or less independently. The two were Niels Henrik Abel and Carl Gustav Jacobi. In 1827, Abel and Jacobi were both in their mid twenties, mathematicians of exceptional power and ability. It is difficult to say with certainty which of the two first be came interested in elliptic integrals. Two letters from Jacobi to the astronomer Heinrich Schumacher dated 13 June and 2 August 1827 contained the earliest notice of Jacobi's work on the subject. These letters, whose primary content was an account of Jacobi's extension of Legendre's recent discover ies on the transformation of elliptic integrals, were published by Schumacher in the September 1827 issue of Astronomis che Nachrichten. In the same month, Abel's first paper on elliptic functions appeared in the second volume of Crelle's journal (1], containing the first explicit inversion of elliptic integrals for its own sake, and the introduction of Abel's three elliptic functions c{J(x) , F(x) , and j(x). Unlike Jacobi, Abel was more interested in applying his new elliptic functions to the problem of dividing the lemniscate into n equal parts, a solution of which is also contained in his 1827 memoir. On 18 November 1827, perhaps influenced by reading Abel's pa per in Crelle, Jacobi wrote a third letter to Schumacher in which he used the inversion of an elliptic integral for the first time to give definitions of three elliptic functions es sentially equivalent to Abel's. This letter was published in the Astronomische Nachrichten that December. Although their motivations were somewhat different, and the notations they used also varied,H at the core of both Abel and Jacobi's work on elliptic functions lay some com mon, fundamental results. They both rewrote Legendre's three kinds of elliptic integrals using the simple substitu tion t = sin e to obtain u=
defined the "sine of the amplitude, " x = sin(am u) = sin ¢, to be the inverse of u=
cos(am u)
(The notation for Jacobi's new sine, cosine, and delta func tions was later abbreviated to snu, cnu, and dnu, respec tively 9) From these definitions, it was not difficult to de rive certain properties and identities, such as
d du
- (sn u) = en u · dn u and sn u · en v · dn v + en u · sn v · dn u sn(u + 0 = ����--�� ����--�� 1 - k2 sn2 u sn2 v
en u · sn K dn u 1 - �sn2 u sn2K ·
1 - k2sn2 u
dt
came
V( l - t2
cn( u + K) = - k '
- k2 t2) - k' 2 t2 )
dn( u + K) =
.
Both Abel and Jacobi (and indeed Legendre) were aware of the analogy between elliptic integrals and inverse trigonometric functions; for example, when k = 0, the in tegral of the first kind is simply the inverse sine function, and K becomes merely 7r/2. However, they differed from their predecessors in taking as the principal subjects of their study, not the elliptic integrals themselves, but the func tions inverse to them. For example, given an elliptic inte gral of the first kind, u, with amplitude ¢ = am u, Jacobi
en u dn u
Similarly, using equivalent addition theorems for the en and dn functions, it could be shown that
In particular, the complete integrals of the first kind be
2��1 ��
(8)
already known to Legendre as formula (6). From here, it was easy to prove that Jacobi's so-function was in fact periodic in 4K Since, by definition snK = 1 and cnK = V1 - sn2K = 0, then by the addition theorem (8),
en u . dn u dn2 u
Y(l - t
(7)
en u · dn u
�� 2) . 2 t� J �(_1_+_n_t�2)--r Y( 1 -=� 1=-== )(:= k� t2:=
K' =
V 1 - sin2 (am u)
Mam u) = V1 - k2 sin2 (am u).
V1 - k2 t2 dt, Y 1 - tz
f f
=
and
dt
K=
�
d 7C:= Y ) := 1 =_=""' ) .. ' 1 =_==: t2.,. t 2:= k2:=
with complementary functions
J -V7C:=1=_==e:t2;;,)(�1=_==:k2:=t2.,.) . ' J
r
sn u dn u
k'
--
dn u
.
Hence, sn(u + 2K) =
cn(u + K) = - sn u. dn(u + K)
Therefore sn(u + 4K) = sn((u + 2K) + 2 K) = - sn(u + 2K) = sn u. None of this was a particular departure from the kind of generalized trigonometric-style results previously ob tained by Legendre, albeit with a revised notation and a
81n the interests of uniformity, I shall largely use Jacobi's notation in preference to that of Abel. 9Jacobi's snu, cnu, and dnu functions were essentially equivalent to Abel's elliptic functions (x), F(x), and f(x), respectively.
© 2008 Springer Sc1ence+ Business Media, Inc., Volume 30, Number 2, 2008
51
slight change of emphasis. Indeed, as we have seen, re sults equivalent to the addition theorems produced by both Abel and Jacobi had already been derived by Legendre, and concepts now recognizable as modern elliptic functions ap pear in formula (6) of Legendre and ( 1 ) of Euler. How then did Abel and Jacobi transform the subject? Up to this point, all functions had been considered from a purely real standpoint. No complex quantities had en tered the subject-although Legendre had on occasion used an imaginary parameter n in his elliptic integrals of the third kind. But in 1 827, both Abel and Jacobi found that they were able to obtain perfectly meaningful results when us ing values from the complex domain in their new func tions. For example, sn(K + iK ) = 1/ k and cn(K + iK ) = 0. From here it was not difficult to prove that these new func tions also had complex periods; for example, the so func tion was also found to have period 2 iK , so that, in gen eral, sn(u + 4 mK + 2 niK ) = sou. They had dramatically extended the scope of the subject, for now not only were their functions defined over the complex plane, but they had been found to be doubly periodic. This characteristic was further exploited by Jacobi in his subsequent work on elliptic functions, particularly his Fun damenta nova theoriae function urn ellipticarum ( 1829), in which he introduced an important new tool, known as theta functions, to refine and develop the theory [8] , [20] . These theta functions, which Jacobi introduced first as infinite products and later as series, were not only rapidly conver gent, but also had doubly periodic quotients. His stroke of genius was the realization that, for all values of the inde pendent variable, elliptic functions could be represented as quotients of theta functions. Jacobi's use of theta functions to construct elliptic functions formed the basis of the new theory contained in the Fundamenta nova. As the first sys tematic exposition of elliptic functions (as opposed to el liptic integrals), Jacobi's Fundamenta nova definitively es tablished the subject as a bone fide mathematical discipline, making 1829 the latest possible date that could be consid ered as the "birthday" of elliptic functions. '
'
'
'
Possible "Birthdays" But what do we mean by "the birthday of elliptic functions" anyway? Well, for the time being, let us regard it simply as the date on which a doubly periodic function defined over the complex plane first appeared in the writings of a math ematician. We can now look back over the previous list of mathematical developments and determine whether they meet our criterion. The first date on our list was 1694, fea turing in the work of Jakob Bernoulli the earliest published example of what we would now recognize as an elliptic integral. Although clearly a nontrivial event, since it does not fit our definition, we cannot regard this date as the birthday of elliptic functions. Similarly, the ingenious work of Fagnano in 1 718, despite its catalytic effect on the sub sequent research of Euler, remains a contribution to the theory of elliptic integrals, not elliptic functions. Turning now to Euler, should we, with Jacobi, view 23 December 1751 as the birthday of the subject, or are the dates of his subsequent contributions on elliptic integrals (27 January 1 7 5 2 and 30 April 1 753), or the year of their
52
THE MATHEMATICAL INTELLIGENCER
first publication ( 1761), stronger candidates? Here, for the first time, it would appear that actual results involving what we would today recognize as elliptic functions are in evi dence, as, for example, in formula ( 1) . However, it is only with a retrospective knowledge of elliptic functions that such observations can be made, as Euler's papers in this area were primarily concerned with elliptic integrals, and there is nothing in them that bears any resemblance to a doubly periodic complex function. That said, however, there is no doubt that Euler's work in this area did mark the beginnings of the theory of elliptic integrals. Which brings us to the not insubstantial work of Le gendre between 1 786 and 1 828; he worked longer than anyone else had on elliptic integrals, establishing a coher ent and systematic mathematical theory, and even giving the subject a name. Moreover, it is certainly possible to note the appearance of elliptic functions in his work. For ex ample, his addition theorem (6) features (what we would now recognize as) Jacobi's three elliptic functions, so, en, and do. Indeed, in passing from equation (5) in elliptic in tegrals to formula (6) in elliptic functions, Legendre was implicitly treating the elliptic integral as the inverse func tion of the amplitude. But this inverse relationship, and these inverse functions, while present in Legendre's work, were not his main focus, nor (with the exception of the oc casional use of an imaginary value in his third kind of el liptic integral) does his work feature complex variables. The amplitude 4> and modulus k are always regarded as real quantities, with the result that Legendre's elliptic functions are always real-valued. Consequently, such elliptic functions that do appear in Legendre's work are never found to be doubly periodic. The realization of the importance of these inverse elliptic functions, the extension of their definition to the complex domain, and the resultant discovery of their double periodicity would have to await the insights of Abel and Jacobi. It is in their publications from late 1827 that we see the full emergence of our definition of an elliptic function as a doubly periodic function over the complex numbers. Here at last, elliptic functions become the principal object of study, with both Abel and Jacobi taking full advantage of the extension of their definition to the complex domain. The publication of Jacobi's Fundamenta nova in 1829 is also significant in two key respects. First, it coincided, sadly, with Abel's premature death in April of that year. Second, and most important, it marked the appearance of a definite theory of elliptic functions, in the modern sense, as doubly periodic functions defined over the complex plane. However, despite this progressive step, neither Abel nor Jacobi used complex variables in a way that would be prac ticed today. Although their complex-valued elliptic func tions resulted from inverting a (sometimes) complex-valued integral, these integrals were never taken over complex paths, and their complex function theory concentrated more on the algebraic derivation of formal results than the more subtle analytic discipline, of which Cauchy was then in the process of laying the foundations. Their theory was there fore far from finished or complete, requiring, in addition to Cauchy's complex analysis, the introduction of topological surfaces by Riemann, and the subsequent introduction of
new elliptic functions by Weierstrass and his followers, be fore it came to resemble the subject as it is presently un derstood. Nevertheless, by 1829, elliptic functions had emerged as a vibrant new area of mathematics in the space of just two years. We have thus considered the contributions of six mathe maticians to the theory of elliptic integrals and elliptic func tions made between 1 694 and 1829. Although all were cru cial to the development of the subject, the first appearance of elliptic functions (understood as doubly periodic functions defined over the complex plane) seems to have been in the memoir by Abel published in September 1827, with their first systematic treatment being in Jacobi's Fundamenta nova of 1829. Which of these two events should then be regarded as the "birthday" of elliptic functions? From a purely chrono logical approach, it seems reasonable that the first appear ance of elliptic functions would be in the paper by Abel from September 1 827. And yet, despite all the evidence, it might still be argued that this conclusion is not completely correct. For, were we to make such a judgment, we would be over looking a development made some thirty years before either Abel or Jacobi wrote a word on the subject-an unpublished contribution by one of their most distinguished senior con temporaries: Carl Friedrich Gauss.
Of course, in retrospect we can see that Gauss's lem niscatic sine function, x = sl(u), where
is really just Jacobi's x ulus k = i, since
Moreover, since
du d = dx dx
u=
LX o
�
du
(sl u)
= dx du
sl(u + v)
(Lx
dt
Y1 - t 4
o
)
1 Yl - x4 '
=
1 = -- =
du! dx
�=
Y 1 - sl4 u.
d dv
d du
sl u -(sl v) + sl v -(sl u) ·
=
·
-------,----=-----
1 + sl2 u sl2 v
d dv
d du
sl u · -(sl v) + sl v -(sl u) ·
and, generalizing the modulus to some arbitrary k (which in turn changes the sl( u) function to the more general Ja cobian sn u) gives the well-known
sn(u + v)
=
d dv
d du
sn u -(sn v) + sn v -(sn u) ·
·
-------
1 - k2sn2 u sn2 v
sn u en v dn v + en u sn v dn u 1 - k2sn2 u sn2 v
�-
·
-
sn u function with imaginary mod
Gauss's addition theorem (9) then becomes
dt
By analogy with the sine function from standard trigonom etry, Gauss defined the "lemniscatic sine" function, x = sl(u), to be the inverse of this integral , with the "lemnis sl2(u). He was then able to catic cosine" as cl( u) = \h deduce a variety of results pertaining to his new functions, such as the following addition theorem, which follows from Euler's 1 753 generalization of Fagnano's theorem. Recall that Euler had solved the lemniscatic differential equation
=
df
V1 - t4 '
then, if x = sl(u),
Gauss's Claim to Priority In addition to his first-rate and highly original published work, Gauss is also remembered for publishing only a frac tion of his many profound mathematical discoveries. Among the many pieces of work left unpublished until well after his death are two manuscripts, one of which was begun on 8 January 1 797, just a few months before his twentieth birthday [ 1 7] , [18], [ 1 6 , vol. 3, p. 493] (see Fig. 2). The man uscripts concern the lemniscatic integral as studied by Bernoulli, Fagnano, and Euler,
X Lo
u=
·
·
·
We thus see how Gauss's addition theorem is connected to those of Euler, Legendre, Abel, and Jacobi. But we may not yet conclude that Gauss's lemniscatic sine was in fact an elliptic function. To do this, we still have to show that it is doubly periodic when defined over the complex domain. Gauss initially proved its periodicity with respect to a con stant he defined, again in analogy to regular trigonometry, as w
z
dx
=
fl
dt
o �-
Since, by his definition of the function, sl(w/2) Gauss's addition theorem (9) we find that
to obtain c
=
x
+ Vl=)fi y� . 1 + x2y2
sl( u}V 1 - sl4(v) + sl( v}V1 - sl4( u) . sl(u + v) = 1 + sl2( u) sl2( v)
(
sl(w) = sl � +
The realization that Euler's x corresponded to Gauss's sl( u), the y matched his sl(v), and the constant c was equivalent to his sl(u + v), resulted in the following addition theorem: (9)
(by (7))
2
�) 2
2sl =
(�) 2
1 - sl4
1 + sl4
(1'-)
(�) 2
=
=
1 , using
0
so that sl(2w)
=
sl(w + w) =
2sl(w) Y 1 - sl4 (w) 1 + sl4 (w)
=
0.
© 2008 Springer Science+ Business Media, Inc., Volume 30, Number 2, 2008
53
A final application of the addition theorem then gives sl(u +
2w)
=
sl(u).
2w,
Having found that one period of sl(u) is Gauss then extended the domain of the function to the complex num bers. Given that the identity
d(it) . dt V1 - t4 Yl -Cit)4 implies that sl(iu) i sl(u), it is easy to deduce that sl(u + 2iw) = sl(u)Yl -sl4(2iw) + sl(2iw)Yl -sl4(u) 1 + s12(u) sJ2(2iw) sl(u)Yl -sl4(2w) + isl(2w)Y1 - sl4(u) 1 + sJ2(u) i2s!2(2w) sl(u)\11"=0 + i 0 Yl -sl4(u) 1 sl2(u) 0 = sl(u). = r
r= = =e-
-
=
·
-
·
·
Gauss thus found that his lemniscatic sine function was doubly periodic, 1 0 with periods 2w and In other words, by our definition, he found that it was an elliptic function. Despite being written in these two manuscripts on lemniscatic integrals were not published until 1876, when they appeared in the third volume of Gauss's collected works [16] . Given that these papers contain an earlier ex ample of functions matching our definition, their status as special cases of functions defined three decades later by Abel and Jacobi, and their relationship to previous work by Bernoulli, Fagnano, and especially Euler, it would therefore seem that to ignore the mathematics contained in the pa pers would disregard a nontrivial contribution to the sub ject that falls well within the historical period of consider ation. Should we then revise our initial conclusion and regard as the "birthday" of elliptic functions? Recall that our definition of "birthday" made no explicit reference to a date of publication. We defined it simply as the date on which a doubly periodic function defined over the complex plane first appeared in the writings of a math ematician; no stipulation was even made that the work be published at all. This of course implies that it may have had no influence on later developments, and indeed, while it was known that Gauss had worked on elliptic integrals, thanks to his correspondence and the little that he did pub lish on the subject, his creation of the first elliptic functions in had no effect on the subsequent evolution of the topic whatsoever. But how can be described as the "birthday" of elliptic functions if the mathematical commu nity at large was not aware of this work for nearly eighty years? This objection brings to mind an analogy with the fun damental early work of Newton on the calculus between and Although via correspondence and the cir culation of certain manuscripts, Newton's claim to priority in the development of an efficient algorithm had been vig orously asserted, nothing was published by him on the sub-
2 iw.
1797,
1797
1797
1666
1797
1672.
Figure 2.
