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3
Non-Euc l i dean Pythagorean Tripl es, A Prob l e m of Eu l er, and Rational Poi nts on 1<3 S u rfaces
ROBIN HARTSHORNE AND RONALD VAN LUIJK
""\ ""\ \ \
jl
1
e discover surprising connections among three seemingly different problems: finding right triangles with rational sides in a non-Euclidean geom etry, finding three integers such that the difference of the squares of any two is a square, and finding rational points on an algebraic surface in algebraic geometry. We will also reinterpret Euler's work on the second problem with a mod ern point of view.
Problem 1: Pythagorean Triples An ordinary Pythagorean triple is a triple ( a, b, c) of pos itive integers satisfying a2 + tJ2 = d-. Finding these is equiv alent, by the Pythagorean theorem, to finding right trian gles with integral sides. Since the equation is homogeneous, the problem for rational numbers is the same, up to a scale factor. Some Pythagorean triples, such as (3, 4, 5), have been known since antiquity. Euclid [7, X.28, Lemma 1] gives a method for finding such triples, which leads to a complete solution of the problem. The primitive Pythagorean triples are exactly the triples of integers (m 2 - n 2, 2mn, m 2 + n 2) for various choices of m, n (up to change of order). Dio phantus in his Arithmetic [6, Book II, Problem 8] mentions the problem of writing any (rational) number as the sum of squares. This inspired Fermat to write his famous "last theorem" in the margin. Expressed in the language of algebraic geometry, the equation x 2 + y 2 z 2 describes a curve in the projective =
4
THE MATHEMATICAL INTELLIGENCER
plane. This curve is parametrized by a projective line according to the assignment (in homogeneous coordi nates) (1)
( m, n)
�
(m 2 - n 2, 2mn, m 2 + n 2) .
The rational points on the curve correspond to primitive Pythagorean triples, which explains why the same param etrization appeared previously. Now let us consider the analogous question in a non Euclidean geometry. One defines a hyperbolic plane by re placing the usual parallel axiom of Euclidean geometry (through a point P not on a line l, there is a unique line m parallel to I) by Hilbert's hyperbolic parallel axiom [5, p. 374]: "Through a point P not on a line l, there are two limiting parallel rays to the line l, not lying on the same line." A limiting parallel is a ray m from P that does not meet I, but such that any other ray r from P, inside the an gle between m and the perpendicular PQ from P to !, must meet l (see Fig. 1). The segment PQ uniquely determines the angle between PQ and the limiting parallel ray m, which is called the angle ofparallelism of the segment PQ, and conversely this angle uniquely determines PQ up to con gruence [5, 40.7.1]. The relations between the sides and angles of a right tri angle are expressed by formulas of hyperbolic trigonome try [5, 42.2, 42.3] in terms of the two angles and the three angles of parallelism of the sides. In particular, if a and b are the legs and c is the hypotenuse of a right triangle, and
m
Q
,�-
r
Figure I. The limiting parallels. if a, b, c are the corresponding angles of parallelism, the hyperbolic analogue of the Pythagorean theorem is sin a . sin b = sin c. To interpret this formula in terms of the sides themselves, we need to know how to measure lengths, and we need the relation between the length of a segment and its angle of parallelism. There is a model of the hyperbolic plane resulting from Poincare as follows. Fix a circle f in the Euclidean plane. The "points" of the Poincare model are the points in the interior of f. The ''lines" of the model are those portions of circles orthogonal to f that lie inside f. Two "lines" are limiting parallels if they meet on r, since the points of r are not part of the model. One can verify that this system is a model of a hyperbolic plane [5, §39]. For any two points, A, B in this model, one defines J.L(AB) (AB, PQri, where P and Q are the intersections of the "line" AB with f as points of the Euclidean plane, and (AB,PQ) represents the usual cross-ratio of the (Euclidean) distances among the points A,B,P, Q (see Fig. 2). If A,B, C are three points in or der on a "line," then J.L(AC) = J.L(AB)J.L(BC). Therefore the distance function in the hyperbolic plane is usually defined by taking the logarithm of J.L(AB). The logarithm introduces an arbitrary constant (its base) into the formulas of geom etry and gives rise to the transcendental trigonometric func tions sinh, cosh, tanh. For our purpose it is more natural to use the multiplicative distance junction J.L itself without taking logarithms. The relation between the multiplicative length of a segment AB and its angle of parallelism a is expressed by a wonderful formula of Bolyai [5, 39. 13]: =
(2)
tan
a
-
2
= J.L(AB) 1 .
Figure 2. The intersection points P and Q of the ''line" AB
with r.
Now consider a right triangle where a,b,c represent the multipli'?tive lengths of the two sides and the hypotenuse, and a, b, c their angles of parallelism. From Bolyai's for mula (2), it follows that 2x . sm x = 1 :x? + for any x. So the hyperbolic Pythagorean theorem becomes1 (3)
2a 1 + a2
•
1
2b
+ b2
=
2c 1 + c2 '
and our problem is to find triples (a, b, c) of rational num bers satisfying this equation. We call these non-Euclidean Pythagorean triples. Clearly, an even number of the num-
'Compare thrs to the more familiar expressron of the Pythagorean theorem usrng the addttive dtstances a,b,c, namely cosh a · cosh b =cosh c.
RONALD VAN LUIJK received his PhD from
ROBIN HARTSHORNE Robin Hartshorne says
the University of California at Berl<.eley in
his lifelong interest in geometry goes back to h1s
2005 in anthmetic geometry-in particular,
K3 surfaces. He is a postdoctoral fellow at the
class in mechanical drawing 1n the fifth grade of elementary school. Both the algebraic geometric
(d
University of British Columbia and Simon
and the non-Euclidean sides
Fraser University, and he has been a visrt1ng
Geometry, Euclid and Beyond) are in v1ew in the
-...-- professor at the Universidad de los Andes in
his recent text
present article. As1de from mathematics and
Bogota; next he will go to Warwick in the
family, he is a mus1cian (flute, piano, shakuhachi)
UK as a Mare Curie Fellow. Aside from
and an avid mountaineer.
mathematics, he likes hang gliding best Department of Mathematics Department of Mathematics
University of California
Simon Fraser University
Berl<.eley,
Bumaby (BC), VSA I S6
USA
Canada
CA 94720-3840
e-mail:
[email protected]<.eley.edu
e-mail:
[email protected]
© 2008 Spnnger Sctence+Bustness Medta, Inc , Volume 30, Number 4 , 2008
5
a,b,c are negative, unless we have a trivial solution abc= 0. After changing signs, we may assume a,b,c > 0. Note also that if we replace any number in a
bers with
non-Euclidean Pythagorean triple by its inverse, we again get a non-Euclidean Pythagorean triple, so we may focus on those triples with a,b,c 2: 1. To keep them nontrivial, we require a, b, c> 1. They are not so easy to find. (Let the reader try before reading further.) Note that we must allow rational numbers in stating our problem, since there are no similar triangles in hyperbolic geometry, or, in arithmetic terms, since the equation is not homogeneous. Indeed, Bjorn Poonen has shown by an el ementary argument that this equation has no solution in in tegers> 1. So our first problem is to find non-Euclidean Pythagorean triples.
Remark 1. There are other similar ways to express the hy perbolic Pythagorean theorem, as described in for instance [13, (14.51-52)] and [16]. These references do not have an arithmetic point of view, though. Problem II: A Problem of Euler
xz .t_ z2 z2 - D,
x2 1 = 2D, z
2 ..L. -1=0. z2
If we set
� .r_� , , = z - q2- 1 z � p2- 1 or p= Vx2- z2/(x-z) and q= Yy2- z2/(y- z), then the second and third equations of (4) are automatically sat (5)
isfied, just as in the parametrization of Pythagorean triples mentioned previously. Now we only need to satisfy the first equation. In terms of p and q we want
4Cp2q2-1)(q� Cp2- 1)2(q2- 1)2 to be a square, and for this it suffices that the numerator be a square. Dividing by 4p2, we are reduced to finding p and q such that
ep2q2- 1) is a square. Setting m =
(�- 1)
q/p, we must find p and m such that
(6) is a square. Obviously, this is a square for p= 1, but then x is undefined, so we set p 1 + s and then seek to make =
(m2- 1 + m2(s4 + 4s3 + 6s2 + 4s)) (m2 - 1) 6
THE MATHEMATICAL INTELLIGENCER
a= m2/(m2-
1 + 4as + 6as2 + 4as3 + as4 a square. There are unique f g E Q[a] such that for w = 1 + Js + gs2 the coefficients in w2 of sk for k= 0,1,2 coincide with the coefficients in (7), namely f= 2a and g= 3a- 2a2. Then, to make (7) equal to w2, we need (8) which is the case, besides for s = 0 (with multiplicity 3), (7)
for
8a4 _ -_;;_s= --::=-:4a2- 8a + 1 Reading backward, take any m * :':::1 you like, set a= m2/(m2- 1), take s as just given, let p 1 + s, q= mp, and then equation (5) will yield x,y,z, up to scal ing, solving the original problem. Since m is arbitrary, this (9)
·
=
gives infinitely many (but far from all) solutions. For example, if we take m 2, then =
Euler, in his Algebra [3, Part II, §236], considers the prob lem of finding three squares (of integers), :x?, y, z2, whose differences x2-y2, x2-z2, y2- z2 should also be squares. A first ad hoc argument yields a single solution (x,y,z) = (697,185,153), which turns out to be the smallest solution possible, ordered by 1x1. Then in §237 he provides a method for finding infinitely many solutions. Since one of the purposes of this article is to reinterpret Euler's method in terms of algebraic geometry, we recall his method here. First, he notes that passing to rational numbers, it is suffi cient to find x,y,z satisfying
(4)
a square. Dividing by (m2-1)2 and writing 1) for simplicity, we need to make
60 , a=-4 , s=-23 3 so
� 949 = , z 420
1:.
- 6005 , z 4947
giving rise to the relatively prime solution X=
1564901,
y
= 840700,
Z
= 692580,
and one can easily verify that
x2-z2 = 14032992, x2- y2 = 13199012, = 4765602. y2-z2 Remark 2. Euler also considers two other problems. In §235 he requires three integers a < b < c, whose sums and differences two at a time are all squares. One can show easily that this problem is equivalent to ours, as the three pairwise sums of the three integers provide a solu tion to our problem, and every solution is of this form [8, section 4]. Euler, however, does not mention this equivalence and appears not to have noticed it. He does see that the squares of a,b,c yield a solution to Problem II, but solutions arising this way tend to be much larger, which is why he treats our problem independently by the method described previously. Several other authors have considered these two problems (see for instance [10], [2] and references mentioned there) . Like Euler, however, many seem not to realize the equivalence of the two prob lems. In §238 Euler requires three squares such that the sum of any two is again a square. This one has a geometric in terpretation in 3-dimensional Euclidean space, to find a rec tangular box (cuboid) with integral edges and integral face diagonals. Our problem can also be interpreted in terms of cuboids. For if we put
It is amusing to verify equation (3) for these triples. The numbers factor, and many of the factors, cancel each other as if by magic.
these are equivalent to
(11) t2 + v2= uz,
v2 + z2= y2,
Thus our problem is equivalent to finding a cuboid with integral edges t,v,z, of which two face diagonals and the full diagonal are integral. Note that (x,u,t) is also a solu tion to Problem II, and it gives the same cuboid as the triple
(x,y,z).
In this connection, it is still an open problem to decide whether or not there exists a peifect cuboid, having all edges, face diagonals, and the full diagonal integral. For more on this problem, see [4, D18], [8], and [17], and the references given there.
Equivalence of the Two Problems We thank Hendrik Lenstra for first pointing out to us the equivalence of problems I and II. Equation (3) and similar equations have been studied in relation to Euler's problem before [8, section 4,5]. Leech also shows Problem II can be used to construct spherical right triangles whose sides and angles all have rational sines and cosines. If we take x,y,z satisfying Euler's problem, and write equations similar to (4), namely
1-..L2. =o , 2 X
z2 = 1-2 0, X
z2 = 1 -2 0, y
then we can parametrize them (inhomogeneously) as in by
..E.= 2a x a2 + 1'
(12)
From L
x
·
z X
2c c2 + 1'
z y
:!.. = :!.. , we then obtain equations y
X
(1)
2b b2 + 1 .
(3) of a non
Euclidean Pythagorean triple. Conversely, such a triple
(a,b,c) will give a solution to Euler's problem through (12). To find a,b,c explicitly, note that for example the first equation in (12), namely 1 -..L2. t2 , 2 xz
A Cycle of Five In his commentary on the work of Lobachevsky, F. Engel noted that to each hyperbolic right triangle, one can asso ciate another triangle in a natural way. Repeated five times, this process returns to the original triangle [9, pp. 346-3471. This association is closely related to the formulas of hy perbolic trigonometry, and forms a parallel to Napier's analogies in spherical trigonometry. Here is the construction. Given the right triangle ABC, with sides a,b,c opposite A,B, C, respectively, angles cr.,/3 at A and B, and a right angle at C, draw the perpendicular to BC at B, find the limiting parallel to AB that is perpendic ular to this new line, thus obtaining F. Draw the limiting parallel from B to A C, intersecting the previous limiting par allel at E. Then the new triangle is DEF with D= B (see Fig. 3). Note that LFBA is complementary to /3, and that it is the angle of parallelism of segment DF. Thus we write DF= /3', where the prime denotes complementary angle, and the bar denotes the correspondence between segments and an gles by the angle of parallelism. Note also that LEBC equals a, and so LFDE= a'. Knowing two of the five quantities (d,ej,8,E) of the new triangle, where the variables denote the obvious lengths and angles, one can compute the oth ers. Thus, if the original triangle has sides and angles (a,!3_c,cr.,f3), the new triangle has sides and angles (b,f3' ,a,a' ,C) (see [5, 42. 5 and Exercise 42.23] for more de tail). It is then easy to verify that this process, repeated five times, comes back to its starting point. It is an amusing ex ercise in hyperbolic trigonometry to compute the new tri angle from the old one. Since the triangle is determined by any two of its five measurements,J!: is enough to compute b (which we already know) and /3'. Here is a recipe.
X
is parametrized by
2a a2-1 a2 + 1' a2 + 1' from which we find a= (x + t)/y. If t is positive, then a> 1, assuming x,y> 0. Similarly we find b,c> 1, and we ob tain a one-to-one correspondence between solutions (x,y,z) to Problem II with x> y> z> 0 and gcd equal to 1 on ..E._ X
X
the one hand and (ordered) non-Euclidean Pythagorean triples (a,b,c) with a,b,c> 1 on the other. So, for example, from Euler's smallest solution (x,y,z) = (697,185,153), we see
(13)
a=
37 , 5
c= 9,
II " I \ \' \ '' ' \ \ .., \
'fu'•A / I
E
-- - ---- - - - --
FI I I
/
c
,_
- ---
D,:;-\ -I I
-' ... \ ' \ \ \ \ \ \ \ \ \ \ \ \
while, for the second example above, we obtain
(14)
a=
1201 350 '
--
b= 97 , 51
c= 30 7
Figure 3. Engel's associated triangle.
© 2008 Spnnger Sctence+Bus1ness Med1a, Inc , Volume 30, Number 4, 2008
7
b 2- 1
Given a and b, set e = a22a -1 b2 + 1 . {3' = e + �LEMMA 3.
--
•
-
Then
0
PROOF. Left to the reader!
The general formulas for the edges of the new triangle in terms of those of the old one are not elegant, but they can be expressed using a recurrence relation thanks to Ly ness (see [10] , and [8, p . 524]). Using Lemma 3 on an explicit example, say the triple
in (13) , we obtain four more triples
( ( ( ( .i!_ 17
7
27
) ) ) )
9 ' 6' 14 '
7
5 21 6 ' 4 ' 16 '
5 41 13 4 ' 1 3' 4 , 37 5 '
13 '
13
as further examples of non-Euclidean Pythagorean triples. Associated to these are further solutions to Euler's problem, namely (x,y,z) =
(697 , 185,1 53), (925,765,756), (3485,3444,3360), (7585,7400,4264), ( 1 5725,9061 ,2405). Remarkably, the transformation of order five can be ex pressed simply in terms of (x,y,z,t,u,v) of ( 1 0) , namely by sending (x,y,z) to (uy,uz,tz) and then dividing by the great est common divisor. Just for fun. we computed the 5-cycle associated to the triple in (14). We find
( ( ( ( (
)
1 20 1 97 30 , , 350 ' 51 7
)
97 47 99 , , 5 1 33 47 ' 47 37 33 ' 23
,
1551 85 1
)
37 73 74 23' 26 , 23 ' 73 26
,
1201 350
,
40 7
)
'
)
.
Algebra-Geometric Interpretation With the methods of algebraic geometry, we can provide a geometric interpretation of the preceding discussion. We work over an algebraically closed field, in this case the complex numbers, so that polynomial equations will have
8
THE MATHEMATICAL INTELLIGENCER
enough solutions. The affine n-space An is the set of all n tuples (x1, . . . , xJ of complex numbers. An affine alge braic variety is the set of common solutions of a set of polynomial equations in the x,. The projective n-space p n is the set of (n + 1 )-tuples (-XQ, . . . , xJ of complex numbers, not all zero, modulo the equivalence relation (-XQ, . . . , xJ (A-XQ, . . . , AxJ for any A E C, A =/= 0. A projective algebraic variety is the set of common solutions of a set of homogeneous polynomials in the x,. In our case, the equations (10) �
describing Euler's problem define a projective algebraic variety X in projective 5-space IP'5 with coordinates (x,y,z,t,u,v). It is an algebraic surface (meaning two com plex dimensions) since there are three equations and their loci of zeros intersect properly. If the equations defining an algebraic variety have ra tional coefficients, it is said to be difined over Q. In this case, it makes sense to ask for rational points on the vari ety, that is, points whose coordinates are all in Q. In the case of a projective variety, this is equivalent to finding in tegral points (whose coefficients are all in Z), since we can multiply all coordinates by their common denominator. Thus, Euler's problem can be rephrased as the problem of finding rational points on the projective algebraic surface X. We have thus gone from Euclidean geometry, through hyperbolic geometry and number theory, back to geome try, but now algebraic (and arithmetic) geometry. For algebraic curves (one complex dimension), the prob lem of finding rational points has been studied in detail. The answer depends on the genus of the curve. The genus g of a (nonsingular) projective algebraic curve is a topolog ical invariant. Regarding the curve as a compact orientable real 2-manifold, it is homeomorphic to a sphere with g han dles. A curve of genus 0, as soon as it has one rational point, is isomorphic to the projective line, with the isomor phism defined over Q. In that case it has infinitely many rational points. For example, the equation x2 + y2 = z2 mentioned previously defines a curve of genus 0 in the pro jective plane. It has a rational point, for example, ( 1 ,0, 1 ) , and s o i s isomorphic to IP'1 and can be parametrized by pairs of integers (m,n) as we have seen above (1). Note however that the equation x2 + y2 + z2 = 0 also defines a curve of genus 0 in IP'2, but this one has no rational points, as it does not even have real points. A curve of genus 1 with a rational point is called an el liptic curve. In this case, one knows that the set of ratio nal points has the structure of a finitely-generated abelian group A. It can be finite or infinite. The rank of the group (defined to be the (i)-dimension of the Q-vectorspace A @il (!)) can be quite large. Noam Elkies has found an el liptic curve of rank 28, but one does not know if there are elliptic curves of arbitrarily high rank. For example, the equation x3 + y3 = z3 defines an elliptic curve in IP'2, which, because of Fermat's Last Theorem for exponent 3, has only three rational points, ( 1 , - 1,0), (1 ,0,1), and (0,1,1). For curves of genus at least 2 , Falting's proof of the Mordell Conjecture tells us that there are at most a finite number of rational points.
For algebraic surfaces (and more generally for varieties of dimension at least 2), little is known about the set of ra tional points. This is a topic of active research: see for in stance the books [1], [11], and [ 1 2], and the review paper [15] by Swinnerton-Dyer. The classification of surfaces is much more complicated than that of curves, and we can not explain it here. Suffice to say that our surface X is what is known as a K3 swface. This class includes nonsingular quartic surfaces in IJJ>3, complete intersections of a quadric and a cubic hypersurface in IJJ>4, and complete intersections of three quadric hypersurfaces in [JJ>5, which is our case. There are some K3 surfaces with no rational points, such as the surface x6 + x1 + x� + x�= 0, which does not even have any real points. There are other K3 surfaces whose set of rational points is dense in the Zariski topology, mean ing that they are not contained in any proper algebraic sub variety. We will see that ours is one of the latter. An open problem is whether there exists a K3 surface whose set of rational points is neither empty nor dense. Our main result is the following.
THEOREM 4.
The set of rationalpoints on the suiface X rep resenting Euler's problem is dense in the Zariski topology.
A good part of the proof is already contained in Euler's calculation. The equation (15)
w2 = 1 + 4as+ 6as2 + 4as3 + as4
obtained from (7) describes a surface in the affine (a,s,w) space. Projection to the a-line creates a fibration whose fibers (for each fixed value of a) are curves of genus 1 in the (s, w)-plane. Since each contains some rational point, for example, (s,w)= (0,±1 ) , they are elliptic curves. The assignment a= m2/(m2 - 1 ) makes the m-line a double cover of the a-line. The inverse image in (m,s,w)-space of the surface (15) is a surface Yfibered in elliptic curves over the m-line. This surface is birational (meaning isomorphic except along proper algebraic subvarieties) to our surface X via the formulas relating x,y,z,t,u,v,p,q,m,s,a,w shown above and the equations
(x + y)(x+ z) tu Thus , our surface is an elliptic K3 suiface, that is, a K3 sur p=
x+z , u
--
q- � v , -
w=
face fibered in elliptic curves over the m-line. A section of this fibration is the image of a map from the m-line back into the surface, shown by m � ( m,S(m) , W(m)), where S and W are rational functions, so that each point of the m line lands in the fiber above that point. Euler's formulas (9) for s and w= 1 + 2as + (3a -2a2)s2 yield such a section, and thus each rational point of the m-line provides a ra tional point of the surface. This is how Euler constructs in finitely many solutions to his problem and how we see that all those solutions lie on a single curve on the surface X, namely the section just described. If we now think of m as an indeterminate instead of a number, equation (15) describes the generic fiber of the fi bration of X over the m-line. This generic fiber is an ellip tic curve E defined over the function field Q( m). A point onE , rational over the field Q( m), is given by (S( m), W(m)) ,
where S and W are rational functions, so a s t o satisfy the equation ( 1 5) . Thus rational points of the generic fiber E over Q(m) are in one-to-one correspondence with sections of the fibration of X over the m-line. What Euler has done is to find one such rational point of the generic fiber E. The new ingredient we add to this picture is the obser vation that after fixing the origin of the group structure to be the point fJ given by (s,w)= (0, 1 ) , the point Q corre sponding to Euler's section is a point of infinite order. This can be checked with standard techniques from elliptic curves (which we omit). Taking multiples of this point yields infinitely many more rational points on the generic fiber E , hence infinitely many more sections o f the elliptic fibra tions on X. The rational points on all these sections are then dense in the Zariski topology on X, since they are not contained in any finite union of curves on X; and this proves Theorem 4 .
Remark 5 . We have taken the origin of the group law on E to be the point (s,w) = (0, 1 ) . By looking at the other ob vious rational point P= (0, - 1) onE (which does not yield solutions to Euler's problem, since then p= 1 and x is un defined), one can verify from the geometry of the curve that Euler's point Q is equal to 2P in the group law. Thus Euler has discovered what we might call the "simplest" non trivial section of the fibration on X. Remark 6. With a little more effort, we can also check that the rational points on X are dense in the usual topology in the space X(IR) of real points of X. The right-hand side of (15) , which equals ap4- a + 1 , has two roots (in s or p) for a< 0 and a > 1, whereas it attains only positive values for 0< a< 1 . Closer inspection of the geometry of the cor responding elliptic curve above a shows that in the case of two roots, the real part (of the projective closure) of that curve is connected. Therefore in that case it has the topol ogy and the group structure of a circle group IR/2, so the multiples of any point of infinite order are dense. In the latter case of no roots, the real part consists of two com ponents, one with w > 0 and one with w< 0. Again the multiples in the connected component containing the ori gin fJ of any point of infinite order are dense in that com ponent, and since there is an automorphism w-- w that switches the two components, the rational points are dense in the whole fiber. By a theorem of Silverman [14], the point Q of infinite order on the generic fiber yields a point of in finite order in the fiber above m for all but finitely many mE Q. The latter are dense in the m-line, so the rational points on X are also dense in X(IR). Having found a dense set of rational points on X, can we find all rational points on X? We believe this is hope less, because special fibers of the fibration can contain ratio nal points that do not come from a section of the fibration. For example, Euler's smallest solution (697,185,153), found by an ad hoc method, lies in the fiber over the point m 13/5, and we have verified that it is linearly independent, in that fiber, from the points coming from Euler's section and its multiples. In fact, we can show that the group of rational points on the generic fiber E is isomorphic to 2 X 2/22 , whereas the rank of the group of the rational points in this
Remark 7.
=
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special fiber is 2. Thus there are many rational points on spe cial fibers that do not come from sections. Moreover, with a computer we checked that there are 1 440 integer solutions (x,y,z) to Euler's problem II with 0 < 1 z < y < x < 107 and gcd(x,y,z) = . Only 5 of these points lie in a fiber in which that point and the point from Euler's section Q are linearly dependent. These are the fibers above
}
{
2_ 289 3267 , 2 ,4 , , . m E 4 240 2209 As mentioned previously, it therefore seems hopeless that we would ever be able to find all rational points. In fact, we do not know of any case where the complete set of rational points on a K3 surface is described to satisfac tion, except when there are none.
Remark 8. Now that we have understood Euler's problem from a modern point of view, what about the problem of perfect cuboids mentioned previously? Here the corre sponding algebraic surface is a surface of general type in IP6. According to a conjecture of Bombieri and Lang, the ratio nal points on a surface of general type should be contained in a finite number of algebraic curves in the surface. In par ticular, if there are any (nontrivial points), they will be much rarer, and more difficult to find. So for the moment, the so lution to this problem still seems out of reach.
Chelsea (1 952}; Ch. 1 5, ref. 28, p. 448 and cross-refs to pp. 446-458; Ch. 1 9, refs. 1 -30, 40-45, pp. 497-502 , 505-507. [3] L. Euler, Algebra , f1rst published in 1 770 as Vo/lstandige Anle1tung zur Algebra ,
English translation: Euler, Elements of Algebra,
Springer 1 984. [4] R. Guy, Unsolved Problems in Number Theory, Problem books in mathematics, Springer-Verlag, 1981. [5] R. Hartshorne, Geometry, Euclid and Beyond , Springer, New York, 2000. [6] T. L. Heath, Diophantus of Alexandria , Cambridge, 1 9 1 0.
[7] T. L. Heath, The thirteen books ofEuclid's Elements, 2nd ed. , Cam bridge University Press, 1 926.
[8] J. Leech, The rational cuboid revisited, Arner. Math. Monthly, 84 (1 977), no. 7, 51 8-533. [9] N. Lobatschefskij, Zwei geometrische Abhandlungen, ed. F. En gel, Teubner, Leipzig, 1 898. [ 1 0] R. C. Lyness, Cycles, Math. Gazette, 26 (1 942), 62, ibid. 29 (1 945), 231 -233, ibid. 45 ( 1 96 1 ), 207-209. [1 1 ] E. Peyre and Yu. Tschinkel, Rational points on algebraic varieties, Progress in MathematiCS 1 99, Birkha.user, 2001 . [ 1 2] B. Poonen and Yu. Tsch1nkel, Arithmetic of higher-dimensional al gebraic varieties ,
Progress in Mathematics 226, Birkhauser, 2004.
[1 3] H. Schwerdtfeger, Geometry of Complex Numbers, Dover Publi cations Inc., 1 979. [ 1 4] J. H. Silverman, Heights and the specialization map for families of abelian varieties ,
J. Reine Angew. Math. 342 (1 983), 1 97-21 1 .
Note. For more technical details, see the preprint version of this article on arXiv. org and the paper [17] .
[1 5] P. Swinnerton-Dyer, Diophantine equations: progress and prob
REFERENCES
[1 6] A. Ungar, The hyperbolic Pythagorean theorem in the Poincare
lems,
[ 1 ] K. Boroczky, Jr. , J. Kollar, and T. Szamuely, Higher dimensional va rieties and rational points,
Springer-Verlag, 2003.
in ref. [ 1 2] , pp. 3-35.
disc model of hyperbolic geometr y,
Arner. Math. Monthly 1 06
(1 999}, no. 8, 759-763.
[2] L. E. Dickson, History of the theory of numbers, Vol. 2, Diophan
[1 7] R. van Luijk, On Perfect Cuboids, undergraduate thesis, 2000,
tine Analysis, Carnegie Institute of Washington (1 91 9) , Reprint by
available at http://www.math.leidenuniv.ni/reports/2001 -1 2 .shtrnl.
Errata: In The Mathematical Intelligencer, val. 30, no. 3, the Years Ago column: Max von Laue's Role in the Relativity Revolution, by David E. Rowe, the caption on page 56 is incorrect. It should be: From four vectors in Prague to tensors in Zurich: Einstein revises the Marx Manuscript. The cover illustration for Tbe Mathematical Intelligencer, val. 30, no. 3, is the work of Nikolaus Witte and Thilo Rorig.
10
THE MATHEMATICAL INTELLIGENCER
Mathematically Bent
The proof is in the pudding.
Colin Adams, Editor
A Subpri m e Lending Marl<et Pri mer 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
ou have probably heard quite a bit about the subprime lending market by now. But basic questions still remain. What is it? Why is everyone talking about it? How can I get in on the action? These are all good questions. In or der to clear up the confusion associ ated with these financial instruments, we will take this opportunity to an swer these and other frequently asked questions.
open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
What Is a Subprime? Of course, everyone has heard of the primes. These are the integers with no nontrivial divisors. They include 2,3,5, 7 , 1 1 , . . . Mathematicians have spent hundreds of years studying their prop erties. How many are there? What is their distribution? How come 2 is the only even one?
But What Is a Subprime?
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Sctence Center, Williams College, Wtlliamstown, MA 01267 USA e-mail: Coltn.C.Adams@wil l iams.edu
12
A subprime is a prime number that is a factor of a larger prime. For instance, 7 is a factor of 85 1 , making 7 a sub prime. Subprimes are the building blocks of the prime numbers. And the prime numbers are the building blocks of the integers, which are themselves the building blocks of the real numbers, upon which all of mathematics is based. So clearly subprimes play a critical role in any and all numerical computations, whether they be elementary additions or complex hyperextended equational manipulations.
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But Then, What Does It Mean to Lend or Borrow a Subprime? Subprimes are not as ubiquitous as one might first suppose. Considering all in tegers up to 1 ,000,000, there are fewer than 10,000 subprimes currently known, not nearly enough to go around. And once a subprime has been discovered, it is patented, and its use is restricted to the patent holder. An individual or corporation that uses the subprime without permission is subject to litiga tion and severe financial penalties. Luckily, a lucrative market in sub prime lending has developed. One can purchase the right to use a particular subprime for a specified time period through a variety of brokerage houses, including Bear Stearns and others.
What Does This Have To Do with Housing? Yes, you have probably heard the word "housing" associated with the subprime lending market. Probably also "mort gage" and perhaps even "mess. " Let's tackle each of these terms in turn. First, "housing." When a number is a subprime other than 2 or 3, its neigh boring numbers cannot be subprimes. For example , although 7 divides 85 1 , its neighbors 6 and 8 do not. More gener ally, if n divides m where nand m are primes, then n 1 and n + 1 cannot. Let's prove this fact. This is some thing mathematicians do. They prove a fact in an irrefutable manner, so that there can be no question about its ve racity. We' ll show that n - 1 does not di vide m. A similar argument applies to n+ 1 . We argue by contradiction. Thus, suppose that both n and n 1 divide m. First note that the n 1 and n can not have a common nontrivial factor. For if k divided both n 1 and n, then k would divide their difference n( n- 1) = 1 . Hence k must equal 1 . But if each of n 1 and n divides m, and they have no common nontriv ial factors, then their product (n- 1 ) n must also divide m . This contradicts the fact m is a prime.
-
-
- -
Of course, you might say, "Wait a minute. I agree that n- 1 and n + 1 cannot divide m, but they might divide some completely different prime. '' Good point. Didn't think you would no tice that. But it's not a problem. Because re member, n is a prime. And so unless n = 2, n is an odd number. This means that for n > 3, n - 1 and n + 1 have to be even and greater than 2. So they can not be primes. So they cannot be sub primes. Problem solved! We call these neighboring numbers the housing for the given subprime. They play a similar role to that played by the housing for an electrical conduit, serving as a protective layer that keeps out weather and rodents. Often, when one leases a subprime, one also leases the housing to go with it.
And What About the "Mortgage"? The term "mortgage" traditionally refers to a contractual agreement to borrow money for the purchase of a domicile or other piece of real estate. The so called collateral is the building or prop erty itself. In the case of the subprime market, there is no real estate. There is simply a number, together with its "house." But by abuse of terminology, the act of bor rowing the money to purchase a lease on a subprime and its house has be come known as a mortgage. These
mortgages are a means to dramatically increase your investment potential. In stead of being limited to the funds you have on hand, you can "mortgage your future" and invest funds that actually
belong to someone else.
And "Mess"? As you know, if you still own a house, it doesn't take long for it to become a mess. The same holds true for these number houses. A lease not only pro vides the rights to n 1 , n, and n + 1 , it also provides the rights to a ll the real numbers in between. That is an un countable collection of numbers. Many of them are given by nonrepeating dec imals that GO ON FOREVER. If this col lection of numbers gets just a little bit out of order, you can imagine the mess that ensues. But don't worry. Things are not as bad as they first appear. For it turns out that the real numbers are well ordered. This means that there is a choice of or dering on the numbers such that a ny subset has a least element. It's not the usual ordering, but so what? If your real numbers get mixed up, just apply this ordering, and find the least element in the entire set. Then remove this element, and repeat the process. In no time at all, you will have your house in order. -
Is There Any Risk? Getting out of bed every morning is a risk. But if you stay in bed, you end up
covered in bed sores. Not to mention a meteor smashing through your bed room ceiling, and off you go to join the dinosaurs. So yes, there is some risk. But keep in mind that these instruments are trusted by brokerage houses that are the bedrock of the entire financial commu nity. If they feel safe and protected, why shouldn't you?
I'm Still a Little Confused About What I Do with a Subprime Once I Lease lt. That's not a question.
What Do I Do with a Subprime Once I Lease It? That's entirely up to you. It's your oys ter. You get to decide. There are es sentially no restrictions. So go ahead, cut loose. Have some fun!
How Do I Sign Up? Legally, you cannot just send us your credit card information. We are sup posed to send you a prospectus which you are then supposed to read. But the truth is that hardly anyone ever reads prospectuses, let alone using the plural of that word. Given the need for fast action, and the fact you look smart, we can forgo the "information" stage. So don't wait. It's not so clear how long this opportunity will remain available.
© 2008 Spnnger Sc1ence+ Bus1ness Media, I n c , Volume 30, Number 4, 2008
13
Vie\N p o int
The G eometry of Paradise MARK A. PETERSON
The Viewpoint column offers mathematicians the opportunity to write about any issue of interest to the international mathematical community. Disagreement and controversy are welcome. The views and opinions expressed here, however, are exclusively those of the author, and neither the publisher nor the editor-in-chief endorses or accepts responsibility for them. Viewpoint should be submitted to the editor-in chief, Chandler Davis.
14
�
�
1 '/
athematics before 1700 presents a peculiar picture. It is difficult to avoid the impres sion that there was a golden age of mathematics in the Hellenistic period of Euclid and Archimedes, that a new mathematical golden age began in the 17th century, the golden age in which we are now living, and that in the long period in between, mathematics for some reason languished. Medieval Arab and European cultures inherited classi cal mathematics, and good mathemati cal minds were undoubtedly at work, but the circumstances put them some how at a disadvantage. Lucio Russo's remarkable book Tbe Forgotten Revolution (1) argues that the damage to mathematics as a collective enterprise was done already in the Ro man period. Thus what later civiliza tions inherited was already somehow maimed, cut off from the problems that gave rise to it. Medieval cultures were in the peculiar condition of being un mathematical cultures in possession of sophisticated mathematics. They pos sessed it in the sense of having the books, studying them and translating them, and even doing some mathe matics, but they had no clear indica tion where this rich subject had come from or what it would be good for. They did not know, in our terms at least, what it was. The story is complicated by excep tions to this sweeping characterization. Certain constructions called geom etria practica, useful in building and com merce, had a continuous existence right through the period, changing hardly at all, as if they were already adequate to their problems, with no need for inno vation. Methods and notations for do ing arithmetic certainly changed, but ancient civilizations had already been good at arithmetic, so the main inno vation here was perhaps the diffusion of arithmetical competence to a large commercial and professional class. Above all, algebra developed, with al gorithms for the solution of polynomial
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equations and systems of equations, in response to problems that arose first in commerce, and then took on an ab stract life of their own. This develop ment looks like normal mathematics, and it is noteworthy that algebra did not have any essential connection to the Greek mathematics of the classical past. Geometry, on the other hand, was moribund. It is as if its high sophisti cation, precisely its roots in the classi cal past, somehow disadvantaged it, and made it almost a dead subject, in spite of its high status. I have stated this view of medieval geometry more starkly than I could jus tify. It is not given as the topic of this article (I am not so ambitious as to un dertake a proper evaluation of it), but as a background against which one ex ample stands out dramatically: Dante's geometry in The Divine Comedy, specif ically in Paradise. Dante seems to have an unusual mathematical gift, but in an unmathematical age this gift finds a peculiar outlet. What Dante does with mathematics may bear out the previous suggestion that late medieval European culture possessed mathematics, but without knowing what it was. Where we expect mathematics to find appli cation in practical, earthly problems, medieval mathematics, apart from geometria practica and commercial arithmetic, did not come with that ex pectation at all. If anything, its associ ations were with astronomy and ce lestial things (2]. It must have seemed natural to Dante to find, as we will see, applications of mathematics in theology! And perhaps it is we who are not sufficiently imaginative. Math ematics, as the ultimate in abstraction, does not come with any prescription for what it might mean, so-theology, why not?
Dante's Universe is S3 It has been noticed by many readers that Dante's universe is topologically S3 [3, 4, 5. 6) . Still, the occurrence of a compact 3-manifold without boundary
in a late medieval poem is so unex pected, and the suggestion seems so implausible, that it might be good to go over the evidence here, lines of the poem that cumulatively leave little doubt. Dante invented concepts that were reinvented long afterward. Paradise represents the ascent of Dante and Beatrice through the spheres of the Aristotelian heaven, concentric with the Earth, beginning with the sphere of the Moon, then Mercury, Venus, the Sun, etc. [7]. In Canto 27, Dante and Beatrice make the ascent from the sphere of the fixed stars to the Primum Mobile, the outermost sphere of the universe, the one that turns all the others. Beyond that is the Empyrean, which has no conventional geometric description, but which Dante must now describe. The exposition begins as early as the end of Canto 22, in lines that are ad dressed to Dante, but are also prepar ing the reader, "Tu se' si presso a l'ultima salute," comincio Beatrice, ''che tu dei aver le luci tue chiare e acute; e pero, prima che tu piu t'inlei, rimira in giu, e vedi quanta mondo sotto li piedi gia esser ti fei . . . ' '
Beatrice began: "Before long thou wilt raise Thine eyes and the Supreme Good thou wilt see; Hence thou must sharpen and make clear thy gaze, Before thou nearer to that Presence be, Cast thy look downward and con sider there
How vast a world I have set under thee . . . " [8] Par. 22: 1 24-129 Dante does look down, seeing "this little threshing floor" the Earth below, surrounded by the heavenly spheres through which he has ascended, e tutti e sette mi si dimostraro quanta son grandi e quanta son ve loci e come sono in distante riparo. All seven being displayed, I could admire How vast they are, how swiftly they are spun, And how remote they dwell . . . Par. 22:148-150 This calling attention in Canto 22 to the sizes and velocities of the heavenly spheres before ascending to the sphere of the fixed stars is a kind of fore shadowing, to be recalled in Canto 28. In Canto 27 Beatrice asks Dante once again to look down, admiring the spheres below, as they ascend from the sphere of the fixed stars to the Primum Mobile, the ninth sphere. Dante is care ful to say of the Primum Mobile Le parti sue vivissime ed eccelse si uniforme son, ch'i' non so dire qual Beatrice per loco mi scelse. This heaven, the liveliest and loftiest, So equal is, which part I cannot say My Lady for my sojourn there deemed best. Par. 27:100-102
That is, this sphere has full rotational symmetry, and the ascent they have chosen is in no way distinguished from any other way they might have come. That S0(3) symmetry is a crucial in gredient of Dante's image, and he does not want it to be missed. At the beginning of Canto 28, Dante sees reflected in Beatrice's eyes a bright Point. Turning, he sees the Point in re ality, surrounded by nine whirling cir cles, moving the more slowly as they are larger. The seventh of these is al ready larger than the rainbow's circle. It emerges in the next lines, where this whole structure is discussed, that these angelic circles are a kind of mirror im age of the heavenly circles below. Mir ror symmetry is already suggested in the way that Dante first sees them, as a reflection. The Point, representing God, the center of the angelic circles, is the mirror image of the center of the material universe, the center of the Earth down below, where Satan is fixed in ice. La donna mia, che mi vedea in cura forte sospeso, disse: "Da quel punta depende il cielo e tutta la natura. " Observing wonder i n m y every feature, My Lady told me what I set below: "From this Point hang the heavens and all nature. '' Par. 28:40-42 The word "depende" used in this way expresses essentially what we mean when we say S3 is IS2, the suspension of the 2-sphere. Dante even describes himself as "sospeso," suspended, per-
MARK A. PETERSON is Professor of Physics and Mathematics on
the Alumnae Foundation at Mount Holyoke College. His interest in Renaissance mathematics goes back to h1s graduate student days in physics at Stanford. He has just completed a book, from which the material of this article is taken, Ga/J!eo's Muse: The Renaissance
Re-invention of Science by the Arts. He also works on geometrical methods for continuum mechanics. Department of Mathematics and Statistics Mount Holyoke College South Hadley, MA 0 I 075-6420 USA
e-mail:
[email protected]
© 2008 Spnnger Sclence+BusJness Media, lnc , Volume 30, Number 4, 2008
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haps suggesting the precariousness of the soul between these theological poles, in addition to the more literal meaning of being "in suspense," al though what is literal and what is fig urative here is hard to pin down. In any case, the picture is the suspension construction, with a sense of "higher" and "lower" imposed on it. Dante is bothered by something that seems wrong to him. The heavenly spheres tum faster the larger they are, but the angelic ones tum slower the larger they are. We have already been alerted to these velocities, but now the matter is brought up explicitly. As Dante puts it, he would like to know why "l'essemplo e l'essemplare non vanno d'un modo," that is, why the pat tern and the copy don't move in the same way. Beatrice laughs "Se li tuoi diti non sono a tal nodo sufficienti, non e maraviglia: tanto, per non tentare, e fatto sodo!" "There's naught to marvel at, if to untie This tangled knot thy fingers are un fit, So tight 'tis grown for lack of will to try. Par. 28:58-60 This is an assertion that we are look ing at new mathematics! Beatrice ex plains that the spheres are ordered by velocity, and that they tum faster the higher they are (in the sense of higher knowledge, higher love, that is, prox imity to God), not the larger they are, another nice way to think of S3. The Primum Mobile may be the largest, but it is only the equator of the universe as a whole, being only midway in the or dering. The smallest circles, closest to the Point, tum really fast, as Beatrice points out. It is clear that Dante invents the no tion of manifold here, in building the universe out of two balls, glued along their common boundary. That is the meaning of "l'essemplo e l'essemplare. " The reader may b e bothered by the frequent use of the word "circle" where the right word would seem to be "sphere," but Dante explicitly says, in another place that we will see later, that he uses the word circle for both, and in general for anything round. In any case he makes clear more than once that
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THE MATHEMATICAL INTELUGENCER
the circles about the Point are actually spheres. For example, "l'essemplo" is the spheres of the conventional uni verse, and it is the model for "l'essem plare," the angelic "circles. " Also, the ho mogeneous uniformity of the Primum Mobile, in which Dante can't distinguish any location, implies that the angelic "circles" are not just off to one side of the Primum Mobile but surround it. A skeptic might reluctantly agree that this looks very much like S3, de scribed in several different ways, in fact, but that it is simply impossible: Dante could not so easily have over come the normal tendency of the hu man mind to regard space as infinite (and Euclidean). I wonder, though. It might have been easier for Dante to in vent S3 than for, say, Immanuel Kant. Aristotle's universe, which was also Dante's universe, was explicitly de clared to be finite, although in Aristo tle's version it went only up to the Pri mum Mobile and no resolution was offered for the puzzle of what lay be yond. From the beginning, therefore, Dante was describing a finite universe. Going on, it appears that he did not have the Euclidean prejudice in favor of infinite straight lines. This objection might not even have occurred to him. His preferred geometrical object was the circle, and a space built out of cir cles might well be S3. The internal evidence of Dante's writing suggests that although he knew Euclid's geometry, and made casual, easy use of it, he does not necessarily regard it as a model for space, espe cially globally. Rather he regards it as a branch of philosophy whose propo sitions are true with peculiar certainty (as Aristotle also regards geometry). There are two Euclidean theorems in Paradise, but neither of them carries a meaning that has anything to do with space. The theorem in Par. 1 3 : 1 0 1-102 is Elements III.31, a triangle inscribed in a semicircle is a right triangle. This the orem occurs as just one of several learned propositions in a list, the other propositions not being from geometry. The list consists, tellingly perhaps, of things that King Solomon did not ask to know when he was granted wisdom. This rather backhanded reference might even be read as slightly dismis sive of geometry.
In Par. 17: 1 3-18 Dante says, ad dressing Beatrice, that she sees the fu ture as clearly as men see that no tri angle can have two obtuse angles. Neither this occurrence nor the previ ous one uses Euclid to describe some thing in space. When Dante asks about his future, he does not mean triangles. Rather, these theorems are cited as ex amples of things that are known with certainty to be true. In short, geometry seems to be less geometrical for Dante than it is for us.
Dante's Geometer Dante discusses the seven liberal arts of the trivium and quadrivium, geom etry being one of them, in his earlier unfinished book of classical learning, Tbe Banquet. From Tbe Banquet II. l3, Geometry moves between two things antithetical to it, namely the point and the circle-and I mean "circle" in the broad sense of any thing round, whether a solid body or a surface ; for, as Euclid says, the point is its beginning, and as he says, the circle is its most perfect fig ure, which must therefore be con ceived as its end.Therefore Geome try moves between the point and the circle as between its beginning and end, and these two are antithetical to its certainty; for the point cannot be measured because of its indivis ibility, and it is impossible to square the circle perfectly because of its arc, and so it cannot be measured ex actly. Geometry is furthermore most white insofar as it is without taint of error and most certain both in itself and in its handmaid, which is called Perspective. [9] This passage cites Euclid, but the sen timents attributed to Euclid are virtually unrecognizable. The implied meaning of geometry in this passage is precise measurement, and the point and the circle are "antithetical" to the certainty of geometry because they can't be mea sured, not at all a Euclidean idea. Nor does Euclid call the circle "most per fect. " The enthusiasm for the circle ex pressed here must be Dante's own. The problem of measuring the circle, given such prominence here, is of course not a problem of Euclid. One is left with the impression, con-
sistent with the two theorems in Par adise, that although Dante knows and respects Euclid, he does not find him very interesting. The passage in The Banquet summarizing geometry essen tially ignores Euclid, even as it cites him. The certainty of geometry seems less interesting to Dante than its op posite, the antithetical point and circle, for he devotes most of this little state ment to them. The unmeasurability of the circle definitely interests him.
Dante and Archimedes D ante returned to the problem of measuring the circle in one of the most astonishing passages he ever wrote, the final image of Paradiso. He is looking at an image of the Trinity, as three cir cles, and staring especially at the sec ond of these, representing the Son: dentro da se, del suo colore stesso mi parve pinta de Ia nostra effige: per che 'l mio viso in lei tutto era messo. Qual e 'l geometra che tutto s'affige per misurar lo cerchio, e non ritrova, pensando, quel principia ond' elli indige, tal era io a quella vista nova: veder voleva come si convenne !'imago al cerchio e come vi s'indova; rna non eran da cia le proprie penne: se non che Ia mia mente fu percossa da un fulgore in che sua voglia venne. A !'alta fantasia qui manco possa; rna gia volgeva il mio disio e 'l velle, si come rota ch'igualmente e mossa, l'amor che move il sole e l ' altre stelle. Par. 33: 1 30-145 Within itself, of its own coloration I saw it painted with our own hu man form: So that I gave it all my attention. Like the geometer, who exerts him self completely To measure the circle, and doesn ' t succeed, Thinking what principle he needs for it, Just so was I, at this new sight. I wanted to see how the human im age
Conforms itself to the circle, and finds its place there; But there were not the means for that, Except that my mind was struck By a flash of lightning, by which its will was accomplished. Here strength for the high imagin ing failed me, But already the love that moves the Sun and the other stars Turned my desire and my will Like a wheel that is turned evenly. I have preferred the unpoetic transla tion here in order to be as literal as possible, for the purpose of a close reading. Notes to this passage always point out the futility of trying to square the circle. They suggest that squaring the circle functions here as a metaphor for the impossibility of understanding the mystery of salvation by Christ's cruci fixion. In the last century or so, notes on this passage even cite Lindemann's 1882 proof that 7T is transcendental! That result cannot be relevant to Dante's intention in this image, but we already have Dante's own opinion in The Banquet that the circle cannot be squared. The message of futility might appear to be unavoidable, in view of the gen eral agreement that what the geometer is trying to do is impossible, but there are subtle problems with this reading. In the first place, it just doesn't sound like Dante to give up. Why would he come to the end of his amazing epic poem and then admit defeat by intro ducing an impossible problem in the very last lines? It isn ' t even so clear that he is defeated. That flash of light ning might indicate the opposite. Typ ical notes suggest that the flash is a metaphor for the acceptance of God's grace, as if the struggle with geome try were over, but the geometrical metaphor seems to continue even af ter the lightning flash, in the image of the turning wheel. Scholarship has re turned to this enigmatic passage again and again, without a wholly satisfac tory conclusion. Dante never mentions the name Archimedes, but Archimedes's little treatise On the Measure of the Circle had been translated several times be fore Dante wrote, from Arabic by both
Plato of Tivoli and Gerard of Cremona, and from Greek around the time of Dante's birth by William Moerbeke. Given Dante's interest in the question of measuring the circle, he would nat urally have sought out this treatise and studied it. It seems impossible that he would not have. It is short and easily copied, especially its Proposition I, which is the relevant one. According to Marshall Clagett [10] , versions of the Gerard translation were widely circu lated, and we will notice evidence be low that this is the version that Dante knew. I believe that Dante used the Archimedes proof as an extended metaphor in the last lines of Paradise for the drama of salvation, as I will now explain. It will follow that Dante un derstood the proof perfectly, and used it with precision. Let me recall the familiar magiste rial argument of Archimedes in On the Measure of the Circle. Archimedes shows by a method of exhaustion that the circle of radius R and circumference C is, in area, neither larger nor smaller than 1RC. Thus it is exactly 1RC. For as sume that the circle is larger than �RC. We construct a sequence of regular polygons inside the circle, beginning with the inscribed square, and doubling the number of sides at each step, as in Figure 1 . More than half the remaining area outside the polygon is incorpo rated at each step into the next poly gon in the sequence. Hence, by our as sumption that the circle is larger than iRC, some polygon in the sequence will also have area larger than 1RC. But this is impossible, because the area of the polygon is the sum of the triangular wedges in Figure 2, namely !z hNb in the notation defined there, and b < R, and Nb < C. Thus the circle is not greater than !zRC. A similar argument using a sequence of polygons outside the cir cle shows that the circle is also not less than iRC. Figure 1 shows the sequence of drawings that anyone would make who actually carried out the constructions of the proof. Only the rightmost figure of those four is found in the manuscripts, and only the Gerard translation manu scripts show the whole polygon [10] . The Moerbeke translations confine the construction lines to the upper left quadrant, so that the visual impression is quite different, although someone re-
© 2008 Spnnger Se>ence+Bus>ness Med>a. Inc . Volume 30. Number 4. 2008
17
Figure I. A sequence of regular polygons is constructed in the circle.
capitulating the process might still draw it as in Figure 1 . I believe that Dante worked with the Gerard translation, and made the figures in the sequence shown in Figure 1 , because the second figure in that sequence, the cross in the circle, must have struck him as crucially significant (pun intended). The ap pearance of the cross in an argument that already seemed to have a tran scendent meaning must have been ir resistible to him. If we examine again the last lines of Tbe Divine Comedy, we see that they follow the Archimedes proof thought for thought. The second circle, painted with man's image, is the Son, and "Ia nostra effige, " our own human form, is the cross, that is, a man stretched out (crucified), exactly the second figure in Figure 1 . The strange word "painted" depicts the geometer's literally adding the lines of the cross to the circle with a drawing instrument. The cross gives rise to the square, then to the octagon, and so forth. The geometer wants to know "how the human image/Con forms itself to the circle, and finds its place there. " That is just the question, geometrically, how the sequence of polygons approaches the circle, and theologically how the human and mea surable becomes the divine and im measurable. The geometer knows that no matter how many sides it has, a polygon still has straight sides and so cannot become the circle, which is curved. It is just Dante's point in Tbe Banquet, that the circle cannot be mea sured "because of its arc . " He vainly seeks the principle, until suddenly there is "a flash of lightning, " which re solves the problem. This is the argu ment of Archimedes that shows how the polygons do become the circle in the limit. That result is finally asserted in Dante's saying that the wheel turns evenly, as only a circular wheel can do, not a polygon.
18
THE MATHEMATICAL INTELLIGENCER
The Archimedes proof makes pre cise sense of so many odd details in these last lines of the poem that it seems quite believable that Dante had this proof in mind as an extended metaphor for the union of human (straight) and divine (curved) . If so, he understood that the limit of a sequence can have a property that no member of the sequence had, in this case the property of being curved, since that is the point of the metaphor and the mys tery to which he is leading us. Like the construction of S3 in Canto 28, it is possible that this final image was understood in mathematical terms by no contemporary of Dante, but there is an odd hint in this latter case that some people did. The first commentary that actually mentions the geometer is Benvenuto da Imola 0 375), who says [ 1 1] :
Et explicat summum conatum suum per unam comparationem elegantis simam de geometra, qui volens men surare circulum colit se tatum sibi; et quamvis autor videatur loqui com-
Area of polygon
=
muniter de geometria, tamen iste ac tus et casus quem ponit maxime ver ificatur de Archimede philosopho; ad quod est praenotandum quod sicut scribit Titus Livius etc. And he explains his highest effort by a most elegant comparison with a geometer, who, wanting to measure the circle, gives himself completely to it; and however much the author is seen to speak generally about geometry, this particular case, which he places most highly, is proved by the philosopher Archimedes, about whom it is well known, as Livy writes, etc. Benvenuto seems to know that there is a proof of Archimedes behind this im ager He does not, however, refer to Archimedes's treatise, but to Roman his tories that tell the story of Archimedes. That suggests that he is not one of the people who has actually seen or un derstood the proof. The next commentators seem to have misunderstood this idea in a
( 1/2) h·circumference
Figure 2. The area of the regular polygon is
�hNb,
where
N is the number of sides.
rather hilarious way. Chiose Vernon ( 1390) [1 1], probably from reading Ben venuto, thinks that the geometer star ing at the circles is Archimedes at the moment of his death at the siege of Syracuse, and inserts that whole story into his commentary. Since this makes a ridiculous ending for The Divine Comedy, and is clearly impossible, the idea was dropped in the next genera tion of commentaries, and all connec tion to the Archimedes proof seems thereafter to have been forgotten. If we restore the idea of the Archimedes proof as metaphor, we see that Dante's image might well represent not the failure of human intellect to com prehend the divine, as it is usually un derstood, but rather something more positive, more like a triumph of the hu man intellect, and more characteristic of Dante himself. Understanding the math ematics behind the image potentially changes its meaning.
Geometry as Philosophy It is noteworthy that although Dante refers to geometry, and even does geometry, in ways that we can recog nize (with some difficulty), the mean ing of mathematics for him is philo sophical. Euclid and Archimedes are philosophers. What we call mathemat ics is, for him, and presumably for his
contemporaries and for his culture, a corner of philosophy having to do with the celestial part of creation, exempli fying a particular kind of truth. In par ticular, mathematics does not deal with messy, earthly problems, or with ter restrial space. Is it credible that a philosophical stance of this kind, even if it is accepted by a whole culture, could change the nature of mathematics so drastically in practice? Restricting the problems that mathematics could address appears to have restricted mathematics itself. In principle, mathematics could be an ab straction that feeds on its own abstract problems. The Romans believed, and therefore later civilizations also be lieved, that the origin of Greek mathe matics was a love of abstraction. If the long medieval period attempted to see whether mathematics could flourish without earthly applications, the an swer seems to be a resounding no. The 1 7th-century revolution in mathematics came when it began addressing ques tions that we now call physics, concrete problems with experiments and data. Even if we formally share the math ematics inherited from the ancients, what we make of it depends on our culture, not simply on the contents of mathematics books. To a surprising de gree, the meaning of mathematics is
� Springer
the language of science
what we think it is, and what we want it to be. REFERENCES
[1] Russo, L., The Forgotten Revolution, Springer-Verlag, Berlin, Heidelberg, New York, 2003. [2] Ptolemy, Almagest, Book I. [3] Speiser, Andreas, Klassische StUcke der Mathematik,
Verlag Orell Fuselli, Zurich,
1 925. [4] Callahan, James, "The curvature of space in a finite universe," Scientific American 235, August, 1 976, 9Q-1 00.
[5] Peterson, Mark, "Dante and the 3-sphere," American Journal o f Physics,
47, 1 979,
1 03 1 -1 035.
[6] Osserman, Robert, Poetry of the Universe, Garden City, NY: Doubleday, 1 995. [7] An amusing riff on these spheres is Osmo Pekonen, "The Heavenly Spheres Re gained," The Mathematical lntelligencer 1 5, No. 4, 1 992, 22-26. [8] Verse translations are those of Barbara Reynolds from Dante's Paradise, Penguin Books, 1 962. [9] Dante's II Convivio (The Banquet), tr. R. H. Lansing, Garland Publishing, New York, 1 990, 72. [1 0] Clagett, Marshall, Archimedes in the Mid dle Ages, Vol. 1 , University of Wisconsin Press, Madison, 1 964. [1 1 ] Over 70 Dante commentanes can be searched online at dante.dartmouth.edu.
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© 2008 Spnnger Sctence+Bustness Media, Inc , Volume 30, Number 4, 2008
19
An I ntrod uction to I nfi n ite H at P rob l e ms CHRISTOPHER 5. HARDIN AND ALAN 0. TAYLOR
l_l
at-coloring puzzles (or hat problems) have been around at least since 1961 (Gardner 196 1 ) , and probably longer. They gained wider public attention with a question posed and answered by Todd Ebert in his 1998 Ph.D. dissertation (Ebert 1 998). The problem was presented by Sara Robinson in the April 10, 200 1 , Science section of Tbe New York Times as follows:
1
Three players enter a room and a red or blue hat is placed on each person's head. The color of each hat is determined by a coin toss, with the outcome of one coin toss having no effect on the others. Each person can see the other players' hats but not his own. No communication of any sort is allowed except for an initial strategy session before the game begins. Once they have had a chance to look at the other hats, the play ers must simultaneously guess the color of their own hats or pass. The group shares a hypothetical $3 million prize if at least one player guesses correctly and no play ers guess incorrectly. The same game can be played with any number of play ers. The general problem is to find a strategy for the group that maximizes its chances of winning the prize. If one player guesses randomly and the others pass, the probability of a win is 1/2. But Ebert's three-player solu-
20
THE MATHEMATICAL INTELLIGENCER © 2008 Spnnger Sc1ence+Bus1ness MeCia, Inc
tion is better: pass if the two visible hats are different col ors, and guess the missing color if they are the same. This strategy yields a win, on average, 3/4 of the time: of the eight possible hat assignments, it fails only on the two in which all three hats are the same color. Elwyn Berlekamp generalized this to n = 2 k - 1 players, using Hamming codes to show the existence of a strategy that yields a win with probability n/( n + 1 ) . Joe Buhler gives an account of this, and further variations, in Buhler (2002). In Spring 2004, Yuval Gabay and Michael O'Connor, then graduate students at Cornell University, produced a num ber of hat problems involving infinitely many players, one of which was (an equivalent of) what we will call the Gabay O'Connor hat problem: Infinitely many players enter a room and a red or blue hat is placed on each player's head as before. Each player can see the other players' hats but not his own. Again, no communication of any sort is allowed except for an initial strategy session before the game begins. But this time, passing is not allowed and each player receives $ 1 million if all but finitely many players guess correctly. There are simple strategies ensuring that infinitely many players will guess correctly. For example, let a player guess red if he sees infinitely many red hats, and guess blue oth erwise. If there are infinitely many.red hats, everyone will guess red, and the players with red hats will be correct; if
there are finitely many red hats, everyone will guess blue, and the infinitely many players with blue hats will be cor rect. The problem, however, seeks a strategy ensuring that all but finitely many-not just infinitely many-are correct, and this is what Gabay and O'Connor obtained using the ax iom of choice. The special case in which the set of play ers is countable follows from a 1964 result of Fred Galvin ( 1965); see also Thorp ( 1 967). Although Galvin's argument and the Gabay-O'Connor argument are similar, they are dif ferent enough that neither subsumes the other; a compar ison will appear elsewhere. As the title suggests, this paper is meant to be only an introduction to infinite hat problems, and as such proceeds in a somewhat expository manner. We have made no at tempt here to say anything of the relevance of hat prob lems to other areas of mathematics, but the reader wishing to see some of this can begin with Galvin and Prikry 0976), George (2007), and Hardin and Taylor (2008). The rest of the paper is organized as follows. In "The Formalism and the Finite, " we set up a general framework for hat problems of the Gabay-O'Connor type, and present a few results in the finite case. "Theorems of Gabay O'Conner and Lenstra" concerns the infinite case, which is our primary interest; we present the Gabay-O'Connor The orem, and a theorem of Lenstra involving strategies that ei ther make every player correct or every player incorrect. In "The Necessity of the Axiom of Choice, " we discuss the ne cessity of the axiom of choice in the Gabay-O'Connor The orem and Lenstra's Theorem; this section requires some ba sic facts about the property of Baire, so a short appendix on the property of Baire appears afterward. Our set-theoretic notation and terminology are standard. If is a set, then is the cardinality of and is the complement of If f is a function, then f i is the restric tion of f to A, and Pc is the set of functions mapping the set P into the set C. If x is a real number, then Lx J is the greatest integer that is less than or equal to x. We let N = (0, 1 , 2, . . . }. The authors thank James Guilford, John Guilford, Hen drik Lenstra, Michael O'Connor, and Stan Wagon for al-
A
A.
IAI
A A
Ac
CHRISTOPHER S. HARDIN, B.A. Amherst
1 998 and Ph.D. Cornell 2005, 1s currently a visiting professor at Wabash College. His pub lications are in the area of mathematical logic,
lowing us to include unpublished proofs that are in whole or in part due to them. Their specific contributions will be noted at the appropriate places. We also thank Andreas Blass for bringing Galvin's work to our attention, and thank the referee for many helpful suggestions.
The Formalism and the Finite The problems we consider will resemble the Gabay-0' Connor hat problem, but we allow more generality: the set of players can be any set, there can be any number of hat colors, players do not necessarily see all other hats, and the criterion for winning is not necessarily that all but fi nitely many players guess correctly. So, a particular hat problem will be described by (i) the set of players, (ii) the set of possible hat colors, (iii) which hats each player can see, and (iv) a rule that indicates, given the set of players who guess correctly, whether or not they win the game. We formally define a hat problem to be a tuple (P, C, V,"W') with the following properties. (i) The set ofplayers P is any set. (ii) The set of colors C is any set. (iii) The visibility graph V is a directed graph with P as the set of vertices. When there is an edge from a to b (which we denote by a Vb or b E Va), we interpret this as meaning that a can see (the hat worn by) b. In par ticular, Va is the set of players visible to a. We are only interested in cases where players cannot see their own hats, so we require that V has no edges from vertices to themselves. (iv) The winning family "W is a family of subsets of P. The players win iff the set of players who guess their own hat color correctly is in "W. A function g E Pc assigns a hat color to each player; we call g a coloring Given a hat problem (P, C, V,"W) , a strat egy is a function S : (P X PC) � C such that for any a E P and colorings g, h E Pc, g l Va =
bj Va
=>
S( a, g)
=
S(a, h).
(1)
We think of S(a, fi) as the color guessed by player a under coloring g. Condition (1) ensures that this guess only depends
ALAN D. TAYLOR received his Ph.D. from
Dartmouth in 1 975, and has been at Union College ever since. He has published six books on set theory, combinatorics, fair division, and
and include (wiTh Taylor) "A peculiar connec
the theory of voting, including Soda/ ChoiCe and
tion between the axiom of choice and pre
the Mathematics ofManipulation (Cambridge
dicting the future," Amer: Moth. Monthly
2005). He still keeps trying to run track
(2008). He enjoys climbing and music. He
meters).
(400
does not own blue jeans. Department of Mathematics Department of Mathematics and
Union College
Computer Science
Schenectady, NY 1 2308
Wabash College Crawfordsville, IN 47933
USA
e-mail:
[email protected]
USA '
e-mail:
[email protected]
© 2008 Spnnger ScM3nce+Busmess Med"', Inc , Volume 30. Number 4, 2008
21
l
i
on the hats that a can see, since g Va h Va means that the colorings g and h are indistinguishable to player a. We will frequently consider strategies player by player; for a E P and a strategy S, we define Sa : Pc� C by Sa(i) S( a, gy, and call Sa a strategy for player a. We say that player a guesses correctly if Sa(i) g(a). We call S a winning strategy if it ensures that the set of players who guess correctly is in the winning family, re gardless of the coloring; that is, {a E P : Sa(i) = g(a)} E W for any coloring g. To illustrate the kinds of questions and answers that arise within this framework, we present two results in the con text of finitely many players. For the first, say that a hat problem is a minimal hat problem if it asks for a strategy ensuring that at least one player guesses correctly, and call such a strategy a minimal solution. Our first result (the sec ond half of which is due, in part, to James Guilford and John Guilford) answers the following question. =
=
=
With k players and 2 colors, how much visibility is needed to guarantee the existence of a minimal solu tion? What if there are k players and k colors?
THEOREM 1 A k--player, 2--color hatproblem has a minimal solution iff the visibility graph bas a cycle. A k--player, k-color hat problem has a minimal solution iff the visibility graph is complete. To prove Theorem 1 , it will help to have a lemma that confirms an intuition about how many players guess cor rectly on average.
LEMMA 2 In a k--player, n-color hat problem, for any par ticular strategy, the average number ofplayers who guess cor rectly is k/n. (/be average is taken over all colorings.)
this, we first note that because V has no cycles, we can as sign each player a rank as follows: a has rank k if there is a directed path of length k beginning at a, but none of length k + 1 . Now, if there is a directed edge from vertex a to vertex b, then the rank of player a is strictly greater than the rank of player b. Thus, a player can only see hats of players of strictly smaller rank. Hence, given any strat egy, we can assign hat colors in order of rank to make everyone wrong: once we have colored the hats of players of rank < k, the guesses of players of rank k are determined , and w e can then color their hats t o make them wrong. Now suppose there are k colors. For the right-to-left di rection, assuming the visibility graph is complete, the strat egy is as follows. Number the players 0, 1 , . . . , k - 1 , and the colors 0, 1, . . . , k - 1 , and for each i, let si be the mod k sum of the hats seen by player i. The plan is to have player i guess i - s, (mod k) as the color of his hat. If the colors of all the hats add to i (mod k), then player i will be the one who guesses correctly. That is, if Co + + ck- J = i (mod k) then c, i - s, (mod k). For the other direction, assume that there are k players and k colors, and assume the visibility graph is not com plete. Let S be any strategy. We must show that there is a coloring in which every player guesses incorrectly. Suppose player a does not see player lis hat (with a =f:. b), and pick a coloring in which player a guesses correctly. If we change the color of player lis hat to match player b's guess, player a will not change his guess, and we will have a coloring in which a and b guess correctly. By Lemma 2, the aver age number of players who guess correctly is k/k 1 ; be cause we have a coloring with at least 2 players guessing correctly, there must be another coloring in which fewer than 1 (namely, zero) players guess correctly. 0 ·
·
·
=
=
Our second result along these same lines is also due, in part, to James Guilford and John Guilford (the n 2 case appears in Winkler 2001). It answers the following ques tion. =
Proof Suppose there are k players and n colors. Let S be any strategy. It suffices to show that any particular player a is correct in 1 out of n colorings. Given any assignment of hat colors to all players other than a, player a's guess will be determined; of the n ways to extend this hat as signment to a, exactly one will agree with a's guess. 0
Proof of Tbeorem 1 . Suppose first that there are 2 colors. The right-to-left direction is easy; assuming that the visibil ity graph has a cycle, the strategy is for a designated player on the cycle to guess that his hat is the same color as that of the player immediately ahead of him on the cycle, while all the others on the cycle guess that their hat color is the opposite of the player immediately ahead of them. To see that this works, assume that the first player on the cycle has a red hat and that everyone on the cycle guesses in correctly using this strategy. Then the second player on the cycle has a blue hat, the third player on the cycle has a blue hat, and so on until we're forced to conclude that the first player on the cycle has a blue hat, which we assumed not to be the case. For the other direction, we show that if there is no cy cle in the visibility graph V, then for every strategy there is a coloring for which everyone guesses incorrectly. To do
22
THE MATHEMATICAL INTELLIGENCER
With k players and n colors, how many correct guesses can a strategy guarantee, assuming the visibility graph is complete? Lemma 2 shows us that, regardless of strategy, the num ber who guess correctly will on average be k/ n. But this is very different from ensuring that a certain fraction will guess correctly regardless of luck or the particular coloring at hand. Nevertheless, the fraction k/ n is essentially the cor rect answer.
THEOREM 3 Consider the hatproblem with IPI = k, le i = n, and a complete visibility graph V Tben there exists a strat egy ensuring that L k!nJ players guess correctly, but there is no strategy ensuring that L k!nJ + 1 players guess correctly. Proof The strategy ensuring that L k/nJ players guess cor rectly is obtained as follows. Choose n X L k!nJ of the play ers (ignoring the rest) and divide them into L k/nJ pairwise disjoint groups of size n. Regarding each of the groups as an n-player, n-color hat problem, we can apply the previ-
ous theorem to obtain a strategy for each group ensuring that (precisely) one in each group guesses correctly. This yields L k!nJ correct guesses altogether, as desired. For the second part, we use Lemma 2. For any strategy, the average number of players who guess correctly will be L k!nJ , and L k/nJ < L k!nJ + 1 , so no strategy can guarantee at least L k/nJ + 1 players guess correctly for each coloring. 0 Theorem 3, and most of Theorem 1 , were obtained in dependently by Butler, Hajiaghay, Kleinberg, and Leighton (2008; see this for a considerably more detailed investiga tion of the finite context). With two colors and an even number of players, Theo rem 3 says that-with collective strategizing-the on-aver age result of 50% guessing correctly can, in fact, be achieved with each and every coloring. But it also says that this is the best that can be done by collective strategizing. In the finite case, this latter observation does little more than pro vide proof for what our intuition suggests: collective strate gizing notwithstanding, the on-average result of 50% can not be improved in a context wherein guesses are simultaneous. The infinite, however, is very different, and it is to this that we next turn.