The first entry on this page from Gauss's mathe
matical diary contains reference to his work on the "Curvam Lemniscatum" and its associated elliptic integral, undertaken on 8 January
1797.
ject until the 18th century, by which time his contributions had already been outdated by more recent continental de velopments-a similar situation to Gauss's work on the lem niscatic functions. Although no one would seriously deny that Newton was one of the founders of the calculus, to la bel Newton's early fluxional research as the absolute "birth day" of the subject would not be without controversy, as such a claim ignores the subject's lack of immediate influ ence and the subsequent independent involvement of Leib niz . 1 1 A similar assertion of Gauss's priority with regard to elliptic functions would likewise overlook the crucial con tributions of Abel and Jacobi. Another argument against claiming priority for Gauss would be the more technical objection that, despite being the first examples of explicit inversion of elliptic integrals as well as featuring the essential characteristic of double
1 0According to Gauss's mathematical diary, this realization appears to have occurred to him on 19 March 1 797. 1 1 An excellent discussion of this point, as well as the status of unpublished work in mathematics, is contained in [23].
54
THE MATHEMATICAL INTELLIGENCER
periodicity over C, his elliptic functions fail to contain a further crucial feature of such functions, namely, the flexi bility of the modulus. As we saw, in Legendre's work, k was usually limited to real values between 0 and 1 , but what truly distinguished Abel and Jacobi from their prede cessors was their willingness to allow any value of k, real or complex, a revolutionary step, and one that totally changed the scope of the subject. In the previously men tioned work by Gauss, of course, the modulus is simply equivalent to the imaginary constant i, with the result that the periodic lattices of the sl function in the complex plane are simply squares of side length 2w. In Abel and Jacobi, each of their elliptic functions has a potentially infinite num ber of possible configurations, depending on the value of k, and this in turn is reflected in a greater generality in the shape of the periodic lattices, which are either rectangles or, more generally, parallelograms. So Gauss's elliptic functions do not conform to the gen erality of the modern definition of an elliptic function. But then, which definition should we use? If we go by that adopted at the beginning of this paper, then there would seem to be no doubt that Gauss's work marks the "birth day" of elliptic functions . On the other hand, if we also in sist on a flexible modulus, then the true birthday would be marked by Abel's paper of 1 827. Then again, if we are look ing for the first (implicit) appearance of mathematical ob jects equivalent to elliptic functions, the birthday of the sub ject can be traced all the way back to formula ( 1 ) of Euler, as Jacobi in effect suggested. Thus we find that the period from 1 694 to 1829 is punctuated with a variety of possible dates, any of which could qualify as possible "birthdays" of elliptic functions, depending on how the subject is de fined. All of this would seem to imply that such mathe matical constructs cannot really be said to have been "born," but rather to evolve over time. As they evolve, so too do their definitions, often with the result that by the time one comes to trace their history, the current definition may bear little or no resemblance to the concept as originally iden tified.
Conclusion Is it possible to answer the question What is the "birthday" of elliptic functions? Yes, but far from uniquely. But does the overabundance of possible answers occasioned by the inherent naivety of the question mean that such lines of in quiry are pointless for the historian? Can questions regard ing the temporal origins of mathematical areas and the re search to which they lead ever be useful or instructive? First, any question that leads a mathematician to think historically about the origins and evolution of his or her subject is a good thing. Indeed, the history of mathematics abounds with examples of mathematical research inspired or influenced by an examination of the origins, or at least the early development, of particular lines of investigation. As examples, Fermat's creation of the rudiments of analytic geometry in the 1 7th century arose in part from an attempt to reconstruct the content of Apollonius's Plane Loci; the
first satisfactory treatment of complex logarithms by Euler in 17 49 was inspired by his reading of the debate between Leibniz and Johann Bernoulli on the matter; and Lagrange's important memoir on the solvability of quartic equations in 1 770 surveyed and codified previous work on the theory of equations by many of his predecessors, including Viete, Tschirnhaus, Euler, and Bezout, as well as setting the stage for subsequent attempts to develop a strategy for the quin tic. There is thus ample evidence that research into the be ginnings and evolution of a mathematical area can actively stimulate current research in the discipline. In the words of no less an authority than Poincare, "If we wish to foresee the future of mathematics, our proper course is to study its history and present condition." 1 2 Second, such questions and answers can even help to uncover obscure contributions to subjects whose history is deemed to be already known. For example, for much of the 20th century the history of the 1 9th-century develop ment of symbolic logic was thought to be fully docu mented-until previously unknown manuscripts by Charles Dodgson came to light in the 1970s; had these manuscripts been published when originally written, they would have contained arguably the most novel contributions to English symbolic logic since the foundational work of Boole and De Morgan in the 1 840s and 1850s. 1 3 Hence, such basic questions as when a subject began can lead to important discoveries concerning the development and early history of a topic. Finally, such questions serve to remind both the histo rian and the mathematician that even the history of such a seemingly objective study as mathematics can never be any thing other than subjective. This is because, once again, everything relies on how one defines the topic under in vestigation. Did group theory begin with Galois's intro duction of the concept in 1 83 1 , with Lagrange's work on permutations in 1 770, or with Weber's axiomatic approach to the subject in 1 895? Do the origins of game theory lie in Von Neumann and Morgenstern's classic work of 1 944, or with previous mathematical analyses of games and strate gies by Borel and Zermelo? Should complex analysis be re garded as having begun with the publication of Cauchy's memoir on complex integration in 1 825, or should it date from the composition of the paper in 1 8 1 4? In either case, what then are we to make of the fact that the Cauchy Riemann equations were known to D'Alembert as early as the 1 750s? Thus, while the continuous evolution of mathe matics makes it essentially impossible to locate (or even define) the "birth" of a mathematical discipline, the multi tude of possibilities arising from such a search result in a whole string of dates, leading to a more complex, but far richer and more nuanced, picture of the prehistory and early development of the subject in question. In short, such questions encourage us as historians and mathematicians to probe deeper into our knowledge and understanding of where our subjects of study came from, how and when they evolved, and why. In so doing, even in the reappraisal of well-trodden paths, we have the po-
12"Pour prevoir l'avenir des mathematiques, Ia vraie methode est d'etudier leur histoire et leur etat present." [25, p. 1 67] 13See [5] and, for commentaries, see [2], [3] , [24].
© 2008 Springer Science+Bus1ness Media, Inc., Volume 30. Number 2, 2008
55
tential to learn more about mathematics, more about its his tory, and more about the nature of the history of mathe matics itself.
varurn irrectificabilium. Novi Commentarii Academiae Petropo/i tanae
6, 58-84. In Opera Omnia , ser. 1 , Vol. 20, 80-1 07.
[1 2] Euler, L. (1 91 2-1 9 1 3) Leonhard! Euler! Opera Omnia , ser. 1, vols. 20-21 , Leipzig and Berlin, B. G. Teubner. [1 3] Fagnano, G. C. ( 1 7 1 8) Metoda per rnisurare Ia lernniscata. In Opere
ACKNOWLEDGMENTS
The author wishes to thank Sloan Despeaux, Ivor Grattan Guinness, Peter Neumann, and an anonymous referee for valu able comments and suggestions made on an earlier draft of this paper.
matematiche,
vol. 2 , 293-3 1 3 .
[1 4] Fagnano, G. C. (1 750). Produzioni matematiche, 2 vols., Pesaro, Gavelliana. [1 5] Fricke, R. ( 1 9 1 3) Elliptische Funktionen. In Encyklopadie der math ematischen Wissenschaften ,
vol. 2, pt. 2, article II B 3, Leipzig,
B. G. Teubner, 1 77-348. REFERENCES
[1 ] Abel, N. H. (1 827) Recherches sur les fonctions elliptiques. Jour
[1 6] Gauss, C. F. (1 863-1 929) Werke, 1 2 vols . , Gottingen, Konig lichen Gesellschaft der Wissenschaften.
2, 1 01-1 81 . In Oeu
[1 7] Gauss, C. F. (1 876a) Elegantiores integralis f dx I \IT=X4 pro
[2] Abeles, F. (2005) Lewis Carroll's formal logic. History and Philos
[1 8] Gauss, C. F . (1 876b) De curva lemniscata. In Werke, vol. 3,
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3 vols., Paris, Huzard-Courcier.
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[23] Mahoney, M. S. (1 984) On differential calculuses. Isis 75, 366-372.
[8] Cooke, R. (2005) C. G. J. Jacobi's book on elliptic functions (1 829).
[24] Moktefi, A. (2007) Lewis Carroll's logic. I n D. M . Gabbay and J.
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THE MATHEMATICAL INTELLIGENCER
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[27] Watson, G. N. (1 933) The marquis and the land-agent: A tale of the eighteenth century. The Mathematical Gazette 1 7, 5-1 7.
Math ematica l l y Bent
The proof is in the pudding.
Colin Adam s , Editor
I
M athe matics Satisfaction Survey
6.
7.
COLIN ADAMS
Opening a copy of The Mathematical
Intelligencer you may ask yourself
uneasily, "What is this anyway-a
mathematical journal, or what?" Or you may ask, "Where am /?" Or even "Who am /?" This sense of disorienta tion is at its most acute when you
open to Colin Adams's column.
Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Wi l l iams College, Williamstown , MA 01267 USA e-ma i l: Colin.C .Adams@wi l l iams .edu
58
hank you for participating in this mathematics satisfaction survey. With feedback from people like you, we can make mathematics the best it can be. 1. Are you satisfied with the letters and symbols used to denote num bers, variables, and functions? Do you think "e" is an important enough number to deserve capital ization? Do you feel that the pre dominance of Greek letters does not represent the diversity of cul tures that participate in mathemat ics? Do you feel that certain letters in the alphabet have been typecast as constant, variable, index, or function? Is this fair or appropriate? 2. Given two functions, how do you decide which is better? Do you have a ranking of all the functions you know in your head? Or is it more of a partial ordering? 3. Are you comfortable with the fact 1 + 1 = 2? Should 1 + 1 = 3? (Note: There would be additional consequences if this change were made.) 4. Is it okay that we call irrational numbers irrational, or do you feel that this has a derogatory connota tion these numbers do not deserve? Or do you wish these numbers would just go away? How about fractions and decimals? We could make those go away, too. 5. When we say we are 25 years old, should that mean we have com pleted 25 years or we are in our 251h year? Do you believe that this
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8.
9.
10.
11.
12.
13.
question belongs on a math satis faction survey? Why or why not? Is it fair that some people are bet ter than others at mathematics? Should we even the playing field? If so, how? When learning mathematics, what is your primary goal? a. To have a quantitative advan tage when competing in the marketplace. b. To appreciate its absolute in trinsic beauty. c. Because my parents make me. I am only twelve years old. d. To make myself more attractive to others. If your answer to 7 was d, how is that working for you? And what area of math works best? Have you ever visited the campus at San Clemente State? What is your impression of the math department there? What is the greatest theorem of all time? Is your decision unduly in fluenced by the fame of the name of the person who proved it? Should you consider, perhaps, the orems proved by lesser known mathematicians? Has mathematics become too large, like a bloated corporation that can't respond quickly to changes in the marketplace? Should we spin off certain fields of mathematics, such as functional analysis, to new fields of endeavor, called, say, advanced arithmetic? Can you do mathematics with mu sic playing in the background? What if the music is coming out of an ad jacent office and it is the theme song from Gilligan 's Island, and it is played over and over again? Have you ever met Adam Cleghorn, the "famous" functional analyst at San Clemente State? What do you think of him? Does he deserve the attention he gets, attention that is denied to other members of his de partment?
14. Do you think that instead of offer
15.
16.
17.
18.
ing a million dollars for the Clay problems, the Clay Institute would generate more interest by offering a Ferarri, or perhaps a time-share in Fiji? If someone comes up with a good idea for a prize, such as the Ferarri and/or the time-share, do you think the Clay Institute should give the money saved from the original million dollars to the per son who came up with the idea? Have you ever fallen asleep during a math talk? Was it because you were so upset you didn't sleep the night before or was it because the speaker was inspiringly dull? Have you ever fallen asleep during a talk by Adam Cleghorn, for either of the previously mentioned reasons? If so, when and where' Should there he more than five math jokes? Does the abelian grape riddle count as a joke? Should the number and type of prizes awarded in mathematics be expanded to "spread the wealth"? How about a prize for penmanship? Or congeniality? Suppose six noncommutative alge braists are milling around on a train track, discussing Maschke's Theo rem, and one functional analyst has his shoelace caught on the rail of the sidetrack. The train is coming and you control the switch. Would you switch the train to the side track, thereby saving six but sacri ficing one? What if everything is the same, but the algebraists wander off the track as they are discussing Wedderburn's Structure Theorem.
19.
20.
21.
22.
23.
24.
25.
Would you still switch the train to the sidetrack? Should we put a moratorium on new results until we can all catch up with the old ones? Should mathematics have sponsor ships as NASCAR does? If so, which oil filter makers should we approach? Do you think one individual should have the right to determine the re freshments at math teas? What if that individual has a bias against frosted cookies of any kind? Have you ever lied to another mathematician' Say, told him that you would keep in confidence an error he discovered in his own pre viously published proof, and then subsequently announced it at a conference after drinking too much at the banquet? What should the penalty be for such behavior? Should the Fields medal be opened up to older ( >40) mathematicians? What if a mathematician will not tell you his or her age? What about a mathematician who is 48, but re ally sees his best results ahead of him' What about a prize for him? Would more people go into mathe matics if it were easier? How can we make it easier if we decide to go this route? Could we give more points for effort, and fewer for results' Do you think there should be a mathematical "prison" for mathe maticians who behave in an un ethical manner? If so, do you think it should he associated with a uni versity' Should the punishment be having to attend colloquia, or not being allowed to attend?
These Next Few Questions Are for the Experts Only 26. Where will mathematics be in 100 years? What will be the big results by then? And how do you prove them' 27. Would you like to use your repu tation to help some mathematicians who are not as successful as you are? 28. Did you know that Adam Cleghorn takes pens home from the Mathe matics Department at San Clemente State and uses them for nonmath ematical purposes? Did you know that he helps himself to coffee in the lounge without tossing a dime in the coffee fund can? Did you know that he didn't have his first girlfriend until he was 27?
Last Question for All Survey Participants 29. In reading this over, it appears you may get the impression that I have a problem with Adam Cleghorn. Nothing could be further from the truth. I have no problem with the man, only with his actions. So the last question is, would you be will ing to sign a petition to revoke Adam Cleghorn's PhD on the grounds that the entire mathemat ics community would benefit? Thank you for taking the time to fill out this survey. We appreciate your feedback and we will pass the infor mation we collect along to the mathe matical authorities for serious consider ation. With your input, mathematics can only get better and better.
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K:OJIMOfOPOB B BOCIIOMMHAHIDIX yq£HMK:OB Pe)J;a.KTOP cocraBHTeJTh A.H. IIIHp�eB, IlO)J;pOTOBKa TeKCTa H.r. Xnw�eHKo MWJMO, MOCKBA 2006
l
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A. KUSHNER
ndrei Nikolaevich Kolmogorov (25 April 1 903-20 October 1 987) was, beyond doubt, among the greatest mathematicians of all time. Thanks, moreover, to the manifold na ture of his creative activities and their universality, he transcends mathematics per se. Kolmogorov was a great scholar, and he was a man who profoundly in fluenced our civilization in general. This book consists of recollections of disciples of Kolmogorov, collected, commissioned, and edited by Al'bert Nikolaevich Shiryaev, who himself is one of Kolmogorov's most accom plished students. The texts were pre-
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pared by N. G. Khimchenko (Rychkova) who worked with Kolmogorov for many years and whose literary talent Kolmogorov highly appreciated. She is a contributor as well. Khimchenko's es say is an artistically written chronicle of Kolmogorov and his circle with lively portraits of individuals and events. Collections of memoirs of people of different ages, temperaments, world outlooks, and political and religious convictions are sometimes confusing and fragmentary. Sometimes a reader feels lost in the mass of details, espe cially when the details contradict one another. This is to some extent un avoidable, since different individuals may see the same events in a different light. Contradictions among conscien tious witnesses are well known in, say, judicial practice. Of course, such con tradictions could not be avoided in the Reminiscences. An attentive reader, however, will obtain from the book a lively and consistent picture of a great scholar, of his school, and of his life in the wider context of the time and soci ety in which he lived. Far from being fuzzy, the picture that emerges has a stereoscopic quality; we see the man and his surroundings from a variety of angles. Every reader will find in the book authors and essays that have a special appeal. The list of contributors is im pressive: A. N. Shiryaev, M. Arata, V. I. Arnol'd, G. I. Barenblatt, Ya. M . Barzdin', Yu. K. Belyaev, A. A . Borovkov, A. V . Bu!insky, B. V. Gne denko, I. G. Zhurbenko, V. M. Zolotarev, L. A . Levin, R. F. Matveev, A. S. Monin, S. M. Nikol'sky, A. M . Obukhov, B . A. Sevast'yanov, Ya. G. Sinai, V. M. Tikhomirov, P. L. Ul'yanov, V. A. Uspensky, N. G. Khimchenko (Rychkova), N. N. Chentsov, A. A. Yushkevich, and A. M. Yaglom. The es says are of high literary quality. They are rather like novellas (the very title of the essay by Shiryaev, "Celestial Attrac tion," speaks volumes). Some articles are closer to mathematical review pa pers, complete with theorems and for mulas; those articles are addressed to
mathematicians. But even in such tech nical essays, a nonmathematician will find many things of interest. Some con tributors include letters, or excerpts of letters, that Kolmogorov wrote to them. Many of Kolmogorov's letters are liter ary gems. There was something Hellenistic about the way Kolmogorov cared for body and spirit. Though never taking part in any sport competitions, Kol mogorov engaged in various physical activities and involved his students in them. These activities included 30-40 km hikes and ski runs, year-round swimming, mountain climbing, long trips on rowboats, and so on. Many per sonal recollections focus on a country house (dacha) in the village Komarovka near Moscow that Kolmogorov shared with his lifelong friend, the great topol ogist Pavel Sergeevich Alexandrov. This house was a kind of oasis, an attraction for young mathematicians who were blessed to be students of Kolmogorov or Alexandrov. This was a starting point of famous hiking expeditions that ended with a shower and a dinner. Both great scholars loved classical music and accumulated a large collection of LPs. Musical evenings at Komarovka that crowned clays of physical and mathe matical activities were engraved in the memories of Kolmogorov's students just as strongly as research discussions. Those "Komarovka" days and hours come alive in many of the book's essays. (See, especially, the contributions of Shiryaev, Uspensky, and Tikhomirov. ) Shiryaev secured the restoration and preservation of the Komarovka estate as a landmark, a memorial to two great mathematicians that lived, worked, and taught there. He knows all too well how difficult this task was. With the exception of number the ory, Kolmogorov made important con tributions to almost all areas of mathe matics, beginning with mathematical logic and foundations and ending with theoretical mechanics and hydrody namics. His grasp of mathematics as a single entity, and a mathematical intu ition that enabled him to predict both the direction of future research and par ticular results, were unparalleled. He personified a rare type of mathemati cian who was able to develop new the ories and solve hard open problems with the same energy and success. In
many cases his contributions and influ ence were decisive. Kolmogorov can be called the Euclid of the theory of prob ability. His book, Foundations of the Theory ofProbability, published in 1933 in German, contains the now univer sally accepted system of axioms for the theory of probability based on measure theory. At the age of just 22, Kol mogorov published the first paper in which intuitionistic logic was subjected to a systematic mathematical treatment. His later work ( 1932) on the interpre tation of intuitionistic logic was another breakthrough in understanding the na ture of mathematical intuitionism and constructivism. The very term "Brouwer Heyting-Kolmogorov (BHK) interpreta tion, '' used in contempora1y mono graphs on mathematical logic, indicates the significance of his contribution. This research, on the border of mathematics and philosophy, was combined with great interest in applied problems, in cluding the numerical analysis of the problems. In applied problems Kol mogorov liked to get clown to concrete calculations, and Reminiscences attests to his exceptional numerical intuition. In Reminiscences, we learn about Kolmogorov the person, Kolmogorov the teacher who created one of the greatest mathematical schools in his tory, and Kolmogorov the lecturer. Nu merous lively episodes narrated by his students show Kolmogorov's approach to teaching and guiding his pupils. It took talent and stamina to be near the great scholar, to be his student. On the other hand, he had an astonishing gift to offer problems and subjects that stim ulated his students to work to the top of their capacity but were not outside their reach. It is interesting to compare the essays by Barenblatt and Uspensky in which they recall their graduate study years. Kolmogorov assigned the same time for their visits to Komarovka and practically simultaneously discussed Uspensky's research progress in math ematical logic and Barenhlatt's in hy drodynamics. In his tenure as the supervisor of the Graduate School of Mathematics at Moscow State University, Kolmogorov closely followed the progress of each PhD student, no matter who was his or her official adviser. Kolmogorov's abil ity to understand every mathematical talk and paper, whatever mathematical
discipline was involved, was legendary. But this "legend" is well supported by Reminiscences. Though I was not a stu dent of Kolmogorov's, I nevertheless had personal contacts with him. On a few occasions he presented my papers to Doklady, and each time he had a short discussion with me. Not only did he know the results of each relevant pa per, but he also knew my previous pub lications. And constructive analysis, my mathematical work area, was far re moved from his own mathematical in terests at that time. No wonder Kolmogorov's school en compassed almost all of mathematics from its foundations to its practical ap plications. Kolmogorov would joke: ''One of my pupils is in charge of the atmosphere and another is in charge of the oceans." Indeed, academician Obukhov was the director of the Insti tute of Atmospheric Physics of the So viet Academy of Sciences, and Monin ( now also an academician) was the di rector of the Institute of Oceanography. Kolmogorov took part as research su pervisor in two long expeditions (one around the world) on the research ship "Dmitrii Mendeleev. " Zhurbenko pre sents a lively account of this important part of Kolmogorov's life. On more than one occasion Kol mogorov invited PhD students to run seminars jointly, especially on new sub jects. Uspensky recalls that Kolmogorov invited him to run a joint seminar on "Recursive Arithmetic" ( 1 953/ 1 9'54). At a meeting of this seminar on 9 Febru ary 1954, in an apparently casual man ner, Kolmogorov formulated the basic definitions and a sketch of what later became the theory of numerations. This was one of many cases, mentioned in the Reminiscences, when seemingly ca sual remarks by Kolmogorov marked the beginnings of very important future developments. ( And how many such deep ideas, expressed in passing, were lost forever!) Kolmogorov was usually so immersed in his enormous intellectual world that he produced an impression of remote ness. Many contributors write about "the Kolmogorov enigma, " the difficulty of coming close to his inner being. "To make a judgment about the inner world of Kolmogorov is as hard as about the inner world of an atom"-writes Us pensky. At the same time, Kolmogorov
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treated everyone who approached him with politeness and respect, be it a col league-academician or a teenage school student. One can see this respect in his answers to letters of correspondents in various walks of life. Uspensky offers two impressive examples: an answer to a letter from a teacher of mathematics of a second-grade school in Smolensk and an answer to a widow who sent to the famous mathematician a volumi nous and rather hopelessly dilettante posthumous work of her husband on set theory. Kolmogorov did not con sider it improper to take care of mat ters that did not require his exceptional qualifications. In both cases he could easily have left the task to his PhD stu dents, but he did not. He had a strong sense of responsibility as a person and a mathematician, and therefore took on burdensome administrative duties both at the Academy of Sciences and at the University. For example, for 5 years he was the Dean of the School of Mathe matics and Mechanics of the Moscow Lomonosov State University (MSU). During his tenure, this school was one of the best in the world, both in re search and in the quality of mathemat ical education. Kolmogorov created three major academic u nits at MSU: the Department of Probability Theory, a Laboratory of Statistics, and later the De partment of Statistics. He chaired the first department for many years and the other two during their first years of ex istence. Soon after the death of A. A. Markov, Jr., (October 1979), Kolmogorov accepted the position of the Chairperson of the Department of Mathematical Logic, and he retained this position to the end of his life. In my view, he saved this important academic unit from being taken over by people who would have changed it, not for the better. A few contributors, especially Us pensky and Khimchenko (Rychkova), share vivid memories of Kolmogorov's lectures to general audiences. I myself clearly remember the enthusiastic re sponse of the audience to his lecture "Automatons and Life," delivered in April 1961 at MSU. The largest available university auditorium could not accom modate the crowd of listeners, so a ra dio broadcast was delivered to listeners outside the hall. It was spring, not only on the calendar, but also spiritually, and to some extent even politically, and the
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subject of the lecture was on the bor der of reality and bold fantasy. Could a machine think and have emotions? The lecture was an element of the struggle for cybernetics, a subject that was un der ferocious attack from communist philosophers and ideologists. Kolmogorov's interest in mathemati cal education went far beyond gradu ate school. He began his professional career as a high-school teacher, and he never lost touch with precollege edu cation. Already in 1936 he wrote to gether with Alexandrov the school text Algebra. In 1 963 he established a board ing school for mathematically gifted children from the provinces and taught there while shouldering a weekly work load of 26 hours. The school was affil iated with Moscow State University. To day the school is a part of the University system, and it carries the name of its founder-The Kolmogorov School. Reminiscences contains an essay by Levin, at one time a student at The Kol mogorov School. Later Levin became Kolmogorov's PhD student. In his later years, Kolmogorov was a driving force behind a bold attempt to reform mathematical education in middle- and high-schools of (the)USSR. This attempt would influence the math ematical education of millions of chil dren. He tried to base the mathemati cal school curriculum on set theory and to combine logical rigor with flexibility and practical applications. One could easily see the scope of the challenge and the difficulty of the task. The at tempt to reform did bring him a lot of worries and personal troubles. His ideas were subjected to severe criticism from many directions. His closest disciples did not support him. For example, Arnold writes in his essay that Kol mogorov was inclined to perceive all school children as genius mathemati cians like himself. That is why, in Arnold's view, his ideas were not prac tically feasible. While reading the Reminiscences, it is exciting to see how, beginning in the 1 960s, Kolmogorov's interests in prob ability, mathematical logic, the theory of algorithms, and dynamical systems found a surprising synthesis in his al gorithmic approach (now known as Kolmogorov Complexity) to the con cept of randomness and information theory. It was his last scholarly and hu-
man accomplishment. It is a striking and dramatic fact that the man who estab lished the now universally accepted measure-based axiomatic approach to probability challenged this very ap proach and offered new concepts, new horizons. Kolmogorov's career was a manifes tation of the unity of culture. Not only was this brilliant mathematician greatly interested in, and knowledgeable about, literature, music, and art, but he also initiated and guided mathematical stud ies of Russian versification. Specifically, in the 1960s he was running a special seminar that attracted both mathemati cians and specialists in literature. His contribution to the theory of Russian verse was highly regarded by prominent experts. Kolmogorov was especially in terested in the statistical analysis of de viations of accents from classical rhythms (iamb, trochee, dactyl, and so on). As Monin notes in his essay, sta tistical analysis creates an individual "statistical portrait" of a given poet. A reader of the Reminiscences, and espe cially of Uspensky's essay, will find a great deal of evidence of Kolmogorov's interest in the development of mathe matical methods in linguistics. He was one of the main driving forces that se cured the creation of a new Division of Theoretical and Applied Linguistics in the School of Philology of Moscow State University. This Division played an im portant role both in research in mathe matical linguistics and in the training of specialists in this new (at that time) dis cipline. Shiryaev and Uspensky both note that Kolmogorov resembled Boris Pasternak in appearance. Moreover, the mathematician and the poet had simi lar patterns of walking and speaking ("mooing"). One could speculate that this similarity extended much farther. Both had a strong sense of responsibil ity for their respective areas of creativ ity, and both treated people in a polite and respectful way. And, most impor tantly, both aspired to reach the very bottom of Truth (I paraphrased here a verse line by Pasternak): one in math ematics and the other in poetry. Arnold recalls a speech about the nature of mathematical talent that Kolmogorov delivered at a gathering of his students in his home. He maintained that tal ented mathematicians are like children,
with children's immediacy and curios ity; the greater the talent, the younger "the child." Kolmogorov estimated his own age to be 13 (and his friend Alexandrov's age to be 16 or 18). If I remember correctly, Anna Akhmatova expressed the same thought in one of her writings. She wrote, of course, about poets, and she assigned to Boris Pasternak the very same age of 1 3 . Kolmogorov lived through hard times: revolutions, civil war, hunger and devastation, Stalin's dictatorship, World War II, waves of great terror. . . . For obvious reasons, all but two of the con tributors (Gnedenko and Nikolsky) could write only about the postwar pe riod, actually mostly the post-Stalin time, when mass murders of innocent people by the criminal political system had come to an end. Still, the ideolog ical pressure was great, and the psy chological traumas suffered by people who could expect to he arrested at any time, 24 hours a day, seven days a week, were fresh for life. Now one, now another person disappeared. The deadly Gulag machine worked nonstop. No wonder that Kolmogorov avoided political discussions and never discussed politics with his disciples. Arnold writes: " . . . like almost every body in his generation who lived through the thirties and forties he was scared of ' them' till the last day of his life." Maybe this remark offers an ex planation of some of Kolmogorov's public actions that traumatized his pupils. One might recall here a letter in Pravda 0974), the official newspaper of the Party, directed against Solzhenit syn and signed by him and Alexandrov. It was clear that neither of the two math ematicians was the actual author of the letter. On the other hand, Kolmogorov never disavowed the letter and never explained his motives to his circle. There were other tragic events. Tikhomirov presents what is perhaps the most comprehensive account about the clash between Kolmogorov and N. N. Luzin, when Kolmogorov slapped his teacher in the face. This outburst pained Kolmogorov to the end of his life. The Luzitania, an incredible school that Luzin created in Moscow in the 1 920s, suffered a probably inevitable, but definitely tragic breach between the teacher and his brilliant pupils. This rift
reached its culmination when Luzin's pupils took part in a political assault on their teacher, an event known as the "Luzin affair ( or the Luzin case)'' 0936). It was a miracle that Luzin was not ar rested and shot. I heard often the ques tion why Stalin did not kill one or an other prominent person when all the logic of the relevant events indicated such an outcome. Well, a tyrant is ex actly a tyrant because he does what he wants to do and not what we expect him to do. "The Luzin affair" is a black page in the history of the Russian math ematical school. It is not discussed at great length in the Reminiscences since it belongs to an earlier time; only Gne denko hrief1y mentioned it. On the other hand, Tikhomirov concludes that, judging by relevant publications, Kol mogorov's involvement was minimal. In the last years of his life Kol mogorov suffered a grave form of Parkinson disease. As the disease pro gressed, he was more and more im mobilized and his speech was gravely impaired. At the same time he was growing blind. It is touching to read about the round-the-clock help and at tention his pupils gave him in those hard times. They were with their teacher until the very end. "We do not choose our time; we are living and dying in it'' as the well known Russian poet Alexander Kush ner puts it. "What an interesting life I had, " Kolmogorov once said to his stu dent Bulinsky. This was how the great mathematician estimated the integral over his lifespan, with all its ups and downs . . . . The Reminiscences adds an invalu able human dimension to the bio graphical volume Kolmogorov in Per _,pective, published by the American Mathematical Society in 2000. I believe that an English translation of the Rem iniscences would be of great benefit to the international mathematical commu nity and to general readers interested in history of Science. To this I would like to add that Rem iniscences is part of a major series of publications issued (or in progress) in Russia in connection with Kolmogorov's centenary. A key item is the 3-volume set "Kolmogorov, " edited by Shiryaev, containing a complete bibliography of Kolmogorov's publications, an ex tended biographical essay by Shiryaev
(vol. 1 ) , a selection of the correspon dence (from the 1 930s and 1 940s) be tween Kolmogorov and Alexandrov (vol. 2), and extracts from Kolmogorov's diaries of 1 943 to 1 945 (vol. 3). A com plete or even partial English translation of this set would, in my view, he of great benefit to historians of mathemat ics, to the members of the mathemati cal community, and to the general public. Department of Mathematics University of Pittsburgh at Johnstown Johnstown, PA 1 5904 USA e-mail: [email protected]
Counting Austral ia In: The People, Organizations, and Institutions of Austral ian Mathematics by Graeme L. Cohen SYDNEY, HALSTEAD PRESS, 2006, $79.95 Australian dollars, ISBN 1920831398. Available from http:;;www . austms.org.au.