Theorems of Gabay-O'Connor and Lenstra We begin with a statement and proof of what we will call the Gabay-O'Connor Theorem. As stated, this result is strong enough to solve the Gabay-O'Connor hat problem and to al low us to derive Lenstra's Theorem (below) from it. (One can use an arbitrary filter in place of the collection of cofinite sets, with essentially the same proof, to generalize the result.)
THEOREM 4 (GABAY-O' CONNOR) Consider the situa tion in which the set P ofplayers is arbitrary, the set C of col ors is arbitrary, and every player sees all butfinitely many of the other hats. Tben there exists a strategy under which all but finitely many players guess correctly. Moreover, the strategy is robust in the sense that each players guess is unchanged ifthe colors offinitely many hats are changed. Proof For h, g E Pc, say h g if (a E P : h(a) =f. g(a)} is finite; this is an equivalence relation on PC. By the axiom of choice, there exists a function
: Pc � Pc such that h, and if h g, then ( h) = (g). Thus, is choos (h) ing a representative from each equivalence class. Notice that for each coloring h, each player a knows the equivalence class [h), and thus (h), because the player can see all but finitely many hats. The strategy is then to have the players guess their hat colors according to the chosen representa tive of the equivalence class of the coloring; more formally, we are letting SJ_ h) = (h)(a). For any coloring h, since this representative
=
=
Theorem 4 is sharp in the sense that even with count ably many players and two colors, no strategy can ensure that, for a fixed k, all · but k players will guess correctly, even if everyone sees everyone else's hat. The reason is
that any such strategy would immediately yield a strategy for 2 k + 1 players in which more than 50% would guess correctly each time, contradicting Lemma 2. The following theorem was originally obtained by Hen drik Lenstra using techniques (described below) quite dif ferent from our derivation of it here from Theorem 4.
THEOREM 5 (LENSTRA) Consider the situation in which the set P ofplayers is arbitrary, le i = 2, and every player sees all of the other hats. Tben there exists a strategy under which everyone's guess is right or everyone's guess is wrong. Proof Let S be a strategy as in Theorem 4. A useful conse quence of the robustness of S is that, for a given coloring h, a player a can determine Stf... h) for every player b. Since we are assuming players can see all other hats, a also knows the value of h(b) for every b =f. a. So, we may define a strategy Tby letting TJh) = SJh) iff lfb E P : b =f. a and Stf... h) =f. h(b)ll is an even number. That is, the players take it on faith that, when playing S, an even number of players are wrong: if they see an even number of errors by others, they keep the guess given by S, and otherwise they switch. To see that T works, let h be a given coloring. When lfb E P : Sb(h) =f. h(b)ll is even, every guess given by T will be correct: the players who were already correct under S will see an even number of errors (under S) , and keep their guess; the players who were wrong under S will see an odd number of errors and will switch. When f b E P : Sb(h) =f. h(b)ll is odd, the opposite occurs, and every guess given by T will be incorrect: the players who would be correct under S will see an odd number of errors and will switch (to the incorrect guess); the players who would be wrong under S will see an even number of er rors and will stay (with the incorrect guess). 0
l
The assumption that everyone can see everyone else's hat in Theorem 5 is necessary. That is, if player a could not see player lis hat, then changing player lis hat would change neither his nor player a's guess, but player b would go from wrong to right or vice-versa, and player a would not. Lenstra's Theorem can be generalized from two colors to the case in which the set of colors is an arbitrary (even infinite) Abelian group. The conclusion is then that, for a given coloring, everyone's guess will differ from his true hat color by the same element of the group. Intuitively, the strategy is for everyone to take it on faith that the (finite) group sum of the differences between the true coloring and the guesses provided by the Gabay-O'Connor Theorem is the identity of the group. (Variants of this observation were made independently by a number of people.) Lenstra's original proof is certainly not without its charms, and goes as follows. If we identify the color red with the number zero and the color blue with the number one, then we can regard the collection of all colorings as a vector space over the two-element field. The collection W of all colorings with only finitely many red hats is a subspace, and the function that takes each such coloring to zero if the number of red hats is even, and one otherwise, is a linear functional defined on W. The axiom of choice guar antees that this linear functional can be extended to the
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whole vector space. Moreover, a coloring is in the kernel iff the changing of one hat yields a coloring that is not in the kernel. Hence, the strategy is for each player to guess his hat color assuming that the coloring is in the kernel. If the coloring is, indeed, in the kernel, then everyone guesses correctly. If not, then everyone guesses incorrectly. Another proof of Lenstra's Theorem, at least for the case where the set of players is countably infinite, was found by Stan Wagon. It uses the existence (ensured by AC) of a so-called non-principal ultrafilter on P-that is, a collection OU of subsets of P that contains no finite sets, that is closed under finite intersections, and that contains exactly one of X and XC for every X � P. Wagon's proof goes as follows. Label the players by natural numbers and call an integer a "red-even" if the number of red hats among players 0, 1 , . . , a i s even. Player a's hat color affects which integers b > a are red-even in the sense that changing player a's hat color causes the set of red-even numbers greater than a to be complemented . The strategy is for player a to make his choice so that, if this choice is correct, then the set of red-even numbers is in the ultrafilter OU. The strategy works because either the set of red-even numbers is in au (in which case everyone is right) or the set of red-even num bers is not in au (in which case everyone is wrong). .
The Necessity of the Axiom of Choice Some nontrivial version of the axiom of choice is needed to prove Lenstra's Theorem or the Gabay-O'Connor Theo rem. Specifically, if we take the standard axioms of set the ory (ZFC) and replace the axiom of choice with a weaker principle known as dependent choice, the resulting system ZF + DC is not strong enough to prove Lenstra's Theorem or the Gabay-O'Connor Theorem, even when restricted to the case of two colors and countably many players. His torically, the precursor to our results here is a slightly weaker observation (in a different but related context) of Roy 0. Davies that was announced in Silverman (1966). The reader does not need any familiarity with ZF + DC; all that must be understood is that, as an axiom system, ZF + DC is weaker than ZFC, and somewhat stronger than ZF (set theory with the axiom of choice removed altogether). To follow our argument, some basic facts about the prop erty of Baire are needed; to this end, the appendix gives a short introduction to the property of Baire. As an aid to in tuition, having the property of Baire is somewhat analo gous to being measurable, whereas being meager (see ap pendix) is somewhat analogous to having measure 0. (The two notions should not be conflated too much, however: the real numbers can be written as the disjoint union of a measure 0 set and a meager set.) Let BP be the assertion that every set of reals has the property of Baire. It is known (assuming ZF is consistent) that ZF + DC cannot disprove BP (Judah and Shelah 1993). (This was established earlier, assuming the existence of a large cardinal, in [Solovay 1970]. ) It follows that ZF + DC cannot prove any theorem that contradicts BP, as any such proof could be turned into a proof that BP fails. We will show that Lenstra's Theorem and the Gabay-O'Connor The orem contradict BP, and thus ZF + DC cannot prove Lenstra's Theorem or the Gabay-O'Connor Theorem. AI-
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though BP is useful for establishing results such as these, one should note that BP is false in ZFC (for instance, ZFC can prove Lenstra's Theorem, which contradicts BP). Throughout this section, we take the set P of players to be the set N of natural numbers, and we take the two col ors to be 0 and 1 . The topology and measure on N{o, 1 } are the usual ones. That is, if s is a finite sequence of Os and 1s, then the set [s] of all infinite sequences of Os and ls that extend s is a basic open set whose measure is z - n, where n is the length of s. Identifying N{O, 1 } with the bi nary expansions of reals in [0, 1], this is Lebesgue measure. The topology is that of the Cantor set. Let Tk be the measure-preserving homeomorphism from N{o, 1 } to itself that toggles the kth bit in a sequence of Os and 1s. Call a set D � N{o, 1} a toggle set if there are infi nitely many values of k for which T,j_D) n D 0. The next lemma is key to the results in this section; its proof makes use of the following observation. If a set D has the property of Baire but is not meager, then there ex ists a nonempty open set V such that the symmetric dif ference of D and V is meager. Hence, if we take any ba sic open set [s] � V, it then follows that [s] - D is meager. =
LEMMA 6 Every toggle set with theproperty ofBaire is mea ger. Proof Assume for contradiction that D is a nonmeager tog gle set with the property of Baire, and choose a basic open set [s] such that [s] - D is meager. Because D is a toggle set, we can choose k greater than the length of s such that T,j_D) n D = 0. It now follows that [s] n D � [s] - T,j_D). But T,j_[s]) [s] , because k is greater than the length of s. Hence, T,j(s] - D). Thus, [s] n D � [s] - T,jD) = T,j[s]) - T,jD) [s] n D is meager, as was [s]- D. This means that [s] itself is meager, a contradiction. D =
=
With these preliminaries, the following theorem (of ZF + DC) shows that Lenstra's Theorem contradicts BP, and hence it cannot be proven without some nontrivial version of the axiom of choice.
THEOREM 7 Consider the situation in which the set P of players is countably infinite, there are two colors, and each player sees all of the other hats. Assume BP . Tben for every strategy there exists a coloring under which someone guesses correctly and someone guesses incorrectly. Proof Assume that S is a strategy and let D denote the set of colorings for which S yields all correct guesses, and let I denote the set of colorings for which S yields all incor rect guesses. Notice that both D and I are toggle sets, since changing the hat on one player causes his (unchanged) guess to switch from right to wrong or vice versa. If D and I both have the property of Baire, then both are meager. Choose h E N{o, 1 } (D U I). Under h, someone guesses correctly and someone guesses incorrectly. D -
In ZFC, nonmeager toggle sets do exist: as seen in the previous proof, if all toggle sets are meager, then Lenstra's Theorem fails, but Lenstra's Theorem is valid in ZFC.
We derived Lenstra's Theorem from the Gabay-O 'Con nor Theorem, so Theorem 7 also shows that some non trivial version of the axiom of choice is needed to prove the Gabay-O'Connor Theorem. However, the Gabay O'Connor Theorem, even in the case of two colors and countably many players, is stronger than the assertion that the Gabay-O'Connor hat problem has a solution: the theo rem does not require that players can see all other hats, and it produces not just a strategy, but a robust strategy. The following theorem (of ZF + DC) shows that any solu tion to the Gabay-O'Connor hat problem, even in the count able case, contradicts BP and hence requires some non trivial version of the axiom of choice.
THEOREM 8 Consider the case of the Gabay-O'Connor hat problem in which the set ofplayers is countably infinite. As sume BP . Then for every strategy there exists a coloring under which the number ofplayers guessing incorrectly is infinite. Proof Assume that S is a strategy and, for each k, let Dk denote the set of colorings for which S yields all correct guesses from players numbered k and higher. Notice that each Dk is a toggle set, since changing the hat on a player higher than k causes his (unchanged) guess to switch from right to wrong. If all the DkS have the property of Baire, then all are meager. Let D be the union of the DkS, and choose h E N{O, 1} D. Under h, the number of people guessing incorrectly is infinite. D -
A set B has the property of Baire if it differs from an open set by a meager set; that is, there is an open set V and a meager set M such that BllV = M (equivalently, B = VJlM), where Jl denotes symmetric difference. A topological space is a Baire space if its nonempty open sets are nonmeager.
THEOREM 10 (BAIRE CATEGORY THEOREM) Every nonempty complete metric space is a Baire space. We do not show the proof here, but it can be carried out in ZF + DC. For the special cases of the reals and Can tor space, the proof can be carried out in ZF. REFERENCES
Steven Butler, Mohammad T. Hajiaghayi, Robert D. Kleinberg, and Tom Leighton. Hat guessing games. SIAM Journal of Discrete Mathe matics
22:592-605, 2008.
Joe P. Buhler. Hat tricks. Mathernatlcal lntelligencer 24:44-49, 2002. Todd Ebert. Applications of recursive operators to randomness and complexity.
PhD thesis, University of California at Santa Barbara,
1 998. Fred Galvin. Problem 5348. American Mathematical Monthly 72: 1 1 36, 1 965. Mart1n Gardner. The 2nd Scientific Amencan Book of Mathematical Puz zles & Diversions.
S1mon and Schuster, New York, 1 961 .
Alexander George. A proof of Induction? Philosophers ' Imprint 7 : 1 -5, March 2007.
Theorems 7 and 8 can be recast in the context of Lebesgue measurability to show that both Lenstra's Theo rem and the Gabay-O'Connor Theorem imply the existence of nonmeasurable sets of reals. However, to show that ZF + DC cannot prove the existence of nonmeasurable sets of reals, one must assume the consistency of ZFC plus the ex istence of a large cardinal (Solovay 1970, Shelah 1 984). Al though this is not a particularly onerous assumption, it is why we favored the presentation in terms of the property of Baire. It turns out that with infinitely many colors, some non trivial version of the axiom of choice is needed to obtain a strategy ensuring even one correct guess; this will appear elsewhere.
Appendix: The Property of Baire 9 A subset
N of a
topological space is nowhere dense if the interior of its closure is empty. A set is meager if it is the union of countably many nowhere dense sets.
DEFINITION
Fred Galvin and Karel Prikry. lnfinitary Jonsson algebras and partition relations. Algebra Universal1s 6:367-376, 1 976. Chnstopher S. Hardin and Alan D. Taylor. A peculiar connection be tween the ax1om of choice and predicting the future. American Math ematical Monthly
1 1 5:91-96, February 2008.
Haim Judah and Saharan Shelah. Baire property and axiom of choice. Israel Journal of Mathematics
84:435-450, 1 993.
Saharan Shelah. Can you take Solovay's inaccessible away? Israel Journal of Mathematics
48: 1 -47, 1 984.
D. L. Silverman. Solution of problem 5348. American Mathematical Monthly
73: 1 1 3 1 -1 1 32, 1 966.
Robert Solovay. A model of set theory in which every set of reals is Lebesgue measurable. Annals of Mathematics 92: 1 -56, 1 970. B. L. D. Thorp. Solution of problem 5348. American Mathematical Monthly
74:730-731 , 1 967.
Peter Winkler. Games people don't play. In David Wolfe and Tom Rodgers, editors, Puzzlers' Tribute, pages 301-31 3. A. K. Peters, Ltd. , 2001 .
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l$@u:i§u@iil¥11@1§4fii,iui§,IN
Sub S h oot ! MICHAEL KLEBER
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 bas an urge to pass them on. Contributions are most welcome.
Michael Kleber and Ravi Vakil , Editors
� y day, Leonidas Kontothanassis works as my colleague at Google. LJ By night, he runs a gaudy carni val booth on the boardwalk outside of town. "Step right up and try your luck! Shoot the sub and win a prize," Leonidas was calling out one fine Fall evening. He was standing in the middle of the booth, surrounded by a moat packed with plastic toy submarines. The subs were of every shape and size, and they circled the moat at all different speeds, propelled by a complex system of cur rents. "On the bottom of one of these sub marines is an X," he assured me. "Find the right sub and the prize is yours. " I eyed him suspiciously-he wasn't the most trustworthy character. "All right, how many chances do I get?" "Let me explain the rules," he an swered, "and then you tell me how many guesses you think you'll need." Leonidas's moat, he pointed out, was divided into n regions, numbered in or der from 1 through n as you walked around the booth. On each shot, I would get to pick not just one subma rine, but one whole region, and we would scoop out all of the subs in that region. If any of them was the one marked with the X, I would win. If not, I could take another shot at the re maining subs-which would all have had time to move to new regions by then, of course. "But how fast do the subs move?" The swirl of motion was dizzying, with subs passing each other right and left. "It's actually very orderly, " he as sured me. "At the beginning, there are exactly n2 submarines, with n in each of the n regions of the moat. Moreover, the subs in a region each move at a dif ferent constant velocity: Every velocity 1 is represented from 0 through n once. Velocity is measured in regions per shot. So you get one shot at each . , and there is a time t 0, 1 , 2, 3, sub with position a + bt for every a and b (mod n)." Now that I knew what I was looking at, I had to admit it was very orderly, -
Please send all submiSSions to the Mathematical Entertainments Co-editor, Ravl Vakil, Stanford Un iversity , Department of Mathematics, Bldg. 380, Stanford, CA 94305-2125, USA e-mail: [email protected]
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right down to the one b 0 submarine anchored, unmoving, in each region. "Do I have to take a shot at each time t, or can I take a small number of shots but space them out?" "I don't have all night," Leonidas replied. "The question is how quickly you can find the X, not in how few shots. Anyway, if I let you have arbi trary breaks, then you could shoot only at times 0, n, 2 n and so on, when all the subs got back to their original lo cations. Too easy!" "You don't sound like you've left anything to chance, you old huckster. Do you really expect me to believe I might get lucky and hit the winning sub on my first shot?" "Of course not," he answered with a twinkle in his eye. "I guarantee, the X will be in the last place you look." =
Greed In how few shots can we hit all of Leonidas's submarines? Certainly we can manage in n2 shots, one per submarine, but, of course, we can do much better. On the other hand, we can't hope to do it in fewer than n shots: There are n2 subs, and we can only hit n of them at a time. In fact our 1 subs, so second shot will only hit n even n shots is unattainable. Also, the n stationary submarines--or, indeed, the n subs with velocity b, for any b will each require its own shot. So n is a lower bound that's too low, n2 is an upper bound that's too high, and the truth lies somewhere in be tween. The greedy algorithm is the first refuge of the lazy: On each shot, we could target the most heavily-occupied region. Ignoring the precisely choreo graphed submarine movement entirely, we know that there will always be a re gion holding at least 1/ n of the re maining subs, by the pigeonhole prin ciple. We can get a bound on the efficiency of this approach by working backwards. The last n submarines will, at worst, take one shot each, or n shots total. Before that stage, when from n + 1 to -
2 n subs remain, we can hit at least two at a time, so we need at most n12 shots to make it through. Likewise, we need at most r n/31 shots to get from 3 n subs down to 2 n, and s o o n . We can be sloppy about rounding and still get within n shots of the true answer. The total number of shots to hit all n2 subs in the worst case is within n of + .!� = n ( 1 + _1_ + n + .!� + .!� +
r
..!. + 3
2
·
·
·
3
·
·
·
n
J
2
+ 1._), or around n In n. In fact, II
it's around n/2 + n In n: Experimen tally, this is how many times you need 1 to perform "multiply by 11 - and round down" to get from n2 to �ero. So we know about how long the greedy algorithm would take against submarines that deliberately spread themselves out to avoid our shots. But Leonidas's aren't doing that. If subs conveniently clump together, the greedy algorithm will find more ap pealing targets, and will finish sooner. My friend Pablo Alvarez wrote a simu lation, and his results are plotted in Fig ure 1 . (In case of ties, he shot at the lowest-numbered region which attained the maximal sub count.)
The greedy algorithm appears to do much better than the worst-case analysis would suggest. Empirically, it seems to take 2 n 1 shots when n is prime or a power of two, and n 3 shots when n is twice a prime. But at n 575, the greedy algorithm takes more than 4 n shots for the first time, and the trend for the worst n does seem to keep growing faster than linearly . From a practical point of view, the greedy algorithm's implementation is not as easy as its description. Suppose n were a million. A computer program calling the shots would need to track which of the trillion submarines in the fleet were still afloat and how they were allocated among the regions. This seems like a lot of work, particularly for the approach that was supposed to be the lazy way out. -
f
-
=
Divisiveness If we make use of the precise move ment of the subs, there is a much simpler algorithm that does better than n In n.
Consider a submarine whose veloc ity b is relatively prime to the number of regions n. In n moves, that sub will visit all the regions, in some order. In particular, it will visit region 1 at some point. So if we shoot at region 1 for n shots in a row, we will hit every sub with gcd( b, n) = 1 . Supposing for a moment that n is even, look at the subs with gcd(b, n) = 2. These subs will eventually visit half the regions, either all the even- or all the odd-numbered ones, completing their route with period n/2. We can therefore hit all of them in n shots as well: Shoot n/2 times at region 1, and then n/2 times at region 2. More gen erally, for any d dividing n, we can hit all the subs with gcd(b, n) = d in only n shots, by taking n/ d shots in a row at each region 1 , 2, . . . , d. To hit all the subs, we can pile the strategies for each divisor d on top of each other. First take n shots at region 1 : We've now hit the relatively prime velocities, but at the same time we've hit half the d = 2 subs as well. Next, fire at region 2 for n/2 shots, wiping
Figure I. Number of shots needed by the greedy algorithm for 1 :s n :s 1000. The lines 2n, 3n, and 4n are shown for refer ence. In practice, greedy performs substantially better on Leonidas's submarines than the order n In n upper bound.
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out the rest of the d = 2 part of the fleet. If n is a multiple of 3 as well, we've now fired more than enough times at regions 1 and 2, and firing n/3 times at region 3 will wipe out the re maining subs with d = 3. If we ignored the question of which d divide n-for example, if Leonidas had only told us n approximately, not exactly-this shows that we could hit all the submarines by firing a volley of nlk consecutive shots at region k, for k = 1 , 2, . . . , n. This may be less ef ficient than the greedy algorithm, where n In n is the upper bound, not the true performance, but at least there's no hard work involved. But if we know n's divisors 1 = d1 < d2 < < dr = n, we can shave off some guesses. Instead of bombarding region k with n/k shots, we can "round down to the nearest divisor," firing only n/d; times, where d;- 1 < k � d;. For ex ample, if n is a multiple of 3 but is not even, we can follow the n shots at re gion 1 with only n/3 shots at region 2, not n/2. When n has many small divisors, the savings over the n In n bound is real if modest. When n is one million and the greedy algorithm's worst-case bound is 14,392,720 shots, the divisor-aware ver sion requires only 1 2,687,500, a 10% savings. The advantage is most telling when n is prime, though: The divisor-aware version requires only 2 n - 1 shots. First, take n shots at region 1, hitting all the submarines whose velocity is nonzero, and then take one shot at each other region, to clean up the stationary subs. This is exactly how the greedy al gorithm behaves for prime n. ·
·
·
Exhaustion I visited Leonidas again the following night. The number of regions at his booth is not prime. "So for a million regions," he summed up, "you have one algorithm that you think will do well, but you don't know how well and it will require a trillion pieces of bookkeeping. And you have another algorithm that's easy to use, but will take more than twelve million tries." Under the flashing board walk lights, neither option seemed par ticularly appealing. Leonidas sent me home to think some more. In conversations with friends and
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colleagues, we produced a few refor mulations. Ignore the n2 - 1 sub marines without the X, and say that there is one sub at position a + bt with unknown a and b. You can think geo metrically: The sub at a + bt is a line in the space 7L. X 7L./n7L., and you're try ing to pick a set of points (So, 0), (s1 , 1), etc. that hit each line. Dually, you can view the a + bt sub at the point (a, b) in 7L./ n7L. X 7L./ n7L., and your goal is to 2 cover the n points with one line of slope 0, one line of slope 1 , one line of slope 2, and so on. The slope of the lines must be considered mod n, which highlights the fact that the shots at time t and t + n are equivalent. But none of these perspectives led to a break through. I managed to enlist some coconspir ators: Rich Schroeppel, Steve Witham, Scott Huddleston, and Edwin Clark all took up the computational hunt. They observed empirically that the n-region problem can be done in 2 n - 1 shots for primes and for prime powers as well-and for powers of odd primes, this turns out better than either the di visor strategy or the greedy algorithm. Rich and Steve ran exhaustive searches for small n. To make this fea sible, they needed to use some sym metry considerations that are not im mediately obvious. Without loss of generality, you may assume that your first shot is at the region numbered 1 : I f it isn 't, just add a constant t o each re gion's number, mod n. More surprising, they observed, you may assume your second shot is at region 1 also: Renum bering region r to r + st at time t is an automorphism of the original problem, transforming the submarines moving at velocity b into ones moving at velocity b - s (mod n). You do get to make a nontrivial choice for your third shot, but without loss of generality, the difference between shots two and three might as well be a divisor of n, thanks to multi plication by a unit mod n. Finally, the subs all have paths which repeat in time n, so the shots at times t, t + n, t + 2 n, etc., are all equivalent, as we mentioned before. Using such considerations and well pruned searches, they showed that 2 n 1 is minimal for prime powers up to n = 9. The sequence of 2 n - 1 shots for n = p k is a version of the one we described for primes above: First shoot n times at
region 1 , then shoot once each at re gions 2, 3, . . . n, in that order. Order didn't matter for primes, but it certainly does for prime powers. I'll present a proof of this construction below. For n = 6, the exhaustive searchers report that the minimum to hit all subs is 10 shots, for example 1-1-1-2-1-3-45-4-6, giving us for the first time a so lution under 2 n - 1 ; for n = 10 the minimum is 17 shots, attained by 1 - 1 1-1-1-2-2-7-2-5-9-6-4-10-3-8-8. Scott and Edwin generalized empir ically observed patterns and figured out how to get the job done in time 2 n 1 for n = 2pk, for odd primes p. That construction and proof is also forth coming, but we need a new point of view first.
Shifty Sweeps To make things easier to discuss, let's define a b-sweep to be a sequence of n shots which hit every submarine with velocity b. For the moment we'll restrict sweeps to be n consecutive shots, though we'll relax that requirement soon. We already mentioned that you can't accomplish this with fewer than n shots, since the n subs with velocity b are par allel-that is, their paths never intersect, so no single shot can hit two of them. There are exactly n! b-sweeps (for a given n and b) : You can hit the n subs in any order. Example 1: The permutations are ex actly the 0-sweeps. To hit the station ary subs, shoot at every site (in any or der). Example 2: The sequence of n shots at region 1 is a b-sweep for every ve locity b such that gcd(b, n) = 1 , one of our first observations. These shots hit many other subs as well, but for no other velocities do they hit all the subs. So one (very limiting) strategy for hit ting all the subs is to partition the set of possible velocities B into, say, B1 U B2, and then find one sequence of shots that is a b-sweep for all b in B1 and an other sequence that works for all b in Bz. If n is a prime, for example, then the two examples above go together to reproduce our old 2 n - 1 solution. (Why 2 n - 1 , not 2 n? Well, sweeps are translation-invariant-that is, you can freely add a constant to each region number: They only care about the ve locities of the subs, not their initial po-
sitions. So if we're going to shoot one sweep and then another, we can trans late one so that the final shot of the first sweep also serves as the initial shot of the second sweep.) Here's a useful lemma: If shooting at the sequence of regions r1 , r2, r3, . . . , rn is a b-sweep, then shooting at r1 + w, r2 + 2 w, r3 + 3w, . . . , rn + nw (all mod n) is a (b + w)-sweep. This is another invocation of the automorph ism used by the exhaustive searchers. It's easy to justify with some mod n arithmetic: The sub with position func tion a + bt gets hit by sweep shot i if a + bi == r, which is the same as a + (b + w) i == r, + iw. Now I can give a simple proof that we can hit all subs in 2 n - 1 shots whenever n is a prime power, say pk. First, fire n shots at region 1 , hitting all subs with velocity prime to n, i.e., not divisible by p. This is the sweep 1, 1 , 1 , . . . , 1 . Now transform it to the sweep 1 , 2 , 3, . . . , n, and, by the lemma, it will hit every sub with velocity b as long as b - 1 is not divisible by p. Consec utive numbers can't both be multiples of p, so every sub will be hit by one of the two sweeps. We can generalize this to n with more complicated prime factorizations. This solution schema just depends on the shifts by which we transform the sweeps. Round 1 : Take n shots at re gion 1 . Rounds 2, 3, . . . , k: Take n 1 shots, adding c2, c3, . . . , ck to the region number before each shot. The k rounds take kn - k + 1 shots, and will hit every submarine with a velocity b such that any of b, b - Cz , b - c3, . . . , b - ck is prime to n. (We assumed c1 = 0 by the symmetry arguments made earlier.) For example, take n to be one mil lion again. If we set c2, c3, c4 to be 1 , 2, 3, this method will let u s clear all the subs in four million shots (well, 3,999,997). These shifts work because in any four consecutive numbers, there is one which is relatively prime to a mil lion, i.e., divisible by neither 2 nor 5. We can't do it using this technique in only three rounds. Two of the shifts, say Cx and c would have the same parity, .Y' and we could always find a b that made Cx + b and Cy + b even and Cz + b a mul tiple of 5, if odd. Indeed, the Chinese Remainder Theorem tells us that this is the general case: No choice of shifts will
do any better than taking c1 = 0, c2 = 1 , . . . , ck = k - 1 for some k. The number of rounds we need, k, is therefore the maximal distance be tween consecutive integers relatively prime to n. Courtesy of Neil Sloane and his Online Encyclopedia of Integer Se quences (http://www .research.att.com/ -njas/sequences/), I learned that this maximal distance is known as the jacobsthal function of n, sequence A04-8669, hereafter j(n). So we now have a very simple al gorithm which will succeed in just un der nj( n) shots. The value of j(n) de pends only on the set of primes dividing n-the powers in n's prime factoriza tion are irrelevant to questions of what's relatively prime-so this approach pro duces solutions that scale up well to large n with only a few prime factors. When n is a prime power, j(n) = 2, and this reduces to our old 2 n - 1 so lution. When n p kq C for odd primes p and q, j(n) = 3: certainly if p divides b, and q divides b + 1 , then neither can divide b + 2 . If, instead, n = 2 kq C , we have j(n) = 4, just as we saw for n of one million. Not bad. But not always great, either. For n = 210 = 2 3 5 7, for instance, this con struction does very poorly. j(n) 10, since no number between 1 and 1 1 is prime to 2 10. So this scheme requires IOn - 9 = 2091 shots, while the divisor aware algorithm uses 1 1 18 and the greedy algorithm only 648. =
·
·
·
=
Time Warp Thej(n) bound is great for prime pow ers, but when n = 2pk, it degrades to a length 4 n - 3 solution. Scott Huddle ston and Edwin Clark figured out how to get the job done in time 2 n - 1, but their solution isn't a concatenation of two sweeps. For n = 10, for example, the sequence of shots is 10-2-10-4-106-10-8-10- X -5-7-5-9-5-1-5-3-5. (The X means you don't fire at all.) Scott figured out the general picture of which this is one example. The key is removing the limitation that sweeps consist of consecutive shots. Useful lemma number two: If shoot ing at regions r1 , r2, . . . , rn at times t1 , t2 . . . , tn is a b -sweep, then shoot ing at those same regions at times wt1 , wt2, . . . , wtn is a b' -sweep for any ve locity b' with wb' = b. When w divides n, multiplication by w mod n is a many-
to-one function, so spreading out the timing of our shots can greatly increase the number of velocities swept. The simplest use of this lemma is transforming our old friend n-shots-at the same-region. Applying the earlier lemma as well, we get Scott's key result, describ ing the effect of a sequence of shots that are linear in both time and space.
THEOREM 1 In the n-region subma rine problem, shooting at region ro + sdr (mod n) at time t0 + sd1, for 1 ::::; s ::::; n, will hit all submarines with all velocities b such that gcd(n,dr - bd 1) = 1 . The rules o f the problem don't look kindly on widely-spaced shots. Indeed, the most extreme application of this the orem, with .:11 = n and dr 1, reminds us that we could hit all the submarines in only n shots total, if only we were al lowed to wait for time n between shots. But now we can improve on this: It's suf ficient to let .:11 be the square-free part of n---let's call it �n) = II� n p. It's still in efficient time-wise, but it's good to know that we can finish the million-region sub marine problem in a million shots and time ten million, by shooting at regions 1, 2, 3, . . . at times 10, 20, 30, . . . These temporally-dilated sweeps can be used efficiently, also: If we want to shoot several of them, we can interleave them and not skip shots. Our old j(n) based approach filled time 2 n by two concatenated sweeps; in Scott's ap proach, you could instead fill time 2 n with two sweeps with .:11 = 2, alternat ing between them. For n of one million, Scott can do just that, hitting all the subs in two mil lion shots: =
•
•
on even-numbered shot 2 t, fire at re gion t; on odd-numbered shot 2 t + 1, fire at region 3t.
Setting .:11 = 2 means every sub's effec tive velocity bL11 is always even; sub tracting either one or three is sure to leave you with a velocity relatively prime to a million. Any n of the form 2 kpC works the same way. Without going into any details, let me also mention that Scott has a way of whittling a few shots off the end of a solution, like the overlapping sweep trick that led to j(n) - n + 1, but more subtle. (It also lets him skip some shots entirely, though in ways that don't re-
© 2008 Spnnger Sctence+Bustness Medta, Inc., Volume 30, Number 4, 2008
29
Questions
duce the total time needed.) Choosing the two 11/s correctly, you could finish in only 1 ,999,999 shots. For a given (!n), the problem be comes one of efficiently arranging and packing sweeps with the same 11/s. In addition to the above time 2 n - 1 re sult for (X n) = 2p Scott reports these successes:
"Bravo! " applauded Leonidas when I described how to clean out a million regions in two million shots. But he was ruthless in his interrogation. "For primes and prime powers, you make it sound like 2 n - 1 is the mini mum possible number of shots. Have you proved that?" No, I admitted-not even when n is prime, which felt par ticularly galling. "And your algorithms have prime powers as their best performance. But didn't the exhaustive search data show other small n needed fewer than 2 n shots, not more?" H e was right again, of course. Should we expect a million regions to take substantially less than two million shots? And I have no rea son to think that products of small primes are inherently harder than other n, just because they are the Achilles' heel of one tactic. The greedy algorithm only needs 68 and 648 shots for n 30 and 2 1 0, respectively, far better than any organized scheme I know. "Do you think there is an algorithm which is at worst linear in n?" Inspect-
,
•
•
•
time 3 n - 2 for (! n) 3pq with primes q > p > 3; time 4n - 3 for (X.. n) = 2 3 5, and, at most, 3 n - 2 shots are needed; time 6n - 5 for (! n) of 2 3 5 7 2 3 5 7 11 and 2 3 5 7 1 1 . 13; time 8 n - 7 for (X.. n) = 2 3 5 7 1 1 13 17 =
·
·
•
·
·
·
·
·
·
·
·
·
·
·
·
·
·
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·
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.
The first three use 111's of 3, 2, and 6, respectively. The fourth uses six sweeps with 111 = 6 and two with 111 = 2, and requires careful work to get them to all align correctly. As (X n) grows to the product of many initial primes, Scott says he would eventually use sweeps with time dilation factors of 30, 6, and 2 .
=
fl Springer
ing the data, it seems tantalizingly pos sible. As my friend and colleague Thomas Colthurst pointed out, a con stant-time bound would let us refor mulate the game: Suppose you are al lowed to fire at up to k regions at a time, for some fixed k. Now can you clean out all n2 subs in time n? And perhaps even k = 2 would do the trick! Leonidas smiled at me and walked away from his booth. "See you tomor row,'' he called back to me. ''We have work to do!"
Thanks I am much obliged to the real Leonidas Kontothanassis for inventing the sub marine-mod- n problem and for letting me put words in his mouth now. Doubtless he had no idea what he was getting into when he posed the ques tion to me. Thanks also to the collab orators mentioned by name for letting me report on their advances, particu larly Scott Huddleston for feeding me quickly when it counted, and to Jessica Polito, Will Brockman, and Thomas Colthurst for many discussions.
.
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30
THE MATHEMATICAL INTELLIGENCER
Stefan Banach Re me mbe red 1n l
•
he readers of the "Mathematical Tourist" column seldom have an opportunity to read about memorials dedicated to mathematicians, as few members of this profession have ever been commemorated by a statue. Among the notable exceptions is Ste fan Banach, whose monument was un veiled in 1 999 in front of the building of the Mathematics and Physics De-
partment of the Jagiellonian University in Krakow in Reymonta Street (Figure la,b). Banach was one of the greatest mathematicians of the twentieth cen tury. In the list of scholars who are most often mentioned in the titles of mathe matical and physical papers published in the twentieth century, his name ranks first, followed by those of Sophus Lie
DANUTA CIESIELSKA AND KRZYSZTOF CIESIELSKI
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe 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 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 Mathematical Tounst Ed1tor, Dirk Huylebrouck, Aartshertogstraat 42, 8400 Oostende, Belgium
·
e-ma 1 l : [email protected]
Figure I a,b. The monument of Stefan Banach.