REVIEWED BY IAN H. SLOAN
his is a fine book, one that de serves a wide readership among mathematicians and their friends. It is a meticulous record of the people and institutions of Australian mathe matics, from before European settle ment to the present day. Although (as the author is at pains to emphasize) this is not a history, it nevertheless docu ments clearly the mathematical interests of most of the 1 000 or so individuals who populate this work. The book takes an engagingly broad view, em bracing all branches of the mathemati cal sciences, including statistics. Lord Robert May (himself a notable product of Australian mathematics, and with a story well documented in the book) comments in an exuberant Foreword that this breadth of view of the subject
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is a particular strength of Australian mathematics. Why might a non-Australian want to read the book? It is, in the first place, surprisingly enjoyable to read. If the de tail is sometimes overwhelming, a math ematically trained reader surely knows how to skip ahead. The book is writ ten as a series of narratives, well cross referenced. The broad stories that emerge from the narratives have lessons and resonances for mathematics in many other parts of the world. One broad theme that emerges, un surprisingly, is that individuals matter; there are some who have really made a difference, whether as stars or as builders of the discipline. The book is current enough to include, among the stars, the 2006 Fields Medalist Terry Tao, born in Adelaide in 1 975, a participant in his first International Mathematical Olympiad at the age of 1 1 , and winner of an IMO gold medal at the age of 1 2 , still the youngest student ever to do so. An early builder, who was also a star, was Horace Lamb, known to earlier generations as the author of Lamb's Hy drodynamics. He came to the Univer sity of Adelaide in 1 876 and stayed for 10 years; during that period he per formed important research on geo physical elastic waves and hydrody namics, despite a massive teaching load. (In 1879 he had 13 contact hours per week in mathematics, plus 7 in physics.) He was followed as Professor of Math ematics (and later of Mathematics and Physics) by W. H. (William) Bragg, who with his son W. L. (Lawrence) Bragg, (born in Adelaide in 1 890) went on to win the Nobel Prize for pioneering work on X-ray crystallography. As builders, Horace Lamb and William Bragg were hard to beat. In the early days of Australian math ematics, the Professor (invariably male) was typically the sole mathematical ap pointment at that level, and often had only one or two other colleagues to as sist in the teaching task. In that situa tion, the influence of an individual could be as easily negative as positive. I was struck by the performance of Syd ney University's second Professor of Mathematics, of whom it was said, at the end of his 25-year career: "Mentally equipped with every gift except ambi tion, he has, as you know, never pub lished a line . "
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The book holds surprises, such as the story behind the selection of Mel bourne University's Professor of Math ematics in 1928, T. M. (Tom) Cherry. The book reveals that a rival for the po sition was the celebrated American mathematician, and father of cybernet ics, Norbert Wiener. Wiener's applica tion was supported not only by G. H. Hardy (who described him as "quite ob viously one of the very best American mathematicians"), but also by a galaxy of European stars, including Harald Bohr, Max Born, Caratheodory, Fn§chet, Hilbert, Lebesgue, de Ia Vallee Poussin, Veblen, and Weyl. Cherry, too, was not without his supporters: they in cluded Littlewood, ]. ]. Thomson, E. T. Whittaker, and E. A. Milne. If the com mittee had made a different decision, would Wiener have thrived as Mel bourne University's sole mathematics professor, responsible for all adminis tration and much of the teaching? Would his appointment have trans formed Australian mathematics? I , at least, have doubts. In the event, the committee ap pointed Cherry over Wiener (perhaps an inevitable decision in those Anglo centric days), and Cherry went on to serve as Professor of Pure and Mixed Mathematics (and later of Applied Math ematics) until 1 963. During those years, he had an immense impact on Aus tralian mathematics, ranging from insis tence on the importance of school mathematics, to helping to found the Australian Academy of Science. I, who as a callow youth was unmoved by Cherry's course on hydrodynamics in the latter's later years, had my eyes opened by the evidence of inspiring leadership, for example of wartime re search at the Division of Aeronautics in Melbourne. G. K. (George) Batchelor, who later founded the journal ofFluid Dynamics, and is recognized as founder of Cambridge University's Department of Applied Mathematics and Theoreti cal Physics, graduated from Cherry's de partment and then served during most of the war years within Cherry's orbit in the Division of Aeronautics. Another of Cherry's graduates from that period was R. H . (Dick) Dalitz, who went on to a glittering career in Physics at Ox ford. An individual who made a difference of a different kind was Peter O'Hallo-
ran, the driving force behind the Aus tralian Mathematics Competition, now Australia's largest mass participation event, with around half a million par ticipants each year, which is an extra ordinary number for a nation with a population of 20 million. His influence was not confined to Australia: the World Federation of National Mathematics Competitions was founded by O'Hallo ran in 1 984. Another theme that emerges from the book is the importance of friends. When the first Australian universities started in Sydney and Melbourne in the middle of the 1 91h century, they were totally reliant for advice on the well es tablished British universities, especially Cambridge. It seems that this assistance was given readily; the first selection committee (sitting in England of course) contained the eminent mathemati cian/astronomers Sir John Herschel and Sir] ohn Airy. If in time many Australians came to believe that the relationship with Cambridge was too dependent, at least in those early days the assistance was surely indispensable in setting stan dards. World events, and especially the Sec ond World War, had a major impact on Australian mathematics. A sometimes moving, sometimes entertaining chap ter is devoted to the role of Australian mathematicians in the war. One notable individual exploit was that of M. N. (Maurie) Brearley, who signed up to train as an air force pilot under a false name. Brearley asserted (although not a Catholic) that his last occupation was training for the priesthood, and he had memorized the book of eye tests to con ceal his colour blindness. Important wartime contributions in cryptography and aeronautics also make interesting if more sober reading. But the biggest influence of the Sec ond World War surely lay in the en richment of Australia's intellectual life by the human upheavals caused by the war and the persecutions that preceded it. In the 1 950s B. H. (Bernhard) Neu mann and his wife Hanna came to the newly created Australian National Uni versity in Canberra. Both had major im pact on Australian mathematics, and Bernhard especially helped to bring Australia into the international commu nity through his involvement with the International Mathematical Union. In his
long life ( he died in 2002) he made ma jor contributions to the mathematical community, not least by attending every conference he could, not caring whether they were "applied" or "pure . " Perhaps his greatest contribution was to attract the great number-theorist Kurt Mahler to Australia. In turn, as is the nature of such things, Mahler created a strong number theory school in Aus tralia. ]. A. (john) Coates, later the PhD supervisor at Cambridge of Andrew Wiles (of Fermat's last theorem fame) was just one of those products. The book does not shy away from tensions and disputes. The most cele brated dispute was that between the Professors of Pure Mathematics and Ap plied Mathematics at the University of Sydney during the decades after the Second World War. Both T. G. Room and K. E. (Keith) Bullen were Fellows of the Royal Society, and both had ca reers of considerable distinction. But they could not agree about the teach ing of mathematics, and eventually the university solved the problem in a time honoured way, by creating separate de partments for each. Room was not the only one who did not get on with Bullen; a recent obituary for L. C. (Les) Woods (later Head of Oxford's Mathe matical Institute) in London's DaiZY Telegraph said that Woods, who joined Sydney University's Applied Mathemat ics Department in 1 954, "left after a blazing row with his professor Keith Bullen." At the root of the disagreement be tween Room and Bullen was a differ ence of view about the proper rela tionship between pure and applied mathematics. Bullen considered that ap plied mathematics should be close to experiment, and that while having ob vious links with pure mathematics, it also has links with disciplines such as physics and engineering. Applied math ematics should not be too close to pure mathematics, lest the applied mathe matician be blinded by theory. (We know Bullen's views, because they are reprinted in the form of a well written article in an Appendix. ) Another dispute covered by the book was the one between two "Applied" Professors at the University of New South Wales, ]. M (john) Blatt and V. T. (Ted) Buchwald. This also had both personal and philosophical ele-
ments: Blatt was a distinguished theo retical physicist ( and a former colleague of Robert May at Sydney University), whereas Buchwald, with a British edu cation, was closer to the Bullen mould. Since they both agreed that "if it has a theorem in it then it isn't applied math ematics, " I, who had to live with them both, could agree with neither. Yet in the long run, disputes of this kind went away, even if they had to wait for death or retirement of the pro tagonists. The Australian solution that has evolved over time is to accept that applied mathematics is a broad church, embracing those who conduct experi ments at one end and those who enjoy analysis and theorems at the other. There no longer seems to be a good reason why applied mathematicians of different persuasion cannot coexist, and cannot cooperate with mathematicians of still other leanings. A last major theme is that organiza tions matter. The Australian scene now has a number of professional societies, for Statistics, Operations Research, and others, but the pure and applied math ematicians are surprisingly united; for the professional body of the applied mathematicians, ANZIAM ( standing for Australia and New Zealand Applied and Industrial Mathematics), although hav ing every outside appearance of being a stand-alone society, in legal and ad ministrative reality is a Division of the Australian Mathematical Society. ANZlAM gained international visibility as the host society for the International Congress of Industrial and Applied Mathematics ( ICIAM), held in Sydney in 2003. In its closing pages, the book makes it clear that the Congress was a triumph for Australian mathematics as a whole; the Australian Mathematical Society was not only the sole underwriter, hut was also a major contributor to the success of the Congress by embedding its Annual General Meeting within the Congress. Whether as separate societies or as one, the mathematical scientists of Aus tralia have in recent years cooperated closely, in the face of declining gov ernment funding and shrinking depart ments. The hook describes a 1 986 strategic review that highlighted the achievements and the problems of Aus tralian mathematical sciences. But the hook was completed before the an nouncement late in 2006 of the findings
of a second strategic review. This time the international reviewers found the outlook for Australian mathematical re search to be "dire" because of declin ing staff numbers in mathematics de partments, shortages of mathematics teachers, and other problems. Although those difficulties are only too real, there is now a glimmer of light: the Australian budget, released in May 2007, an nounced a 50% increase in the relative funding for undergraduate students in mathematics subjects, bringing the funding to the same level as comput ing. That change to the funding model was a key recommendation of the strategic review. It can hardly he doubted that the pressure applied by the two strategic reviews, and the pres sure applied by many friends within and without Australia, was a key factor in changing the climate for mathematical science in Australia. The next edition of this fine book will surely have more to report. University of New South Wales School of Mathematics and Statistics UNSW Sydney NSW 2052 Australia e-mail: [email protected]
Our Lives: Encounters of a Scientist by Istvan Hargittai BU DAPEST, AKAD E MIAI KIADO , 2004, 264 P P . ,
30 EUR, I S B N 10: 9630581019, ISBN 13: 978-9630581011
IA.;
REVIEWED BY PETER LAX
stvan Hargittai is a distinguished structural chemist. His book, Our Lives: Encounters of a Scientist, is a curious mixture. Each of the 19 chap ters starts with a description of a meet ing with a Nobel-Prize-winning chemist, physicist, or biologist, from various na tions. Half of them are Americans. These are not interviews but brief his tories of the laureates; included is a sketch of their personalities and some of the dramatic events in their lives,
jU·
]mo
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such as their struggles to obtain an ed ucation for themselves. There are clear, nontechnical descriptions of the work by each of the laureates, and implica tions of their discoveries. But then the author strays from his subject by free association, to describe the work of other scientists, whose work is related to that of the scientist featured in the chapter, or whose lives have been through similar turns. Almost compulsively, the author returns to tragic events in his own life, the death of his father in 1 942 in a forced labor battalion on the Russian front, and the deportation in 1 944 of the rest of his family for slave labor in Austria. Hargittai devotes many pages to chronicling antisemitism: a law passed in Hungary after the First World War re stricting the number of Jewish students at universities, the increasingly harsh re striction imposed by laws in the late 1 930s, and the bestialities of German, Hungarian, and Ukranian Nazis during the Holocaust. There was plenty of an tisemitism in America, too, in the 1 930s. Herbert Hauptmann had trouble getting into graduate school, as did his collab orator Jerome Karle, who was told by the Dean of the graduate school at Har vard: "We have enough Jews in Mass achusetts; I am not going to add one from New York. " Hargittai himself suffered from dis crimination under the communist regime. Since his father had been a lawyer, a bourgeois profession, he was labeled "class alien," and he was not permitted to continue his high school studies. It was only after the revolution of 1956 that this restriction was lifted. Admission to the university was another hurdle; Hargittai was able to overcome it by sheer perseverance, but being a class alien, he had to pay tuition; for students from the peasant and working class, attendance was free. Hargittai received part of his educa tion in Moscow; it had a strong impact on him. In later years he visited Moscow frequently. He writes knowledgeably about the Soviet scientific scene, about Kapitza, Landau, their clashes with the system under Stalin, about antisemitism in the Soviet Union, and the deadening hand of bureaucracy. Although the pos sibility of studying with masters of mathematics and science in Moscow
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and Leningrad was one of the rare com pensations of living under Soviet occu pation, relatively few Hungarians took advantage of it. Hatred of the Soviet Union, and the harsh living conditions discouraged most; Hargittai was an ex ception. Another exception was the out standing mathematician Alfred Renyi who had studied in Leningrad. Renyi described how he beat the bureaucracy; when he was getting nowhere with one official, he demanded to speak to his "nachalnik" (boss). When he couldn't convince the nachalnik, he demanded to speak to his nachalnik, and so on until he got to a sufficiently intelligent person. Although Hargittai has deep interest in symmetry, a highly mathematical sub ject, there is not much about mathe matics in this book. There are brief dis cussions of Vera S6s, Paul Turan, and Paul Erdos; the latter had the foresight to get out of Hungary in 1 938. In fact, three of the scientists dis cussed in this book have made highly original uses of mathematics. 1 . Philip Anderson invented "Anderson localization," a deep property of the spectrum of a class of differential operators. 2. Herbert Hauptman and Jerome Karle solved the problem of determining the structure of crystals by x-ray dif fraction. One of their key ideas was using the classical inequalities of Otto Toeplitz on the Fourier trans form of positive mass distributions. 3. Eugene Wigner had a lifelong love affair with mathematics. He used random matrices to describe com plicated molecules and determined the expected distribution of their spectra. This has developed into a major subject, with unexpected con nection with other parts of mathe matics. Wigner also is the author of an intriguing philosophical discus sion on "The Unreasonable Effec tiveness of Mathematics in the Phys ical Sciences." A fourth scientist interviewed by Har gittai, the biochemist Marshall Nieren berg, was enormously proud until the end of his life that, when he was a stu dent at Columbia, he had won the Put nam Competition. I warmly recommend this book to one and all; it is a lively read.
Department of Mathematics Courant Institute of Mathematical Sciences New York University 251 Mercer Street New York, NY 1 001 2 USA e-mail: [email protected]
Leonhard Eu ler: Life, Work and Legacy Edited by Robert E. Bradley and
C.
Edward Sandifer
STUDIES IN TH E H I STORY AND PHILOSOPHY OF MATHEMATICS, VOL. 5., AMSTERDAM: ELSEVIER,
2007, 540 pages, 137 EUR, ISBN 13: 978-0-444· 52728·8, ISBN 10: 0-444-52728-1 REVIEWED BY KIM WILLIAMS
nitiatives commemorating Euler were not lacking during the year 2007, the 300th anniversary of his birth ( 1 5 April 1 707). Although conferences come and go, books remain. One book likely to become an important reference for future Euler studies is Leonhard Euler: Life, Work and Legacy, edited by Robert E. Bradley and C. Edward Sandifer. Bradley and Sandifer are, respectively, president and secretary of the Euler So ciety (based in the United States), founded in 2001 to promote research on Euler's work and work inspired by him, by organizing meetings, seminars, and hosting a website (http://www. eulersociety.org). The editors also par ticipate in the valuable online Euler Archive (http:!/www. math.dartmouth. edu/�euler). As they note in their In troduction, organized efforts to make available all of Euler's papers began in the early years of the 20th century, fol lowing the 200th anniversary of Euler's birth. Successive anniversaries (250th anniversary of his birth in 1 957, and 200th anniversary of his death in 1 983) kept enthusiasm for research high. Thus we arrive at his 300th birthday with more material than ever before. The present book is particularly valuable, because the sheer volume of works by Euler himself (886 books, pa-
pers, memoirs, notebooks, and almost 3000 surviving letters) makes ap proaching this mental giant a daunting task. The object of the book, commis sioned by Elsevier for their series "Stud ies in the History and Philosophy of Mathematics," was to provide "a more or less complete picture" of Euler's works and times. That the editors man aged to do this in a volume of 540 pages is commendable. Twenty-five papers (including the Introduction) touch on most of the major aspects of Euler's re search in mathematics (analysis, rigid bodies, calculus of variations, the loga rithm function, series, geometry, num ber theory, combinatorics, graph theory, differential geometry), the areas of spe cial interest to which he turned his pow erful attention (astronomy, cyclotomy, probabilities, polyhedra) , areas in which Euler's work constitutes a pre history of future fields (kinematics, vec tor mechanics, topology, and quantum mechanics) and elements of his life and thinking that greatly influenced him (the major events in his life, the Russian mi lieu, the prestigious positions of the academies, his religion and philosophy, and how he was received). There is even a paper about images of Euler. It is greatly to the editors' credit that they were able to engage authors competent in topics without overlap, so that each paper constitutes a single point of ref erence for its particular subject. It is this that gives the book its character as a comprehensive guide to Euler and makes it valuable to both those who wish to know more about Euler as a whole and those who wish to begin a study of one particular topic. What the editors did not do was pro vide a guide to the book. The Intro duction is, for the most part, limited to a discussion of the book in the context of previous anniversary editions. Only the last three paragraphs discuss the book's philosophy toward the history of mathematics and its organization. This is a shame, because surely the editors had to have put a great deal of thought into what topics to address, which au thors to approach, and guidelines for authors that reflect the book's overall object. They mention in the Foreword, "we soon found that the Euler Society alone could not provide the 'complete picture' . . '', leading to a collabora-
tion with European scholars, but the ed itors don't provide details. An account of how they crafted this book-for, col lecting and editing a group of papers to represent an overarching concept is as creative an act as writing a mono graph, perhaps more so, because if done well the result is a harmonious choir rather than a cacophony of soloists-would have helped a great deal in how the reader should approach the book. Instead they provide only the most skeletal outline-biographical and historical papers in the beginning, arti cles about Euler's mathematical and sci entific works in the middle, and articles about Euler's influence on future math ematicians in the end-and they leave the reader to navigate Euler's vast ocean more or less alone. Ronald Calinger's biography of Euler ("Leonhard Euler: Life and Thought," pp. 5-60) , which opens the book and is justifiably the longest single paper, ably describes his life and career, and what he was working on at the various stages of his life, which provides a con text for the technical papers that follow. Peter Hoffmann ( " Leonhard Euler and Russia, '' pp. 6 1 -73) describes the cul tural climate of 18th-century St. Peters burg and Euler's special relationship to that city, and subsequent efforts by the Russians to safeguard and catalogue the extensive body of Euler's works and correspondence that remained in Rus sia after his death. Ronald Calinger and Elena Polyakhova ( ''Princess Dashkova, Euler, and the Russian Academy of Sci ences, " pp. 7S-95) discuss the intellec tual climate that Euler helped to create in the Russian Imperial Academy before his death in 1 783, and his legacy under the direction of the formidable Princess Dashkova . While the Russians consid ered Euler an integral part of their sci entific patrimony, other schools had to come to terms with his legacy as well. Ivor Grattan-Guinness ( "On the Recog nition of Euler among the French, 1 790-1830") discusses the reactions to Euler's contributions to mathematics in the 50 years after Euler's death, at a time when the French dominated the field. Euler's stolid personality and evident visual handicap made him the object of derision of sophisticated intellectuals in Frederick II's court. Papers by Wolfgang Breidart ("Leonhard Euler and Philoso-
phy," pp. 97-108) and Florence Fasanelli ("Images of Euler," pp. 1 09-120) discuss the interesting issue of how Euler was viewed, both metaphor ically and literally. Breidart makes clear the influence that Euler's unwavering Christian faith exerted-for better or worse-on his personal as well as pro fessional, and even intellectual, life. The papers at the book's heart are those dedicated to his work. What these papers make clear is that if Euler's life was dense with work, his work was equally dense with life. The picture that emerges from these technical papers is a picture of the man as well as the sci entist. In "Euler and Applications of An alytical Mathematics to Astronomy" (pp. 1 2 1-145), Curtis Wilson traces Euler's contributions to astronomy, from the calculus of trigonometric functions through his lunar theories. A comple ment to Wilson's paper is that of Kim Plofker ("Euler and Indian Astronomy, " p p . 1 47-166) about Euler's brief inter est in the Sanskrit calendar. It shows Euler's willingness to explore unknown fields (in this case, astronomy) at the behest of a colleague . Another example is when Euler began to examine a Dio phantine problem posed by Christian Goldbach. Jeff Suzuki ("Euler and Num ber Theory: A Study in Mathematical In vention, " p. 383) shows that, although at the time that Euler became interested in it, number theory was given scant re gard as a field of mathematical study, his interest in it, and the results that he achieved, inspired others to carry it for ward. But Euler's choice of problems to pursue was also conditioned by the re quirements of his patrons. In discussing Euler's work on probability, David Bell house ("Euler and Lotteries," pp. 385-394) attributes Euler's interest in the subject of lotteries to a request by Fred erick II. Another request asking help from Euler's powerful attention to re solve a particular problem came from the mayor of Danzig in 1736, who asked Euler to send a solution to the Kbnigs berg bridge problem. The history of the problem and Euler's solution is de scribed by Brian Hopkins and Robin Wilson ( "The Truth about Konigsberg," pp. 409-420). Euler's reply to the mayor is a combination of gentility ( he twice addresses the mayor as "most noble Sir") and impatience; Euler at first not
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only does not see the relationship of the problem to mathematics, he con fesses to knowing nothing about the "geometry of position. " But he never theless became interested enough to produce a solution, the first contribu tion to graph theory. Many of the papers show Euler's comradeship with and generosity to other scholars. Curtis Wilson repro duces the letter from Euler to Clairaut regarding the concordance of inequali ties of lunar motion and Newtonian the ory: " . . . I felicitate you on this happy discovery, and I even dare to say that I regard this discovery as the most im portant and the most profound that has ever been made in mathematics" (p. 1 39). Rudiger Thiele ("Euler and the Calculus of Variations," pp. 235-254) re ports on the care that Euler took not to subtract any of the credit for the dis covery of the calculus of variations from the young Lagrange. Even when credit was properly due him for priority of a discovery, Euler appears not to have been overly concerned about claiming it; Robert Bradley cites a letter from Euler to James Stirling giving credit to Maclaurin for the work on series: "I have very little desire for anything to be detracted from the fame of the cele brated Mr. Maclaurin since he probably came upon the theorem for summing series before me" (p. 256). Edward San difer ("Some Facets of Euler's Work on Series, " pp. 279-302) traces Euler's work on series in depth, establishing Euler's priority over Maclaurin by some 6 years. There are certainly enough well-known examples of bitter disputes over priority in the history of the sci ences, such as that between Johann Bernoulli and his son Daniel over hy draulics, to show that Euler's generos ity is exceptional. But not all of Euler's relationships with his peers were problem-free. Bradley ("Euler, D'Alembert and the Logarithm Function, " pp. 255-277) uses Euler's lack of egotism vis a vis Maclau rin to preface his account of Euler's dif ficult relationship with D'Alembert. The debate between the two over the loga rithm function lasted through 1 5 years of private correspondence before be coming public; another conflict con cerns D'Alembert's demand for ac knowledgment for priority of theories discussed by Euler in four of his mem-
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oirs; they quarrelled still further over the vibrating string problem. It is thus some what surprising to read of the almost affectionate first face-to-face meeting between Euler and D'Alembert in Berlin in 1 763, after almost 20 years of what must have been an irritating relation ship for both. Ronald Calinger reports on Euler's rhapsodic praise of their friendship in a letter of October 1 763 to Christian Goldbach: "Our friendship is perfect . . . " (p. 45). This is probably as good an example of the courtly man ners of the 18th century as it is of Euler's magnanimity. In showing the differences between Euler's textbooks on analysis (Integral Calculus, Differential Calculus, and the first part of Introductio), Victor Katz ("Euler's Analysis Textbooks, " pp. 2 1 3-233) shows how much Euler was a man of his times, not only because he treated contemporary concepts (infini tesimals, which were to give way to the concept of the limit) but because his textbooks were oriented toward a class of students that disappeared as a result of the influence on society of the French Revolution and Napoleon. His involve ment with the world of the academies is another reflection of his times. Perhaps the most fascinating rela tionship of all is that of Euler to his own ideas. We see him time and time again taking on a problem, formulating a so lution, and then returning, sometimes many years later, to rework it. Thus Cur tis Wilson reports of his work on lunar cycles that he published a first theory in 1 746, and another 26 years later, in 1 772. Ed Sandifer writes, "More than 25 years pass between the time that Euler has his first ideas related to Bernoulli numbers and the time when Euler sees that relationship" (p. 283). Homer White ("The Geometry of Leonhard Euler," pp. 303-321) shows how Euler wrote his first paper on reciprocal trajectories, and then returned to them several times in his career. Brian Hopkins and Robin Wilson ("Euler's Science of Combina tions, " pp. 395-408) discuss Euler's pa pers on partitions, the first in 1 74 1 , a chapter in Introductio in 1 748, another paper in 1 750, still another in 1 768. Karin Reich ("Euler's Contributions to Differential Geometry and its Recep tion, " pp. 479-502) contrasts Euler's first work on surface theory, in an appen dix to Introductio in 1 748, and his
"spectacular results" 15 years later. Re garding Euler's return to a problem of deriving the formula of moments, which he had already accomplished some 20 years earlier, Sandra Caparrini writes, "It is curious that Euler returned to the same problem without citing his previ ous derivation, yet this case is by no means unique. Evidently he had for gotten what he himself had achieved in 1 763" (p. 467). Had he actually forgot ten, or, since we find examples of this over and over, was the recreation of a solution a necessary part of his creative process? Of course, there is no way to know, but as curious of mind as he was, as wide-ranging and prolific as were his works, Euler apparently never suffered from the conceit of believing that a so lution that he had found to a particular problem was the solution. For Euler, there is never an attitude of "been there, done that"; going back was just as prof itable as going forward. A body of work as dense as that left by Euler naturally becomes the founda tion for future studies. In many cases, a more-or-less linear development of ideas from Euler through his successors can be traced. Dieter Suisky ("Euler's Mechanics as a Foundation of Quantum Mechanics, " pp. 503--5 25) places Euler's work o n clas sical mechanics in the context of progress made in the field beginning with Galileo, Descartes, Newton, and Leibniz, and then shows how Euler's work can be used to understand and derive Schrodinger's ba sic quantum mechanical equation. Olaf Neumann ("Cyclotomy: From Euler through Vandermonde to Gauss, " pp. 323-362) begins with Euler, then traces the evolution of his ideas from Euler's contemporary, Vandermonde, to Gauss. Part of the immense richness of Euler's work, apart from his far-ranging interests, is that it contains results of which he seems not to have been aware, thus providing fertile ground for historians whose understanding of later developments is acute enough to allow them to find hidden gems in Euler that presage future developments in science, and perhaps indeed whole fields of in quiry. Discussing a prehistory of vector calculus in Euler's work, Sandra Ca parrini ("Euler's Influence on the Birth of Vector Calculus, " pp. 459-477) writes, " . . . there are many reasons to believe that Euler did not fully under stand the vectorial character of the en-
tities and the operations that occur in his purely algebraic calculations" (p. 459). Caparrini examines two memoirs of Euler's to make this vectorial char acter clear, then traces future develop ments of Euler's formula through the work of Laplace, Prony, Poinsot, Pois son, Bordoni, and Cauchy. Teun Koestler ( "Euler and Kinematics," pp. 167-194) shows another example where Euler's work constitutes a pre history of a future field. After providing an excellent introduction to the modern science of kinematics, Koestler finds in Euler's work ideas that preceded the identification of this field some 50 years after Euler's death. Koestler writes. "Euler's focus was not on kinematics, which is part of the reason why Euler missed some results that we find rather obvious" ( p . 1 92). Yet another example of Euler's results forming part of the pre history of a later field of inquiry is brought to light by David Richeson ("The Polyhedral Formula," pp. 421--439); although Euler's polyhedral formula had been discussed for 1 00 years, "no one noticed the topological significance of Euler's formula" ( p. 436). That significance is now clear. But along with results too far-reach ing for Euler to recognize, there were also errors. Stacy Langton ( " Euler on Rigid Bodies," pp. 1 95-2 1 1 ) shows how Euler achieved a correct result in his demonstration of the equations for mo tion on rigid bodies only by compen sating with a second error. Euler erro neously concluded that both Jupiter and Saturn were accelerating, thanks to faulty algebra; this conclusion is de scribed in detail by Curtis Wilson (pp. 1 24-133). There were some embarrass ing practical failures when Euler erro neously calculated the required water pressure in the pipes and pumps that powered the fountains at the royal res idence of Frederick II in Potsdam. Ronald Calinger addresses this issue, writing, ''He made a few errors. These and his scarce lapses in rigour have got ten his work portrayed as 'happy go lucky analysis' or as reckless. This judgement seems to stress the infre quent errors and might apply to another with less intuition but seems mistaken for him" (p. 42). The only real flaw in this book is the index. It includes electromagnetism but not elasticity; undulatory mechanics but
not quantum or vector mechanics; An dre Wei! but not Voltaire. Yale Univer sity is listed, but the Berlin and Russian academies are not. References to Diop trica are listed as appearing on pages 54 and 69. but the reference appear ance on page H2 is not listed. One won ders what criteria were given to authors for keywords and how to indicate them. This flaw is unfortunate indeed, for a complete and useful index is a very im portant addition to such a book. Riidiger Thiele's closing remark is worth reproducing: We conclude with a question of Adolf Kneser posed at the Euler con ference in 1907: "Why do we rum mage in rubble for some antiqui ties?'' He and I give the same answer: "To enrich the ars inveniendi, to ex plain the methods by excellent ex amples, and last but not least to ap preciate the intellectual company. " Euler's intellectual company, and that of the authors of these papers, is as good as it gets. Via Cavour, 8 1 01 23 Turin (Torino) Italy e-mail: [email protected]
Mathematics and Common Sense: A Case of Creative Tension hv
Philip I Dauis
WELLESLEY, MA, A. K. PETERS, 2006, US $34.95, ISBN 1-56881-270-1
REVIEWED BY KAREN SAXE
y now, many of us have come to know Davis's pieces as insightful, accessible, and often deeper-than they-first-appear works about mathe matics. Mathematics and Common Sense is akin to Davis's earlier The Mathemat ical Experience, written jointly with Reuben Hersh and awarded the Amer ican Book Award in 1 983. I have used the earlier hook in a class for under graduates looking to fulfill their distrib ution requirement. I was thus quite in-
terested to read this newer offering. My first questions upon picking up such a book are: For whom is it written? For the general public? For my sort of lib eral arts students? Or, rather, is it meant for undergraduate math majors, or grad uate students in mathematics and work ing mathematicians? This book has something for all of these people. Dif ferent readers will take away ditferent messages, and will be interested in dif ferent chapters. It is an achievement that the quality of the writing can work for so many different audiences. One of Davis's gifts is writing about deep ques tions, while keeping the prose light and effective. Professor Emeritus Davis has been at Brown University for most of his career, and he has published many papers on complex variables, numerical analysis, and approximation theory. His hooks on these subjects (e.g., Intetpolation and Approximation and The Schwarz Function and Circulant Matrices) are well-regarded. He is also an award winning expositor of mathematics (he is a recipient of the Mathematical As sociation of America's Chauvenet Prize). Finally, and it is in this category that the current book falls, he has written on the philosophy of mathematics, and the role of mathematics in society. To excel as a writer in all three categories is no small feat. In all these efforts, we owe him much for attempting to expunge the con tempt for mathematics and mathemati cians that persists in the United States. So what exactly is this book about? In the preface, Davis explains that the impetus for writing the book came, at least in part, from a list of questions about mathematics that a certain Christina had sent him. This is followed by two sections of Christina's questions, and his answers to them. These sections are followed by 33 chapters. each touching on something to do with how mathematics is connected to, or per ceived by, "society" (whatever that might be). Some of these chapters are adaptations of previously published es says. I enjoyed the opening sections with Christina. She is clearly an astute ob server of mathematics, and of some mathematicians. The first question she asks is: What is mathematics? Now, this is not a new question; we all know a big book published first in 1 9 4 1 [ 1 ] with