© 2008 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 30, Number 4, 2008
31
Figure 2. Grodzka Street. The house where Banach lived in his youth is the first building on the left.
and Bernhard Riemann. This is mainly due to the fundamental importance of the concept of "Banach space" in math ematical analysis. Banach is regarded as the founder of functional analysis. One could ask why Banach's mon ument was placed in Krakow, if almost all his mathematical results were ob tained-from 1920 until his death in
1945-in Lvov. Nevertheless, Banach had strong connections with Krakow as well. It was in Krakow, in St. Lazarus Hos pital in Kopernika Street, that Banach came into the world on March 30, 1892, and was baptized four days later. He was born out of wedlock and took his surname from his mother, Katarzyna
Banach. His father, Stefan Greczek, was a soldier. Banach never saw his mother later in life. Entrusted into the care of Franciszka Plowa, a laundry owner, a few months after birth, he was brought up by her and by her daughter Maria Plowa (later Puchalska). They lived in Krakow in building No. 71 (now renum bered 65) in Grodzka Street (Figure 2), at the foot of the Wawel Castle. In 1902, after finishing elementary school, Banach entered Gymnasium (secondary school) No. 4 in Krakow formerly a branch of St. Anne Gymna sium No. 1 , the oldest gymnasium in Krakow and in Poland (dating back to 1588)-situated at Na Groblach Square. The branch was housed in Podwale Street in "Goetz Building" (Figure 3a,b), named after the brewery owner Jan Al bin Goetz-Okocimski, from whom it was leased. In the same year of 1 902 the branch became an independent gymnasium and got its own number. Being a "classical gymnasium, " it did not put a particular emphasis on sci ence in its curriculum. On completion of his secondary ed ucation in 1910, Banach decided to study engineering in Lvov. He was in terested in mathematics, but considered it to be a nearly complete science to which very little was left to be added. He moved to Lvov, but studies at the Technical University did not appeal to
Figure la,b. The Goetz Building, where Gymnasium No. 4 was housed.
32
THE MATHEMATICAL INTELLIGENCER
him and, upon the outbreak of World War I, Banach returned to Krakow to enrich his mathematical knowledge by independent study. Reportedly, he at tended lectures by Stanislaw Zaremba at the Jagiellonian University. He also engaged in many discussions with his long-time friends Witold Wilkosz and Otto Nikodym. In 1916 an event crucial for the de velopment of mathematics took place. Hugo Steinhaus, by that time already a well-known mathematician, was staying in Krakow for some time. During an evening walk in the Planty Gardens in the city center, he heard the words "Lebesgue integral. " At that time it was a recent idea known almost exclusively to specialists. Steinhaus was intrigued. He joined the conversation between two young people, who turned out to be Banach and Nikodym. In particular, he told them about a problem he had been working on for some time. When, to his great surprise, his new acquain tance brought him the solution a few days later, Steinhaus realized that Ba nach had a superb mathematical talent. Although Steinhaus could boast many outstanding results, he would later say that his greatest mathematical discovery was the discovery of Banach. Banach would regularly meet Stein haus in Krakow, and their mathemati cal discussions continued. His first sci entific paper ([2]) was published in 1918, following a presentation at the meeting of the Academy of Arts and Sci ences in Krakow. On April 2, 1919, the Polish Mathe matical Society was established. Among the 16 mathematicians present at the Constituting Session held on the premises of the Philosophy Seminar at St. Anne Street 12 in Krakow was, of course, Stefan Banach. In 1 920 Banach got married in the church in Karmelicka Street in Krakow. The church belonged to St. Stephen Parish. His wife was Lucja Braus, who he first met through Steinhaus, at the flat where Steinhaus lived. In the years of Banach's childhood, Krakow and Lvov were in the Austro Hungarian Empire (however, the Pol ish language was spoken there). After World War I , Poland regained in dependence, and Krakow and Lvov were again in Poland: In 1 920 Banach moved to Lvov again. With support
from Steinhaus, he received an assist antship at the Lvov Technical University. In 1920 he was granted the doctoral degree, and on July 22, 1920, was ap pointed professor at the Jan Kazimierz University of Lvov. For more informa tion on Banach's achievements in his Lvov times, his life, and some anecdotes see [3], [4), [5], and [7).
"
AK WI '-# frtf. ,
A place that played a special role in mathematical life in Lvov was the Scot tish Cafe. Mathematicians would meet there to eat, drink, talk mathematics, formulate and solve problems. Discus sions at the Scottish Cafe led to a large number of outstanding results over many years (see [8)). In 1931 Banach's fundamental
I ltrakowie, dnia 9
osenca
1 945
r,
Do Pana Dra Stetena B a n a o h a Proteeora Uninr.;rtetu n L'I'Owie
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uprzejaie , te koaiaja ao obaadz nia 111-ciej latadr:r
t:rld na l)'dziale Yilozofi0S1111l Uniwora7tetu Jagielloilaki�
go, uohwalila wnioaok o przedat wienie Pana Proteaora , cbdata
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/
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,
Figure 4. The official letter to Stefan Banach inviting him to the Jagiellonian Uni versity, signed by Franciszek Leja (1895-1979), an outstanding specialist in complex analysis. (Courtesy of the Jagiellonian University Archives.)
Figure 5. The Scottish Cafe in 2005 (now a bank).
© 2008 Spnnger Sc1ence+Bus1ness Med1a. Inc • Volume 30. Number 4. 2008
33
Figure 6. The Banach portrait in the assembly hall in Gymnasium No. 1 . monograph on functional analysis, "Op eracje liniowe'', was published. Its French translation, "Theorie des opera tions lineaires" [1], appeared the fol lowing year as the first volume in the series "Mathematical Monographs . ., For many years it was the fundamental book on functional analysis, until the famous monograph [6] was published. It should be noted that the term "func tional analysis" only came into use in the 1 940s. In Banach's day, this branch of mathematics was known under an other name, the "theory of linear oper ators." The story goes that, upon pub lication, Banach's monograph, Theory of Linear Operations, was displayed in some Lvov bookshops on shelves for medical books. History repeats itself: Some years ago the Polish translation of the famous book, "Does God Play Dice?" by Ian Stewart, could be found in one of Krakow's bookshops in the religious book department. After the end of World War II in 1 945, the Allies decided that the area
34
THE MATHEMATICAL INTELLIGENCER
containing Lvov would become a part of Ukraine, USSR, so a large number of Poles living there decided to move. Ba nach was offered a Chair at the Jagiel-
Ionian University in Krakow (Figure 4), which was to be created especially for him. It would have been the third math ematical Chair at this university. Banach accepted the offer but never took the Chair, as he died in Lvov on August 3 1 , 1945. In the 1 970s the premises of the for mer Scottish Cafe housed an ordinary restaurant. In the 1990s, Ukraine gained independence. Lvov was included in its territory, as a city of the former Ukrain ian Soviet Republic. There are many nice cafes in Lvov now. Unfortunately, in the place where the Scottish Cafe had been there is now a bank (Figure 5). We have heard that the authorities of the university (now named Ivan Franko University) wanted to buy this building because of its glorious mathematical tra dition, but it turned out to be too ex pensive. Let's g o back to Krakow. The idea to commemorate Banach by a statue was put forward by the Krakow Branch of the Polish Mathematical Society in 1992, on the occasion of the lOOth anniversary of his birth. The mathematicians wanted to place the monument in the Planty Gar dens, where Banach was "discovered" by Steinhaus. Unfortunately, their initia tive was turned down by the local au thorities in Krakow. Finally, the monu ment, designed by Ma}gorzata Olkuska, was unveiled in front of the Mathemat ics Institute on August 30, 1999, on the occasion of the Session of the Polish Mathematical Society marking its 80th
Figure 7. A Banach Street sign in Krakow.
anniversary. The monument was un veiled jointly by the Rector of the Jagiel lonian University, the President of the Polish Mathematical Society and Ba nach's daughter-in-law, Professor Alina Filipowicz-Banach, an ophthalmologist. Banach's only son, Stefan Banach, Jr. , a neurosurgeon, had died a few months before the ceremony. In Krakow, there are very many gym nasiums now, but none of them is a continuation of Gymnasium No. 4. However, Banach is commemorated in Gymnasium No. 1 (now called the Nowodworski Gymnasium). Displayed in the assembly hall of this school are the portraits of some of its most illus trious students, including figures of worldwide renown, like the writer Joseph Conrad. Since 1998 Banach's portrait has also been there (Figure 6). About 20 years ago several new streets in Krakow were named after fa mous mathematicians. One of these is Banach Street. All the above-mentioned
places connected with Banach are in the city center, within walking distance from one another. However, Banach Street (Figure 7) is far from the center, on the outskirts of the city. It is a cross road of the main road leading to War saw, so everyone driving to Warsaw can see a street sign with Banach's name. However, one might be left with an im pression that, rather than a "Banach Street,'' this place would be more ap propriately described as a Banach space.
[4] K. Ciesielski, Lost Legends of Lvov 2: Ba nach's Grave, Math. lntelligencer 1 0(1 988) no. 1 , 5D-51 . [5] K. Ciesielski and Z. Pagoda, Conversation w1th Andrzej Turowicz, Math. lntelligencer 1 0(1 988) no.4, 1 3-20. [6] N. Dunford and J.T. Schwartz, Linear Op erators,
lntersc1ence Publishers, New York,
vol. I, 1 958.
[7] R. Katuza, Through A Reporter's Eyes. The Life of Stefan Banach,
Birkhauser, Boston,
1 996. [8] R.D. Mauldin (ed.), The Scottish Book.
REFERENCES
Mathematics
[ 1 ] S. Banach, Theone des operations lineaires,
Birkhauser, Boston, 1 981 .
from
the
Scotttsh
Cafe,
MonografJe Matematyczne 1 , Warszawa, 1 932.
Pedagogical University 1n Krakow, Mathemat
[2] S. Banach, H. Steinhaus, Sur Ia convergence en moyenne de series de Fourier, Bull. Int. I'Acad. Sciences de Cracovie, Senes A, Sci ences Mathemattques,
ics Institute Podchor9Zych 2, 30-084 Krakow, Poland e-mail: [email protected]
1 91 8, 87-96.
[3] K. Ciesielski, Lost Legends of Lvov 1 : The
Jagiellonian University, Mathematics Institute
Scottish Cafe, Math. lntelligencer 9(1 987)
Reymonta 4, 30-059 Krakow, Poland
no.4, 36-37.
e-ma1l: Krzysztof.Ciesielski@im. uj.edu.pl
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35
QiifW·i. [.i
David E. Rowe , E d itor
The Doub le Life of Fe l ix H ausdorff/ Paul Mongre WALTER PURKERT Mathematical Institute Bonn University Beringstr. 7 0531 1 5 , Bonn,
Germany
e-ma1l: [email protected] de
]
EDITOR'S NOTE Nearly every mathematician is familiar with the name Felix Hausdorff, and yet until fairly recently little was known about his life. Despite an extraordinar ily productive career, perhaps unparal leled in intellectual and creative breadth, this multitalented figure only received scant attention from biogra phers or historians of mathematics. The circumstances of his life and tragic death, even the broader scope of his mathematical work, were forgotten, cast into the shadows. That situation changed dramatically, however, in the 1990s when a team of scholars began work on a unique endeavor in the an nals of mathematical publishing: to pro duce a Hausdorff edition containing his Gesammelte Werke in nine volumes. Its projected structure and contents are: Band I: Biographie. Hausdorff als akademischer Lehrer. Arbeiten uber geordnete Mengen Band II: Grundzuge derMengenlehre ( 1 914) Band III: Mengenlehre ( 1927, 1935). Arbeiten zur deskriptiven Mengen lehre und Topologie Band IV: Analysis, Zahlentheorie
Algebra
und
Band V: Astronomie, Optik und Wahrscheinlichkeitstheorie Band VI: Geometrie, Raum und Zeit Band
VII:
Philosophisches
Werk
(Sant' Ilario ( 1 897). Das Chaos in kosmischer Auslese ( 1 898). Essays zu Nietzsche) Band VIII: Literarisches Werk (Ek stasen, Der Ar:zt seiner Ehre, Essays) Band IX: Korrespondenz
Send submissions to David E. Rowe, Fachbereich 08-lnst1tut fur Mathematik, Johannes Gutenberg Un iversity, 055099 Mainz, Germany.
36
This edition will ultimately contain reprints of all of Hausdorffs published astronomical and mathematical works along with detailed commentary. The volumes will also present for the first time a considerable number of carefully edited unpublished texts from a vast manuscript collection now housed at the Bonn University Library. Yet this is
THE MATHEMATICAL INTELLIGENCER © 2008 Spnnger Sctence+ Bustness Medta, Inc
not all; as evident from the list of vol umes, the edition will also contain Hausdorff's literary and philsophical works, written under a pseudonym, many of which enjoyed high critical ac claim in their day. The initial plans for this project grew out of activities in Bonn that took place in 1992, the fiftieth anniversary of Haus dorffs death. To mark that occasion Professor Egbert Brieskorn organized a special commemorative exhibit de voted to Hausdorffs life and work (Brieskorn 1 992). Alongside this he also arranged a colloquium out of which emerged the essays in Brieskorn (1996). Brieskorn has continued to collect and assemble documents related to Haus dorff's biography for many years, and these documents will eventually appear in volume 1 of the edition. He was also instrumental in raising funds to finance the cataloguing of Hausdorffs papers, work carried out by Walter Purkert from 1 993 to 1995. This catalogue (Findbuch) is now available online at www. aic.uni-wuppertal.de/fb7/haus dorff/findbuch.asp. It can also be used for searches, for example to determine whether, and if so in which documents, a person or concept of interest happens to appear. In order to pursue the broader ob jectives of this editorial project, it was necessary to bring together scholars representing a broad spectrum of ex pertise. In fact, the novelty of this ven ture is reflected by the backgrounds of the contributing editors who have been or are still associated with this project: 16 mathematicians, four historians of mathematics, two literary scholars, one philosopher, and one astronomer. Their nationalities are also diverse, rep resenting Germany, Switzerland, Rus sia, the Czech Republic, and Austria. Since January 2002 the Hausdorff edi tion has been taken on as an official project of the Nordrhein-Westfalischen Akademie. Editorial responsibility for the edition as a whole lies with Egbert Brieskorn, supported by Friedrich Hirzebruch, Reinhold Remmert, Walter Purkert, and Erhard Scholz. Springer-
Verlag has assumed responsibility for publishing the Hausdorff edition, and five of its nine volumes have already appeared: volumes IV (2001), II (2002), VII (2004), V (2005), and III (2008). These handsome books, which ought
to be found in every major mathemat ics library, might even lead some to brush up their German and try to sa vor the mathematical and literary de lights they contain. As an added in ducement, Walter Purkert, who has
managed the meticulous daily work of this difficult undertaking from the be ginning, offers the following com pelling portrait of Hausdorff's career and his extraordinary intellectual ac complishments. D.E.R.
The Double Life of Felix Hausdorff/ Paul Mongre by Walter Purkert Translated by H i lde Rowe and David E. Rowe
n a 1921 review of Hausdorff's prin cipal work, Grundziige der Mengen lehre (1914), the American mathematician Henry Blumberg wrote: It would be difficult to name a vol ume in any field of mathematics, even in the unclouded domain of number theory, that surpasses the Grundzuge in clearness and preci sion. (Blumberg 1 92 1 , 1 1 6) Compare that statement with another remark about Hausdorff in a letter from Paul Lauterbach, writer and translator, written to the musician and Nietzsche scholar, Heinrich Koselitz (pseudonym: Peter Gast): A Dionysian mathematician! That sounds incredible; but let him send something to you and we will wa ger that there is something about him to be experienced. 1 "Dionysian" refers to Dionysius, the Greek god of wine, fertility, but also of ecstasy and the intoxicating, irrational, ecstatic elements necessary for experi encing the world or the creative process. An individual who writes books of mathematics of such unsur passed clarity and precision, on the one hand, and is considered to be Dionysian, on the other, surely must lead a remarkable double existence and Hausdorff was just such a man. As Felix Hausdorff he was an important mathematician whose work has re mained relevant and influential up to the present day; as Paul Mongre he was a man of letters, a philosopher, and a
critical essayist, a figure whom the jour nalist Paul Fechter recalled in 1 948 in his autobiography, Menschen und Zeiten, as ''one of the most remarkable individuals to appear in the first decades of the twentieth century" and who "has wrongfully been forgotten by the younger generation." (Fechter 1948, 1 56) Naturally, in this double life many visible and invisible threads became in-
tertangled, and these must be retraced to understand properly the man and his work. Felix Hausdorff was born in Breslau on November 8, 1868. His father, a jew ish businessman named Louis Haus dorff ( 1843-1896), moved in the Fall of 1 870 with his young family to Leipzig, where he managed various companies including linen and cotton
Figure I. Felix Hausdorff working in his home study in Bonn during the early 1920s.
'Paul Lauterbach. Letter to Heinrich Koselltz from 30 December, 1 893. Goethe- und Sch1llerarch1V We1mar, 1 02/4 1 7 .
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shops. He was an educated man who at age 13 had obtained the title Morenu 2 He wrote several papers, including a long one on the Aramaic translation of the Bible from the perspective of Talmudic Law, which appeared in the Monatsscbrift fiir
Gescbichte und Wissenscbaft des ju dentbums. For many years Louis Haus dorff was involved with the "Deutsch Israelitischen Gemeindebund, "3 and was made a member of the executive committee because its presiding officer thought it would be desirable to have the decidedly conservative position, for which Mr. Louis Haus dorff was known, to be represented on the committee.4 In an 1 896 obituary of Louis Hausdorff, the Gemeindebund recalled: His great and noble heart beat warmly for the affairs of his fellow believers. At the same time, he was a devoted, self-sacrificing father in the true Jewish sense; in the same manner, his bountiful acts of charity corresponded to the most beautiful tradition of our people. (Ibid. , Nr. 44, 1896.) Felix Hausdorff's mother, Hedwig (1848-1902; she was called Johanna in various documents) , was a member of the widely dispersed Jewish family Tietz. From one branch of this family came Hermann Tietz, the founder of the first department store and later the principal owner of a chain of depart ment stores "Hermann Tietz. " During the period of the National-Socialist dic tatorship, the firm was "aryanized" un der the name BERTIE. We do not know how Felix Haus dorff was reared as a child, but it seems likely he had a strict religious up bringing. In a report to the executive committee of the "Deutsch-Israelitis chen Gemeindebund" his father said, The center of Judaism is not found in the sermon, nor in the religious services. Its true focus is much more to be found in the religious life of the family.
How Felix Hausdorff reacted to his up bringing is also unclear, but evidence points to sharp differences with his fa ther's views. In one of his aphorisms, he later wrote: Whoever invented the fable of the happiness of childhood forgot three things: religion, upbringing, and the early phases of sexuality. (Hausdorff 1897a, 254) In another aphorism, he writes , in re gard to the rearing of children in his day, But the method is still the same to day: exterminate, hinder, cut off, deny, restrict, prohibit-it was a fun damentally negative, privatistic, pro hibitive method or rearing, improv ing, punishing-eradicating instead of creating, amputating instead of healing. (Hausdorff 1897a, 62) The results of Felix Hausdorff's reli gious training were the opposite of what his father wanted to achieve: Hausdorff gave up practicing the Jew ish faith. He became an agnostic who critically disputed the tenets of Jewish religion just as he did the Christian. Still, he was never baptized, a religious rite that would have offered him consider able advantages. Hausdorff's educational background was in many ways typical for a child from a middle-class family with high as pirations. For three years he attended the former second Biirgerschule in Leipzig; afterward, beginning in 1 878, he went to the Nicolai Gymnasium. This school had an excellent reputation as a humanistic educational institution. Hausdorff was an outstanding pupil, the best in his class over many years, and he often was given the honor of reading the poems he had composed in Latin or German during school va cations. In his graduating class of 1887 he was the only pupil to receive the cumulative grade of "I." The focus of the gymnasium education was on clas sical languages, which comprised ap proximately 45% of the obligatory cur riculum. Hausdorff was required, for
example , in the final examination for graduation to write a Latin essay on the theme: "Cupidius quam verius Cicero dicit res urbanas bellicis rebus antepo nendas esse" (freely translated: "it cor responds more to Cicero's interests than the truth when he states that mat ters of public welfare have priority over those of warfare"). (jabresbericbt des Nicolai-Gymnasiumsfiir dasjahr 1887, X-XI) The choice of field for his uni versity studies may well have been a difficult one for the multitalented Felix Hausdorff. Magda Dierkesmann, a stu dent in Bonn from 1 926-1932 who was often a guest in Hausdorff's home, re ported many years later: His versatile musical talent was so great that it was only due to the urg ing of his father that he gave up his plans to study music and become a composer.5 By the time he graduated, the decision had been reached (though we do not know what prompted it): in the annual report of the Nicolai Gymnasium for 1 887, next to the list of graduates, one finds a column giving the "future field
Figure 2. Hausdorff as he appeared dur ing his tenure in Greifswald, 1913-192 1 .
2Morenu be1ng Hebrew for "our teacher"; this title was conferred on those who qualified to teach as rabbis. 3Th1s orgamzatlon was founded after the creat1on of the German Re1ch to represent the 1nterests of German Jews w1th1n the new state 4Mittheilungen des Deutsch-lsraelit1schen Geme1ndebundes, Nr. 5 (1 878). The conservatives ma1nta1ned a strict line with respect to conventional relig1ous practices. They fought, for example, to have Jewish pupils freed from attending the Gymans1um on the Sabbath so that they could attend the Synagogue, or at a m1mmum that they be freed from wnting tests on these days. 5(D1erkesmann 1 967, 51 -52) In a conversation with Egbert Brieskorn, Frau D1erkesrnann assured him that she was told th1s dwectly by Hausdorff
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Figure 3. The elderly Hausdorff re
mained active mathematically up until his suicide in 1942.
of study," which for Felix Hausdorff was "natural sciences. " (jahresbericht
des Nicolai-Gymnasiums fiir das jahr 1 887, XVI) Between 1 887 and 1 89 1 , Hausdorff studied mathematics and astronomy in Leipzig, though with interruptions of one semester each to study in Freiburg (SS 1888) and in Berlin (WS 1888/1889). He had exceptionally broad interests and took courses in mathematics, as tronomy, physics, chemistry, and ge ography. He also attended lecture courses in philosophy and history of philosophy, languages and literatures, and on the history of socialism and the labor movement. One of his electives was a course on the scientific founda tions of belief in a personal God, and another dealt with the relationship be tween mental disorders and crime. He also attended lectures by the Leipzig musicologist Paul on the history of mu sic. Hausdorff's early love for music re mained with him all his life. Already as a student in Leipzig he had a special affinity for and excellent knowledge of the music of Richard Wagner. Later, he often invited friends to his home for musical evenings and regaled them at the piano.
During his last semesters as a student in Leipzig, Hausdorff worked closely with Heinrich Bruns 0848-1919), pro fessor of astronomy and director of the astronomical observatory. Bruns, a stu dent of Karl Weierstrass, was known above all for his work on the three body problem and on optics (Bruns's Eikonal). He gave Hausdorff a disser tation topic on the refraction of light in the atmosphere (Hausdorff 1891). This work was followed by two further pub lications on the same subject, leading up to Hausdorff's Habilitation for which he submitted a study on the extinction of light in the atmosphere (Hausdorff 1895). These early astronomical works by Hausdorff were-their excellent mathematical presentation notwith standing-of no further consequence. As it turned out, Bruns's principal idea was unworkable (astronomical obser vations of refraction near the horizon were required, which, as Julius Bauschinger thereafter showed, were impossible to obtain with the necessary exactitude). Beyond this, new technol ogy opened the possibility of obtaining atmospheric data directly by means of test balloons, thereby obviating the need for difficult calculations of these data based on refraction observations. With the Habilitation, Hausdorff could begin his academic career as a Privatdozent in Leipzig. He offered a wide range of courses, but alongside his teaching and research he also con tinued to pursue literary and philo sophical interests. This brought him into contact with a circle of notewor thy writers, artists, and publishers that included Hermann Conradi, Richard Dehmel, Otto Erich Hartleben, Gustav Kirstein, Max Klinger, Max Reger, and Frank Wedekind. During the period from 1 897 to 1904-the high point of his own literary and philosophical cre ativity-he published eighteen of the twenty-two works that appeared under his pseudonym, including a volume of poems, a play, a book on epistemol ogy, and a volume of aphorisms. The book of aphorisms was the first among Hausdorff's works to appear under the pen name Paul Mongre. He entitled it Sant' !!aria. Gedanken aus
der Landschaft Zarathustras (Hausdorff 1897a). The choice of pseudonym al ready suggests his orientation: a mon gre-after my own taste. This reflected an individuality, spiritual autonomy, and a rejection of prejudices and con formity in political, social, religious, or other spheres of human affairs. The subtitle, "Gedanken aus der Landschaft Zarathustras" stems from the circum stance in which Hausdorff completed his book while recuperating on the Lig urian coast near Genoa, the same lo cale where Friedrich Nietzsche wrote the first two parts of Also sprach Zarathustra; the subtitle also naturally suggests a strong spiritual affinity to Ni etzsche. In a preview of Sant' Ilario that appeared in the weekly magazine Die Zukunft, Hausdorff explicitly acknowl edged his debt: On this blissful coast [ ] I followed the lonely paths of Zarathustra ' s cre ator-wonderful, narrow paths along banks and cliffs that have no room for moving an army. If one should thus wish to count me among Nietzsche's followers, then let this serve as my own confession. 6 Hausdorff did not attempt to copy Nietzsche let alone surpass him; as one reviewer put it, there is "not a trace of mimicking Nietzsche. " Hausdorff posi tions himself, so to speak, next to Nietzsche in an effort to release his in dividual thoughts and to gain the free dom needed to question conventional norms. He also maintained a critical dis tance to Nietzsche's later works. In his essay on Nietzsche's Der Wille zur Macht, a book compiled from various fragments in the Nietzsche-Archiv, he wrote: Nietzsche glows like a fanatic. If his moral order based on breeding were to be established drawing on our modern knowledge of biology and physiology, that could lead to a world historical scandal compared to which the Inquisition and witch trials would appear like merely harmless confusions. (Hausdorff 1902, 1336) Yet Hausdorff took his critical standard from the younger Nietzsche, from the gracious, moderate, under·
·
·
6(Hausdorff 1 897b, 361 ) . For details on Hausdorff's relationship to Nietzsche, see Stegma1er (2002) as well as the h1stoncal introduction to Hausdorff (2004)
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longing or posessing, in science or culture; fruitful is everything that oc curs less than twice, every tree growing in its soil and reaching up to its sky, every smile that belongs to only one face, every thought that is only once right, every experience that breathes forth the heart strengthening smell of the individ ual! (Hausdorff 1897a, 37) The year 1898 saw the appearance of Hausdorff's critical epistemological study-again under the pseudonym Paul Mongre Das Chaos in kosmischer Auslese (Chaos in cosmic selection). Its critique of metaphysics resulted from Hausdorff' s effort to come to terms with Nietzsche's idea of eternal recurrence. His aim is nothing less than to destroy permanently every type of metaphysics. Regarding the world in itself, a tran scendental world core as Hausdorff calls it, we know nothing and can know nothing. We must take "the world in it self'' to be undetermined and indeter minant, a mere chaos. Our world of ex perience, our cosmos, is a result of selection, which we have always invol untarily undertaken and continue to un dertake according to our possiblilities of knowledge. Starting from that chaos, there are any number of other orders, other cosmoi, that could be conceived, but from the world of our cosmos there is no possibility of drawing conclusions regarding a transcendental world. -
Figure 4. The International Conference Center, University of Greifswald, is named
for Felix Hausdorff. Photo by Stan Sherer. standing, free spirit Nietzsche and from the cool, dogma-free, systemless skeptic Nietzsche [ ] (Ibid, 1338) Any attempt to describe the contents of a volume of aphorisms would be sense less, but in order to say at least some thing about it, one can point to two ideas that Hausdorff takes up over and again: first, he expresses a deep skep ticism with regard to all forms of tele ology and, even more, ideologies or theories for improving the world that claim to know the true meaning and purpose of humanity. As exemplars of this, consider these two excerpts from the first and third Aphorisms: The world is so full of outrageous nonsense, cracks, fragmentation, chaos, "free will"; I envy those whose good, synthesizing eyes are able to see the world as the un folding of an "idea, " a single idea. (Hausdorff 1897a, 4) If not truth itself, then surely the be lief in holding truth is, to a danger ous degree, antagonistic to life and murderous for the future. Not one of those who deluded themselves that they were blessed with the truth hesitated for a moment to pronouce the grand finale, or the great day, or some other end point, turning point, or climax for humanity, and every time this meant that all future hu manity was to be molded by their image, their stamp, and their nar rowness. (Hausdorff 1 897a, 6) ·
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This raises the question of the rela tionship between individuals and soci ety. For Hausdorff, as for Nietzsche, the individual is no mere figure within an historical process that subordinates his individuality to a higher order. On the contrary, individuals, especially those who are creative, should be placed in the center and their rights should be defended. From Aphorism 35: Fruitful is anyone who calls some thing his own, whether making or enjoying, in speech or gesture, in
Figure 5. Hausdorff's home at Am Graben 5, Greifswald, with mathematical tourists. Photo by Stan Sherer.
We will have to show the full di versity of both worlds and the untenability of any reasoning from empirical consequences to transcen dental premises [· · ·] and to do so in a comprehensive generality that also goes beyond Kant's result in a prac tical way [· · ·] (Hausdorff 1 898, 4) As methodology, he proposes, [· · ·] we have [· · ·] simply to deter mine those transcendental variations that leave a given empirical phe nomenon unchanged. (Hausdorff 1 898, 9) In Chaos in kosmischer Auslese he at tempted to carry out this program for the categories of time and space. Consider the following passage from his Leipzig inaugural lecture Das Raumproblem. By studying a map one can never determine the form of the original space without knowing the method of projection used to obtain it. Thus [ · ·] our empirical space is just such a physical map, an image of the ab solute space [absolute in the sense of transcendental]; but [· · ·] we do not know the method of projection and so we cannot know the origi nal. The two spaces are related by means of an unknown and unde termined correspondence, a com pletely arbitrary point transforma tion. Still, the empirical space maintains its value as a means of ori entation; we are able to find our way with this map and we can commu nicate with those who also possess this map; the distortion never enters our consciousness because not only the objects but we ourselves and our measuring instruments are uniformly affected by this. [ · ·] If this viewpoint is correct, then it must be possible for the preimage to undergo an arbitrary transforma tion without changing the image: [· . · F ·
·
The simplest such transformation would be a uniform shrinking or ex panding of the transcendental space by a constant factor. But Hausdorff was
concerned with arbitrary transforma tions, which means that the transcen dental space must remain completely undetermined and indeterminant such a space is thus a senseless con cept, scientifically speaking. Hausdorff worked intensively on the space prob lem for many years; in the winter se mester 1 903 to 1904 he offered a lec ture course in Leipzig on Zeit und Raum (Time and Space) (NL Hausdorff: Kapsel 24: Fasz. 71), in which he spoke of his passion for this problem. The fundamental concept of a topological space, which he later created, was con ceived in order to accomodate practi cally every situation in which "spatial ity," in the topological sense, plays a role. This concept was probably influ enced by his philosophical reflections on the space problem. It is especially striking that in Chaos in kosmischer Auslese, a philosophical study, Hausdorff brought in elements from the very newest mathematics, set theory. This unique but also problem atic aspect surely made the work's re ception more difficult. In 1 904 the periodical Die neue Rundschau published Hausdorff's one act play Der Arzt seiner Ehre (The Sur geon of his Honour). This earthy satire dealt with duelling and the conven tional code of honor of aristocrats and the Prussian officers' corps. By Haus dorff's day, such forms of chivalry were becoming outmoded in bourgeois so ciety. The Hamburger Echo on 15 No vember 1 904 noted in a review, Mongre has the courage to show du elling in the light that it deserves. He treats it as comedy, about which one can agree over a glass of wine so long as one is not chained like a vain fool to the fashion demon of "honor" . Der Arzt seiner Ehre was Hausdorff's greatest literary success. Between 1 904 and 1 9 1 2 it was performed over 300 times on stages in Berlin, Brunswick, Bremen, Breslau, Bromberg, Budapest, Dusseldorf, Dortmund, Elberfeld, El-
bing, Frankfurt, Furth, Graz, Hamburg, Hannover, Kassel, Cologne, Koenigs berg, Krefeld, Leipzig, Magdeburg, Muhlhausen, Munich, Nuremberg, Prague, Riga, Strassburg, Stuttgart, Wien, Wiesbaden and Zurich.8 Haus dorff was anything but an obscure play wright. In June 1 9 1 2 he attended a ban quet at the Hotel Esplanade in Berlin held in honor of Frank Wedekind, ar riving in the company of Max Rein hardt, Felix Hollander, and Arthur Ka hane, the creme de Ia creme of the Berlin theatrical scene.9 We must content ourselves with these few glimpses of Hausdorff's lit erary and philosophical works without touching on his poetry volume Ek stasen (1900) or his essays, true pearls in this literary genre, about which see Vollhardt 2000. Most of the essays ap peared in the periodical Neue Deutsche Rundschau (Freie Buhne) (later re named Die neue Rundschau (Freie Buhne)), the then leading literary jour nal about which was said: "Remember, that your life will pass, even if you made it into the Neue Rundschau (Gedenke, Mensch, dass Du vergehst, auch wenn Du in den Neuen Rund schau stehst). " After the Second World War, Haus dorff's philosophical writings were to tally forgotten, as were his literary works. One might conjecture that anti Semitism and the cultural barbarism of the Nazi distatorship contributed to this neglect. Up until the Nazi era there was still a public awareness of Hausdorff as a philosopher and writer, as this entry in the 1931 edition of the GraBen Brockhaus shows (dates and localities of his life have been omitted): Hausdorff, Felix, Mathematician and Writer. He is the author of: Grundzilge der Mengenlehre ( 1 914), and under the pseudonym Paul Mongre the epistemological study "Das Chaos in kosmischer Auslese" (1898), the works "Sant' Ilario. Gedanken aus der Landschaft Zarathustras" ( 1897) and "Ekstasen"
7(Hausdorff 1 903, 1 5) How such transformations m1ght affect physical properties rema1ns open here In a posthumous (unfortunately undated) fragment "Transforma . tionsprincip" Hausdorff wrote about th1s: "That the physical content also m1ght take part 1n the transformation needs to be considered more carefully. That is perhaps not so simple Perhaps 1n th1s respect the principle is even ObJeCtionable-an 1dea I find attract1ve now that I've noticed that others (Po1ncare) have taken up th1s pnn Ciplel! l " (NL Hausdorff. Kapsel 49: Fasz. 1 079, Bl 3 ) 8For the rev1ew c1ted above a.nd the informat1on about these performances I thank Udo Roth, Munich. 9Frank Wedekind. Gesammelte Bnefe Hrsg von Fritz Strich. Bd. 2, Munchen 1 924, p 269, for th1s reference I thank Ariane Martin, Mainz
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(1900). He has close affinities to the fundamental ideas of Nietzsche; he rejects all metaphysics and regards the world of experience as a seg ment drawn by consciousness out of a lawless chaos. Regarding his philosophical contribu tions, W. Stegmaier wrote in the preface to volume VII of Hausdorffs Gesam
melte Werke, The more I delve into Felix Haus dorff's writings, the more they com mand my respect: for their clarity, their honesty, their noble modesty, their intellectual independence, and above all for their astonishing pre sent-day relevance. Perhaps now the time has come, after one hundred years, that they can lead to fruitful philosophical orientation as they so deserve. (Hausdorff 2004, VII.) In 1899 Hausdorff married Charlotte Goldschmidt, the daughter of a Jewish physician, Siegismund Goldschmidt, from Bad Reichenhall. This man's step mother, incidentally, was the famous feminist and preschool pedagogue, Henriette Goldschmidt. In 1900 the Hausdorffs' daughter Lenore (Nora), their only child, was born; she survived through the Nazi era and died in 1991 in Bonn. In December of 1901 Hausdorff was appointed as an unofficial associate professor (aufSerplanmafSiger Extraordi narius) at Leipzig University. In sub mitting the faculty's proposal for this appointment, which contained a very favorable assessment given by his col leagues and composed by Heinrich Bruns, the Dean added the following remark:
Figure 6. The plaque outside Haus dorffs home in Greifswald. Photo by Stan Sherer.