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this question as its title. But though the words are the same, the question is dif ferent. Courant and Robbins do philos ophize a bit, but philosophy is really a minimal indulgence relegated to the in troductory chapters; in their book they are aiming, primarily, to teach some math to some nonmathematicians. Davis aims to start us on a road think ing about such questions as: How does mathematics fit in the more general con figuration of knowledge? What is the impact of mathematics, and not just ap plications; what impact does mathe matical abstraction and universality have on the way that humans think and construct knowledge? What is and has been the image of mathematics, since 1 945? What is its social history and im pact on institutions? We are now think ing philosophically, historically, and an thropologically, about mathematics, and are now ready to examine Christina's questions further. The remaining 33 chapters of the book do just that. The first chapter is, again, titled "What is Mathematics?" Chapter 2, "Mathematics and Common Sense: Relations and Contrasts," begins with a brief discussion of Common Sense, with subsections on topics such as Language Dependence, Gender De pendence, and Professional Depen dence. I especially liked the bit on lan guage, seeing how the phrase "common sense" is translated into several lan guages, and what its implications thus are. Davis then tells us what philoso phers have to say about common sense and draws some parallels with mathe matical thinking. In the third chapter, Davis describes examples in which common sense interacts with and im pacts the process of doing mathematics (e.g., this happens in "packing prob lems" in which one is asked to figure out how many objects of a given shape can fit inside a specified container). Af ter this chapter, the cohesiveness of the chapters dissipates, though the connec tion between mathematics and common sense recurs throughout the book. For example, Davis discusses (p. 8 1 ) the fact that mathematical entities sometimes do not accord with our intuition, or com mon sense. The Banach-Tarski paradox is one good example, as is the angst that was felt for so long about imagi nary numbers. He also points out (in Chapter 1 9) that problems with language
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could impact our common sense or, at least, the sense with which we discuss mathematics. For example, ambiguity can flavor math. There are many words (e.g., "regular") that have several math ematical meanings, and there are some objects (e.g., invertible matrix = nonsin gular matrix = regular matrix) that have several names. Can we use the language to sway our common sense? What about uses of analogy and metaphor in math ematics? Later in the book there are a few chapters about how our common sense plays a role in our understanding of probabilistic notions. This is quite a natural discussion. "The Linda problem" is given as an example of how normal, intelligent thinkers can get so jumbled with their thinking: Linda is 3 1 years old, single, out spoken, and very bright. As a stu dent she was deeply concerned with issues of discrimination and social justice and also participated in anti nuclear demonstrations. Choose the statement that is more likely: 1 . Linda is a bank teller. 2. Linda is a bank teller who is active in the feminist movement. (This is a simplified version of a prob lem posed by Amos Tversky and Daniel Kahneman, which surfaced in the early 1 980s and appears in [2].) Alarmingly, 86% of individuals in the study chose the second statement as the more likely of the two to be true. In this case, com mon sense has been swept aside in fa vor of some sort of 'moral' sense. In any case, those of us who have had the fun of teaching an elementary statistics course know that common sense is of ten ignored by our students. Of the remaining chapters, my fa vorites are: • Chapter 1 1 , "Deductive Mathematics," is a nice introduction to the notion of mathematical proof. Euclid's Elements is introduced, and a brief discussion of axioms and formal language is un dertaken. The question of what con stitutes a proof-what is an accept able proof and who gets to decide whether or not a proof is acceptable raises its head for the first time here. Davis returns to this in later chapters. • Chapter 1 7 , "If Mathematics Says 'No' Does It Really Mean It?, " is a stan dard topic, also good for our un dergraduate math students. • Chapter 20, "Mathematical Evidence:
•
•
•
Why Do I Believe a Theorem?, " is one of the longer chapters in the book. It starts with examples of mathematical statements that we may, or may not, believe. For ex ample, do you believe that a nega tive times a negative always yields a positive? Do you believe that the Pythagorean Theorem is true for all right-angled triangles? Do you be lieve that there is an infinite number of twin primes? You probably be lieve all of these statements, but have different reasons for your beliefs. Sometimes our beliefs are based on proof, sometimes on computational or visual or statistical evidence, sometimes on analogy, sometimes because there is a lack of contradic tion so far. Again, good for our un dergraduate math students. Chapter 22, "The Decline and Resur gence of the Visual in Mathematics, " picks up on one o f the reasons for belief discussed in Chapter 20. This is an important and timely discussion. The twentieth century was a period of rigor, abstraction, and desire for universality in mathematics. Comput ers make it easier to generate (po tentially very complicated) images, permitting a use of vision as a tool. The focus here is on the visual and its connection to (and impact on) the analytic or formulaic understanding of a problem. At the end of this chap ter, Davis discusses the recent appli cations of mathematics in animation, a subject he returns to later. In Chapter 23, "When Is a Problem Solved?," Davis gives the Clay crite ria and then discusses the socially constructive nature of mathematics. This is a good chapter for our stu dents and for us as teachers as we read their proofs and explanations, and as we tell students our expecta tions for such. Chapter 30, "Mickey Flies the Stealth: Mathematics, War, and Entertain ment," and Chapter 3 1 , "The Media and Mathematics Look at Each Other, " are fun. The first is mostly about modeling, simulation, anima tion, and the role that the US mili tary has played in the development of computer and video games. A must read for parents of teenage boys! The second provides lots of references for novels, movies, plays
about mathematics or with mathe maticians in leading roles. The chap ter closes with a short discussion on the disconnect that often exists be tween "our world" and what is por trayed in these venues, and what might be done to remedy this. • Chapter 32, "Platonism vs. Social Constructivism," is a short chapter, and the title really tells it all. There is the assertion at the beginning that math is a branch of science. Well, is it? What makes it so? We are like sci entists in that we are trying to un derstand the structure of something and, sometimes, we engage in ex perimentation. But are we trying to understand the structure of some thing that we ourselves created and choose to represent in a certain way? Criticisms? My main one is that the chapters are too loosely connected and are too short. Some, for example Chap ter 6, "Are People Hard-Wired to Do Mathematics?," is only three pages long, and Chapter 10, "Category Dilemmas," is a mere two pages. Davis acknowl edges the loose connection in the pref ace, so I am not criticizing his intent, but it would be a better hook if he 'd put more effort into connecting each es say to the next. Many of the chapters left me wanting more, hut then came the next chapter. The connections with common sense, promised in the title, are devel oped less frequently as the book pro gresses. Again with Chapter 6, there could be an interesting discussion on "hard-wiring" and common sense, hut this is not developed. A more minor crit icism is that sometimes I wanted a ref erence where none was given. For ex ample, on page 62 we learn that the IQ is no longer given as a single number, but as an 8-tuple. It would have been nice to have a reference, perhaps a web reference, where one could learn about these eight measurements . And, on page 1 13 we read that there is a tablet dating to around 1 700 Be that provides a good hexadecimal approximation to Vz, again with no reference. Final notes of praise. Many of the es says include references to some inter esting books, articles, and websites. Ex ploring these was an unexpected delight. I love to think about the sorts of ques tions raised by Davis and Christina: How does my chosen field lie in the body of
human knowledge and thought? What makes it unique as a subject and as a social endeavor? How have events in his tory changed the course of mathematics, and how have developments in mathe matics changed the course of human his tory? I thank Davis for prompting me to think of these questions again. In short, this book was fun to read, and I recommend at least a perusal to undergraduate math students, graduate students in mathematics, and working mathematicians. Teachers of under graduate math students should consider assigning certain essays as readings for their students. It gives a healthy view of our subject. REFERENCES
1 . R. Courant and H. Robbins, What is Math ematics? An Elementary Approach to Ideas and
Methods ,
Oxford University Press,
1 941 . 2. A. Tversky and D. Kahneman, "Judgments of and by representativeness" in Judgment under uncertainty: Heuristics and biases,
Kahneman, D . , Slavic, P . , and Tversky, A. (eds), Cambridge University Press, 1 982. Department of Mathematics and Computer Science Macalester College St. Paul, MN USA e-mail: [email protected]
The G reat Pile Debate
Colin Adams us. Thomas Garri(y, Moderated hy Edward Burger MATHEMATICAL ASSOCIATION OF AMERICA, 2006, US $24.95, ISBN 10: 0883859009, ISBN 13:
978-0883859001 REVIEWED BY PAMELA GORKIN
everal years ago I was ap proached by the provost of my university, who asked if I would give a short presentation to a group of alumni on the teaching of calculus and how it has changed. Feeling quite hon ored, I agreed to do so. I set to work, preparing an elementary talk that dis played limits numerically. I showed how to compute ��= l 1/2" by filling in
the area of a square with sides of length 1 with colorful little rectangles. I had nifty applications. Before I gave my pre sentation, I gave the talk to my two chil dren, then ages 7 and 1 1 . "Oooh, I get it," shouted my 7-year-old as I pulled out my chartreuse rectangle, "you're go ing to fill in that whole square . " After the calculator demonstration, my 1 1year-old piped up, "Those numbers are getting really close to 1 ! " Confident that I had created a talk at the right level, I presented it to a room full of well dressed, financially well-endowed peo ple. At the end of my presentation, a young woman approached me, and placed her hand gently on my arm. "I just have to tell you, " she grinned, "I didn't understand a single word you said." Since then, I have given a lot of thought to what I would do differently. Thus far, my solution has been to avoid such presentations. But knowing how to talk to people with no mathematical background is a valuable skill, espe cially if they have a stake in your uni versity. You may be asked to speak at a prospective student clay, to nonmath ematical colleagues about mathematics, or to parents and their students at a Phi Beta Kappa induction ceremony. For example, I discovered from the Williams College website that the school hosts a "Family Day" with opportunities for par ents to participate in life at Williams Col lege with their sons or daughters. Al though many parents might be loathe to participate in their child's college life, quite a large crowd of parents showed up for the event sponsored by the Williams mathematics department, The Great Pile Debate, now a minor motion picture in DVD format. This DVD is exactly what its name implies: a debate about the virtues of the numbers e and 17'. In this case, Ed ward Burger is the moderator of the de bate. As you may know, Burger is one of the three winners of the 200 1 MAA National Haimo Teaching Awards, and he is the author of several books, in cluding (with Michael Starbird) the award-winning The Heart ofMathemat ics. More important to this review, how ever, Burger has performed stand-up comedy. The two debaters are Colin Adams and Thomas Garrity. Colin Adams (also one of the winners of the MAA National Haimo Teaching Awards)
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is the author of the very funny column "Mathematically Bent" that appears in this very journal, 7be Mathematical In telligencer. He is the coauthor of sev eral books, including (with Joel Hass and Abigail Thompson) one of my fa vorite calculus books, How to Ace Cal culus: 7be Streetwise Guide. Adams is, perhaps, best known for the lectures that he gives disguised as Mel Slugbate, a cheesy real estate agent with no taste in clothing (sorry Mel). In this video, Adams makes the case for 7T. The sec ond of the two debaters is Thomas Gar rity, who is the author of All the Math
ematics You Missed [But Need to Know for Graduate School} and one of the three winners of the 2004 MAA National Haimo Teaching Awards. He is an an tics man: according to a letter about the nomination written by Adams, "Garrity once taught an entire class hopping on one foot. " Garrity appears here on be half of the number e. After a very clever introduction by the moderator, the debate takes off. One interesting feature of the video is that the viewer will learn as much about Adams and Garrity as he or she will about 7T and e. This is, of course, per fectly appropriate for an event like "Family Day." Much of the rest of the video reminded me of the joke about the top 10 reasons why e is better than pi (reason #10: "e is easier to spell than pi"), or the top 10 reasons why pi is better than e (reason #10: "e is less chal lenging to spell than pi"). Adams presents his case first, cleverly
documenting a few mathematical reasons why pi is better than e. Garrity, his voice rising and falling dramatically as he makes his point, extols the virtues of the number e. Adams comes back with ref erences to properties of pi. Garrity re sponds by talking about apples. Good natured insults are traded. Rules of debate are broken. At the end of the de bate, the audience even gets to vote on the winning number. As I watched the debate, it became clear to me that the audience was having a much better time than my audience did when I showed them my colorful little rectangles. The DVD, produced by the Mathe matical Association of America, is 40 minutes long and demands, mathemat ically, very little from the viewer beyond the basic facts a calculus student would learn about these two numbers. This
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makes it the perfect video for a high school calculus class on those days when the teacher is absent and the sub stitute teacher would really rather be teaching art class. It is also ideal for an apres calculus-AP exam party. On the other hand, I decided not to show this to my own students, because I thought they would learn less from it than 40 minutes of class time are worth. I did, however, recommend it to my colleagues and to my students to view on their own time. Why? Because 7be Great Pile Debate presents an excellent recipe for a talk to a general audience: Just come up with a brilliant idea, think of an original way to present it, find funny people to help you, keep the mathematics to a minimum, get the au dience involved, and enjoy yourself. Who knows? Maybe someone will make a movie about you. Department of Mathematics Bucknell University Lewisburg, PA 1 7837 USA e-mail: [email protected]
Mathematik fur Sonntagnachmittag ( Mathematics for
Sunday Afternoon) By George G . Szpiro Zurich: Nzz Libro, 2006, 219 PP. , CHF 38.00, ISBN 10: 3038232254 ISBN 13: 978-3038232254
REVIEWED BY MARTIN AIGNER
I
t is common that mathematicians find it difficult to communicate their sub ject to the general public. They envy physicists for their mythical concepts that have found their way even into everyday language. When listening to a lecture on black holes, dark matter, or parallel universes, the audience usually doesn't understand a word and might even suspect the very speculative na ture of the argument. And yet the crowd feels it is present at the revelation of some awesome truth. Things are not so smooth for mathematics. Even wonder fully expressive terms such as imaginary -· .
numbers or singularities arouse only mild interest. Only using the big bang as an example of a singularity arouses undivided attention. So, how to write about mathematics and reach a wide audience? One sure way is to appeal to the esoteric-minded, the 7T-ologists, neopythagoreans, or the butterfly-hurricane world explainers. Some authors compile an impressive ar ray of technological advances in which mathematics plays a key role, with primes, the RSA scheme, and cryptog raphy being the current favorites. Some "human touch" books (Turing and the Enigma, Ramanujan and Hardy) have found a large readership. And, of course, there have been several valiant attempts to portray mathematics as a cultural force. The wonderful books 7be Mathematical Experience by Davis and Hersh, and Mathematics: A Very Short Introduction by Gowers, are recent ex amples, acclaimed by critics and bought by the public. George Szpiro chooses a different approach, charming mathe matical small talk, and he clearly is a master at that, almost without peer among German writers. With a twinkle in his eye, he calls his collection of 50 short vignettes Mathematicsfor Sunday Afternoon, as if to invite the reader to muse about mathematics over a cup of coffee before moving on to weightier matters. After having read a few pieces, you realize that Szpiro is a mathematicians' journalist. As a trained mathematician, he knows what he is talking about, and he writes about it in a prose of cool el egance. Some of the pieces have a his torical touch, for example, the essay on the fascination of numbers with which the book begins; others discuss recent developments, such as the famous Green-Tao theorem on primes in arith metic progressions. Some articles rise to abstract heights (as the piece on the ax iom of choice), whereas others are very down-to-earth, as witnessed by such ti tles as " Warum giht es Se:i?" (Why is There Sex?), "Das Leben wird wieder kiirzer' (Life is Getting Shorter Again), or "Verbrecherjagd mit Kopfchen statt Fausten " (Chasing Criminals with Brain Instead of Fist). Sometimes one wishes to learn a bit more about the mathe matics, but in general the relation be tween mathematics and journalism is just right.
One of my favorite pieces is a one page jewel entitled "Der letzte gemein same Ahne" (The Last Common Ances tor). Genealogists have observed that frequent intermarriage must have oc curred not only in small communities, but on a grander scale. Every person alive today has, theoretically, 240 an cestors 40 generations back, around the year 1 000-roughly a trillion. But the total world population at that time was only on the scale of a billion! Szpiro re ports that a computer model simulating the mating habits in different parts of the world produced the astonishing re sult that the last ancestor common to all living people lived around the year 300 Be. What's more, every person today seems to hail from the same group of people 169 generations back, around the year 3000 Be. All the other families have apparently died out. This is vin tage Szpiro: charming, informative, un expected, and a perfect story for next Sunday afternoon. George Szpiro is a wanderer be tween worlds. He studied mathematics and physics in Zurich, obtained his PhD in Jerusalem, and later turned to jour nalism. Since 1 987, he reports from Jerusalem for the renowned Neue Zurcher Zeitung, and his regular col umn on mathematics is eagerly awaited by a faithful and growing readership. With this book you have a chance to join the readership ranks-don't miss it. Freie Universitat Berlin lnstitut fUr Mathematik II Arnimallee 2 D-1 41 95 Berlin Germany e-mail: [email protected]
Crimes and M athdemeanors by Leith Hathout WELLESLEY, MA, A. K. PETERS, 2007,
xi + 196
PP.,
US $14.95, ISBN 13: 978·1·56881-260-1 (pbk.), ISBN 10: 1-56881-260-4 (pbk.)
REVIEWED BY JOHN J. WATKINS
I
t's time for Sherlock Holmes, Hercule Poirot, Inspector Maigret, Adam Dal gliesh, and even Charlie Eppes to
step aside; there is a new kid on the block. And I do mean kid. He is a slen der, 1 4-year-old boy and crime fiction's newest and most brilliant detective. Ravi lives with his parents in Chicago and, when he isn't playing basketball or studying math, he is busy helping his father and the father of a school chum who conveniently happen to be a Chicago district attorney and police chief, respectively-fight crime on the streets of The Windy City. And while he's at it, Ravi also solves a few minor misdemeanors he comes across at school and during his summer vacation. Ravi is the fictional alter ego of Leith Hathout, the author of Crimes and Mathdemeanors, who is himself a high school student and a true lover of math ematics. It was in fact Hathout's love of math that inspired him to write this de lightful collection of stories for his friends and other young people as a way to spark in them the very same en thusiasm he feels for the subject. He and Ravi indeed make a dynamic duo; Leith has uncovered fourteen absolutely wonderful mathematical problems most of them are previously well known, although a few are entirely his own creation-and he has spun about each problem a highly entertaining tale of mystery for his young sleuth Ravi to unravel. My favorite moments in each story come at just the stage when: the police chief is about to arrest the wrong per son, and Ravi says "Hold on!" ; or all the adults are completely stumped and Ravi calmly announces, "The murderer is . . . "; or Ravi turns to his father saying, ·'Dad, I'd rethink that case if I were you," at which point Hathout abruptly calls the action to a complete halt and invites readers to solve the crime. At this mo ment in each story we know precisely everything Ravi knows and (theoreti cally) we should be able to figure out the solution, too. Hathout then simply leaves the rest of the page blank. This is an inspired literary device. Readers can decide for themselves how long they want to pause to work on the prob lem before continuing with Hathout's analysis on the next page. The problems vary nicely both in level of difficulty and in subject matter. As noted by Martin Gardner, who con tributed a highly complimentary blurb on the back cover, these problems for
the most part require only high-school mathematics. One problem does need a bit of calculus, and a few concepts from high-school physics show up once or twice. Hathout's explanations of Ravi's solutions are clear and lively and, more important, he is also adept at com municating how much fun mathematics can be. This is an excellent book for young readers. This is also an excellent book for teachers. I plan to use several of these stories in the near future in a discrete math course: stories about basketball and industrial espionage for the con cepts of permutations and combina tions, and a story called "A Day at the Racetrack" for the topic of derange ments. The next time I teach number theory, I'll be sure to use "Moon Rock" as an interesting variation alongside my usual treatment of Diophantine equa tions. And any discussion of falling bod ies in a physics class would be greatly enlivened and enriched by inclusion of the story "Caught on Film." The stories themselves are simply in genious. I'm not suggesting that Hathout is in the same league as Chekhov as a writer of short fiction, but his stories do provide charmingly play ful settings for the real point of each story, which is to display an intriguing math problem. I suspect that most read ers will forgive this youthful author for dialogue that is sometimes stilted, and for occasional lapses in plot such as wa termelons that lose half their weight af ter only a few hours. in the sun, or for a middle school principal who not only approves, but suggests, an outing to the racetrack where students get to bet on the horses. Hathout's stories may not always be plausible, but they work as intended, to make the mathematics be hind them enjoyable. This is quite an accomplishment, and is more than has been achieved by most mathematics writers. Warning-! will now discuss several of the stories and math problems in considerable detail, so some readers of this review may prefer to stop reading at this point. I found, however, that knowing the problem ahead of time al lowed me to savor all the more the way in which Hathout chose to handle both the story and the mathematics. A good case in point is the last story, "A Snowy Morning on Oak Street" ; I knew the
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problem as soon as I saw the illustra tion (the book is beautifully illustrated by Karl Hoffman) of a snowplow in a heavy snowstorm. This problem has long been known at Colorado College as the "Snowplow Problem . " I first came across it 30 years ago when I taught dif ferential equations from a marvelous book by my colleague, George Sim mons. This problem first appeared, as it happens, in 1 937 as a problem in the American Mathematical Monthly [2] on the very same page as an equally in triguing geometry problem proposed by a 24-year-old P. Erdos from Budapest, Hungary. Here is George's version: It began snowing on a certain morning, and the snow continued to fall steadily throughout the day. At noon a snowplow started to clear a road at a constant rate in terms of the volume of snow removed per hour. The snow plow cleared 2 miles by 2 P.M. and 1 more mile by 4 P.M. When did it start snowing? What Hathout likes about this prob lem, and it is a common theme in sev eral of them, is that there doesn't seem to be nearly enough information pro vided to solve it. That's why this has been one of my favorite problems for many years; students have to grapple with it for a very long time, before they eventually are able to solve it. If you haven't seen it before, I urge you to try it. It does require a little calculus, and so it is a fitting choice for the last chap ter of the book. For this problem Hathout constructs a somewhat far fetched story involving a jewel thief named Jimmy "Pickles" Graziano. Since Ravi is good friends with the snowplow driver, he is able to learn the crucial data about times and distances. This in turn enables him quickly to evaluate a couple of integrals, and he can then figure out exactly what time it started to snow. This is all his district attorney father needs to know in order to deal with the notorious "Pickles." The first story in the book, "A Mys tery on Sycamore Lane," is also one where the nature of the math problem quickly reveals itself. In fact, at the point in the murder investigation when the host of the party where the murder took place is explaining to Ravi that he had asked his guests, including his wife, how many hands each of them had
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shaken as they were leaving the party, I knew immediately that this crime was going to be about an extremely well known problem in graph theory. Then, when 3 people tested positive for gun powder residue, I reached for a pen cil-I wanted to solve this one before Ravi did! Hathout does a nice job of explain ing exactly how Ravi solved this mur der. Since the host says each of the 5 people gave him a different answer, and the host's wife says she shook 4 hands, Ravi quickly can see that the wife must be lying. This is because the 4 guests were also couples, so 4 is the largest number of hands anyone would have shaken. Thus, the 5 answers the host received had to have been 0, 1 , 2, 3 , and 4. S o , i f the wife shook 4 hands, it was one of the guests who shook 0 hands, but that is impossible since the wife shook hands with all 4 guests. Therefore, the wife lied. (Exactly why Ravi never considers the possibility that the host could be lying is left unclear, but we must simply ignore this logical inconsistency, otherwise there would be no math problem left to discuss.) Hathout then goes case by case show ing the wife could not have shaken 0, 1, or 3 hands either. Therefore, she shook 2 hands. Once Ravi knew this, and since it also fit the gunpowder residue data, the police chief was able to extract a confession from the wife. Hathout could have shortened this proof considerably-and his proof did get somewhat repetitive-by using sym metry to eliminate half the cases. There is an underlying "hand-shaking graph" here (couples, of course, never shake hands), and so any feasible graph in this problem produces a complemen tary feasible graph; hence, if the wife can't shake 4 hands, she can't shake 0 hands (similarly, if she can't shake 3 hands, she can't shake 1 hand). Case by case arguments can almost always be shortened. Alternatively, Hathout could have provided a much easier proof by concentrating on the graph and not on the wife. Such a proof hinges on the idea that since all answers are different, the graph must be unique. There is a vertex of degree 4, and it must be paired with a vertex of degree 0 ("paired with" means the two vertices represent a couple). Then, the vertex of degree 3 is one of the remaining ver-
tices (it doesn't matter which), and since its edges are now forced, it must auto matically be paired with a vertex of de gree 1 . The graph is now done, and the host clearly has degree 2. Hence, his wife also has degree 2 . After Ravi has solved this crime and gone to bed, Hathout does a very good thing, which he also repeats with other stories: he adds an "extension" where he goes on to discuss a generalization of this problem. In this case, he proves that no matter how many couples come to a party, as long as the host receives a different answer from everyone, his wife will have shaken exactly as many hands as the number of couples invited to the party. Hathout proves this re markable theorem by induction, and claims this is the "best way" to do it. I heartily support the idea of showing young readers the method of induction, but induction rarely offers much insight into what is really happening. In this case, a constructive proof not only pro vides a genuine feeling for what is go ing on, but also results in a significantly stronger theorem as well. In fact, it is even entirely sufficient to give a proof for a specific instance of this stronger theorem, say, for 10 couples, and show that the person who shook 8 hands must be paired with the person who shook 0 hands (since the other 9 peo ple already have at least one hand shake). Then the person who shook 7 hands (it is best to draw a graph at this point) clearly must be paired with the person who shook 1 hand. Then the person who shook 6 hands, well you get the idea by now. Finally, at the end, you have constructed a unique graph, and this graph has a special property, namely, all couples have degrees adding to 8. Since the host received a different answer from everyone, the host has to be in the couple whose de grees are both 4; hence, his wife also has degree 4. So, it turns out that what is lurking behind the mathematical mys tery on Sycamore Lane is a beautiful graph. Another story I particularly enjoyed, but this time because the mathematics, or rather the physics, was surprising to me, was "Caught on Film," which in volved two steel balls dropped from the top of the Sears Tower. These steel balls were, quite by chance, seen on film se riously injuring an actor, the one ball
striking the actor while the other is clearly seen 30 feet above it still in midair. Ravi is able to show-and this is what was really quite amazing to me-that when the marbles left the hand of a "spiky-haired" teenager 1431 feet above, they were only 2 inches apart vertically. And so, contrary to sev eral eyewitness accounts, this teenager had to be the sole perpetrator of this badly misguided prank. Hathout presents a very elegant so lution to this problem using the basic distance formula s = .!..gt 2 for falling 2 bodies. Moreover, he is pleased to point out that his solution does not depend on the reader needing to know that g = 32 ft/sec 2 . Many students, however, do know this number and would easily come up with a perfectly straightfor ward solution using this fact. Also, many students tend to think like physicists and might solve the problem another way entirely. The first ball had a 2 inch head start which effectively meant it started with an initial velocity of 3.27 ft/sec. Since the whole trip for both mar bles took about 9.46 sec, that means the marble with the extra velocity at the be ginning gained 3 . 27 ft/sec X 9.46 sec = 30.89 ft on the way down, a much eas ier argument than Ravi's. More impor tantly, it also explains why 2 inches at the top can turn into 30 feet at the bot tom. Hathout missed a golden oppor tunity to cash in on the extremely clever use of film in his story. He did point out that the constant g = 32 ft/sec2 con veniently cancelled itself out in his equations, but failed to explain why it had to disappear. This is the whole point about gravity and acceleration; they work the same way on the earth and the moon. The film crew could have been recording the steel balls drop on the moon and it would have just looked like slow motion. This was a perfect chance to demonstrate the fun damental way in which gravity, time, and distance are all intertwined. Such criticisms aside, Hathout has chosen problems that have genuine depth, problems that will intrigue and fascinate young readers and at the same time stimulate teachers of mathematics who have the experience to plumb these problems for their full potential. I eagerly await the next volume of sto ries that undoubtedly will spring from the author's creative young mind. Per-
haps in Still More Crimes and Mathde meanors he will even be willing to di vulge Ravi's last name. And he should consider spinning a story around this variation of the snowplow problem that appeared in the Month(v in 1 952: It had started snowing before noon and three plows set out at noon, 1 o'clock, and 2 o'clock, respectively, along the same path. If at some later time they all came together simultaneously, find the time of meeting and also the time it started snowing.
A Piece of J ustice by jill Paton Walsh ST. MARTI N ' S PRESS, 1995, 182 PP., US $15.95, ISBN 0-312-29252-X
The Cambridge Theorem by Tony Cape CLAREMONT BOOKS, 1990, 380 PP. , AUST. $12.95, ISBN 0-670-90787 I
REFERENCES
1 . G. Simmons, Differential Equations, with Ap plications and Historical Notes ,
McGraw Hill,
1 972, 1 991 .
2. J. A Brenner and W. B. Campbell, E275, The American Mathematical Monthly 44,
10 (1 937), 666-667.
no.
3. M. S. Klamkin and L. A Ringenberg, E963, ibid.
59, no. 1 (1 952), 42.
Avenging Angel by Anthony Appiah ST. MARTI N ' S PRESS, 1990, 207 PP., US $16.95, ISBN 0-312-05817-9
G hostwal l<
Department of Mathematics and
by Rebecca Stott
Computer Science
&
Colorado College
SPIEGEL
Colorado Springs, CO 80903
ISBN 978-0-385-52106-2
USA
REVIEWED BY MARY W. GRAY
email: [email protected]
The Three Body Problem by Catherine Shaw ALLISON
&
BUSBY, 2004, 286 PP., US $9.95,
ISBN 0-7490-8347-6
Flowers Stained with M oonl ight bv Catherine Shaw ALLISON
&
BUSBY, 2005, 314 P P . , US $9.95,
ISBN 0-7490-8208-9
The Library Paradox by Catherine Shaw ALLISON
&
BUSBY, 2006, 315 P P . , £18.99,
ISBN 0-7490-8293-3
GRAU, 2007, 304 PP., US $24.95,
ooking for a birthday gift for someone who has everything, like maybe King Oscar II of Sweden? To honor his sixtieth birthday a prize competition was organized, in which several problems were posed for solu tion, including the three-body problem. Most mathematicians know that Poin care's prize-winning partial solution helped establish his reputation, even though the publication of his erroneous original submission had to be recalled at his expense. But the title of Cather ine Shaw's The Three Body Problem refers not only to the problem itself but also to the dead bodies of three math ematicians seeking to gain fame (and fortune, although not much, compared with today's Clay awards) by winning the competition. Enter Vanessa Duncan, a sleuth of whom we are to hear more, who has a school for young girls in Cambridge. An interesting source for her quite progressive instruction is a puzzle from Lewis Carroll's "A Tangled Tale. " Vanessa's fellow lodger is a Cam bridge mathematician, Arthur Weather burn. Because Arthur was the last to see
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two of his fellow mathematicians before they mysteriously died, he is accused of their murder as well as that of a third colleague, all of whom were victims of the same poison. It also appears that all of the dead men were competing for the Birthday Prize. Vanessa discovers that papers of Ak ers, one of the victims, may contain a clue that will exonerate the imprisoned Arthur, of whom she has grown quite fond. Off she goes to Belgium to seek out Akers's sister, to whom his papers were sent. Vanessa is accompanied by Emily, one of her young pupils, who has decided to travel secretly to the con tinent to unravel a family mystery. This supposedly occurs around 1890, so the likelihood of such an undertaking on the part of two young, na'ive, middle class females is fairly slim, but their ad ventures make a great story. As a result of the revelations from the papers that Akers's work on the eponymous problem may have been stolen, Vanessa and Emily are off to Stockholm to seek the help of Mittag Leffler. Well-received by the mathe matician, the adventurers wait in sus pense while he approaches King Oscar himself, for the seal on the submitted solutions must be prematurely broken in order to reveal the identity of the real murderer. The solutions were to be submitted anonymously, with an epi gram as identification. Poincare's was "Nunquam prxscriptos transibunt sidera fines" (Nothing exceeds the limits of the stars), but he also sent a signed cover
ing letter [ 1 ] . Meanwhile, back in Cambridge, Arthur is on trial and things are look ing grim. But Vanessa's last-minute re turn from Stockholm with the evidence she has procured through the help of Mittag Leffler and King Oscar dramati cally saves him. Although it is true that the denoue ment can be foreseen, Shaw gives the reader a merry tale, full of atmosphere if not a lot of suspense. The Cambridge mathematical scene is engagingly por trayed, complete with such luminaries as Arthur Cayley and Grace Chisholm, about to depart for Germany because of Cambridge's exclusion of women from the pursuit of advanced degrees. Although not much substantive mathe matics is discussed, an interesting side light is Cayley's attempt to gain accep-
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tance for his reform plan for school mathematics. His pronouncement that "those who take decisions and argue the value of teaching methods of math ematics had better be those who mas ter the subject" is not well received by some of his less research-oriented col leagues. They claim that Cayley has a closed mind that does not welcome new ideas. And many of us thought that the debate over mathematics education reform-rising to the level of "Math Wars" in the United States-was a re cent phenomenon! We hear also of Sonya Kovalevskaya, whom we are told Vanessa just missed during her sojourn in Stockholm but whose accomplish ments stir Vanessa's feminism. In real ity, in a letter to Mittag-Leffler, Ko valevskaya had presciently expressed concern over the difficulties the com petition was likely to encounter [ 1 ] . The story i s told in the form o f let ters to Vanessa's sister, a device that is a bit stilted but generally works well. There is a useful brief appendix pro viding some background to the three body problem, the mathematicians mentioned, and the work of Lewis Car roll. But we have not heard the last of Vanessa. Shaw's second venture, Flow ers Stained with Moonlight, suffers from a common second-novel slump, per haps occasioned by a departure from the mathematical scene insofar as the actual crime is concerned. Because Vanessa has acquired a reputation for problem-solving, she is drawn into the difficulties of a local family. The mother who seeks Vanessa's help lives with her daughter, whose much older husband has recently died in mysterious circum stances, and a friend of the daughter. Very early on, the solution to the mys tery surrounding the household be comes obvious. More interesting than the principal plot is the introduction of a subsidiary character, the "Prussian amateur mathematician," G. Korneck. Korneck was a real person who devoted himself to trying to solve Fermat's Last Theorem, but about whom little else is known. To please feminists, we hear much of the work of Sophie Germain on Fermat's Last Theorem (although contrary to what Vanessa says, the use of the name M. LeBlanc was not enough to admit her to the Ecole Polytech nique). Also described are the efforts at
proof of the theorem by Lame and Cauchy, and Kummer's refutation. The device of letters to Vanessa's sis ter is used once again, but unfortunately is abandoned toward the end of the book in favor of a tedious and unnec essary excerpt from the journal of the murderer. A useful background to Fer mat's Last Theorem is contained in an appendix. As a mystery, Shaw's second book is mostly a failure; at least for her, mathematicians seem to be more inter esting as victims and suspects than are more prosaic characters. Flowers turns out to be an aberration as Shaw is back in form in Tbe Library Paradox. Here the title again has dou ble meaning-it refers to the crime scene and to one implementation of Russell's paradox. Vanessa, now the wife of Arthur and the mother of twins, has established a reputation as a part time private inquiry agent. An appeal from a mathematician at King's College brings her to London, where the un raveling of the mystery takes her deep into London's orthodox Jewish com munity. It is harder to judge the au thenticity of Shaw's portrayal of this set ting than that of the somewhat similarly insular Cambridge mathematical circle. However, Shaw is skilled at making her characters and the atmosphere of a Purim celebration in the East End come alive. Again a young mathematician is in carcerated when the authorities deter mine that the only resolution to the paradox of a variant of a locked-room murder is that he must be lying about having seen a rabbi on the scene and hence must himself be guilty of the crime. The dead man was a notorious anti-Semite, and we hear much of the Dreyfus affair, although not of the role of mathematics in the trial and aftermath [5]. Here the feminist aspect comes from Emily, Vanessa's former pupil, who has followed Charlotte Scott to Queen's Col lege London to study mathematics be cause, unlike Cambridge, Queen's tol erates if not welcomes women. In Paradox Shaw abandons her format of Vanessa's writing to her sister in favor of the more effective device of having her draw from her own diary. As in the earlier volumes, there is an appendix explaining the mathematical setting. Let us hope that we will hear more of Vanessa Duncan and her adventures,
the more connected with mathematics the better. Radmila May claims that Oxford is the murder capital of the world [8]. Al though Cambridge may not be able to compete in general with the setting for the works of Dorothy Sayers, Colin Dex ter, or Veronica Stallwood, it seems to be the leader in murders connected with mathematics (but see The O:iford Murders [4]). In addition to the work of Catherine Shaw, Jill Paton Walsh has written a Cambridge mystery centered on a mathematician. In A Piece ofjus tice, the fictional Gideon Summerfield made his reputation on the discovery of Penrose-like tilings involving heptagons. Three successive biographers ceased to work on his biography, each in myste rious circumstances that may or may not have involved murder. The fourth, Fran Bullion, a graduate student to whom a senior Cambridge academic has con signed the latest attempt, does not give up easily. Her ally is the protagonist in Walsh's mystery series, Imogen Quy, college nurse and quilter. Investigation by Fran finds that Summerfield discov ered the heptagon, nonrepeating pat tern on a quilt in an isolated farmhouse in Wales. Up until then his widow had-even to the extent of resorting to murder-made sure that this blow to her spouse's reputation was not re vealed. Perhaps the device of a folk art origin of the tilings seems more believ able in light of recent revelation of Pen rose-like patterns in the fifteenth-cen tury Darb-! Imam shrine in Isfahan and in Uzbekistan, Afghanistan, Iraq, and Turkey [13]. Another pair of mysteries has loose connections with mathematics at Cam bridge and in particular with the Cam bridge society known as the Apostles, whose members included G. H. Hardy, Bertrand Russell, A. N. Whitehead, James Clerk Maxwell, C. P. Snow, and John Maynard Keynes. One might ex pect The Cambridge Theorem to involve a mathematical result, real or imagined, original or stolen. However, the title refers merely to the paradigm used by the victim, a student of mathematics, in analyzing events shrouded in contro versy. First came what might be called the "Kennedy Theorem, " a detailed analysis of the assassination. refuting the lone gunman explanation. The "the orem" at the heart of the mystery con-
cerns the identity of the legendary "fifth man" of the British spy scandal involv ing Burgess, MacLane, Philby, and Blunt. There is little actual connection to mathematics in The Cambridge Theo rem except for references to the work at Bletchley and to the fate of Alan Tur ing. However, it is a good mystery, harking back to the days of the Cold War and skillfully invoking the closed atmosphere of academic Cambridge. The most interesting character, how ever, is Sergeant Derek Smailes of the Cambridge CID , who comes up with the solution to the puzzle in spite of the town-gown antipathy and personal in volvement hindering his investigation. Smailes is the protagonist of several other mysteries by Cape-unfortunately not involving mathematics. The title of the other Apostles mys tery, Avenginp, Angel, refers to the role of an alumnus of the society (called an "angel") in solving the mystery. Al though David Viscount Glen Tannock, the first victim in this tale, was exam ining the papers of mathematician Gre gory Ransome. there is little actual mathematical content in the mystery, not surprising given that David is said to "know next to nothing about math ematics. " The angel in this case is a cousin of David, Sir Patrick Scott, a bar rister similarly ignorant although with a Bletchley past. Although David's interest is in the nonmathematical aspects of Ransome's career, he does come across an inter esting result. Godfrey Stanley, a math ematician and Apostle, had earlier served as Ransome's scientific executor, resulting in a major result, the Ran some-Stanley theorem, on which Stan ley's reputation was based. That not much was produced later in his career seems not to have bothered Stanley, as he asserts the often-heard belief that "mathematical achievement is, like beauty, a thing of youth. " The roles of Ransome and Stanley were character ized by Stanley as Watson and Crick, whereas in reality Mozart and Salieri provide a better analogy of the rela tionship, particularly if one subscribes to the theory that Salieri poisoned the man whose talent so far eclipsed his own. However, misappropriation of the work of Ransome, rather than mere envy, is at the heart of this mystery.