The faculty considers itself, how ever, duty bound to inform the Royal Ministry that the present proposal was not approved by all members in the meeting on the 2nd of No vember this year, but rather by a vote of 22 to 7. The minority who voted against Dr. Hausdorff did so because he is of the Jewish faith. 1 0 This amendatory remark illuminates at a glance the open anti-Semitism that was especially on the rise across the entire German Empire after the finan cial crash (Griinderkrach) that followed its founding in 1 87 1 . Leipzig was at the center of the anti-Semitic movement, in which students played a large role. This may well have been one reason why Hausdorff never felt particularly com fortable teaching there; another reason was the strong sense of hierarchy among the full professors (Ordinarien), who tended to disregard their junior colleagues. Later in Bonn Hausdorff commented retrospectively in a letter to Friedrich Engel: In Bonn one has the feeling, even as a junior faculty member (Nicht Ordinarius), of being formally ac cepted, a sense I could never bring myself to feel in Leipzig [an der Pleisse] (Letter from 21 February 1 91 1 . NL Engel, UB GiefSen, Hand schriftenabteilung) Hausdorffs mathematical research covered unusually broad terrain: he wrote papers on such diverse topics as optics (Hausdorff 1896), non-Euclidean geometry (Hausdorff 1899), hypercom plex number systems (Hausdorff 1 900b), insurance mathematics (Hausdorff 1897c), and probability theory (Haus dorff 1 90 1 b). The last two works con tain several noteworthy results that were not without influence. In Haus dorff ( 1 897c) he introduced the vari ance of an insurer's losses as a mea sure of risk. Today theories of individual risk have given way to col lective risk theories, but variance of loss nevertheless remains a fundamental quantity for the evaluation of insurance plans with fixed coverages and premi ums. (In this paper Hausdorff also pre sented a first correct proof of the The orem of Hattendorff.) For various types
of life insurance he calculated the vari ance of loss, and these results were taken up immediately afterward in the textbook literature. In Hausdorff C l 901b) he called special attention to the concept of conditional probability, a notion of fundamental importance that had only been used implicitly up until then. He also introduced the new terminology ("relative probability") along with a suitable notation for it. 1 1 In this same paper (and independent of Thiele) he dealt with semi-invariants and gave highly simplified derivations of the Gram-Charlier series of Type A. His example of a sequence of inde pendent, identically distributed random variables X1 , X2, . . . with density cp(x) = ! e- ix l , for which
n Zn = L a,.Xk with ak = k� !
THE MATHEMATICAL INTEWGENCER
(k+ !) 7T
does not converge to a normal distri bution, provided the motivation for P . Levy t o formulate a n interesting con jecture about the decomposition of the normal distribution into two indepen dent components. This conjecture from the early 1 930s was proved in 1936 by H. Cramer (for details, see Hausdorff 2005, 579-583). Hausdorffs principal field of re search, however, soon became set the ory, especially the theory of ordered sets. Initially it was his philosophical interests that led him to Cantor's ideas (see Hausdorff 2002, 3-5). In the sum mer semester of 1 901, Hausdorff of fered a lecture course on set theory; this was nearly a first in Germany, only Ernst Zermelo's course in G6ttingen the previous semester preceded it. (Cantor himself never offered lectures on set theory in Halle.) It was in the context of teaching this course that Hausdorff made his first discovery in set theory: the type class T(�0) of all countable order types has the power � of the con tinuum. He soon found, though, that this theorem had appeared in Felix Bernstein's Dissertation-and carefully noted in the margin of his manuscript: Presented on 27 June 190 1 . Disser tation of F. Bernstein received on 29 June 1 90 1 . (NL Hausdorff: Kapsel 03: Fasz. 12, Bl. 37)
'0Archiv der Universitiit Leipzig, PA 547. The full report is reproduced in Beckert and Purkert (1 987), 231-234. 1 1 Kolmogoroff later adopted this notation (Pa(A)) in his book Grundbegriffe der Wahrscheinlichkeitsrechnung (1 933).
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1
Hausdorff undertook a thorough study of ordered sets, motivated in large part by Cantor's continuum problem, which poses the question of finding the place occupied by � = 2Ko in the sequence of the � "'_ 1 2 In a letter to Hilbert dated 29 September 1 904, he said this prob lem "had plagued me almost like an obsession" (NL Hilbert, Niedersachsis che Staats- und Universitatsbibliothek zu Gottingen, Handschriftenabteilung, Nr. 1 36). He thought that the theorem card ( T (�0)) � offered a new strat egy for attacking the problem. Cantor had long conjectured that � = � 1 , but it had only been proved that � 2: � 1 , where � 1 represents the " number" of possible well-orderings of a countable set. It turned out that � is the "num ber" of all possible orderings of such a set, which naturally led to the study of special types of orderings that were more general than well-orderings but less general than arbitrary orderings. This was precisely what Hausdorff did in his first set-theoretic publication (Hausdorff 1901a) in which he studied "graded sets" (gestufte Mengen). Today we know, from the results of K. Godel and P. Cohen, that this strategy for solv ing the continuum problem had no more chance of attaining its goal than Cantor's approach, which tried to gen eralize the Cantor-Bendixson theorem for closed sets to the case of arbitrary uncountable point sets. In 1 904 Hausdorff published the re cursion formula that now carries his name: for every nonlimit ordinal J.L =
�� "' = � JL �� � 1 · This formula, together with the concept of cofinality that he later introduced, served as the foundation for all further results on the exponentiation of alephs. Hausdorff's precise knowledge of the problematics of recursion formulae of this type enabled him to detect an er ror in Julius Konig's lecture presenta tion at the 1904 International Congress of Mathematicians in Heidelberg. Konig claimed to have "proved" that the con tinuum cannot be well-ordered, which
would have implied that its cardinality is not an aleph, a result that evoked considerable interest. It was only re cently discovered that it was Hausdorff who uncovered this error. 1 :3 Between 1906 and 1909 Hausdorff published his fundamental works on ordered sets (Hausdorff 1906, 1907a, 1907b, 1 908, 1909). For this theory, his concepts of cofinality and coinitiality were of key importance. If A is an or dered set and M C A, then A is said to be cofinal (coinitia!) with M if for every a E A there exists an m E M such that m 2: a ( m ::5 a). Thus, for example,
{ � }mE N { � }nE N·
(0, 1 ) is cofinal with coinitial with
m 1
and
These concepts
carry over to order types: for example, the type A associated with the set of real numbers under its natural ordering is cofinal with the type w of the nat ural numbers. An ordinal number is called regular if it is not cofinal with a smaller ordi nal number, otherwise it is called sin gular. Hausdorff called the smallest number in each of Cantor's number classes an initial number (An fangszahl) 1 4: w = w0, w1 , w , . . . , ww, 2 Ww+ l , . . . All Wa+ l are regular, but Ww = lim,. w, is cofinal with w and thus an example of a singular initial num ber. Hausdorff asked whether there ex ist regular initial numbers with a limit number as index, a query that served as the point of departure for the the ory of inaccessible cardinal numbers. He realized that, were such numbers to exist, they would have to be of " exor bitant size . " 1 5 In connection with this theory Haus dorff proved the following fundamen tal theorem: for every dense ordered set A without boundary there exist two uniquely determined regular initial numbers w�, wTf such that A is cofinal with w� and coinitial with w� (where signifies the inverse ordering). This the orem offers a sensitive instrument for characterizing gaps and elements in or*
dered sets, as Hausdorff showed. If, for example, the ordered decomposition A = P + Q represents a gap-that is, P has no greatest and Q no smallest ele ment-then there exist two uniquely determined regular initial numbers w�, wTf such that P is cofinal with w� and Q is coinitial with w�. Hausdorff called the pair (w�,w�) = : c�Tf the character of the gap. By this means, he obtained from the decomposition A = P + {a} + Q a uniquely determined character for the element a, though here one must al low for characters of the type (l ,w�), (w�, l), or (1, 1). Thus, in the set of ra tional numbers (with the natural or dering), all gaps and elements have the character Coo · If W is a given set of characters (for elements and gaps), for example W= leao, Col , cw, c22l, the question naturally arises whether there exists an ordered set having precisely W as its set of characters. A necessary condition for W is relatively easy to find, but Hausdorff succeeded in showing that this condition is also sufficient. For this purpose one needs a large reservoir of ordered sets, and this Hausdorff was able to create using his theory of gen eral ordered products and powers (see Hausdorff 2002, 604-605 . ) In this reser voir one finds such interesting structures as Hausdorff's 7Ja normal types. Cantor had already considered the type 7J = 7Jo of the rational numbers in their natural ordering. He discovered that this type is universal with respect to the type class T(�0) of all countable order types, that is, for every countable order type J.L there exists a subset in 7J that has the type J.L. Hausdorff's 7Ja type accom plishes the same thing for the type class T(�a). The question whether there ex ist 7Ja+l sets with the least possible car dinality �a+l then leads to the question whether 2K = �a+l holds. It was in this context that Hausdorff raised the gen eralized continuum hypothesis for the first time. His 7Ja sets also served as the point of departure for the notion of sat urated structures, which has since played a major role in model theory. 1 6 ,
1 2Hausdorff's reftecttons on ttme as background for hts study of order structures are taken up by E. Scholz tn his article Log1sche Ordnungen 1m Chaos Hausdorffs fruhe Be1trage zur Mengen/ehre (in Brieskorn 1 996, 1 07-1 34). 1 3Detatled tnformatton can be found tn Hausdorff (2002, 9-1 2) and tn Purkert (2004). Further tmportant source material on this story can be found in Ebbinghaus (2007).
14Today these are identified with the cardtnal numbers: K0, K 1 , . Kwo Kw+ 1 • . . . 1 5See Hausdorff (2002) and th'e commentary by U. Feigner, pp. 598-601 . 160n this, see the essay by U. Feigner. 01e Hausdorffsche Theone der Tfa-Mengen und 1hre Wirkungsgeschlchte, tn Hausdorff (2002, 645-674).
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Hausdorff's general products and powers also led him to the concept of partially ordered sets. It turned out that the final gradations of sequences and functions he had been intensively studying were partial orderings. His proof of the existence of (w1 ,wi) gaps in the maximal ordered subsets of these semiordered sets is among his deepest results in set theory. Hausdorff showed that every ordered subset of a partially ordered set is contained in a maximal ordered subset; this theorem is known today as Hausdorff's maximal chain theorem ("Maximalkettensatz''). Not only does the theorem follow from the well-ordering theorem (resp. the axiom of choice), it was later shown to be equivalent to both of themP In 1908, Arthur Schoenflies had pointed out that the more recent theory of ordered sets (that is, the extensions of this theory undertaken after Cantor) were almost exclusively due to Haus dorff (Schoenflies 1908, 40.) Indeed, de velopments within set theory immedi ately after Cantor have, in comparison with foundational issues, received com paratively little attention in the historical literature with the exception of the work of Zermelo. This applies in particular to the contributions of Hausdorff and Ger hard Hessenberg. In the summer semester of 1910, Hausdorff was appointed to a position as official associate professor (plan mafSiger Extraordinarius) at Bonn Uni versity. As mentioned previously, he found the academic atmosphere in Bonn far more to his liking than that in Leipzig. There he had not taught any courses in set theory since 190 1 , even though this was his primary field of research. After arriving in Bonn, he immediately gave a course on set theory, which he repeated in the summer semester of 1912 in a re vised and expanded form. It was during that summer that Hausdorff began work on his magnum opus, Grnndzuge der Mengenlehre. He completed it in Greifs wald, where he began teaching as a full professor (Ordinarius) in the summer se mester of 1913; his book appeared in print in April 1914.
Set theory, as this area of mathemat ics was understood at the time, included not just the general theory of sets but also point sets and the theories of con tent and measure. Hausdorff's work was the first textbook that dealt systematically with all aspects of set theory in this com prehensive sense and which provided complete proofs in a masterful form. Moreover, it went well beyond the pre sentation of known results: it contained a number of significant original contri butions by its author, which can only briefly be described here. The first six chapters of the Grnndzuge deal with general set theory. Hausdorff begins by setting out an al gebra for sets that includes some new concepts that would prove influential (Differenzenketten, rings and fields of sets, and 0-0"-systems). These introduc tory paragraphs on sets and their oper ations also contain the modem set-the oretic concept of a function; here we encounter, so to speak, many of the in gredients that form the modem language of mathematics. There follows in chap ters three to five the classical theory of cardinal numbers, order types, and or dinal numbers. In the sixth chapter on "Relations between ordered and well ordered sets (Beziehungen zwischen geordneten und wohlgeordneten Men gen)'' Hausdorff presents, among other things, the most important results from his own researches on ordered sets. The chapters on "point sets"--one might better say topology--exude the spirit of a new era. Here Hausdorff pre sents for the first time, beginning with his axioms for neighborhoods, a sys tematic theory of topological spaces, to which he added the separation axiom known today by his name. This theory arose through a comprehensive synthe sis involving the work of other mathe maticians as well as his own reflections on the space problem. The concepts and theorems from classical point set theory in �" are extended-so far as this is pos sible-to the general case, where they are subsumed into the newly created general or set-theoretic topology. Yet in the course of carrying out this "transla-
tion work," Hausdorff created a number of fundamentally new constructions for topology, such as the interior and clo sure operations, while developing the fundamental concepts of open set (which he called a "Gebiet") and com pactness, a concept he took from Frechet. He also established and devel oped the theory of connectedness, in troducing in particular the notions of "components" and "quasi-components." He further specialized general topologi cal spaces by means of the first and sec ond Hausdorff countability axioms. The metric spaces comprise a large class of spaces that satisfy the first countability axiom. These were introduced in 1906 by Frechet, who called them "classes (E)"; the terminology "metrischer Raum" is derived from Hausdorff. In his Grundzuge, he gave a systematic pre sentation of the theory of metric spaces, to which he added several new concepts (Hausdorff metric, completion, total boundedness, p-connectedness, re ducible sets). Frechet's work (Frechet 1906) had received little attention; it was through Hausdorff's Grundzuge that metric spaces became widely familiar to mathematicians. 18 Both the chapter on mappings as well as the final chapter of the Grundzuge on measure theory and integration are impressive for the generality of their ap proach and the originality of the pre sentation. Hausdorff's laconic remarks pointing to the significance of measure theory for probability would prove to be highly insightful. The final chapter also contains the first correct proof of Borel's strong law of large numbers. 1 9 Finally, the appendix contains the single most spectacular result in the whole book, namely, Hausdorff's theorem that one cannot define a finitely additive measure, invariant under congruences, on all bounded subsets in �" for n 2: 3. Haus dorff's proof is by means of a famous paradoxical decomposition of the sphere, for which it is necessary to in voke the axiom of choice.20 In the course of the twentieth cen tury it became standard practice to place mathematical theories on a set-
1 7Regard1ng this theorem and s1m1lar results of C. Kuratowsk1 and M. Zorn, see the commentary by U Feigner in Hausdorff (2002, 602-604). 1 8Deta1led commentaries on Hausdorff' s contributions to general topology and the theory of metnc spaces can be found 1n Hausdorff (2002, 675-787). 1 98. D. Chatteq1 g1ves commentary on measure theory and integration 1n the Grundzuge in Hausdorff (2002, 788-800), see also Chatterji (2002). 200n the histoncal 1mpact of Hausdorff's sphere paradox, see Hausdorff (2001 , 1 1 -1 8); see also the article by P. Schreiber 1n Brieskorn (1 996, 1 35-148), and the monograph by S. Wagon (1 993)
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THE MATHEMATICAL INTELLIGENCER
theoretic and axiomatic basis. The cre ation of axiomatically grounded theo ries, as for example in general topology, served among other things to expose the structural elements common to var ious concrete situations or special ar eas and to place these in an abstract theory that subsumed them as special cases. By so doing, there is a consid erable gain in simplicity, unity, and ul timately in economy of thought. Haus dorff placed special emphasis on this viewpoint in his Grundziige (Hausdorff 1 9 1 4 , 2 1 1). In this respect, the topo logical chapters in the Grundziige rep resent a pioneering achievement that paved the way for the development of modern mathematics. This modern conception of the essence of mathe matics, made manifest through this new methodological orientation, was con ceived by Hausdorff many years before he composed the Gnmdzilge, indeed well before the appearance of the rel evant works of Frechet ( 1906) and F. Riesz ( 1907, 1908). An important im pulse in this direction surely came from the Grundlagen der Geometrie, which David Hilbert published in 1899. In Hausdorffs lecture course on Time and Space, held during the winter semester of 1 903 to 1904, he remarked about mathematics: Mathematics stands completely apart not only from the actual meaning that one attributes to its concepts but also from the actual validity one as cribes to its propositions. Its unde finable concepts are arbitrarily cho sen objects of thought, its axioms are also arbitrary, though chosen so as to be free from contradiction. Math ematics is a science of pure thought, just as is formal logic (NL Hausdorff: Kapsel 24: Fasz. 7 1 , Bl. 4). And about space in particular, Thus: space is a logical construction, namely it includes all propositions that follow logically from the arbitrar ily chosen axioms, whereas the con cepts employed are arbitrarily chosen objects of thought (Ibid., Bl. 3 1 ) .
I n light o f these quotations, one might wonder why Hausdorff did not under take to secure the ultimate foundations, "das Fundament des Fundamentes" ( Grundzilge, 1) by developing set the ory on an axiomatic basis. He was, of course, familiar with Zermelo's axiom atization. But he regarded this theory as only provisional: By stipulating suitable conditions E. Zermelo undertook the [ ·] neces sary attempt to curtail the processes leading to a boundless construction of sets. However, these highly astute investigations cannot yet be re garded as complete, and since an in troduction to set theory along this path would surely lead to great dif ficulties for beginners, we prefer here to admit the naive concept of set, taking due account of the re strictions necessary in order to cut off the path leading to the para doxes. (Hausdorff 1914, 2) It surely did not escape Hausdorff's no tice that Zermelo's concept of ''definite property" (definite Eigenschaft) lacked precision (see Feigner 1979, 3-8 and 49-91). In the remainder of the Grundzilge, he avoided foundational questions. 2 1 The Gnmdziige der Mengenlehre appeared at the dawning of the First World War. When it broke out, in Au gust 1 9 1 4, scientific life in Europe was affected in the most dramatic ways. Un der these circumstances, Hausdorffs book had little impact for five to six years. After the war, a new generation of researchers began to take up the many suggestive impulses it contained, especially for topology, now a central field of interest. The reception of Haus dorffs ideas was enhanced by the founding in 1 920, of a new journal in Poland, Fundamenta Mathematicae. This was the first mathematical journal specializing in the fields of set theory, topology, the theory of real functions, measure theory and integration, func tional analysis, logic, and the founda tions of mathematics. Within this spec·
·
trum of interests, general topology oc cupied a central place. Hausdorff's Grundziige was cited with great fre quency beginning with the very first is sue of Fundamenta Mathematicae. In the 558 articles (excluding the three written by Hausdorff himselD that ap peared in the first twenty volumes be tween 1920 and 1933, no fewer than 88 referred to the Grundziige. Here one must also take account that Hausdorffs concepts had become so commonplace that one finds these in several articles in which he was not explicitly cited. Hausdorffs Grundziige had a simi lar influence on the Russian topologi cal school founded by Paul Alexandroff and Paul Urysohn. This is evident from his correspondence with Alexandroff and Urysohn (after Urysohn's early death with Alexandroff alone) as well as from Urysohn's Memoire sur les mul tiplicites Cantoriennes (Urysohn 1925/ 1926), a work the size of a book in which Urysohn set forth his theory of dimension, citing the Grundziige no less than sixty times. The demand for Hausdorff's book continued until well after the Second World War, as attested by the three Chelsea reprints that ap peared in 1949, 1 965, and 1978. In 1916, Hausdorff and Alexandroff solved (independently) the continuum problem for Borel sets22: Every Borel set in a complete separable metric space is either at most countable or has the power of the continuum. This re sult generalizes the theorem of Cantor Bendixson, which makes the same as sertion for closed subsets in !R n. Earlier in 1903, W. H. Young extended this theorem to linear G0sets (Young 1 903), and in 1914, Hausdorff proved it for Go
2 1 0n these matters, see P Koepke· MetamathematJsche Aspekte der Hausdorffschen Mengentehre, 1n Bneskom 1 996, 71-106). There one f1nds an Interesting par allel between set-theoretic relatJvJsm and epistemological relat1v1sm 1n Chaos m kosm1scher Aus/ese. 22(Hausdorff 1 9 1 6), (Aiexandroff 1 91 6) The not1on of a "Borel set" 1n the modern sense was introduced by Hausdorff 1n the Grundzuge. Schoenllies had used the term "Borel sets" merely for the case of G8-sets 23(Aiexandroff and Hopf 1 935,' 20). For further 1nformat1on see the commentary of V Kanove1 and P. Koepke in Hausdorff (2002, 779-782) and 1n Hausdorff (2008,
439--442).
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(Dimension und aufleres MajS) (Haus dorff 1 9 1 9a). This publication is still highly relevant; it has probably been cited more often in recent years than any other research paper from the decade 1910 to 1 920. Here a few tech nicalities are required. Let au be a sys tem of bounded sets in !Rq such that each set A C !Rq can be covered by the union of at most countably many sets UE au with diameters d( [/) < E (E > 0 arbitrary). Let A(x) be a continuous, strictly monotone increasing nonnega tive function on [ 0 oo) . Then for A C !Rq Hausdorff introduces ,
L�(A) = inf
�'
and
A(d(U.)), A C
�
u.
d(U,) <
L'\A) = lim L� (A). dO
'}
This LA(A) is today called the Hausdorff measure for the function A(x). Haus dorff assigned to a set A the dimension [A] if
A fundamental and difficult question now arises: for a given function A do there always exist sets A C !Rq having dimension [A]? Hausdorff was able to show that this is indeed so for every strictly monotone increasing, every where concave continuous function A(x) : [ O,oo) � [O,oo) with A(O) = 0 and limx_.oo A(x) = oo. In the case where A(x) = xP, p positive real, one obtains the usual concepts associated with Hausdorff measure and Hausdorff di mension. The Hausdorff dimension of a set A is then the number a for which a
: = sup{p > 0: J!Pl(A) = oo) = inf{p > 0: J!P)(A) = 0),
where L(P) = LA and with A(x) = :xf'. Hausdorff's concept of dimension is a finely-tuned instrument for charac terizing and comparing sets that are "highly jagged." The concepts in Di mension und iiufleres Mafl have been
applied and further developed in nu merous areas, for example, in the the ory of dynamical systems, geometric measure theory, the theory of self-sim ilar sets and fractals, the theory of sto chastic processes, harmonic analysis, potential theory, and number theory.24 Unfortunately the boom of interest in "fractal theory" has often led to misun derstandings and m1smterpretations about Hausdorff's conceptions.25 The University of Greifswald was a small Prussian provincial university of merely local importance. Its mathemat ics institute was small, and in the sum mer semester of 1 9 1 6 and the follow ing winter semester, Hausdorff was the only mathematician teaching there! Thus his teaching activities were almost completely dominated by elementary courses. His situation improved markedly from a scientific standpoint when he went to Bonn in 1 9 2 1 . Here he had the opportunity to expand his teaching to a wide number of themes and to lecture over and again on his current research interests. Particularly noteworthy, for example, is the lecture course he offered in the summer se mester of 1923 on probability theory,26 in which he placed this theory on ax iomatic and measure-theoretic founda tions, ten years before the publication of A. N. Kolmogoroff's Grundbegri.ffe der Wah rscheinlichkeitsrechnung. In Bonn, Hausdorff found in Eduard Study and later Otto Toeplitz colleagues who were not only outstanding mathema ticians but who also became good friends. During this second period in Bonn, Hausdorff produced important work in analysis. In Hausdorff 0921), he de veloped an entire class of summation methods for divergent series which to day are known as Hausdorff methods. 27 The classical methods of Holder and Cesaro are special cases of these Haus dorff methods. Each such Hausdorff method is given by a sequence of mo ments; in this context Hausdorff pre sented an elegant solution of the prob-
lem of moments for a finite interval that bypasses the theory of continued frac tions. In Hausdorff ( 1 923b) he dealt with a special moment problem for a finite interval (subject to certain re strictions on the generating density cp (x), for example that cp (x) E LP[0, 1]). Hausdorff spent many years working on criteria for the solvability and de termination of moment problems, as evidenced by hundreds of pages left in his posthumous papers. 28 Hausdorff made a major contribu tion to the emergence of functional analysis in the 1920s with his extension of the Fischer-Riesz theorem to LP spaces in Hausdorff (1923a). There he also proved the inequalities named af ter him and W. H. Young29: If an are the Fourier coefficients of /E Lq(0, 2 7T), q � 2,
1..
+
.!.
= 1 , then
(� i �)h ( 00
p
an
q
�
l l
1 27T
)
21T � L lfl q dx . 0
If le
The Hausdorff-Young inequalities served as the point of departure for wide-ranging new developments (see the commentary by S. D. Chatterji in Hausdorff (200 1 , 182-190). In 1927 Hausdorff published his book Mengenlehre as the second edi tion of the Grundztlge. In reality this was a totally new book. In order to ap pear in the Goschen series, it was nec essary to provide a far more restricted presentation than in the Grundztlge. Thus large parts of the theory of or dered sets and the sections on measure theory and integration had to be dropped. "Even more regrettable than these omissions"-according to Haus dorff in his prefacewas the need to save further room
240n the historical impact of D1mens1on und aul3eres Mal3, see the articles by BandVHaase and Bothe/Schmeltng 1n Brieskorn (1 996, 1 49-183 and 229--252) as well as the commentary by S D Chatterji in Hausdorff (2001 , 44-54, and the literature ctted theretn) 25About this, see K. Steffen· Hausdorff-D1mens1on, regulare Mengen und totai lrregulare Mengen, 1n Brieskorn (1 996, 1 85-227). 26NL Hausdorff: Kapsel 2 1 : Fasz 64, repnnted tn 1ts enttrety wtth detatled commentary tn Hausdorff (2005), 595-756 271n Hardy's classtcal study (Hardy 1 949) he devotes an enttre chapter to Hausdorff methods. 280n the enttre complex of these published and unpublished works, see Hausdorff (2001 , 1 05-1 71 , 1 91 -235, 255-267, and 339--313) 29Voung had proved these for the special case p = 2n, n = 2, 3,
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in point set theory by sacrificing the topological standpoint, despite its at tractions for many readers of the first edition, and instead confining the discussion to the simpler theory of metric spaces, [ · · ·] (Hausdorff 1927, 5-6). Some reviewers of the work expressly regretted this circumstance. As a form of compensation, however, Hausdorff offered an up-to-date presentation of the state of research in descriptive set theory. This insured that his new book received almost as strong a reception as had the Grundziige, especially in Fundamenta Mathematicae. It became a highly popular textbook and ap peared again in 1 935 in an expanded second edition, which was reproduced by Dover in 1 944. An English transla tion was published in 1 957 with new printings in 1962, 1978, and 1 99 1 . A Russian edition came out in 1 937, al though this is not really a true transla tion; parts of the book were reworked by Alexandroff and Kolmogoroff in or der to bring the topological standpoint to the foreground.3o In 1 928 Hans Hahn wrote a review of the Mengenlehre (reprinted in Haus dorff 2008, 416-417). Possibly Hahn al ready sensed the dangers of German anti-Semitism. He ended his review with these words: This in every respect masterful pre sentation of a difficult and haz ardous subject is a work of the type written by those who have carried the fame of German science around the world, a work of which the au thor as well as all German mathe maticians may be proud. (Hahn 1 928, 58) Like most German academics, Haus dorff never engaged directly in politi cal activity. His views were far more liberal, however, than most of his col leagues. After the First World War he joined the newly founded German Democratic Party (DDP), which for a brief time represented a sizable leftist liberal constituency in the Weimar Re public. Several leading Jewish politi cians and intellectuals were drawn to the DDP, including Albert Einstein, but
its popularity quickly evaporated dur ing which time its more conservative wing took control. Although never an active member, Hausdorff dropped out of the DDP altogether in the mid-1 920s; the party languished on during the years that followed, becoming virtually irrelevant by the time that National So cialists began their dramatic surge. One year before they came to power, Hausdorff expressed his opin ion of the Nazi movement in a letter to Hilbert. The occasion was the latter's seventieth birthday, so the comment was perhaps phrased in a more ironic tone than it might have been otherwise. Hausdorff had been contemplating an appropriate honorific title for Hilbert, sit?ilar to the one given to Gauss in the twilight of his career: the Prince of Mathematicians. So he explained to Hilbert that since the title princeps mathematicorum has already been be stowed, I would have suggested call ing you the dux mathematicorum were it not that today the title duce Fuhrer is so discredited by those who offer to lead the German people on the basis of a license (Fuhrerschein) that they have bestowed on themselvesY With the assumption of power by the National Socialists, anti-Semitism became an official state doctrine. Haus dorff was not directly affected in 1 933 by the notorious "law to restore the civil service," because he had already been a German civil servant since before 1914. His teaching activity was, how ever, apparently affected by activities undertaken by Nazi student func tionaries. In his manuscript for his lec ture course Infinitesimalrechnung III held during the winter semester of 1 934 to 1 935 he noted on page 16: "Inter rupted 20 November" (NL Hausdorff: Kapsel 19: Fasz. 59). Two days later, on 22 November 1934, the "West deutsche Beobachter" reported in an article entitled "Party educates the Po litical Students'' that "during these days" a working conference of the Nazi Stu dent Union was taking place at Bonn University. The focus of their work dur ing this semester was the theme of "race and folklore. " These circum-
stances make it likely that Hausdorffs decision to break off his lectures was connected with this political activity. At no other time in his long career, ex cept for the brief period of the Kapp Putsch, did he ever cancel a lecture course . On 3 1 March 1 935, after some back and forth, Hausdorff retired as an emer itus professor in Bonn. For his forty years of successful labor in German higher education he received not a word of thanks from the then respon sible authorities. He continued to work on indefatigably, publishing not only the newly revised version of his book Mengenlehre but also seven papers on topology and descriptive set theory, all of which appeared in two Polish jour nals: one in Studia Mathematica, the others in Fundamenta Mathematicae. This work is reprinted in volume III of the Gesam melte Werke (Hausdorff 2008) with detailed commentary. In his final publication (Hausdorff 1938), Hausdorff showed that a con tinuous mapping from a closed subset F of a metric space E can be extended to all of E (allowing for the possibility that the image space can also be ex tended). In particular, a homeomor phism defined on F can be extended to a homeomorphism on all of E. This was a continuation of earlier investiga tions published in Hausdorff 0919b) and Hausdorff 0930). In Hausdorff (1919b) he gave a new proof of the Tietze extension theorem, and in Haus dorff 0930) he showed that if E is a metric space and F C E closed, and if on F a new metric is given that leaves the original topology invariant, then this new metric can be extended to the entire space without altering its topol ogy. In Hausdorff (1935b) he studied spaces that fulfill the Kuratowski clo sure axioms, except for the axiom de manding that the closure operation be idempotent. He called these "gestufte Raume" (today they are usually known as closure spaces); he used them to study relations between Frechet's limit spaces and topological spaces. The unpublished papers in Haus dorffs Nachla!S also show how he con-
3D'The complete text of the Mengenfehre is reprinted in Hausdorff (2008, 41-351 ) Background and reception to the work appear in an historical introduction (1 -40), and the text 1tself rece1ves detailed commentary (352-398) regard1ng mathematical as well as historical matters. 31 Hausdorff to Hilbert, 21 January 1 932, Cod Ms D H1lbert 452c, N1edersachsische Staats- und Universitatsb1bliothek Gottingen
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tinued not only to work on but to fol low the most recent developments in areas that interested him during these ever more difficult times. A major source of support came from Erich Bessel-Hagen, who remained a faithful friend of the Hausdorff family through out their ordeal. Bessel-Hagen brought books and journals from the mathe matics library, which Hausdorff, as a Jew, was no longer allowed to enter. Several articles would not suffice to list all the perfidious laws, decrees, or dinances, and other legalistic machina tions designed to discriminate and iso late the Jews and to deprive them of their property and rights. Historians have counted them though: up to the November 1938 pogrom there were more than 500 such proclamations. One wonders why Hausdorff, an inter nationally recognized scholar living un der such conditions, did not attempt to emigrate during the mid 1 930s. The an swer can only remain conjectural: in Bonn he had his home, his library and the possibility to work, some true friends, and although he was always a skeptic, even he would not have con sidered it possible that the Nazi regime would destroy the economic founda tions established by elderly people in the course of their long lives, or that ultimately they would pay with their lives. The November pogrom, which came to be known as the Night of the Broken Glass (Reichskristallnacht), with its open brutality made all this quite evident and clear. Hausdorff, now over 70, at last made an attempt to em igrate. Richard Courant wrote to Her mann Weyl on 10 February 1939: Dear Weyl, I just received the en closed short and very touching let ter from Professor Felix Hausdorff (which please return), who is sev enty years old and whose wife is sixty-five years old. He certainly is a mathematician of very great merit and still quite active. He asks me whether it would be possible to find a research fellowship for him.32 Weyl and John von Neumann provided letters of recommendation that were
presumably sent to American institu tions and colleagues. Weyl emphasized Hausdorff's many accomplishments and contributions to mathematics, call ing him "A man with a universal intel lectual outlook, and a person of great culture and charm . " These efforts of Weyl and von Neumann were, how ever, evidently unsuccessful. From several sources, in particular the letters of Bessel-Hagen, we know that Hausdorff and his family were forced to undergo a number of hu miliations, especially after November 1 938.33 In mid-1941 the Nazi govern ment began to deport the Jews in Bonn to the monastery "Zur ewigen Anbetung" in Bonn-Endenich, from which the nuns had been expelled. From there they were then transported to the extermination camps in the east. In January 1942, Felix Hausdorff, his wife, and her sister Edith Pappenheim, who lived with them, were ordered to resettle in the internment camp in Bonn-Endenich. On 26 January, all three took their own lives with an overdosage of Veronal . Their last rest ing place is the cemetery in Bonn-Pop pelsdorf. Some of Bonn's Jewish citizens may still have had illusions about the camp in Endenich; Hausdorff had none. Er win Neuenschwander found Haus dorff's farewell letter to the Jewish lawyer Hans Wollstein in the papers of Bessel-Hagen,34 from which we cite the beginning and end: Dear Friend Wollstein! By the time you receive this letter, we three will have solved this prob lem in another way-the way you always tried to dissuade us from. The feeling of safety that you pre dicted would be ours once the dif ficulties of moving had been over come has not come about at all. On the contrary: Even Endenich Is perhaps not yet the end (das Ende nich)! What has happened to the Jews in the last months awakes justified anx iety in us that we will no longer be
allowed to experience bearable con ditions. After expressing his gratitude to friends, and with great composure in formulat ing his last wishes regarding his funeral and last will, Hausdorff wrote further: Excuse us for causing you troubles even after death; I am convinced that you will do what you can (and that is perhaps not very much). Ex cuse us also for our desertion! We hope that you and all our friends will experience better times. Your truly devoted, Felix Hausdorff This last wish of Hausdorff's was not fulfilled: the lawyer Wollstein was mur dered in Auschwitz. Hausdorff's library was sold by his son-in-law and sole heir Arthur Konig. His posthumous papers were preserved by a friend of the family, the Bonn Egyptologist Hans Bonnet, who later wrote about their further fate (Bonnet 1967). Hausdorff's papers, he wrote, were not yet saved, for in Decem ber 1 944 a bomb explosion de stroyed my house and the manu scripts were mired in rubble from a collapsed wall. I dug them out with out being able to pay attention to their order and certainly without saving them all. Then in January 1945 I had to leave Bonn [· · ·]. When I returned in the summer of 1 946 almost all the furniture had dis appeared, but the papers of Haus dorff were essentially intact. They were worthless for treasure hunters. Nevertheless, they suffered losses and the remaining scattered pages were mixed together more than ever. The once well-ordered cosmos had become a chaos. (Bonnet 1 967, 76 052)) The late Professor Gunter Bergmann from Munster performed a great service by carefully ordering the surviving 25,978 pages of the Hausdorff Nach lass. In 1 980 he transfered the now se cure results over to the Bonn Univer sity library. Bergmann also published some of the preserved papers in two facsimile volumes (Hausdorff 1969).
32Veblen Papers, Library of Congress, Container 31 , folder Hausdorff We thank Reinhard Siegmund-Schultze, Kristiansand, for maktng a copy of th1s letter avatlable. He was unable to find Hausdorff's original letter. 33Neuenschwander, E.: Felix Hausdorffs letzte Lebensjahre nach Ookumenten aus dem Bessei-Hagen-Nachla/3, in Bneskorn (1 996, 253-270). 34Bessei-Hagen, Untversttiitsarchiv Bonn Pnnted in Brieskorn (1 996, 263-264 and in facsimile, 265-267).
48
THE MATHEMATICAL INTELLIGENCER
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Acad. Sci. Paris 1 62, 323-325. Alexandroff, P., Hopf, H. 1 935. Topologte.
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Math. Semesterberichte 49, 1 29-
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Chatterj1, S. D. 2007. The Central Limit Theo rem a Ia Hausdorff
Expositiones Math. 25,
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Springer,
Berlin, etc.
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lungen der Kon1gl. Sachs. Ges. der Wiss. zu Leipzig. Math.-phys. Classe 53, 460-475.
Eichhorn, E., Thiele, E.-J. 1 994. Vorlesungen zum Gedenken an Felix Hausdorff.
Helder
mann Verlag, Berlin. Fechter, P 1 948. Menschen und Zetten. Ber telsmann, Glltersloh.
Hausdorff, F. 1 901 b. Beitrage zur Wahr scheinltchkeitsrechnung.
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Rendiconti del Circolo Mat.
di Palermo 22, 1 -74. tronomischen Strahlenbrechung
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Die
Machtigkeit
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Math. Annalen 77,
43Q-437. Hausdorff, F. 1 9 1 9a. Dtmension und auBeres MaB.
Math. Annalen 79, 1 57-179.
Hausdorff, F. 1 9 1 9b. Ober halbstetige Funk tionen und deren Verallgemeinerung
Math.
Zeitschrift 5, 292-309. Hausdorff, F. 1 92 1 . Summatwnsmethoden und Momentfolgen I, II
Math. Zeitschrift 9,
1 : 74-1 09, II: 280-299.
Hausdorff, F. 1 923a. Eine Ausdehnung des Parsevalschen Satzes uber Fourierreihen.
Wiss. zu Leipzig. Math.-phys. Classe 53,
Hausdorff, F. 1 923b. Momentprobleme fur em
Hausdorff, F. (P. Mongre). 1 902. Der Wille zur Neue Deutsche Rundschau (Freie
Hausdorff,
F.
1 903.
Das
endliches Interval/
Math. Ze1tschrift 1 6,
22Q-248. Hausdorff, F. 1 927. Mengenlehre, zweite,
BOhne) 1 3(1 2), 1 334-1 338. Raumproblem
(Antrittsvorlesung an der Universitat Leipzig,
Hausdorff, F. 1 891 . Zur Theone der as
New York, 1 949, 1 965, 1 978. Hausdorff, F. 1 9 1 6.
Math. Zeitschrift 1 6, 1 63-1 69.
Macht
Buchges. , Darmstadt. Frechet, M. 1 906. Sur quelques points du cal
Verlag Veit & Co, Leipzig. 476 S. mit
handlungen der Konigl. Sachs. Ges. der 1 52-1 78.
Feigner, U. (Hrsg.). 1 979. Mengenlehre. Wiss.
cui fonctionnel.
Hausdorff, F. 1 900b. Zur Theorie der Systeme
Hausdorff, F. 1 901 a. Uber eme gewisse Art
51 (1 27)-54(1 30). Ebbinghaus, H.-D. 2007. Ernst Zermelo. An
phys. Klasse 31 , 295-334. Hausdorff, F. 1 9 1 4. Grundzuge der Mengen
Borelschen Mengen
Leipzig. 2 1 6 S. komp/exer Zahlen.
2 1 5-234. Lebensbtld.
Ber. uber die
gehalten am 4 . 7 . 1 903). Ostwalds Annalen
neubearbeitete Auflage. Verlag Walter de Gruyter & Co. , Berlin. 285 S. mit 1 2 Figuren.
Hausdorff, F. 1 930. Erwetterung einer Homoo morphie
der Naturphilosophie 3, 1 -23. Hausdorff, F. (P. Mongre). 1 904a. Der Arzt
Fundamenta Mathemat1cae 1 6,
353-360. Hausdorff, F. 1 935a. Mengenlehre, dritte Au
Konigl. Sachs. Ges. der Wiss. zu Leipzig.
seiner Ehre, Groteske
Math.-phys. Classe 43, 481 -566.
(Freie BOhne) 1 5(8), 989-1 01 3. Neuheraus
flage. Mit einem zusatzlichen Kapitel und einl
gabe als: Der Arzt semer Ehre Kom6dte in
gen Nachtragen. Verlag Walter de Gruyter &
Hausdorff, F. 1 895. Ober dte Absorption des Ltchtes in der Atmosphare
(Habilitations
schrift). Ber. uber die Verhandlungen der
Die neue Rundschau
emem Akt mit emem Epilog
Mit 7 Bildnis
Co., Berlin. 307 S. mit 1 2 F1guren. Nachdruck:
sen, Holzschnitte von Hans Alexander Muller
Dover Pub. New York, 1 944. Englische Aus-
© 2008 Spnnger Sc1ence+Bustness Medta, Inc , Volume 30, Number 4, 2008
49
gabe: Set theory. Obersetzung aus dem
Hausdorff, F. 2008. Gesammelte Werke Band
Deutschen von J. R. Aumann, et al , Chelsea
111· Deskriptive Mengenlehre und Topolo
Pub. Co., New York 1 957, 1 962, 1 978, 1 991 .
gle.
Hausdorff, F. 1 935b. Gestufte Raume. Funda
Fundamenta Mathematica 30,
Hausdorff, F. 1 969. Nachgelassene Schriften. Ed. : G. Bergmann, Teubner,
Stuttgart 1 969. Band I enthalt aus dem NachlaB die Faszikel 51 0-543, 545-559,
Hardy, G. H. 1 949. Divergent Series. Oxford Jahresbericht des Ntcola1-GymnastUms fUr das Jahr 1887.
Stadtarchiv Leipzig, Bestand
598-658 (aile Faszikel sind im Faksimile
DMV 69, 54 (1 30)-62 (1 38).
II. "Grundzuge der Mengenlehre
" Springer
Hausdorff, F. 2004. Gesammelte Werke. Band VII. Ph1/osoph1sches Werk.
Springer-Verlag,
Hausdorff, F. 2005. Gesammelte Werke. Band lichkeltstheone.
Optik und
Wahrschein
Springer-Verlag, Berlin, Hei
� Spring er
Stegmaier, W. 2002. Ein Mathemat1ker m der Philosoph
Nietzsche-Studien 31 , 1 95-240.
tlpllcites Cantoriennes.
Fundamenta Math.
7, 30-1 37; 8, 225-351 . zur Kulturwissenschaft? D1e llteransch-es saylstlschen Schriften des Mathemat1kers Fel1x Hausdorff ( 1868- 1942)
Vorlaufige
Bemerkungen in systemat1scher Abs1cht
In: Huber, M . , Lauer, G. (Hrsg.): Nach der
In: Se1s1ng, R . , Folk
erts, M., Hashagen, U. (Hrsg.): Form, Zahl, Ordnung,
Boethius Bd. 48, Franz Steiner Ver
Riesz, F. 1 907 D1e Genes1s des Raumbe griffs.
Math. und Naturwiss. Berichte aus Un
beiten
strakte Mengenlehre
aturwissenschaft
zw1schen
H1stonscher
Anthropologie, Kulturgeschichte und Me dlentheone
Max Niemeier Verlag, Tubingen,
S.55 1 -573. Wagon, S. 1 993. The Banach-Tarski Paradox Cambridge Univ. Press, Cambridge. Young, W. H. 1 903. Zur Lehre der mcht abgeschlossenen Punktmengen.
I, 1 1 0-1 61 .
R1esz, F. 1 908. Stetigke1tsbegnff und ab Att1 del Congr. lnter
naz. dei Mat., Roma 2, 1 8-24.
delberg.
Ges. der Wiss. zu Leipzig, Math.-Phys. Klasse 55, 287-293.
s p n n g e r.co
Spri n g e r a nd Society Pu b l ish i n g The selection of a publisher is a critical decision for scholarly and profession a l Societies. Our flexib l e a pproach, our record of i n novation, and our long history as publisher of respected journals make Spri nger the p referred publ ishing partner of some of the most renowned scholarly Societies i n the world. We invite you to take a closer look at how Spri nger ca n su pport your publishing activities
springer.com/societies
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50
THE MATHEMATICAL INTELLIGENCER
Berichte
Ober die Verhandlungen der Konigl. Sachs.
.
t h e la nguage o f science
2.
Soz1algeschlchte-Konzepte fur eine Liter
garn 24, 309-353. In his Gesammelte Ar
Berlin, Heidelberg.
DMV,
Woh/ordnung--d1e spektakularen Ere1gnisse
lag, Wiesbaden, S.223-241 .
Verlag, Berlin, Heidelberg.
der
Erganzungsband, Teubner, Leipzig.
auf dem lnternationalen Mathematikerkon greB 1904 m Heidelberg.
Springer-Verlag, Berlin, Heidelberg. Hausdorff, F. 2002. Gesammelte Werke. Band
V· Astronom1e,
Jahresbericht der
Purkert, W. 2004. Kontinuumproblem und
IV. Analysis, Algebra und Zahlentheorie
den Punktmannigfaltigkeiten.
Jahresbericht
Vollhardt, F. 2000. Von der Soz1algeschichte
Nicolai-Gymnasium. Lorentz, G. G. 1 967. Das mathematische Werk von Felix Hausdorff
Hausdorff, F. 2001 . Gesammelte Werke. Band
von
II.
Urysohn, P. 1 925/1926. Memo1re sur les mul
561 -577, Band II die Faszikel 578-584, druck wiedergegeben).
Lehre
Teil
Landschaft Zarathustras Felix Hausdorff a/s
56-58. Univ. Press, Oxford.
40-47. 2 Sande.
Hahn, H. 1 928. F Hausdorff, Mengenlehre Monatshefte fOr Mathematik und Physik 35,
menta Mathemat1cae 25, 486-502. Hausdorff, F. 1 938. Erweiterung emer stet1gen Abb1ldung.
Springer-Verlag, Berlin, Heidelberg.
Schoenflies, A 1 908. Die Entwickelung der
J u l ia Sets that are Fu l l of H o l es KIMBERLY A. ROTH
l
n the nearly three decades since Benoit Mandelbrot cap tured the popular imagination with his first striking frac tal images, computer-generated pictures such as those shown in Figure 1 have become commonplace. As can be imagined, behind the pretty pictures lies some equally beau tiful mathematics. The particular fractal shown in Figure 1 is a Julia set. Julia sets have been studied since the first ex plorations in complex dynamics by Gaston Julia and Pierre Fatou at the turn of the twentieth century. Despite this fa miliarity, there is much that is yet to be discovered about these sets. One of the basic questions about a set is whether it takes up any space, that is, whether it has Lebesgue measure greater than zero. It is known that most Julia sets do not take up space in this sense, yet there is a striking variation in how "large" these sets appear when we look at the im age. To quantify this difference in size, we will turn to the concept of dimension, in particular, box dimension. Vari ous notions of dimension are often used to describe the geometric structure of a fractal. In fact a fractal is some times defined as an object with nonintegral dimension. The possibilities for the dimension of a Julia set range all the way from zero-dimensional fractal "dust" to the whole 2-dimensional Riemann sphere. Aside from some restricted classes of functions, it is surprisingly difficult to determine much about the measure or dimension of a given Julia set. With the exception of those that are the whole Riemann sphere, all Julia sets whose measures have been calculated
have measure 0; however, it has been conjectured that there are Julia sets of positive measure. If a set in the plane has box dimension less than 2, it must have Lebesgue measure zero (Schleicher 2007), so looking at dimension can also give answers about measure. I will examine one way of showing that a Julia set has box dimension less than 2. We define a property called "porosity," which means the Julia set has holes at every scale; in other words, no matter how close we look at our Julia set, it is full of holes. Good introductions for undergraduates can be found in an informal setting in Lesmoir-Gordon, et a!. (2000) and in a more formal one in Devaney 0992). A good graduate level introduction to complex dynamics can be found in Carleson and Gamelin 0 993).
What is a Julia Set? A julia set is a set of points ]1on the Riemann sphere, which arises in a certain way: for a function f with its iterates fn, n = 1 , . . . , oo, f.t is the closure of the set of repelling pe riodic points, that is, periodic points such that points in a neighborhood around them leave the neighborhood under iteration. We often study classes of polynomials or rational functions. For a polynomial f, there is a simpler definition: the Julia set !J is the boundary of the set of points which go to oo under iteration by f Every Julia set also defines two other sets. The Fatou set F1 is the complement of the Julia set. The filled-in
© 2008 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 30, Number 4, 2008
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julia set K1 is the union of the Julia set with the bounded components of the Fatou set. In the pictures, the filled-in Julia set is usually the region colored black. Julia sets often display a complexity much greater than would be expected from the given generating function; as we see in Figure 1 , even a supposed simple map such as z2 1 has a complicated Julia set. For a more docile example, consider the polynomial z2. In the complex plane, all numbers with modulus greater than 1 go to in finity since the modulus is squared by every iteration. This means that the Julia set for z2 is the circle of radius 1 centered at the origin. The Fatou set is everything but the circle. The filled-in Julia set is the disk of radius 1 centered at the origin. In the picture generated by z2 - 1 in Figure 1, the filled in Julia set is the region colored black, the actual Julia set is the boundary of the black region (the coloring-book out line of the black region), and the Fatou set is the interior of the black region along with all the colored regions. Pictures of most Julia sets are complicated and beauti-
-
Figure I. The Julia set for f(z)
=
z2
-
I.
ful, as can be seen in Figures 1 , 2, 5, and 6. Figures 5 and 6 show only the Julia set.
Dimension Complicated objects such as fractals often have nonintegral dimension, but there are many different notions of dimen sion. I will use the upper box dimension. The upper box di mension of a set S is log N(S, r)
dimBS
field of study is complex dynamics, and she especially enjoys teaching chaos and fractals to undergraduates. Her hobbies include wheel-thrown pottery
Department of Mathematics Huntingdon, PA 1 6652 USA e-mail: [email protected]
52
THE MATHEMATICAL INTELLIGENCER
(�)
Figure 2. A Julia set caJied a "rabbit," f(z)
KIMBERLY A. ROTH, following her undergraduate work at Oberlin Col
Juniata College
log
,
where N(S, r) is the minimum number of filled squares of side-length r required to cover S. The filled squares are called boxes. The logarithms are used here because we wish to calculate the rate at which the number of boxes needed to cover the set increases with respect to the scale. Lower box dimension is defined just as the upper box di mension, with the upper limit replaced with a lower limit. Therefore lower box dimension is always less than or equal
lege and a year at the Budapest Semesters in Mathematics, completed her doctoral work at Penn State under the supervision of Greg S wi�tek Her
and singing in the Juniata College Choral Union.
= )�0
=
z2
-
.12 +
.
66i
.
to upper box dimension. When the limit exists in the strict sense, lower and upper box dimensions are equal, and the dimension of the object is just called box dimension.1 To illustrate box dimension, I will begin with a simple example that is not a Julia set, the Sierpinski gasket. This example is chosen because it is affine self-similar, which means it contains scaled copies of itself within the set, and all of the scale factors are the same.
of coverings. However the Box Counting Theorem identi fies classes of sequences for which we can do this includ ing this one (Pesin 1997, 40). Finding box dimension for something such as the Sier pinski gasket is not difficult. Unfortunately, for most Julia sets, box dimension is not as easy to calculate. This diffi culty comes from the fact that Julia sets are generally not affine self-similar and so do not yield to the techniques used on the Sierpinski gasket.
The Sierpinski Gasket
To generate a Sierpinski gasket, start with a solid equilateral triangle. Next, take the midpoints of the sides and draw line segments between them, and throw out the inner triangle they create. Now, you have three smaller solid triangles. Then, take the midpoints of their sides and apply the previous process. The Sierpinski gasket is the limit of repeating this process ad infinitum. Let us assume for the sake of conve nience that the sides of the original triangle have length 1 . Figure 3 shows the results of the first few iterations of this process and the box covering for them. Side length of box
Number of boxes to cover
I
I
1 /2
3
1/4
9
1/8
27
Table
1.
Box dimension
=
l im n --> co
log log
3n 2"
=
log log
3 2
.,
1 .58
Now we will compute box dimension. In Table 1 , we list a sequence of side-lengths and the number of boxes of this side-length needed to cover the gasket. Pictures of the first three coverings are shown in Figure 3. We then take the limit of our sequence and derive box dimension 1 . 58. We might wonder at this point if the calculation we have completed in Table 1 was quite legal, since we are con cerned with the limit overall and not just of one sequence
Porosity Porosity may be used to estimate dimension. We start with a compact set S and a length 0 < r < 1 . For z E S, we take any square of side-length r containing z, then we choose suf ficiently large N so that we can subdivide the square into N2 subsquares of side-length r/N with one of these subsquares disjoint from S. This disjoint square is our "hole. " There may be several, but one will suffice (see Fig. 4). If there is an in teger N > 0 for which one can perform this subdivision for any z E S and any 0 < r < 1 , then the set is porous (or shal low). We call N the constant of porosity. (There is an equiv alent definition of porosity using disks rather than squares. A set is porous if we can find circular holes in our set at every scale, that is, within any disk a hole is guaranteed to be big ger than a constant multiple of the original disk size.) The definition can be modified in several different ways. The notions of mean porosity and E-mean porosity were developed as extensions of porosity by Koskela and Rohde (1997) and loosen the requirements on the uniformity of the size of the holes within certain limits. I introduced the notion of nonuniform porosity as a weakening of porosity (Roth 2006). This retains the holes at every scale while com pletely eliminating uniformity. Porosity and Dimension
Why do we care about porosity? Because porosity implies that upper box dimension is less than 2, which implies that the Lebesgue measure is zero. If the constant of porosity,
Figure 3. Box coverings for the first three rows of Table I. Note in the rightmost picture that the center box is not needed for the covering.
" 1 Most results in the field are stated in terms of Hausdorff dimension, which 1s less than or equal to lower box dimension However it is not necessary to state results 1n these terms for th1s paper
© 2008 Spnnger Sc1ence+ Bus1ness Media, Inc , Volume 30, Number 4, 2008
53
and so
side-length r
--
dimBS :S
log
Ndn
log Nn
= d < 2.
0
For a complete proof, see Martio and Vuorinen 0987). How Do We Show Porosity?
a
hole
Figure 4. Illustration of definition of porosity. N, is exactly calculated, which it often cannot be, a better bound for dimension can be provided. See Martio and Vuorinen (1987) for details. Similar bounds exist for mean porosity and e-mean porosity, although the bounds are slightly different (Koskela and Rohde 1 997). Unfortunately nonuniform porosity does not provide a dimension bound, although it does imply Lebesgue measure zero.
THEOREM 1 A porous set in the Riemann sphere has upper box dimension less than 2 .
PROOF
IDEA .
Let's call our porous set S. We want to estimate
N_S, r), the number of boxes needed to cover the set.
First, we cover S with squares of a side-length r, where r is a number greater than zero and less than one. We can assume we have used a finite number of squares, since S is compact by the definition of a porous set. We will call that number of squares k. Now, examine a specific square. Divide it into N2 subsquares, where N is the porosity con stant. By the definition, there must be at least one square that contains no points of S, as in Figure 4 . We do not need that square to cover S, so throw it out. Repeat this process on the rest of the k squares in our cover. Now, we r have k(N2 - 1) squares of side-length '
N
We repeat the process, subdividing our cover of squares of side-length ....!..._ After subdividing and throwing out Jv.
squares not needed to cover S, we now have squares of side-length
�· N
k(N2 - 1) 2
If we repeat this process n
. times, we have k(N 2 - l) n squares of side-length that N 2 - 1 = Nd for some d < 2 . S o the upper box dimension log N(S, r)
dimES = }!p0
54
log
(�)
THE MATHEMATICAL INTELLIGENCER
= lim
n -. oo
r
N
" . Note
Let's discuss the simplest example. The Julia set of j(z) = z2 is the unit circle centered at the origin. At any point z on the unit circle we take a box of side-length r centered at z. The circle takes up less than half the area of the box, because the tangent line to the circle at z lies outside the circle and divides the box into two parts with equal area. Using N = 3 guarantees a hole when you subdivide. Since this is true for all r at z, the Julia set is porous there. This is true for all points on the circle, and the constant of poros ity remains the same for each point. For a general Julia set, showing porosity is more com plicated. First we must find holes at some point in the Ju lia set. Once we have porosity for some z E ]fi we find a conformal mapping that allows us to duplicate these holes at every point. For rational functions expanding on a Julia set, the proof of porosity is easy. The idea was known for a long time. Just pull back large-scale holes to all small scales by iteration of inverse branches of f (Przytycki and Urbanski 2001). The main difficulty in this approach is keeping the holes reasonably close to the same size while pulling back. If we distort our holes too much under the pull-back we cannot replicate the porosity. To keep the distortion of our holes bounded, we use the Koebe One-Quarter Theorem and the Koebe Distortion Theorem (Carleson and Gamelin 1993). Of course, guaranteeing that we have one constant of poros ity for all of the holes is a much more complicated matter that I will not discuss here.
Julia Sets that are Full of Holes Several types of Julia sets have been shown to be porous. Most of these results have been published relatively re cently, and the Julia sets considered are subjected to re strictive hypotheses to make the mathematics easier. I will
consider a few related sets, although I will not discuss the conditions on the sets in detail. For other known results on porosity of Julia sets, see Geyer 0999), Jarvi 0995), Przy tycki and Rohde 0998), Przytycki and Urbanski ( 2001 ), Sul livan ( 1 983), and Youngcheng (2000). Our first Julia set that is full of holes is the rabbit shown in Figure 2 (Przytycki and Urbanski 2001). It belongs to a class of Julia sets for nonrecurrent and parabolic Collet-Eck mann rational functions. The second group of porous Julia sets are the polyno mials R.. z) = fi27T'8z + ZZ, which were shown to be porous for irrationals (} of bounded type (McMullen 1 998) . The Julia set fp for (} =
1 CVS - 1 ) is shown in Figure 5 .
The third group o f porous sets i s not quite a Julia set. (z - 3) ]e is a subset of the Julia set fR of %.. z) = fl-7rrr ;i2 , 1 - 3z where T is a number chosen so that the rotation number is (} (see Figs. 6 and 7). The set ]e is important because of its relationship to the previous class of Julia sets fp. ]e is porous for irrationals of bounded type (McMullen 1 998), and for other irrationals it is a close call: ]e for the function (z - 3 ) . . any 1rrauona . . ? "mr(O) z:1 , 1s nonun11orm , where (} 1s e·c 1y 1 - 3z porous (Roth 2006). Despite all that is known aboutj0, it is unknown whether ]R is porous. The problem is finding the initial point on the Julia set where the set is porous. If such a point could be found, the techniques from the previous results could be applied to find holes at every scale. Mathematically speak-
ing, Julia sets are part of a "young" field, and it is easy to find questions that are still not answered.
Conclusion Many of the most fundamental properties, such as measure and dimension, remain unknown for most Julia sets. Al though there are Julia sets that are the whole Riemann sphere and so have dimension two and positive measure, no other Julia sets of measure bigger than zero have been found. Shishikura's surprising result 0 998) shows that there are other Julia sets of dimension 2, which makes it appear possible that there are other Julia sets of positive measure. Proving that a Julia set is full of holes, or porous, pro vides a bound on the upper box dimension, but this has so far been possible only for special classes of Julia sets. Mean porosity and mean E-porosity, both found in Koskela and Rohde 0997), provide better dimension bounds; nonuniform porosity (Roth 2006) implies measure zero, but is not known to provide dimension bounds. These notions can be used in some cases when it is not possible to prove porosity. In the end, we do not know in general which Ju lia sets are porous and which are not. In fact, for ]R, little is known about its dimension or measure. There is much left to explore. ACKNOWLEDGMENTS
The figures in the paper were generated using several differ ent programs. Figures 1 and 2 were generated using Fractal Explorer. Figures 5 and 6 were generated using the software of Curt McMullen (2001). Figure 6 comes from Peterson (1996). The Sierpinski gasket was generated using BRAZIL Fractal Builder. REFERENCES
L. Carleson and T. Gamelin. Complex Dynamics. Universitext, Springer Verlag, New York, 1 993. R. Devaney, A First Course in Chaotic Dynamical Systems. Westview Press, Perseus Books Group, Boulder, 1 992. L. Geyer, Porosity of parabolic Julia sets. Complex Variables Theory Appl.
39 (1 999), 1 91 -1 98.
P. Jarvi, Not all Jul1a sets are quasi-self s1m1lar. Proc. Amer. Math. Soc.
Figure 6. The Julia set, }R, for R(z) T
6
=
e27T1" z2 1!..:-_ll, where
I - lz
1 25 (1 995), 835-837 . P. Koskela and S. Rohde, Hausdorff dimension and mean porosity.
is a number chosen so the rotation number is =
t (vS -
1 ).
Math. Ann.
309 (1 997), 593-609.
N. Lesmoir-Gordon, W. Rood, and R. Edney, lntroducing Fractal Geom etry.
Icon Books Limited, Cambridge, 2000.
C. McMullen, Self-similarity of Siegel disks and Hausdorff dimension of Julia sets. Acta Math. 1 80 (1 998), 247-292. C.
McMullen,
Julia Sets of Polynomials and
Rational
Maps.
http://www.math.harvard.edu/-ctm/programs.html, 2001 . 0. Martio and M. Vuorinen, Whitney cubes, p-capacity, and Minkowski
content. Expo. Math. 5 (1 987), 1 7-40.
Y. Pes1n, Dimension Theory in Dynamical Systems: Contemporary Views and Applications.
Chicago Lectures in Mathematics. University of
Chicago Press, 1 997. C. L. Peterson, Local connect1v1ty of some Julia sets containing a cir cle with an irrational rotation. Acta Math. 1 77 (1 996), 1 63-224.
Figure 7. A subset of the previous Julia set, we'll call it }e, for 6 (Vs I) (from Peterson 1 996) =
1
-
.
F Przytycki and S. Rohde, Porosity of Collet-Eckmann Julia sets, Fund. Math .
1 55 (1 998), 1 89-199.
© 2008 Spnnger Sc>ence+ Bus>ness Media. Inc . Volume 30, Number 4, 2008
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F. Przytyck1 and M. Urban-ski, Porosity of Julia sets of non-recurrent
M. Sh1sh1kura, The Hausdorff dimension of the boundary of the
and parabolic Collet -Eckmann rat1onal functions, Ann. Acad. Sci. Fen
Mandelbrot set and Julia sets. Annals of Math. 1 47 (1 998), 225-
nicae
267.
26 (2001), 1 25-154.
K. Roth, Non-uniform porosity for a subset of some Julia sets, Com
D. Sullivan, Conformal Dynamical Systems, Geometric Dynamics (Rio
plex Dynamics: Twenty-Five Years After the Appearance of the Man
de Janeiro, 1 981), Lecture Notes in Math. , vol. 1 007, Springer-Ver
de/brat Set,
American Mathematical Society, Contemporary Math.
lag, Berlin, 1 983, 725-752. Y. Youngcheng, Geometry and dimension of Julia sets. The Mandel
396 (2006), 1 53-1 68. D. Schleicher, Hausdorff dimension, 1ts properties, and its surprises, Amer. Math. Monthly
� Springer
the language of science
brot set, theme and variations, ture Note Ser.
1 1 4 (2007), 509-528.
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THE MATHEMATICAL INTELUGENCER
pp. 281 -287, London Math. Soc. Lec
27 4, Cambridge Univ. Press, Cambridge, 2000.