Ransome was apparently a man of many interests. Keynes is said to have praised his work as "some elegant stuff about taxation" in his eulogy after Ran some's early death in the Alps, although algebraic topology and applications of topology to neural networks are listed among his other interests. The tale has a little of everything, from an expose in Private Eye to several possible Russian spies, threatening notes enCJypted us ing the RSA algorithm, and references to monster groups and the Frechet con jecture. The device of a threat to the ques tionably-acquired reputation of a math ematician also entered into Oxford based fiction in a recent episode of the TV series " Inspector Lewis, " the suc cessor to the classic "Inspector Morse" oeuvre. That Isaac Newton was perhaps not a noble character is hardly news, espe cially in light of his role in the Royal Academy's report on the calculus pri ority issue [2] . To picture him as a sus pect in the murder of five men in pur suit of a Trinity fellowship, however, is left to Rachel Stott in Ghostwalk. Even for those not fond of ghost stories, the specter of a figure in the red robes of a Lucasian professor of mathematics haunting the streets of twenty-first-cen tury Cambridge produces a certain fris son. Ghostwallis narrator, Lydia Brooke, has agreed to complete The Alchemist, the story of Newton that was intended to "challenge the myth of Newton as a lone genius, working completely in iso lation . " The focus of this work's origi nal author, Elizabeth Vogelsang, was on the intricacies of seventeenth-century alchemical networks and Newton's de pendence on them, until she discovered suspicious deaths connected with Trin ity College in the 1660s, deaths with which she thought he might have been connected in some way. Newton's seri ous engagement in alchemy has been remarked before by his biographers. generally accompanied by disclaimers, statements of incredulity, or the wish to make it all disappear if the evidence were not so strong. For example, we find Richard Westfall embarrassed and confounded by Newton's alchemy: Since I shall devote quite a few pages to Newton's alchemical inter ests, I feel the need to make a per-
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sonal declaration . . . . I am not my self an alchemist, nor do I believe in its premises. My modes of thought are so removed from those of alchemy that I am constantly uneasy in writing on the subject, feeling that I have not fully penetrated an alien world of thought. Nevertheless, I have undertaken to write a biogra phy of Newton, and my personal preferences cannot make more than a million words he wrote in the study of alchemy disappear. It is not inconceivable to most historians that twentieth-century criteria of ratio nality may not have prevailed in every age. Whether we like it or not, we have to conclude that anyone who devoted much of his time for nearly thirty years to alchemical study must have taken it very seri ously-especially if he was Newton [ 1 2]. Elizabeth's view of Newton's varied interests was more nuanced, asserting that he saw no separation of natural from supernatural, the material world from the spiritual. In hope of commu nicating with Newton across the cen turies, she acquired a prism thought to have been purchased by Newton at Cambridge's famous Stourbridge Fair, only to be told by her psychic contact that glass constitutes unsuitable mater ial for transmitting discourse with spir its. She, and later Lydia, however, per sist in their quest for communication with Newton and his contemporaries. The seventeenth-century deaths that intrigued Elizabeth included the poet Cowley, two Trinity scholars, a draper's son, and, presaging the twentieth-cen tury spy scandal, a "fifth man . " Another key figure, Mr. F, Elizabeth managed to identify as Ezekiel Foxcroft, mathemati cian and alchemist, fellow of King's Col lege, Cambridge, and translator of the Rosicrucian document Chymical Wed ding. Foxcroft is described by Lydia as John the Baptist to Newton's Messiah, but neither he nor Newton was an es pecially admirable character. Foxcroft's own ambitions in mathematics and in alchemy were apparently sacrificed to aid what he recognized as Newton's greater talents. The connection of the seventeenth century events to twenty-first-century Cambridge suggests itself to Elizabeth and then to Lydia as three deaths oc-
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cur on the same calendar days as the earlier ones in eerily similar circum stances, including a fall down a stair case. Lydia was recruited to finish the work of Elizabeth by her son, who is Lydia's recurrent lover. But entwined with the alchemy, the mystery, and the love affair, are events attributed to an imal-rights activists, characterized by the legend "NABED, " also appearing in the works of Newton and, in good British mystery fashion, possible involvement of Scotland Yard and MIS. Extracts on Newton's work with col ors-his favorite was red-are included in an appendix to Ghostwalk, together with his remedies for sickness and his famous list of his sins. One recipe from Newton's notebook now in the Pierpont Morgan Library: "Take some of the clearest blood of a sheepe & put it into a bladder & with a needle prick holes in the bottom of it then hang it up to dry in the sunne; & dissolve it in alum water according as yo have need. " What then o f the suggestion that Newton's 1667 election as fellow was questionable? Up until then, he had not distinguished himself academically and had given a poor performance in his fellowship vive voce exam by Isaac Bar row. His unimpressive performance, it has been suggested, was because Bar row examined him on Euclid while Newton had skipped Euclid and had read Descartes's geometry instead. How ever, because of the deaths of three fel lows and the removal because of in sanity of another, and the fact that there had been no elections to fellowships in the preceding two years, there were nine fellowships to be filled. It was cru cial that Newton obtain a fellowship in order to avoid having to go back to the farm at Woolthorpe. His biographers generally have attributed his success to luck; one might think that the actual concurrence of events was so unlikely as to require Newton himself, even if entirely innocent of involvement in the series of deaths, to consider whether there might be another explanation. Will we ever know whether his chances were enhanced by nefarious activities of his supporters? As Lydia becomes immersed in her task of completing The Alchemist, we read with her the account by Elizabeth of Newton's progress and Foxcroft's role:
He set Newton initiatory tasks to test his mettle and to strengthen his al chemical powers. Under Ezekiel's influence, Newton came to believe that he could do anything, that everything he did was sanctioned by divine authority, that nothing could stop the flood of knowledge pass ing through him-secrets about light, colour, gravity, numbers. As the conduit of divine knowledge he was untouchable. Ezekiel had said so. Interspersed with the account of New ton's alchemy, we find asides describ ing his more familiar work: In 1665-66, Newton scratched out the fundamentals of what would come to be called the calculus. In 1 665, in his rooms in Trinity, Newton proved that white light was made of colours and took to his bed, temporarily blinded. In 1 666, working by candlelight late into the night, Newton devised a method of calculating the exact gra dient of a curve, a method which would come to be known as differ entiation. Sometime in 1665 or 1666, some where between a garden in Wool sthorpe and a garden in Trinity, Newton carved out the rules of grav itation . . . [no apple!) In 1 668-69, Newton, with the help of his friend Wickins, installed an elaborate experimental apparatus in his rooms and constructed the very first functioning reflecting telescope. In 1 669 Isaac Newton was appointed Lucasian Professor of Mathematics. In 1669, Newton wrote up "De Analysi," another milestone in the road towards the calculus. In 1 67 1 , dressed in his red robes, Newton unveiled his telescope to the men of the Royal Society in Lon don. It caused a sensation. After another fatal plunge down a Trinity staircase, in Stott's somewhat surprising denouement we get her ac count of who killed whom and why, in both the seventeenth and twenty-first centuries. In an author's note she dis tinguishes fact from fiction. She reports that all the historical characters are real
and the circumstances of the deaths of the five men in the seventeenth century were recorded in the contem porary diary of Alderman Samuel Newton (no relation). In weaving together the events of the two periods she tells us much about patronage, conspiracies, murder, and scientific progress, while painting a captivating picture of Cambridge in both eras. But, in the end, as Stott tells us, the narrative is speculative. Whether it is also factual will, she declares, never be known. Cambridge can claim other academic-based mysteries; not only mathematicians can be victims and murderers. Nora Kelly has produced In the Shadow of Kings [6] and Bad Chemistry [7] , Christine Poulson Murder is Aca-demic [9] and Stage Fright: A Cambridge Mystery [ 1 0] , and Dorsey Fiske Aca-demic Murder [ 3] , which has a couple of math ematicians as peripheral characters. Laura Principal, Michelle Spring's investigator in a mystery series, is based in Cambridge, but only Nights in White Satin [ 1 1 ] gets in side the university. So we see that as we walk the streets, courtyards, and even staircases of either Oxford or Cambridge we may find a site where a murderer lurked.
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REFERENCES
(I ] J . Barrow-Green, Poincare and the Three Body Problem, Prov idence: American Mathematical Society, 1 997.
[2] C. Djerassi and D. Pinner, Newton 's Darkness: Two Dramatic Views,
London: Imperial College Press, 2003.
[3] D. Fiske, Academic Murder, New York: Critic's Choice Paper
lication is available at www.springerlink.com. Please use the appropriate URL and/or DO! for the article (go to Linking Options in the article, then to OpenURL and use the link with the DOl-). Articles disseminated via www.springerlink.com are indexed, abstracted and referenced by many ab stracting and information services, bibliographic networks, subscription agencies, library networks, and consortia. While the advice and information in this journal is believed to be true and accurate at the date of its publication, neither the authors, the editors, nor
backs, 1 980.
the publisher can accept any legal responsibility for any errors or omissions
(4] M.W. Gray, " Review of The Oxford Murders, " The Mathemati
that may be made. The publisher makes no warranty, express or implied,
29, no. 3 (2007), 77-78.
with respect to the material contained herein. Springer publishes advertise
[5] D. Kaye, "Revisiting 'Dreyfus': a More Complete Account of a
ments in this journal in reliance upon the responsibility of the advertiser to
cal lntelligencer,
comply with all legal requirements relating to the marketing and sale of prod
Trial By Mathematics," Minnesota Law Review, Vol. 91 , No . 3 ,
ucts or services advertised. Springer and the editors are not responsible for
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claims made in the advertisements published in the journal. The appearance
[6] N. Kelly, In the Shadow of Kings, Scottsdale: Poisoned Pen Press, 1 984. [7] N. Kelly, Bad Chemistry, Scottsdale: Poisoned Pen Press, 1 993. [8] R. May, "Murder most Oxford, " Contemporary Review, Vol. 227, No. 1 61 7, pp. 232-239, October 31 , 2000.
[9] C. Poulson, Murder is Academic, New York: Thomas Dunne Books, 2002.
[ 1 0] C. Poulson, Stage Fright: A Cambridge Mystery, New York: St. Martin's Minotaur, 2005. [I I ] M. Spring, Nights in White Satin , London: Ballantine Books, 1 999. [1 2] R. Westfall, Never at Rest: Isaac Newton, Cambridge: Cam bridge University Press, 1 983.
(1 3] J. N. Wilford, "In Medieval Architecture, Signs of Advanced Math," The New York Times, p. D2, February 27, 2007.
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Robin W i l so n
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(trapezoids). This Malaysian stamp rep resents a lunar module.
The P h i lamath's A l phabet - Q
Quadrivium
Quadrant A quadrant is an instrument that was used for navigational purposes. Like the sextant, which is based on an angle of one-sixth of a circle (60°), the quadrant is based on one-fourth of a circle (90°). To measure an object's altitude, the ob server views it along the top edge of the instrument and the position of a movable rod on the circular rim displays the desired altitude.
Quadrilateral Stamps As well as the usual rectangular stamps, several countries have issued 4-sided stamps of other shapes, notably rhom buses, parallelograms, and trapezia
relativity. This equation explained elec tron spin and led Dirac to predict the existence of antiparticles.
Quetelet
The four mathematical arts of Ancient Greece, described at length by Plato in his Republic, were arithmetic, geome try, astronomy, and music. Plato con sidered them as providing the finest training for those holding positions of responsibility in the state. In combina tion with the other three liberal arts of grammar, rhetoric, and logic, and based on such texts as Euclid's Elements and Ptolemy's Almagest, the quadrivium comprised the curriculum of the me dieval universities in Europe.
Adolphe Quetelet 0796-1874) was su pervisor of statistics for Belgium, and he pioneered techniques for taking the na tional census. His desire to find the sta tistical characteristics of an "average man" led to his compiling the chest measure ments of 5732 Scottish soldiers and ob serving that the results were distributed normally. Taken with earlier studies of life annuity payments by de Witt and Hal ley, Quetelet's work helped to lay the foundations of modem actuarial science.
Quantum Theory
Quipu
Several of those who contributed to the development of quantum theory have been featured on stamps: Planck, Ein stein, Bohr, de Broglie, Schrodinger and Heisenberg. This stamp commemorates Paul Dirac, who effectively completed classical quantum theory by deriving an equation for the electron that, unlike those of Schrodinger and Heisenberg, was consistent with Einstein's theory of
Around 1 500, the Incas of Peru invented the quipu for recording and conveying numerical data and other statistical in formation. The quipu consists of a main cord to which many thinner knotted cords of various colours are attached; the size and position of each knot cor respond to a different number in a dec imal system, and the colours convey other types of information.
Quadrant Quadrilateral stamp
Quadrivium (arithmetic and geometry)
EXFILBRA1Z
Please send all submissions to the Stamp Corner Editor, Robin Wilson, Faculty of Mathematics, Computing and Technology
Quantum theory
PERU
The Open University, M i lton Keynes, MK7 6AA, England e-m a i l : r.j.wi [email protected]
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Quetelet
Quipu
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