la§'h§'.Jtj
Osmo Pekonen ,
Editor
I
The Architecture of M odern Mathematics by jose Ferreir6s andjeremy Gray (eds.) NEW YORK. OXFORD UNIVERSITY PRESS, 2006, 442 PP, US $69.50 ISBN 0198567936 REVIEWED BY ANDREW ARANA
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book ofyour choice; or, if you would welcome being assigned a book to review, please write telling
us
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his collection of essays explores what makes modern mathematics "modern," where "modern mathematics" is understood as the mathe matics done in the West from roughly 1 800 to 1970. This is not the trivial mat ter of exploring what makes recent mathematics recent. The term "modern" (or "modernism") is used widely in the humanities to describe the era since about 1900, exemplified by Picasso or Kandinsky in the visual arts, Rilke or Pound in poetry, or Le Corbusier or Loos in architecture (a building by the latter graces the cover of this book's dust jacket). Though it is hard to say precisely what modernism is, or what distinguishes it from other eras, Gray at tempts a definition in his closing essay in this collection:
I
Modernism can be defined as an au tonomous body of ideas, pursued with little outward reference, main taining a complicated, rather than a naive, relationship with the day-to day world and drawn to the formal aspects of the discipline. (p. 374)
Column Editor: Osmo Pekonen, Agora Centre, 40014 University of Jyviiskylii, Fmland e-mail: osmo. [email protected]
This is a good start. Gray mentions modern algebra, topology, and logic as examples fitting this description, and explains why they fit. These characteristics, though, fit high-profile examples of mathematics before this era also. Ancient geometry as in Euclid's Elements seems to have been pursued as an autonomous body of ideas, at least as far as I understand what this means. D'Alembert's work on differential equations, for instance on
the vibrating-string problem, was criti cized for failing to model empirical re ality adequately, though d'Alembert dis puted this: hence d'Alembert's work had a complicated, rather than naive, relationship with the day-to-day world. Lastly, Euler and Lagrange, among many others in the eighteenth century, were drawn to the formal aspects of analysis. Gray's definition of modernism could be tightened to disqualify these examples (and to be fair, his essay in dicates some ways to do this, as I'll soon point out). But if we are going to make a case for the continuity of modern mathematics with modernism, we must look beyond the quoted definition for another account. We can see how the essays in this collection contribute toward answering what makes modern mathematics mod ern if we instead view modernism as a crisis concerning foundations. Let me explain. All of the artists and architects mentioned previously looked for a new way of practicing their art, because the old ways had been discredited or had ceased to speak to them. This loss opened up many new possibilities. Each experimented with form and content. The results were radically new, and many found them alien upon first con tact. Think of people asking if Kandin sky's spirals of color are really "art." What would it be to really be art? Mod ern artists realized many answers to this question that would not have seemed open in earlier times. This sense of openness, in contrast to a time in which some options would have seemed in escapably "correct," is what I want to call a foundational crisis. As has often been remarked, math ematics in the early twentieth century underwent a foundational crisis. This is usually said to be a result of the para doxes in set theory, which threw into question whether mathematics was con sistent. I agree that this was a crisis of sorts, but there was another crisis con temporaneous with this one that had wider reach. This wider foundational crisis mirrors the crisis in art just dis cussed. There had been consensus in
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the past on what makes a mathemati cal theory, such as arithmetic, true: it was true if it described the way things really were. By the turn of the twenti eth century, this view had lost much of its credibility. It now seemed open whether there were any true mathe matical theories, and if so, there were a variety of possible answers to this question. As with art, I want to call this sense of openness in mathematics a foundational crisis. Viewing modernism as a foundational crisis in the sense de scribed here provides a sharper answer to what makes modern mathematics modem. The modern turn in mathe matics happens in parallel with the modem turn in the arts, both trying to progress despite an awareness that old orders that used to underwrite their on tologies and values had been discred ited. The essays in this collection address this modernist foundational crisis in mathematics in a variety of ways. The essays concern the years following 1800, when non-Euclidean geometries were beginning to receive attention. These geometries gave new urgency to the problem of what it was for a math ematical theory to be true. Which is the real geometry of space? There was con sensus in the past: a geometry was true if it described the way things really were. Until the mid-eighteenth century, there were two main explanations of how this worked: either the description of reality was a result of abstraction from nature, or an expression of Pla tonic forms and their ordering. From ei ther standpoint, Euclidean geometry was thought to be a true description of space. During the eighteenth century, an other explanation gained currency. Gray describes this view as follows: mathematics is "what is presented by idealized common-sense." (p. 390) On this view, "every rational person can recognize a straight line when they see one. " The true geometry is thus the one acknowledged by all rational people; naturally this was thought to be Eu clidean geometry. Around the tum of the nineteenth century, Kant offered a more sophisticated version: roughly, the world appears to us the way it does (for instance, as having unified objects) be cause our minds structure it to appear that way. We can't help but experience
58
THE MATHEMATICAL INTELLIGENCER
the world as we do, but whether the world really is as we experience it is unanswerable. A theory of space is true, on this account, if it expresses the struc ture that our minds are constrained to experience space as having. (Kant seems to have thought that Euclidean geometry expressed this structure.) During the nineteenth century, as mathematicians became aware of non Euclidean geometries, it no longer seemed obvious that space really is Eu clidean, or that the mind structures spa tial experience as Euclidean. The foun dational crisis for mathematics begins here. One option is to conclude that no geometry is the "true" one. Instead, there are different geometries and none is any more true than the others. Some geometries suit our individual situations and present purposes better than oth ers, and our situations and purposes can change. This view is similar to the philosopher Friedrich Nietzsche's "per spectivalism" on moral and scientific matters: there is no single "true," "God's eye view" of morality and the world, only biased individual perspectives. To see things more clearly we should learn to view morality and the world from many different perspectives . As Moritz Epple explains in his fascinating essay in this volume, one mathematician who explicitly took up this Nietzschean ban ner in his work was Felix Hausdorff. Hausdorff lived a double life in print, publishing as a mathematician under the name Felix Hausdorff, and as a Ni etzschean philosopher under the name Paul Mongre. Hausdorffs view, which he called "considered empiricism, " was that mathematics is useful for construct ing axiomatic theories that "model" em pirical phenomena. Each theory repre sented a "perspective" on the empirical matter it concerned. Whether a theory is good is a practical question, to be evaluated based on how well the the ory describes, explains, and predicts data. Since the empirical data may be consistent with several different mathe matical theories, each theory should continue to be developed; Hausdorff thought this was the case with dimen sionality and with the ongoing devel opment of various axiomatic geometries. Relativism regarding the truth of mathematical theories is a radical de parture from the past. It constrains
mathematical activity to the construc tion of theories, without any preten sions to "getting it right." On this view, there is no ·'right" way to think of space, no single true analysis of the concepts of circle or line or continuity. Relativism can extend even into the allegedly "foundational" areas of arithmetic and set theory. There are many axiomatic theories of arithmetic that we can study freely for their mathematical structure, but we must not confuse any of them with the "real thing," for there is no "real" thing. Similarly, there are many different set theories; the relativist can claim that none is absolutely true. As Epple explains, Hausdorff became in terested in a set-theoretic approach to topology in seeking a continuous model of space and time, which led him to Cantor's point-set analysis of the con tinuum. Yet he thought this was just one perspective on the continuum. Alfred Tarski's work on logical con sequence, carefully discussed by Paolo Mancosu in this collection, gave more tools to the technically-minded rela tivist. Today we follow Tarski in saying that a sentence u is a logical conse quence of a set of sentences � if every interpretation or "model" of all the sen tences in � is also a model of u. But the "every" in this analysis of logical consequence raises a question. Suppose we are asking whether a sentence in the language of arithmetic is a logical consequence of the axioms of arith metic. Should we consider just models with the intended domain N, or do we consider models with other domains also? The more radical conception is to vary domains widely, perhaps out of skepticism that the notion of an "in tended domain" makes any sense. Man casu argues that in 1936 at least, Tarski avoided the more radical option. I find Mancosu's argument convincing, but more work is being done on this topic, and there may be new arguments worth considering. In any case, this article is an excellent starting point for under standing this active area of research. Gray notes that an even more radi cal relativism arose in the early twenti eth century: The Modernist foundations of math ematics ultimately dispensed with the idea that the subject matter of logic was the correct rules of rea-
son-those that would be followed by any undamaged mind. A part of logic does consider such rules, but it seemed ever more obvious that the logic needed to create genuine mathematics is not a candidate for even an idealized description of the way people think. Not only geome try, not only the conception of num ber, but eventually any simple minded association of logic with correct thinking was made anew. (p . 396) Is there a principled reason to reject relativism, and maintain the traditional view? Because this question is a live one, I call the ongoing situation a foun dational crisis. Epple and Gray's essays explore (without advocating) this crisis directly, whereas Mancosu's essay bears on this issue without addressing it explicitly. Other essays respond to this view, in one way or another. Two essays concern, in different ways, Hilbert's attempt to reconcile a Kantian approach with these new mathematical developments. Wilfried Sieg considers ongoing developments in the spirit of "Hilbert's program" in proof theory. Hilbert thought mathematical methods could be divided into two categories, the "real" and the "ideal. " In his early work, Hilbert echoed the traditional view that the natural and real numbers, and the points of Euclidean geometry, are real, whereas imaginary numbers and points at infinity in projective geometry are ideal; later, he identified the finitary mathematics of the natural numbers as real and the infinitary meth ods of higher mathematics as ideal. In drawing this real/ideal distinction, Hilbert was echoing Kant's distinction between constitutive and regulative principles, where the former are real ized in experience, and the latter are not, but instead are tools for organiz ing our thoughts concerning experi ence. Theorems proved by real meth ods were contentual, whereas theorems proved by ideal methods were useful tools for theorizing, but lacked content. Hilbert hoped to show that every the orem provable by ideal methods could be proved by real methods. Sieg agrees with the received view that Hilbert's program is dead, as a re sult of Godel's second Incompleteness theorem. But he thinks ongoing work
in proof theory might salvage some thing like it, if the base theory is "ac cessible"-that is, if it has "a unique build-up through basic operations from distinguished objects," so that it consists of "principles that are evident. " For then the base theory would be contentual and thus capable of yielding knowl edge. Sieg doesn't offer criteria for eval uating when an operation is basic, or for when a principle is evident. Instead, he raises this as a project for future work. Though there are important Kantian elements in Hilbert's thought, Hilbert re jected the details of Kant's views on in tuition as "anthropomorphic garbage. " Nevertheless, intuition and experience, and in particular visualization, played an important role in Hilbert's thought. Leo Corry explains that Hilbert believed that we are guided in our formulations of axiomatic theories of geometry by in tuition and experience, and that Hilbert continued to believe this even as he came to understand general relativity. Although in the past Hilbert had thought Euclidean geometry was the true geometry of space, he recognized that general relativity cast doubt on this. This was no problem for Hilbert's views about axioms and experience, because as our experience changes, so should our axiomatic theories. Such a view, though, seems to leave open the ques tion of whether mathematics can "get it right" when describing the world, or if instead it is just our way of describing things, which can only be judged prag matically by what those models can do for us. That is, Hilbert's view as de scribed by Corry leaves open the pos sibility of relativism for geometry. These matters troubled Hilbert's stu dent Hermann Weyl, whose fascinating views Erhard Scholz discusses. Like Hausdorff and Hilbert, Weyl thought mathematical activity was largely a mat ter of producing systems of symbols. But Weyl was no relativist. According to Scholz, Weyl thought that mathematics did more than offer mere tools for the formation of mathematical models of processes or structures, in a purely pragmatic sense. A good mathematical theory of nature . . . expressed, if well done, an aspect of transcendent reality in "symbolical form. " (p. 296)
That is, a good mathematical theory must include a " metaphysical belief in some transcendent world core , " so that the meanings of the symbols used in ordinary practice are not merely the stipulations of individual practitioners (as the relativist would have it), but in stead are rooted in a transcendent re ality; otherwise "no meaningful com municative scientific practice would be possible . " Weyl was inspired by post Kantian German philosophy, particu larly the work of Wilhelm von Hum boldt, Martin Heidegger, Karl Jaspers, and Ernst Cassirer. Following their writ ings, Weyl considered our use of sym bols analogous to the use of tools by carpenters and other craftsmen. The raw materials, the abilities of the car penter, and the item the carpenter wants to build, all put demands on what makes a tool for that task good. Similarly, Weyl thought mathematical language is a tool for the mathemati cian, and it is not entirely up to the mathematician to determine what makes that tool good. Scholz does not explain what Weyl thought were the analogues of the raw materials, etc., for the carpenter that would constrain the goodness of mathematical tools, or how this constraining would work. I agree with Scholz that Weyl's idea is more a plan for future work than a complete solution. But it is a fascinat ing idea, and one that is worth devel oping. (In his fine article, Jean-Pierre Marquis also takes up the idea of tool production and use in mathematics, ar guing that some mathematical theories, specifically homotopy theory, are worth knowing only because of their practical value for work in other sub ject-matters of intrinsic interest, even if these theories are not themselves of in trinsic interest.) Let us turn now to the essays on Gott lob Frege. Frege was a key instigator of the contemporary Anglo-American ap proach to philosophy, in which the log ical analysis of language is of central importance. Though Frege is mostly studied by philosophers nowadays, he was very much a mathematician: his doctorate was in mathematics; he was employed in the ]ena mathematics de partment; he regularly taught courses in complex analysis, elliptic functions, and potential theory. Within the philosophy of mathematics, he is best known for
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his "logicist" project, the goal of which was to show that all the truths of arith metic and analysis (though not geome try) were really truths of logic. Frege, too, was concerned about the problem of relativism, particularly for the concept of number. As he wrote in the introduction to his Foundations of
Arithmetic, Yet if everyone had to understand by this name ["the number one"] whatever he pleased, then the same proposition about one would mean different things for different peo ple,-such propositions would have no common content. (Frege, p. i) Like Hilbert and Weyl, Frege was con cerned that if there is no single correct answer to what a number is, then the intersubjectivity of arithmetic and analysis would fail. He accounted for his answer's correctness by appealing to his logicism, that is, to his view that the laws of arithmetic are reducible to the laws of logic, which are laws of thought and thus are common to every rational person. Nevertheless, Frege recognized that human practices played a role in mak ing explicit the sense of the number concept. Michael Beaney's essay in this volume carefully explores how Frege went about "elucidating" the basic con cepts of arithmetic and analysis, draw ing on "our common conceptual her itage" (p. 53) to make explicit what these concepts really are. (In her en gaging article on twentieth-century French philosophy of mathematics in this volume, Hourya Benis Sinaceur's description of Jean Cavailles' project of unwinding the historical ''dialectic" of mathematical concepts suggests paral lels with Frege's idea of elucidation, al though these parallels are not explored here.) In his essay, Jamie Tappenden care fully situates Frege within the nine teenth-century struggle in Germany over how best to think about complex analysis. This was (roughly) a struggle between two camps. Weierstrass and his followers thought complex analysis should be "arithmetized," meaning in particular that analytic functions should be defined as functions representable by power series. By contrast, Riemann and his followers favored representa-
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THE MATHEMATICAL INTELLIGENCER
tion-independent definitions, which in practice meant defining analytic func tions as those satisfying the Cauchy-Rie mann equations. Riemann advanced the term "geometric" for this kind of ap proach to analysis, but this wasn't a stretch: he encouraged visualization in analysis, developing the notion of a Rie mann surface to help. He even en couraged physical reasoning in analy sis, using Dirichlet's principle freely even though his evidence for it was based on potential theory. (Riemann's own reflections on these matters were quite rich, provocatively engaging philosophical matters, as Ferreir6s doc uments in his delightful essay.) Tappenden thinks it is wrong to see Frege as a Weierstrassian. Instead, he argues, Frege's work should be seen within the Riemannian tradition, where a central aspiration was the identifica tion and clarification of concepts-in Frege's case, of the concept of number. Weierstrass holds that there can be no dispute about the kind of thing that counts as a basic operation or concept: the basic operations are the familiar arithmetic ones like plus and times. Nothing could be clearer or more elementary than explanation in those terms. Series representa tions count as acceptable basic rep resentations because they use only these terms. By contrast, the Rie mannian stance is that even what is to count as a characterization in terms of basic properties should be up for grabs. What is to count as fundamental in a given area of investigation has to be discovered. (p. 1 1 2) Tappenden argues that Frege's attempt to provide a logicist definition of num ber should be understood as an in stance of this general Riemannian quest. To understand better what this quest is all about, I want to pose and answer two questions about these Riemannian definitional "quests. " First, does the Rie mannian think there must always be a "right" definition, and if so, what makes it right? Second, how does the Rie mannian think we are to know when the "right" definition has been found? On the first question, I think Tappen den's discussion is inconclusive. When
he says that the Riemannian thinks what is fundamental in an area has to be dis covered, does he mean that the Rie mannian always thinks there is a fact about this to be discovered? If the an swer is "no , " then the Riemannian is a relativist. As to the second question, Tappenden's answer is that for the Rie mannian, definitions prove their cor rectness by their "fruitfulness," for in stance in organizing our practice well or in playing a role in important further research. Tappenden discusses this view in more detail in other work, but we can address it without leaving this volume by turning instead to the essays on the Riemannians Richard Dedekind and Emmy Noether, by Jeremy Avigad and Colin McLarty, respectively. Avigad writes about Dedekind's Rie mannian approach to developing ideal theory, which he took to mean in prac tice avoiding computation as much as possible. Dedekind instead adopted the axiomatic, set-theoretic approach famil iar to us from contemporary algebra. His work in turn influenced Noether and subsequently a central strand of twentieth-century algebra and algebraic geometry. McLarty presents an overview of how Noether brought this contemporary approach to topology. Continuing Dedekind's Riemannian quest to avoid computation in algebra, Noether took a "purely set-theoretic" approach that was, in her words, "in dependent of any operations" (p. 193). Instead of studying addition or multi plication of the elements of a ring, for instance, she proposed studying partic ular subsets and homomorphisms pre serving the structure of those subsets. In Riemann's terms, she saw these struc tural properties as the "internal charac teristic properties" of rings, rather than the computational properties that she thought were merely "external. " This approach gives special value to homo morphism theorems, as McLarty ably documents. Thus both Dedekind and Noether had views on what the "right" defini tions are in algebra and algebraic topol ogy. How did they think we were to know when we'd found those right de finitions? Avigad suggests some answers for the case of Dedekind, and I want to consider three of these. First, Avigad suggests that Dedekind thought the right definitions in algebra would avoid ·
elements "extraneous" to algebra. This suggestion just pushes the question back, into what it is to be extraneous to algebra. Second, Avigad suggests Dedekind thought the right definitions in algebra would unify the domain be ing defined; as he puts it, "A single uni form definition of the real numbers gives an account of what it is that par ticular expressions are supposed to rep resent" (p. 178). But why should we ex pect that the right definitions will be uniform, rather than having lots of case distinctions? It would be nice if that were so, but wishing doesn't make it so, unless what makes a definition right is that it's the one we want. Correct de finition as wish-fulfillment: if this were Dedekind's view, he would have been a relativist. Fortunately, there is a third possibil ity. Avigad suggests that Dedekind thought the right definitions for a do main would yield properties familiar from other domains. For instance, in ideal theory, Dedekind's "overall goal [was] to restore the property of unique factorization, which [had] proved to be important to the ordinary integers" (p. 171). Then many results following from unique factorization in the integers could be carried over to ideal theory. This is surely an important labor-saving technique. But why should we think that this technique leads to the right de finitions for a domain? There would have to be something "inevitable" about those properties if this technique were to avoid being another type of rela tivism. And indeed Dedekind seems to have thought certain properties were in evitable. Like Frege, Dedekind thought the familiar laws of arithmetic are laws of logic, and he seems to have believed that laws of logic are laws of thought; thus, we can't help but arrive at the properties we do in arithmetic because of the way our minds are constrained to think. Furthermore, he thought that this made inevitable properties in higher mathematics also: as he wrote in his 1 888 essay "Was sind und was sollen die Zahlen?," "every theorem of algebra and higher analysis, no matter how re mote, can be expressed as a theorem about natural numbers-a declaration I have heard repeatedly from the lips of Dirichlet." Thus Dedekind resorted to logicism to solve the fou'ndational cri sis.
I've addressed these matters at length because they help clarify the unity of the subject matter of this essay collec tion. Each essay documents a reaction to the problem of relativism, a problem I argue is central to understanding modernity, not just in mathematics, but in our culture generally. The essays are uniformly a joy to read, and the bibli ography is ample, giving interested readers an extensive springboard for further exploration. I recommend the book highly. [Thanks to my colleague Amy Lara for helpful comments on an earlier draft.] Department of Philosophy Kansas State University Manhattan, KS 66506-2602 USA e-mail: [email protected]
REFERENCES
Gottlob Frege, The FoundatiOns of Anthmetic, translated by J. L. Austin, second revised edi tion, Northwestern University Press, 1 994.
Leonhard Euler. Ein Mann1 mit dem man rechnen kann Leonard E uler. A Man to Be Reckoned With comic album by Andreas K. Heyne, Alice
K.
Heyne (text) and Elena S.
Pini (illustrations); English edition translated by Alice K. Heyne and Tabu Matheson BASEL: BIRKAUSER VERLAG, 2007, HARDCOVER 45 PP.,
18.60 ISBN: 978-3-7643-7779-3,
(GERMAN EDITION);
19.90 ISBN 978-3-7643-
8332-9 (ENGLISH EDITION) REVIEWED BY JOHAN STEN
here are not many mathemati cians, or intellectuals in general for that matter, to whom a comic
book has been devoted. If anyone de serves to be commemorated with such an honor, surely it is Leonhard Euler 0707-1783) on the occasion of his 300th anniversary. The challenge facing such a project is how to present the life and work of the principal character in an interesting and appealing fashion: Euler is more known for his immense scientific out put and his love for peaceful family life. But, the authors show, Euler's life was in fact quite eventful. Throughout his career, Euler was in the service of the rulers of Prussia or Russia, which were in a state of war and political turmoil. Euler had to cope with complicated per sonalities, not least his employer Fred erick, the king of Prussia. Moreover, Euler's interests ranged far beyond pure mathematics and theoretical physics. As many panels suggest, he was deeply in volved in solving engineering problems, such as designing naval vessels, study ing projectile motion, and designing telescopes (the "high-tech" of the time). On the other hand, the mathematical formulas and theories we all know and admire are completely absent. One wonders whether it would have not been possible to flash up some of the celebrated expressions. The first 12 pages of the comic fo cus on Euler's prodigious childhood, his studies of the Classics in Basel, his ac quaintance with the Bernoulli dynasty, and his private studies of mathematics under the auspices ofJohn Bernoulli the elder. Pages 13-19 .describe events in chaotic St. Petersburg during Euler's first stay there 1727-1741. During this pe riod of concentrated scientific investi gation in various fields, which could be called his formative years, Euler also suffered from a grave illness which would cost him the sight of his right eye. Pages 20-37 illuminate Euler's stay in Berlin in the service of the Prussian Academy of Sciences, 1741-1766, dur ing which he earned his reputation as the Princeps Mathematicorum of his time. Of this period, the book highlights the Silesian wars, Frederick's caprices and misbehavior against Euler, as well as Maupertuis's presidency of the Acad emy and his priority affair concerning the Principle of Least Action. Finally, pages 41-45 are devoted to Euler's second St. Petersburg period, 1 766-1783, during which he composed
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nearly half of his scientific oeuvre. The move to St. Petersburg was a result of an invitation by the empress Catherine the Great, who is pictured as a gracious protector of the arts and sciences, but who also displayed an unusual appetite for men. A special event captured in this section (and on the cover of the al bum) is the dramatic shipwreck suffered by Euler's family and household on their journey to St. Petersburg on the Baltic sea. Also illustrated is the great fire of St. Petersburg in 1771, which de stroyed a large part of that city, and from which Euler (then totally blind) and his manuscripts were neatly res cued. The last scene of the album shows a team of contemporary scientists, some easily identifiable, occupied with the editing of Euler's legacy of manuscripts and letters. In general, the quality of Elena Pini's drawings is good and her palette is qui etly colorful. Occasionally, however, the characters are not easy to identify. As for whom such a comic is suit able, I am in doubt: Is it for school children, for undergraduate university students, or for academic scholars? Of these alternatives, I would opt for the third one. I fear that a school child showing interest in the sciences and the accomplishments of the great heroes of mathematics would be disappointed, and I also doubt whether he or she would be very much amused by it. It is not that the album lacks humour, but the wit is so subtle that it is impossible to grasp without at least some familiar ity with the life, science, and politics of the Enlightenment era. Fortunately, the authors have recognized this by col lecting some helpful "appendices" list ing Euler's life and work, contemporary rulers, scientists, and artists, and by ap pending footnotes explaining some dis joint events and quotations in German (often in a Swiss dialect) or Russian. Ad ditionally, as a challenge, the authors have left several anachronisms (and er rors) for the readers to find (the right answers are promised to be announced on the Euler tercentenary website). I, for my part, take this opportunity to point out some less trivial inaccura cies on pages 34-35, where Euler is first pictured in the company of the mar grave Heinrich von Brandenburg Schwedt. As suggested by the picture, the two men were brought together by
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THE MATHEMATICAL INTELLIGENCER
their love for music. In fact, in the scene, the margrave tells Euler that he very much admires his theory of music (alluding to Euler's Tentamen Novae Tbeoriae Musicae, 1739) and that he considers him to be worthy of teaching his "lazy" daughter something, not nec essarily mathematics. Of course, Euler is a man to be reckoned with, and in the next picture, the margrave's daugh ter, the princess Sophie Charlotte, is seen at her desk crowded with piles of Euler's letters on various philosophical subjects, looking rather desperate while the voice behind the door tells her not to go out before she has read every let ter. In the last picture of this episode, the princess cries out: "This will take years. I may as well go to a nunnery. " Now, what the reader may not know is that this is precisely what happened. The eldest daughter of margrave Friedrich Heinrich von Brandenburg Schwedt and princess Leopoldine of Anhalt-Dessau, whose name in reality was Friederike Charlotte Leopoldine Luise von Brandenburg-Schwedt ( 17451808), would indeed ascend the throne of the protestant nunnery of the city state of Herford (Germany) and rule as its last princess-abbess. Besides, nowhere have I seen evidence that the princess was not diligent. On the con trary, based on the engaging style of the Letters (and I mean of course the orig inal Lettres a une Princesse d'Allemagne
sur divers sujets de Physique et de Philosophie, written 1760-1762 and
published 1768-1772; not the numerous translations that have suffered from un fortunate alterations and omissions), I infer that Euler felt that his pupil was genuinely interested in science. More over, in the Letters there is a passage where Euler recalls an encounter with the margrave's family six years earlier and where Euler's scientific tutoring of the margrave's two daughters had been going on in Berlin already some time before 1760, when (following a move of the margrave's family to Magdeburg) the lectures were pursued in written form. To conclude, I would say that this intellectual comic album is a nicely il lustrated companion to the existing bi ographical literature on Euler. As a work of art it is charming, but unfortunately not quite easily accessible. However, if the book manages to create a serious
interest in the life and achievements of a great mathematician and a human spirit, then it has appropriately justified itself. Technical Research Centre of Finland P.O. Box 1 000 FIN-02044 VTT Espoo, Finland. e-mail: [email protected]
GraBmann (Vita mathematica) by Hansjoachim Petsche BASEL: BIRKHiiUSER, 2006,
58, ISBN 978·3·
7643·7257-6; ISBN 3-7643-7257-5 REVIEWED BY GERT SCHUBRING
� iographies
of mathematicians constitute a key dimension of L mathematical historiography and its literature. Active mathematicians en joy sharing the hopes and disillusions, the successes, sufferings, contests and rivalries of famous mathematicians. Such biographies need not necessarily contain essentially new material-as did, for instance, Laura Toti Rigatelli's biography of Evariste Galois [12] , with her findings on the political motivations for Galois's "sacrifice" of his life. A fresh presentation of knowledge already known that addresses complementary contexts or integrates hitherto separated traditions or research approaches will find interested readers, too. One will therefore open this biography of Her mann GrafSmann with the expectation of additional insights, even if it does not much exceed Friedrich Engel's classical biography of 1 9 1 1 . And even if one does not share the author's opinion that GrafSmann was "one of the most extra ordinary research personalities of the nineteenth century" [ 1 , p. xv], GraB mann is distinguished in fact by sev eral characteristics-autodidact, outsider, idiosyncratic terminology-which make him attractive historically. A dimension complementing traditional biographical literature is the detailed study of the political and social context of Stettin, the town in Pomerania which deter mined the largest part of GraBmann's life. Yet this analysis is based on rather raw-boned categories of political his-
tory (see, for instance, Figure 1 on p. xxi). The present book by Hans-Joachim Petsche, a philosopher at Potsdam Uni versity, is essentially his PhD thesis, which he defended in 1979 at the Piid agogiscbe Hocbschule in Potsdam, then in the German Democratic Republic. One is charmed to see that the author has refrained from deleting the quota tions of Karl Marx and Friedrich Engels in his original version of 1979 from his new version, published after the demise of the GDR. An important question that must be raised by republication after a long pe riod of time concerns the more recent literature on GraBmann published be tween 1979 and 2006. Except for some reordering, all chapters have been adopted from the original 1979 thesis in a more or less unmodified way. For in stance, the chapter analysing GraB mann's mathematical achievements has remained unchanged although there are, among other relevant studies, the book by Zaddach [13] and a number of contributions in the Proceedings of the International GraBmann Conference of 1994 [9]. Disregarding international lit erature, the author even claims that GraBmann "nowadays is a largely un known mathematician" [1, p. xiiil. Like wise, as regards GraBmann's biography and his Nachlass, there are-contrary to the author's assertion [ 1 , p. 1 05]-a number of new documents which have since been unravelled and which are important for the methodology issue (see below) and for Grassmann's ideas on the teaching of language [4]. Given the intense research on the history of nineteenth-century mathe matics since the 1970s, one reads with surprise that this history still waits to be investigated [1, p. xiii]. Given GraB mann's, in fact, extraordinarily innova tive mathematical conceptions, one of the key interests of the historiography of mathematics, and of many mathe maticians, has always been to find their sources. The major obstacle has been GraBmann's own discretion with regard to his sources. This silence is actually a common characteristic of the publica tions presented by many Prussian Gym nasium teachers of the first half of the nineteenth century: They preferred to proudly pretend to intellectual original ity. Any attempt towards progress on
this issue would have to start by ad mitting ignorance and by presenting reasonable tentative hypotheses in or der to attain some approximate result. What is Petsche's approach to this key methodological question? Instead, he starts from an alleged certitude. More strongly than in his original thesis of 1979, he claims to be able to "deter mine" the factors conducive to the emergence of GraBmann's novel scien tific conceptions [1, p. xvii]. Regarding the philosophical-method ological factor, the author is convinced that Schleiermacher's Dialektik was most decisive. Since Schleiermacher's Dialektik presents a recurrent topic in the literature about GraBmann, this needs some discussion. Engel was the first to ascribe to Schleiermacher a key influence on GraBmann, in his exten sive biography of 191 1 . Engel, who had intensely searched for sources, even in archives, had found two curricula vitae which GraBmann had composed for dif ferent exam purposes, and gave a num ber of quotes from both [3]. The sec ond CV of 1833-not of 1834, as Engel wrote erroneously-written for GraB mann's application to be admitted to the first exam for future pastors, referred ex plicitly to Schleiermacher, the then lead ing Protestant theologian and major au thority within the Prussian Protestant church. While the applicant claimed he understood nothing of Schleiermacher's lecture on Dialektik in his second term, he professed to have been able to ap preciate the latter's lecture in the last term (on psychology) as a means for heuristics, to find the "Positive" on one's own [3, p. 2 1 f.]. Without reflecting on the purpose and the occasion of this text, and a possible theological mean ing of the "Positive, " Engel understood this CV to be an objective source, de ducing from it his own general assess ment that it was Schleiermacher who had crucially influenced GraBmann's mathematical approaches [3, p. 28]. Albert C. Lewis, in his study of 1977, took up more systematically the ques tion of the conceptual basis of GraB mann's ideas. He dwelt upon Engel's assertion of the decisive character of Schleiermacher's influence, trying to show that Schleiermacher's lectures on Dialektik had indeed provided that con ceptual basis. Using Engel's quotes from the 1833 CV, Lewis no longer mentions
this text's specific nature, dissociating it from a concrete purpose and date [5, p. 109, n. 24]. Although admitting that the Dialektik contains but few references to mathematics, Lewis maintains that mathematics "is presumably the subject of the Dialektik the same as any other branch" of knowledge [5, p. 1 12] . Lewis tried to establish such a relation be tween the Dialektik and GraBmann's key opus, the Ausdebnungslehre of 1844, the so called A 1 . Strangely enough, the bulk of his study consists in a methodological analysis of the A1 itself, with few references to Schleier macher's Dialektik. Lewis's principal ap proach is to highlight opposed mean ings and dualities in GraBmann's approaches in constructing mathemati cal knowledge-with the tacit assump tion that dialectics is a method of pro ceeding by opposites [5, p. 1 1 2 and 160f.J. This seems to constitute a funda mental misunderstanding of Schleier macher's approach. Usually Schleier macher's key opus is interpreted along the very well known lines of Hegel's Dialektik. Instead, one must be aware that Schleiermacher exhibits an entirely independent approach and thinking. He understood dialectics according to the original Greek meaning of the term: as a dialogical way of establishing the cer tainty of knowledge. Settling disputes between different views of individuals by means of language, of dialogue, and by applying general rules, dialectics has the task of establishing an organism of knowledge, to act as a Wissenscbafts lehre. Starting from common knowledge and achieving certain knowledge by di alogue of the partners clarifies what "knowledge" means and how in it thinking and experience are related (see [7, p. 1 16ff.]). Petsche, in his PhD thesis of 1979, although apparently not aware of Lewis (1977), proceeded analogously. Con vinced of Schleiermacher's determina tive function and based on Engel, he collected the few hints that GraBmann had read Schleiermacher and trans formed them into proof of influence. In the present book version, Petsche has confirmed and elaborated this view of essential determination by Schleierma cher-without, however, demonstrating it concretely in an analysis of GraB mann's work. He restricts himself to as-
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serting a "masterfully implicit application [of Schleiermacher's Dialektik] in Her mann's Ausdehnungslehre' [ 1 , p. 173]. Even in Chapter 4, expressly devoted to the philosophical concepts of the A1, there are but few general references to Schleiermacher's ideas [1, p. 252f.; 270f.]. Actually, Victor Schlegel-GraBmann's first biographer and one who knew him personally-gives an entirely different assessment of Schleiermacher's impor tance for GraBmann. While attributing to Schleiermacher a considerable influ ence, he reports it as a deviation from mathematics. According to Schlegel, GraBmann-after having finished in 1 840 his thesis on the tides, the basis for his later Ausdehnungslehre--did not continue elaborating his new mathe matical conceptions, but diverted him self to philosophical studies of lan guage: The Dialektik of his venerated mas ter Schleiermacher published shortly before attracted him too mightily and tugged him temporarily into a new current, for which he worked jointly with his brother Robert. Due to this current, they elaborated the next year (184 1 ) a philosophical grammar; they used its achieve ments in putting into writing the 'Grundriss der deutschen Sprach lehre ' [outline of German grammar] and the 'Leitfaden fUr den deutschen Unterricht' [manual for German lan guage instruction], both published in 1842. Schlegel stated that "eventually," about Easter 1 842, GraBmann "returned to mathematics with all his forces" [8, p. 4f.]. In my own research for the GraB mann Nachlass [10], I succeeded in find ing the two CVs. The first one, of 1831 for the teacher examination, mentioned Schleiermacher just once and had Greek philology as its focus. It is evident, hence, that the CVs-just as any his torical text-must be interpreted in their proper context. Petsche dismisses this remark [ 1 1 , p. 60f.] by claiming that "the sincerity of his soul" would have ex cluded any "instrumental use" by GraB mann in writing a text [ 1 , p. 279, n. 148]. Actually, there are two texts by GraB mann with differing emphases, a fact
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THE MATHEMATICAL INTELLIGENCER
which makes the requirement of a con textualized interpretation imperative. I am preparing a publication of the two curricula vitae. One recent research study [6] the au thor does mention is the assessment of the arithmetic textbook that the broth ers Hermann and Robert GraBmann published in 186 1 . Contrary to a widely accepted view (see [2]), the author does not understand it as a moment in the movement of axiomatizing arithmetic, but as a constructivist approach [ 1 , p. 2 2 1 ff.] . He does not make explicit, however, his conception of construction and to what extent it excludes an ax iomatic approach. Here again, all de pends on whether the interpretation by an overarching Schleiermacher frame work is acceptable or not. This book underscores the dimen sion of the methodological challenge in understanding historical texts in mathe matics, even for a biography.
[8] Victor Schlegel,
Hermann Grassmann: sein
Leben und seine Werke
(Leipzig: Brock
haus, 1 878). [9] Gert Schubring {ed .}, Hermann Gunther GraBmann (1809- 1877): Visionary Mathe matician, from
Scientist and Scholar: Papers
a
Sesquicentennial
Conference
(Boston Studies in the Phtlosophy of Sci ence, Vol. 1 87). (Dordrecht: Kluwer Acad emic Publishers, 1 996). [ 1 0] Gert Schubring, "Remarks on the Fate of Grassmann's NachlaB", In: [9], 1 9-26. [1 1 ] Gert Schubring, "The cooperation be tween Hermann and Robert Grassmann on the foundations of mathematics," In: [9], 59-70. [1 2] Laura Toti Rigatelli, Matematica su/le bar ricate: vita di Evariste Galois
(Firenze: San
soni, 1 993). [1 3] Arno Zaddach, GraBmanns Algebra in der Geometrie:
mit Seitenblicken
wandte Strukturen
auf ver
(Mannhetm: BI-Wis
senschafts-Verlag, 1 994). Gert Schubring
REFERENCES
[1 ] Hans-Joachim Petsche, GraBmann (Basel: Birkhauser, 2006).
Fakultat fOr Mathematik Universitat Bielefeld Postfach 1 00 1 3 1
[2] L. G. Biryukova; B. V. Biryukov, "On the ax
D-33501 Bielefeld
iomatic sources of fundamental algebraic
Germany
structures: the achtevements of Hermann
e-mail: [email protected]
Grassmann
and
Robert
Grassmann,"
[Russian] Modern Logic, 7 (1 997), no. 2, 1 31 -1 59.
[3] Friedrich Engel,
Grassmanns Leben: Nebst
einem Verzeichnisse der von Grassmann ver6ffentllchten Schriften und einer Ober sicht
des
handschriftlichen
Nachlasses
(Leipzig: Teubner, 1 9 1 1 ). [4] Erika Hultenschmidt,
"Hermann
Grass
mann's contribution to the construction of a German 'Kulturnation' -Scientific school grammar between
Latin
tradition and
French conceptions," In: [9] , 87-1 1 3. [5] Albert Lewis, "Hermann Grassmann's 1 844
Arthur Cayley Mathematician Laureate of the Victorian Age by Tony Crilly THE JOHNS HOPKINS U NIVERSITY PRESS,
Ausdehnungslehre and Schleiermacher's
BALTIMORE, MARYLAND, 2006, HARDCOVER,
Dialektik," Annals of Science
US$69.95, XXII + 610 PP, ISBN 0.8018-8011-4
34 (1 977),
1 03-1 62. [6] Mircea Radu, Nineteenth century contnbu tions to the axiomatizat1on of arithmetic: a historical reconstruction and comparison of the mathematical and philosophical ideas of Justus GraBmann, Hermann and Robert GraBmann, and Otto Holder.
PhD thests,
Btelefeld Untverstty, 2000.
[7] Wolfgang Rod, Dialektische Philosophie der Neuzeit.
Vol.
1:
Von
(MOnchen: Beck, 1 97 4).
Kant
bis
Hegel
REVIEWED BY HENRY E. HEATHERLY
I
ony Crilly has written a definitive biography of Arthur Cayley. Not only is Cayley's mathematical and scientific work discussed thoroughly, but careful attention is given to his pro fessional career, to his personal life, and to his many colleagues and friends. First and foremost, Cayley was a mathematician. He also did important
work in astronomy and physics, as well as making significant contributions to a widely used legal manual. Cayley was one of the most prolific mathematicians of all time, publishing well over nine hundred papers [2]. Arthur Cayley was born on August 16, 1 82 1 , at Richmond upon Thames, a genteel town outside of London. His family was visiting England at the time, on home vacation from St. Petersburg, Russia, where his father, Henry Cayley, was a merchant. Arthur lived his early childhood in St. Petersburg, until his family returned to England when Arthur was 10. There Arthur first attended school, having been tutored at home while in Russia. At age 1 4 he was en rolled in the senior department of King's College in London. There he received a comprehensive mathematical educa tion. Cayley was well prepared for the next step in his formal education, Trin ity College, Cambridge University, which he entered in 1 838. There he was "A Cambridge Prodigy," as Crilly em phasizes with the title of his second chapter. Cayley's course of study lasted for ten terms, culminating in an honors degree after taking the six-day Tripos exams in January 1842. In this highly competitive exam, Cayley scored well above all the other competitors, earn ing the title of "Senior Wrangler." Soon afterward he became the First Smith's Prizeman. The Smith's Prize examina tion was another week-long schedule of competitive tests, 1 13 questions of fered over a five-day period. Even while preparing for these ar duous exams, Cayley was doing math ematical research. His first published paper appeared in May 1 84 1 . This pa per, "On a Theorem in the Geometry of Position, " related algebra to geome try and made use of determinants, though Cayley did not use that term. In terestingly, the young author made the contribution anonymously, "from a cor respondent." With this outstanding undergraduate record, Cayley was an obvious choice for a Cambridge fellowship. Unique among Cambridge colleges at the time, Trinity required fellowship aspirants to take further competitive exams in the classics as well as in mathematics. In September 1842, Cayley took these ex ams. Twenty-two candidates competed
for seven fellowships. Cayley did well on a diverse array of topics, from the philosophy of Plato, Aristotle, and Locke to finding the variation in the moon's orbit. Cayley was selected as a Trinity fellow in October 1842. After visiting Switzerland and Italy, the new Cambridge Fellow turned to re search with delight. He quickly pub lished papers on geometry, elliptic func tions, mechanics, and algebra. He began a long correspondence with George Boole on n-dimensional deter minants and other algebraic topics. Cay ley became familiar with the new sub ject of quaternions, introduced by Hamilton in 1843, and in December 1844 submitted his first paper on that topic, making connections between ro tations and quaternions. In January 1845 Cayley discovered the eight dimen sional, nonassociative, noncommutative algebra that became known as "the Cay ley numbers . " He published these re sults in a postscript to a paper on el liptic functions [1). Here Cayley's habit of publishing his results quickly served him well, for John Graves had discov ered this algebra in December 1843 and had communicated this to Hamilton. Hamilton (1844) noted that Graves's "octaves" are nonassociative, an at tribute that Hamilton felt detracted from their worth. Graves did not publish his results until after Cayley's paper had ap peared [3). Also in 1845, Cayley began publishing papers in what would even tually become a life-long work, invari ant theory. His early work on the sub ject received a tepid response in England. But Cayley's reputation as a mathematician was rising as his publi cation record grew in size and diver sity. Of the 500 or so students who en tered Cambridge with Cayley, more than half went into the Church of En gland after graduation. The next most popular career choice was the law. In April 1846, Cayley took the decisive step onto this second path: He entered Lincoln's Inn as pupil barrister. At that time the four legal inns in London also served an educational function. Each of these inns had a library, and the pupil barristers were supervised by working barristers. The young Cayley was a pupil of the leading conveyancy coon sui and barrister, ]. H. Christie. Cayley rapidly gained expertise in this area, but
he also continued working at mathe matics. Sometime in 1847, Cayley met another mathematician who was work ing as a barrister only a short walk away from Lincoln's Inn, James Joseph Sylvester. They became friends and mathematical confidants. It was a close relationship that lasted until the end of Cayley's life, but never resulted in a joint publication. Fortunately for historians, a considerable amount of the written cor respondence between the two has been preserved. Many of these letters can be found in [5). Cayley led a measured, tranquil life, developing in his legal career and grow ing in mathematical power and renown. A common bond of mathematics dom inated his immediate circle of corre spondents, which included Sylvester, Boole, Kirkman, Salmon, and Hermite. Cayley maintained contact with Cam bridge by taking the role of examiner for the regular Trinity College internal examinations. After three years as a le gal pupil he received full membership in the Society of Lincoln's Inn. He was a barrister. In the 1850s, Cayley's mathematical scientific reputation flourished. In 1852, he was elected a Fellow of the Royal Society of London. In 1857 he became a fellow of the Royal Astronomical So ciety, and in 1859, he was awarded the Royal Medal of the Royal Society of Lon don. His professional interest in astron omy had begun in 1 855, and his first paper on that subject, on lunar theory, appeared in 1857. The subject of lunar motion was one to which he would re turn. During the decade, he had made substantial contributions to the theory of permutation groups, further devel oped the notion of a matrix, and ex plored the algebra of matrices. In both group theory and matrices, Cayley made connections with Hamilton's quaternions. In one of his 1858 papers, he gave what became known as the Cayley-Hamilton Theorem for n-by- n matrices. However, he only gave a proof in the 2-by-2 case, writing that he had "not thought it necessary to un dertake the labor of a formal proof. " This hand-waving argument drew criti cism from Boole. A lack of rigor also appeared in Cayley's attempt to prove the fundamental theorem of algebra in 1859. There he showed the desired al gebraic result was equivalent to a geo-
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metrical statement which he said was "a matter of intuition. " Once Cayley had convinced himself of the truth of a mathematical statement, he would often verify it in some elementary cases and then just assert the general statement. By the beginning of 1859, Cayley's need for an academic position, and his suitability for one, was clear to Cayley and to his friends. Many of the latter al ready held suitable positions: Thomson, Stokes, and Boole held chaired profes sorships, while Salmon and Sylvester were lecturers. Cayley had unsuccess fully applied in 1856 for the chair of natural philosophy in Aberdeen, and tried to secure a position at the pro posed, but never formalized, "Western University of Great Britian," in Wales. In 1863, the Sadlerian Chair in pure mathematics was established at Cam bridge. Cayley applied for this chair and was elected to it in June of 1863. Perhaps the most important conse quence of his secure academic position was that Cayley could settle down. In September 1863, he married Susan Mo line. This was a lifelong and tranquil marriage, which produced two off spring, Henry in 1870, and Mary in 1872. Geometry dominated Cayley's early life as a professor. His inaugural lecture at Cambridge (November 3, 1863) was on analytical geometry. As early as 1846, he had published work on n-dimen sional analytic geometry. In his career he published a large number of papers on a wide variety of geometric subjects. In a short note published in 1865, Cay ley introduced the idea of non-Euclid ean geometry to English readers. In this paper, Cayley referred to Lobachevsky's 1837 paper in Grelle'sjournal, mention ing it as a "curious paper" with certain parts as "hard to be understood. " This serves to illustrate the attention Cayley paid to the mathematical literature, in cluding that in the major continental journals. During this period Cayley was influenced by the work of Chasles and Plucker. He had most in common with Plucker, sharing a belief in the primacy of projective geometry and a commit ment to the analytic method. Cayley's ever growing scientific rep utation, his distinguished university po sition, and his reputation as a man both wise and prudent led to his being elected to several leadership positions in scientific organizations: President of
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the London Mathematical Society (1868-1870); President of the Cambridge Philosophical Society (1869-1870); Pres ident of the Royal Astronomical Society (1872-1874); and President of the British Association for the Advancement of Science (1883). He was awarded the (first) De Morgan Medal of the London Mathematical Society in 1884. Cayley's mathematical and scientific interests remained diverse. He contin ued his great work on invariant theory, which included his monumental 10 memoirs on quantics, the last appear ing in 1878. That year he also stated the representation theorem for groups which we know as "Cayley's Theorem." He addressed most of the then-current topics in algebra, did work in differen tial equations, and continued his long standing work on elliptic functions. The latter led to the only full-length book Cayley published: An Elementary Trea tise on Elliptic Functions, 1876. At a meeting of the London Mathe matical Society in June 1878, Cayley raised a question that puzzled mathe maticians thereafter for almost a cen tury: "Has a solution been given of the statement that in coloring a map of a country, divided into counties, only four distinct colors are required, so that no two adjacent counties should be painted in the same color?" He followed up this query in 1879 with a paper on this "Four Color Problem" in which he ascribes the problem to De Morgan. There Cayley stated, "I have not suc ceeded in obtaining a general proof. " Cayley's interest in graph theory goes back to the 1850s. It was originally mo tivated by questions from organic chem istry. This Jed him in 1857 to introduce the idea of a "tree" in graph theory. In the 1870s, Cayley worked on the prob lem of enumerating "unrooted trees. " These mathematical problems were closely connected to chemistry, and Cayley's theory successfully predicted the existence of some alcohols before they were known to exist in nature. Commentary from some of Cayley's Cambridge students who later became mathematicians of note and the official Cambridge records give one a good idea of what Cayley did in his professional lectures. Until 1886 he was required to give only a single course of lectures for one term each year. The topic for his course varied from year to year and re-
fleeted his current research interests. A five-year sample is as follows: 1877, algebra; 1878, solid geometry; 1879, differential equations; 1880, theory of equations; and 1881, Abel's theory of theta functions. From George Andrew Forysyth, who attended the 1879 lec tures, we have that "old notes were never used a second time," and that the lectures were on Cayley's latest research. Cayley played an important role in the development of many mathemati cians and scientists. Less well known, perhaps, is that he "was a significant fig ure in the movement towards women's education. " He was the first president and chair of the college of what is now Newnham College, one of the first two colleges for women at Cambridge. His best-known female student was Char lotte Scott. She regularly attended Cay ley's lectures, and he helped her later in her mathematicaVprofessional career. (For more on the career of Charlotte Scott, see [4].) In 1892, Cayley's general health de teriorated, very likely due to the long term effect of cancer. Even in this de bilitated state, during the last three years of his life Cayley published 40 papers embracing almost the whole field of pure mathematics. During his last sum mer, he wrote a short monograph on the Principles of Book-Keeping by Dou ble Entry and a paper on quaternions. Until the end, Cayley lived the life of the mind, wholly devoted to intellectual endeavors. Arthur Cayley died at 6 p.m. on Saturday, January 26, 1895. The work under review contains much useful and interesting supple mentary material. This includes a lengthy list of Cayley's community of scholars and friends, each given with their link to Cayley. A nine-page glos sary of mathematical terms is provided, together with an extensive section of supplementary notes and a substantial bibliography. There are 24 pages of photographs, portraits, and illustrations. These include a portrait of Cayley as Se nior Wrangler and photographs of him at ages 35 and 69. The book is remarkably free of er rors. The mathematical typo in the ex ponent of e in note 54 on page 528 is a rare exception. This is a scholarly work of the highest quality. It should be in every university library, and I rec ommend it to all who wish to delve
deeply into the life of Arthur Cayley and mathematical life in nineteenth century Britain. REFERENCES
(1] Arthur Cayley, "On Jacobi's elliptic func tions and quatemions," Philosophical Mag azine
26 (1 845), 208-21 1 .
(2] Arthur Cayley and A. R. Forsyth, (eds.), The Collected Mathematical Papers of Arthur Cayley,
1 4 vols., Cambridge Univ. Press,
Cambridge, 1 889-1 898. [3] John Graves, "On the theory of couples," Philosophical Magazine
26 (1 845), 3 1 5-
320. (4] Patricia Clark Kenschaft, "Charlotte Angas Scott (1 858-1 931 )," 1n Women of Mathe matics ,
Louise S. Grinstein and Paul J.
Campbell, (eds.), Greenwood Press, West port, CT, 1 987, 1 93-203. [5] Karen Hunger Parshall, James Joseph Sylvester: Life and Work m Letters,
Claren
don Press, Oxford, 1 998. Mathematics Department University of Louisiana, Lafayette Lafayette, LA 70504-1 0 1 0 e-mail: [email protected]
H e l mut H asse u nd Emmy Noether. Die l
192�1935 by Franz Lemmenneyer and Peter Raquette (eds.) GOTIINGEN, UNIVERSITATSVERLAG GOTIINGEN, 2006. SOFTCOVER, 301 PP.,
32,00, ISBN 978-3-
938616-35-2 REVIEWED BY WINFRIED SCHARLAU
I
---, oday, Emmy Noether is known even to nonmathematicians, at least by name: Research programs, stipends, research centres, visiting pro fessorships, lecture buildings, and schools are named for her. This posthu mous reputation, however, is in almost paradoxical contrast with the difficulties she encountered during her life. As a woman, as a Jewess, and" also because of her radical-left political convictions,
she never succeeded in finding her feet in the German university system, not even when she was already world fa mous as a scientist. She never had a tenured position in Germany, but rather had to make her living by temporary teaching assignments. Fortunately, she emigrated to the United States (for one century the land of refuge for the per secuted of all countries!) before the Nazi takeover of power. She died, completely unexpectedly, in 1935 before she could build a new circle of mathematical ac tivity around her. How different was the life of Helmut Hasse! After sensational youth work (the local-global-principle for quadratic forms) he was called to Halle in 1925 to be the youngest full professor in Ger many at that time. Calls to Marburg and Gottingen soon followed: There he was confronted with the goal of re-erecting mathematics ruined by the Nazi politics. Due to his conservative political views he was acceptable to the rulers, but there can be no doubt that his princi ple concern was mathematics. As a long-standing editor of Crelle's journal, he exerted a great influence on the de velopment of mathematics in Germany, but he could not stop its decline in the Nazi period. The field of research that established Emmy Noether's fame was abstract "modern" algebra. In Gottingen she as sembled around her a remarkable group of brilliant students to whom she generously left many ideas for further development. Hasse's principal fields of research were algebraic number theory, class field theory, and algebraic func tion fields. He wrote a considerable number of influential textbooks. All to gether, his work seems today less "mod ern" than that of Emmy Noether, but perhaps he is underestimated a little bit, since mathematics has turned to the "great" theories rather than to concrete individual problems. The present book contains as a bio graphical introduction van der Warden's well-known obituary for Noether which appeared in the Mathematiscbe An nalen in 1 935, and a (somewhat aug mented) report by Leopoldt on Hasse's scientific work, which had appeared in Crelle's journal on the occasion of his doctoral golden jubilee. Then follows the principal part, 79 letters (and post cards) from Noether to Hasse over 10
years, 1 925-1935, together with very ex tended and careful comments by the ed itors. Only three letters from Hasse to Noether appear here; the rest are lost. Finally, there is the correspondence of several mathematicians about Emmy Noether's premature death, and a short exchange of letters between Hasse and Fritz Noether concerning her estate. Reading this, one is reminded of the tragic fate of Fritz Noether, who sought refuge in the Soviet Union and perished in Stalin's prisons as an alleged German spy. The book is augmented by ex tended indices of names, keywords, ref erences, and short biographies (1 to 1 0 lines) o f the mathematicians mentioned in the text. The themes of the correspondence come from four large fields: axiomatic algebra, class field theory, central sim ple algebras and their arithmetics, and function fields. The first is mentioned only occasionally; for the last, Noether follows, mostly with admiration, Hasse's progress (proof of the Riemann conjec ture for elliptic curves) without con tributing very much herself. Therefore, most interesting are the contributions to the other two fields where Noether's and Hasse's interests come closest together, namely, at the meeting point of the the ory of algebras and class field theory. As is well known, Hasse, Noether, and Brauer were the first to prove the local global-principle for central simple alge bras (or the Brauer group), and, in con nection with this, the fact that over an algebraic number field every such alge bra is cyclic. This is perhaps the most important result of their cooperation. The reader witnesses in this correspon dence the first appearance of Galois co homology in class field theory. The ed itors' precise and informative comments greatly facilitate the understanding of the relevant sections. It is also very inter esting to follow Hasse's attempts to erect local class field theory (even though his own letters are not available). The foun dation of this theory (without resorting to global means) is perhaps one of Hasse's lasting achievements. Overall, the exchange of letters is closer to Hasse's interests than to Noether's. That is perhaps not an accident, because Emmy Noether, more than most, was able and willing to adjust to the inter ests of others and to stimulate and to encourage them.
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Though mathematics is the focus of the present correspondence-it is the thread that connects such different characters as Hasse and Noether many personal matters are touched upon too. Three motifs occur again and again. First, Emmy's concern for her many students, their mathematical and personal well-being: Even when she herself was forced to emigrate, she sympathised with their fate. Second, her concern about mathematics in Got tingen and in Germany; and finally, her admirable optimism and her firm belief in a better future. The present book is a perfect ex ample of how historical mathematical sources should be reviewed and pre sented. History of mathematics has to orient itself on the principal mathe matical developments of the relevant period. The editors were confronted with the particular difficulty that one half of the correspondence is missing. One would expect, therefore, that some sections would be more or less in comprehensible. Through careful de tailed work, however, the editors have reconstructed the connections so that practically everything can be under stood and put into perspective. They deserve the thanks of everybody inter ested in the modern history of number theory. Mathematisches lnstttut Universitat MOnster EinsteinstraBe 62 481 49 MOnster Germany e-mail: doro.scharlau@t-online. de
Two Cultures: Essays in H onour of David Speiser by Kim Williams (ed.) BASEL, BOSTON, BERLIN, BIRKHAUSER VERLAG, 2005, 201 PP., US$59.37, ISBN 10-3-7643-71862, ISBN 13-978-3-763-7186-9 REVIEWED BY AMIR ALEXANDER
,.....--, clecticism is no vice in a festschrift dedicated to the life and accom plishments of a person rather than
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to a single overarching theme. This is especially the case in a collection hon oring David Speiser, a man whose in terests span the great divide between the "two cultures" of the humanities and the sciences, and whose investigations delved deeply into each. A highly suc cessful theoretical physicist, Speiser was also the general editor of the collected works of the Bernoulli mathematical clan and wrote on issues ranging from the history of science to medieval ar chitecture. It is only fitting that essays in his honor by friends, colleagues, and students would be just as diverse, as in deed they are in this volume. But Speiser's friend, Kim Williams, who put together and edited the col lection, aspires to more than eclecti cism in Two Cultures. The volume's underlying theme, she tells us, is "in terdisciplinarianism," and although she never explicitly defines the term, her introduction makes it clear what she has in mind. Interdisciplinarianism is more than just working and writing in different areas: It is an investigation of the interconnections between dif ferent fields of study, their interde pendence, and shared history. Instead of taking for granted conventional dis ciplinary boundaries, interdisciplinari anism questions them and uses ap proaches developed in one field to illuminate another. I am wholly in agreement with Ms. Williams about the merits of such an approach and share her concerns about the intellec tual narrowness of disciplinary over specialization. I am not sure, however, whether this volume, for all the many interesting essays it contains, truly ad dresses the challenge set to it by its editor. Two Cultures is a collection of 15 es says divided into four sections entitled "The Sciences," "The History of Sci ence, " "The Arts, " and-somewhat in congruously-"Nuclear Arms. " The last of these seems to have been added to accommodate the authors' interests and has little to do with the book's cen tral theme. But the other three, as well as being representative of Speiser's broad range of interests, clearly form a conceptual whole: "The Sciences: is representative of one culture. "The Arts" of the other, with the " History of Science" forming a middle ground that requires familiarity with both sides of
the divide. Whether this scheme facili tates the building of bridges between the two cultures or merely restates the divide depends on the essays them selves. The five contributions included un der "The Sciences" are by Speiser's physicist colleagues, and they discuss specific scientific questions from a somewhat broader perspective than would be possible in an ordinary sci entific publication. Jean-Pierre Antoine's essay "David Speiser's Group Theory" discusses Speiser's own contribution to group theory and its application to par ticle physics, and David Ritz Finkel stein's "Whither Quantum Theory" pro vides an overall perspective on the current state of physics and quantum theory in particular. Laszlo Grenacs's "The Direct Determination of Induced Pseudoscalar Current" combines a memoir and institutional history of the Centre de Physique Nucleaire in Louvain, with a technical exposition of a series of experiments undertaken there. The most fascinating article of the group, and possibly of the entire col lection, is Giuseppe La Rocca and Luigi Radicati di Brozolo's "In Praise of Asymmetry," which discusses the work of Pierre Curie. Curie, according to the authors, had learned his mathematics from Camille Jordan, and was therefore one of the few physicists of the age who could express his work in precise math ematical terms. Curie was deeply inter ested in the mathematical symmetries underlying physical phenomena, and most particularly in the point at which these symmetries ultimately dissolve. It is only the breakdown of perfect sym metry, he insisted, which makes physi cal phenomena possible. This convic tion, according to the authors, became a guiding theme of his experimental program. Of the essays in Two Cultures, La Rocca and di Bozolo's piece is among those that come closest to a true inter disciplinary approach. It shows how, for Curie, advanced mathematics was not only a necessary tool for his work in physics, but was almost indistinguish able from it. And though it never strays from the confines of the "science" side of the great divide, the article does open the door to a serious cultural study of mathematical perfection and symmetry
and its role in early twentieth-century thought. The last essay in the "Science" sec tion, Donal Hurley and Michael Vandyck's "An Observation about the Huygens Clock Problem," develops and expands the theory of two-pendulum clocks. In its use of modern methods and notation to deal with a classical problem, it is no different from some of the essays in the following section, ded icated to ''The History of Science. " In particular, Frans A. Cerulus's " Daniel Bernoulli and Leonhard Euler on the Jetski" takes a similar approach, using the tools of modern physics to present the work of the two great eighteenth century geometers on the principles of propulsion used today to power jetskis. While this methodology is undoubtedly valuable for recovering the technical content of historical treatises, it does more harm than good for the cause of "interdisciplinarianism. " By translating old texts into a modern format, the au thors unwittingly impose present-day disciplinary boundaries onto a recalci trant past, implying that our current cat egories are the "correct" ones. The pos sibility of challenging the specialized disciplines of the present by looking at a historical setting in which very differ ent categories prevailed is effectively erased. Giulio Maltese, in his "On the Changing Fortunes of the Newtonian Tradition in Mechanics, " shows con siderably more historical sensitivity when tracing the evolution and chang ing fortunes of Newton's approach to mechanics, and in particular his second law of motion. The same can be said of Patricia Radelet de Grave's fine essay ''Studies of Magnetism in the Correspondence of Daniel Bernoulli," which is a careful textual study of Bernoulli's correspondence that never strays far from its sources. Piero Villa gio's "On Enriques's Foundations of Mechanics" is an intriguing essay on Federigo Enriques's attempt to axiom atize mechanics. Enriques's unconven tional suggestions and his criticism of Ernst Mach could potentially be used to anchor a broader interdisciplinary study of early twentieth-century con ceptions of the foundations of physics, but Villagio limits himself to the work of his main protagonist. The last essay in the "History of Sci-
ence" cluster is Sandra Caparrini's "On the Common Origins of Some of the Work on the Geometrical Interpretation of Complex Numbers," which deals with the question of simultaneous in dependent discoveries in mathematics. Caparrini rejects as vague the explana tion that "the discovery was in the air, " suggesting instead that more often than not the different discoverers drew from a common source. He makes a strong case that many of the numerous math ematicians who independently devel oped the geometrical representation of complex numbers (the "Gauss-Argand Plane'') in the early nineteenth century drew on a single work-Lazare Carnot's Geometrie de position (1803). In "The Arts" section, Bernd Linde mann's "An Unusual Sacra Conver sazione by Giovanni Bellini" aims to determine the correct iconographic ti tle of an unconventional painting by Bellini. While this may be a legitimate question in the context of academic art history, it is also an example of the narrow disciplinary focus that Williams professes to challenge. In contrast, the other two essays in the group deal with music, a field that in different times in its history has been associated with each of the "two cul tures, " and is therefore a natural choice for this collection. Sigmund Levarie, in his interesting article "Architecture and Music," com pares the ratios of a musical octave with those used in Medieval and Renaissance architecture. A slightly broader per spective would make the point that these similarities are byproducts of the Renaissance view that both music and architecture reflected the fundamental harmonies of the universe. The other article, Alessio Ageno's "Ancient Astro logical and Musical Analogies in the Re naissance: Palladia's Villa Rotunda and a Geometric Construction by Leonardo, " demonstrates precisely this point by bringing in the intellectual world of the Renaissance-music, astrology, and geometry-into the analysis of a partic ular architectural design. Though fo cused on a single structure, it is an essay that shows the potential of cross disciplinary studies to shed new light on old questions. Does Tbe Two Cultures as a whole succeed in demonstrating the power of interdisciplinary studies, as Williams
clearly hopes? Like the essays them selves, the end result is a mixed bag. Some essays, most notably La Rocca and di Bozolo's essay on Pierre Curie and Alessio Ageno's article on Palladia's Villa Rotunda, do indeed use a cross disciplinary approach to make persua sive and original arguments. But taken together, this eclectic collection does as much to point to the hazards and chal lenges of an interdisciplinarianism as it does to its benefits. To reap the bene fits of an interdisciplinary approach it is not enough to bring together studies from different fields and present them in a single volume . The various disci plines have been going down separate roads for centuries now, and a great deal of preparatory work needs to be done before they can begin to speak to each other once more. Much of this work is historical, and Williams seems aware of this in her in troductory discussion of the history of architectural theory. In this intriguing essay, Williams recounts how architec tural creations in the Renaissance were viewed as microcosms whose features and proportions carried meaning and imitated the structure of the universe. By the nineteenth century, this inte grated view was replaced by a self-con tained architectural theory that set its own internal standards, to the exclusion of other factors. In her critique of con temporary architectural theory, Williams suggests that the deliberate exclusion of external considerations from architec ture has gone too far, and she seeks to recover some of the integrated mean ings that had characterized an earlier age. In essence, she looks back at older frames of reference and the manner in which they were replaced, and uses her insights to challenge our current disci plinary scheme. Such an approach, I believe, can be both effective and beneficial for cross ing disciplinary boundaries and bring ing insights from one field to bear on another. It is unfortunate, therefore, that Williams makes little use of it in edit ing the volume as a whole. As it stands, Tbe Two Cultures is not really an inter disciplinary collection but rather a mul tidisciplinary one in which representa tives of the "two cultures" stand side by side but have little to say to each other. As a result, this volume does as much to restate and entrench the great divide
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between science and the humanities as it does to bridge it. Department of History University of California at Los Angeles Los Angeles, CA 90095-1 473 USA e-mail: [email protected]
Mathematics and the Divine. A H istorical Study edited by Teun Koetsier and Luc Bergmans AMSTERDAM, ELSEVIER, 2005, HARDBOUND, 716 PP., US $250, ISBN-13: 978-0444-50328-2, ISBN-10: 0-444-50328-5 REVIEWED BY JEAN-MICHEL KANTOR
"God is like a skilful Geometrician." -Sir Thomas Browne (1605-1682), Religio Medici I, 1 6 "As God calculates and executes thought, the world comes into being." -Gottfried Wilhelm Leibniz (1646-1716), Samtliche Schriften und Briefe, 1923, ser. VI, vol. 4A, p. 22 "Par Dieu j'entends un etre absolu ment infini, c'est a dire une sub stance consistant en une infinite d'at tributs dont chacun exprime une essence eternelle et infinie." -Baruch Spinoza (1632-1677),
Ethics n recent years, science and religion have been opposed in numerous books and articles [1], but let's face it, there is one science which, since its (unknown) origins, has been closely and positively connected to religion: mathematics. To explain this proximity one notices common features, for ex ample, a claim for universality, for the eternity of concepts, and certainty of truths [3]. But some other explanations can be given as well. In this 700-page anthology of 35 in dependent chapters written by 30 au thors, the editors, Luc Bergmans, a Dutch cultural historian with a special
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interest in religion, and Teun Koetsier, a historian of mathematics from Ams terdam, have gathered a loose potpourri on the theme of "religion and mathe matics. " Some articles address minor topics, such as numerological observa tions in Chinese magic squares, Michael Stiefel's biblical numerology (an ances tor of the bible code), or the significans movement in Holland in the late nine teenth century. 1 But some articles con tain interesting analyses of parallels in the religious feelings and prescientific or scientific work of major figures, such as Kepler, Newton, Euler, Leibniz, and Cantor. The order of the chapters is essen tially chronological and relates to the Western world. The historical times considered are: • the early pre-Greek period; • the Greek antiquity and its medieval and renaissance heirs; • the birth of modernity and of the sci entific revolution. We mention only in passing the chapters connected to art, religion, and mathematics, exemplified by such ma terial as the sempiternal divine propor tion, or the "Geometry of the Divine, " a s described by Proclus ( a neoplaton ist of the Plotinian school) and repre sented in Constantinople's St. Sophia by the emperor Justinian. The heart of the book consists of examples taken from classical figures from the middle ages and the beginning of modern times with the birth of science as we under stand it today. Historians have written on the positive influence of Christian ity in the development of modern sci ence [6], and we can test their theories here with some examples, well known (Nicolas de Cusa, Lull, Kepler) or not. C. de Pater, in a remarkable article on Newton, relies on many previous Newtonian studies to distinguish the attitude of Newton, clearly opposed to Cartesian mechanics. In short, New ton needs, more than Descartes, God's help to establish distant action. Another striking article in this book, by E . Sylla, concerns Gregory of Rimini (in the mid fourteenth century) deciding how many angels can dance on the head of a pin, a question dismissed as a scholastic stu pidity in the seventeenth century. Here we have a direct contact between reli gion and mathematics in progress (premises of analysis).
One central agent of the connection between mathematics and religion is the concept of infinity (but it is not the only one!). From its first appearance under the name of "apeiron" with Anaximander of Miletus (610-546 Be), to the recent work of Hugh Woodin [9], this is a permanent theme in mathe matics-H. Weyl even wrote that math ematics is "the science of the Infinite" [8]-but the theme is also permanent in the philosophy of mathematics, and the word End is not yet written. This is a fascinating story that has inspired philosophers and theologians, poets and mathematicians. One can follow the birth of the concept, corresponding to attributes of God (or space or time) with mathematics filling more and more space through the centuries, until the Cantorian parthenogenesis between mathematics and religion (but still with a trace of its origins with the theolog ical Absolute to escape the paradox of the set of all sets). Religious and pre scientific thoughts are intimately mixed in the view of infinity in the sefirot of kabbalistic thinkers in medieval Spain ("Is the Universe of the Divine divid able?" by M.-R. Hayoun). In a different style, the fascination of the work of Cardinal de Cusa (14011464; see the Chapter by ]. M. Counet) relies on the variety of the new ideas he brings, from the geometrical repre sentation of the Infinite in his theology to the symbolic role of mathematics in his famous "learned ignorance. " A second theme running through many chapters of the book is the search for a global vision uniting mathematics and religion. This can be found first in the school of Pythagoras, the object of "The Pythagoreans," an interesting study by Reviel Netz. Netz suggests, through an analysis of the mystery of the Pythagorean cult, that religion and mathematics might be able to interact, because they share some way of "ra tionalizing mystery" through analogies and metaphors. The global unity of mathematics with religion is central in Plato's work, and in his followers' such as Plotinus and Proclus, but also much later in modern times (de Cusa, Leib niz's philosophical system). This global vision is still present in modern times, for example, in the unique vision of Fa ther Pavel Florenski (see the study of S . Demidov and the detailed study o f [2]),
which includes the philosophy of divine wisdom in the Orthodox tradition, the connections between physical and spir itual worlds expressed though inverted perspective, or even the use of imagi nary numbers. Mathematics can also serve as an in spiration in the proof of the exis tence of God, or as a "Staircase lead ing to God." Descartes for example proves the existence of God through its essence, and this relies on the "essence" of infinity. The Cartesian "essences" will play a role in mod ern philosophy of mathematics (see for example Godel's mathematical ontological proof of the existence of God, "IF it is possible for a rational omniscient being to exist THEN nec essarily a rational omniscient being exists. '' (see [5), and also Godel's un published manuscripts). The twentieth century is hardly rep resented in the book. There should have been a Chapter on Godel's Pla tonism as a kind of modern religion, on the intuitionist school with Brouwer and Hermann Weyl's philosophical views, the mystical interpretations of the Pythagorean texts by Simone Wei! as seen in her conversations with her famous mathematician-brother Andre [7), and the extraordinary mystical views of Alexander Grothendieck [4). In conclusion, the obvious defects in the selection of topics are balanced by a very rich volume, with some in teresting analysis of a historical char acter. There is a need for further de velopments on the subject, its recent aspects, and its philosophical dimen sions. REFERENCES
[ I ) Conference, "Science and Belief," 2006, http://beyondbelief2006.org/. [2] Loren Graham, Jean-Michel Kantor, "A comparison of two cultural approaches to mathematics, France and Russia, 1 8901 930," ISIS (March 2006) and www.math. jussieu.fr/�kantor/isis.pdf. See also Nam mg God, Naming Infinities, Mysticism and Mathematical Creativity.
Harvard University
Press, to appear. [3] lvor Grattan-Guinness, "Christianity and mathematics:
kinds of link,
and
songes ou Dialogue avec le Bon Dieu," http://www .grothendieckcircle. org/. [5] Kurt Godel, Collected works Volume Ill, The Clarendon Press, Oxford University Press, Oxford, p. 403.
[6] Alexander Kojeve, "The Christian Orig1n of Modern Science," St. John 's Review Win ter (1 984) 22-26. [7] Simone Weil, Oeuvres completes , Galli mard, Paris, 1 988. [8] Hermann Weyl, The Open World (God and the Universe, Causality, Infinity) , Yale,
1 933,
Repnnt Oxbow Press, 1 989. [9] Hugh W. Woodin, "The continuum hypoth esis. 1 . , " Nottces Amer. Math. Soc. 48 (2001) 567-576. Part II. Notices Amer.
Math. Soc.
48 (2001 ) 681 -690.
Universite de Paris VII 75251 Pans Cedex 75005 France
Altogether the book consists of 1 4 chapters, each starting with a newspa per advertisment or a short story lead ing to some implausible idea. In the course of the chapter the problem is discussed and solved in full detail. Gen eralizations or related problems are dis cussed as well. For example, have you ever heard of a body rolling uphill? A double cone on two inclined rails can do it (of course, the angles in the model have to be in proper relation). Or of a solid of finite volume but infinite surface area and, vice versa, of infinite volume and finite surface area? Look at Torricelli's trumpet and Huygens's and de Sluze's drinking vessel. And as for dimension problems: The two-dimensional unit hypersphere is a disk of radius 1 , and we know from geometry classes that its volume, which in this case is its area, is 3.141. . . . Similarly, the volume of the three-di mensional unit hypersphere is 4. 188 . . . . What about dimensions n 2: 3? I will not present here the respective formulae (which you'll find discussed in full detail in Havil's book) but only mention a surprising fact: The volume takes a maximum close to dimension five! That is: the five-dimensional unit hypersphere has volume 5.263, 15 which is greater than not only the respective volumes in dimensions 1 ,2,3, and 4, but also, those in n = 6,7,8, . . . , etc. Indeed, the volume as a function of the continuous variable n attains its maximum at 5.277 . . . and then tends to zero with increasing n. Some perplexing facts arising from probability are included too. The 'Birth day Paradox' asks for the likelihood of two individuals sharing the same birth day. It is obvious, ignoring leap years, that among 366 people at least one rep etition occurs. However, among only 23 people there is a 50:50 chance of at least 2 coincident birthdays! Do you know what triskaideka phobia means? It is the Greek com pound made from tris, 'three' ; kai, 'and'; deka, 'ten'; and phobia, 'fear'. So we learn in the chapter named 'Friday the 13th' and numbered, of course, 1 3 . Here w e find some unbelievable prop erties of our Gregorian calendar: the 1 3th of a month is more likely to fall on a Friday than on any other day of the week; there is at least one Friday
1T =
e-ma1l: [email protected]
t1T =
Nonpl ussed ! Mathematical Proof of Implausible ideas by julian Havil PRINCETON, NEW JERSEY, OXFORDSHIRE, UNITED KINGDOM, PRINCETON U NIVERSITY PRESS, 2007, 208 PP. US$24.95,
14.95, ISBN: 13: 978-0-691-
12056-0 REVIEWED BY CHRISTINA BIRKENHAKE
Alice laughed: 'There's no use try ing', she said; 'one can't believe im possible things'. 'I daresay you haven't had much practice', said the Queen. 'When I was younger, I always did it for half an hour a day. Why, some times I 've believed as many as six impossible things before break fast'. 'Where shall I begin', she asked. 'Begin at the beginning', said the king, 'and stop when you get to an end'. (Lewis Carroll)
the
rare occurrences after 1 750," Physis XXXV II (2000), Nuova Serie, Fasc. 2, 467-500.
[4] Alexander Grothendieck, "La clef des
----, his is Julian Havil's invitation to his intriguing collection of im plausible ideas.
�r=
© 2008 Spnnger Sc1ence+Bus1ness Media, Inc , Volume 30, Number 4, 2008
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the 1 3th in every year but never more than three. These occur in February, March, and November in non-leap years, and in January, April, and July in leap years. The only possibility for Friday the 1 3 in consecutive months is February and March, and this can only occur in a non-leap year; it will hap pen next in 2009, 2 0 1 5 , and 2026. The
first day of a new century can never fall on a Friday, Wednesday, or Sun day. All this follows from Gauss's cal endar formula. In this way, Havil develops some everyday phenomena into intriguing mathematics. The mathematics needed is at high-school level, comprehensible for teachers and students. I recommend
.f1 Springer
the language of science
the book to everyone interested in en tertaining mathematics. Un1versitat Erlangen-Nurnberg Department Mathematik Bismarckstrasse 1 1 /2 D-91 054 Erlangen Germany email: [email protected]
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THE MATHEMATICAL INTELLIGENCER
'?'j@ii,j .I9·JQ:t§l
Robin Wilson
The P h i lamath's A l phabet - S Scalene triangle: Many countries have issued triangular stamps. The triangles are usually equilateral or isosceles right angled (either way up), and examples of scalene triangles, where all the an gles and sides are different, are rare. An early stamp of this type was issued by Colombia in 1 869.
Schickard: The first mechanical cal culating machines appeared in the 1 7th century, following the invention of logarithms by John Napier and the development of the slide rule. In the
Scalene triangle
\
1 620s an early machine was described by Wilhelm Schickard in letters to Johannes Kepler, but it was never built.
Schrodinger: Twenty-five years after Max Planck's original quantum hy pothesis, physicists still had no coher ent understanding of quantum theory, but in 1925 Erwin Schrodinger learned of Louis de Broglie's suggestion that particles can also behave like waves, and found the appropriate partial dif ferential equation to describe these waves.
Seki: Takakazu Seki, also known as 'Seki Kowa', was the first mathematician to investigate determinants, a few years before Leibniz (who is usually given pri ority) contributed to the subject. In 1683 Seki explained how to calculate deter minants up to size 5 X 5, and the Japa nese stamp shows his diagram for cal-
Schickard
culating the products that arise in the evaluation of 4 X 4 determinants.
Snowflake: The delicate structure of a snowflake has six-fold rotational symmetry-rotation by 60° leaves the pattern unchanged-and no two snowflakes have been found that are exactly the same. Their hexagonal form was recognised by the Chinese in the second century BC and was later investigated by Johannes Kepler, Rene Descartes, and others.
Stevin: The Flemish mathematician Si mon Stevin ( 1 548-1620), wrote a pop ular book De thiende [The Tenth] that explained decimal fractions, advocated their widespread use for everyday cal culation, and proposed a decimal sys tem of weights and measures. He also wrote an important treatise on statics that included the first explicit use of the triangle of forces.
Seki
Schrodinger
Please send all submissions to the Stam p Corner Editor,
Robin Wilson, Faculty of M athematics, Computing and Technology The Open Un iversity, M ilton Keynes, MK7 6AA, England e-mail: r.j .wilson@open .ac.uk
88
Snowflake
THE MATHEMATICAL INTELLIGENCER © 2008 Spnnger Sc1ence+Bus1ness Med1a. Inc
Stevin
![The Mathematical Universe [M.U.H.]](https://epdf.mx/img/300x300/the-mathematical-universe-muh_5ba7f346b7d7bc7847901bf7.jpg)
