ERIC C. R. HEHNER
From Booean Agebra to Unified Agebra
oolean algebra is simpler than number algebra, with applications in programming, circuit design, law, specifications, mathematical proof, and reasoning in any domain. So why is number algebra taught in primary school and used routinely by scientists, engineers, economists, and the general public, while boolean algebra is not taught until university, and not routinely used by anyone? A large part of the answer may be in the terminology and symbols used, and in the explanations of boolean algebra found in text books. This paper points out some of the problems delaying the acceptance and use of boolean algebra, and sug gests some solutions. Introduction
This paper is about the symbols and notations of boolean algebra, and about the way the subject is explained. It is about education, and about putting boolean algebra into general use and practice. To make the scope clear, by "boolean algebra" I mean the algebra whose expressions are of type boolean. I mean to include the expres sions of propositional calculus and predicate calculus. I shall say "boolean algebra" or "boolean calculus" inter changeably, and call the expressions of this algebra "boolean expressions". Analogously, I say "number algebra" or "number calculus" interchangeably, and call the expressions of that algebra "number expressions". Boolean algebra is the basic algebra for much of computer science. Other applications include digital circuit de sign, law, reasoning about any subject, and any kind of specifications, as well as providing a foundation for all of mathematics. Boolean algebra is inherently simpler than number algebra. There are only two boolean values and a few boolean operators, and they can be explained by a small table. There are infinitely many number values and number operators, and even the simplest, counting, is inductively defined. So why is number algebra taught in pri mary school, and boolean algebra in university? Why isn't boolean algebra better known, better accepted, and bet ter used? One reason may be that, although boolean algebra is just as useful as number algebra, it isn't as necessary. Informal methods of reckoning quantity became intolerable several thousand years ago, but we still get along
© 2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 2, 2004
3
with informal methods of specification, design, and reasoning. Another reason may be just an accident of edu cational history, and still another may be our continuing mistreatment of boolean algebra. Historical Perspective To start to answer these questions, I'm going to look briefly at the history of number algebra. Long after the in vention of numbers and arithmetic, quantitative reasoning was still a matter of trial and error, and still conducted
3 goats and 20 chickens to be divided equally between his 2 sons, 8 chickens, the solution was determined by iterative approximations, prob
in natural language. If a man died leaving his and it was agreed that a goat is worth
ably using the goats and chickens themselves in the calculation. The arithmetic needed for verification was well understood long before the algebra needed to find a solution. The advent of algebra provided a more effective way of finding solutions to such problems, but it was a dif ficult step up in abstraction. The step from constants to variables
is as large as the step from chickens to num
bers. In English 500 years ago, constants were called "nombers denominate" [concrete numbers], and variables were called "nombers abstracte". One of the simplest, most general laws, sometimes called "substitution of equals for equals",
x=y�fx=fy seems to have been discovered a little at a time. Here is one special case [20]: In the firste there appeareth
2
nombers, that is
14x+15y
equalle to one nomber, whiche is
71y .
But if you
marke them well, you maie see one denomination, on bathe sides of the equation, which never ought to stand. Wherfore abating [subtracting] the lesser, that is that is, by reduction,
lx=4y .
15y
out of bathe the nombers, there will remain
Scholar: I see, you abate
15y
14x =56y
from them bathe. And then are thei equalle still,
seyng thei wer equalle before. According to the thirde common sentence, in the patthewaie: If you abate even [equal] portions, from thynges that bee equalle, the partes that remain shall be equall also. Master: You doe well remember the firste grounds of this arte. And then, a paragraph later, another special case: If you adde equalle portions, to thynges that bee equalle, what so amounteth of them shall be equalle. Each step in an abstract calculation was accompanied by a concrete justification. For example, we have the Commutative Law [0]: When the chekyns of two gentle menne are counted, we may count first the chekyns of the gentylman hav ing fewer chekyns, and after the chekyns of the gentylman having the greater portion. If the nomber of the greater portion be counted first, and then that of the lesser portion, the denomination so determined shall be the same. This version of the Commutative Law includes an unnecessary case analysis, and it has missed a case: when the two gentlemen have the same number of chickens, it does not say whether the order matters. The Associative Law [0]: When thynges to be counted are divided in two partes, and lately are found moare thynges to be counted in the same general! quantitie, it matters not whether the thynges lately added be counted together with the lesser parte or with the greater parte, or that there are severalle partes and the thynges lately added be counted together with any one of them. As you can imagine, the distance from
2x+ 3 =3x+ 2
to
x= l
was likely to be several pages. The reason for all
the discussion in between formulas was that algebra was not yet fully trusted. Algebra replaces meaning with symbol manipulation; the loss of meaning is not easy to accept. The author constantly had to reassure those readers who had not yet freed themselves from thinking about the objects represented by numbers and vari ables. Those who were skilled in the art of informal reasoning about quantity were convinced that thinking about the objects helps to calculate correctly, because that is how they did it. As with any technological advance, those who are most skilled in the old way are the most reluctant to see it replaced by the new. Today, of course, we expect a quantitative calculation to be conducted entirely in algebra, without reference to
thynges.
Although we justify each step in a calculation by reference to an algebraic law, we do not have to
keep justifying the laws. We can go farther, faster, more succinctly, and with much greater certainty. In a typi cal modem proof we see lines like
4
THE MATHEMATICAL INTELLIGENCER
Arar=(bab-1Y=barb-1=ar br=Arbr=(AbY=(a-1baY=a-1bra (a1-1b1)2=a1-1b1a1-1b1=a1-1Cb1a1 -1)b1=a1 -1(p,a1 -1b1)b1=JLa1-2b12 (a1-1b1y=JL1+2+... +(,·-1)a1-rb{=JL1+2+. . +(r-1)=JLr(r-1)12 These lines were taken from a proof of Wedderburn's Theorem (a finite division ring is a commutative field) in
[15] (the text used when I studied algebra). Before we start to feel pleased with ourselves at the improvement, let me point out that there is another kind of calculation, a boolean calculation, occurring in the English text be tween the formulas.In the example proof
[15] we find the words "consequently", "implying","there is/are", "how
ever","thus", "hence", "since", "forces", "if ...then", "in consequence of which", "from which we get","whence", "would imply", "contrary to", "so that","contradicting"; all these words suggest boolean operators. We also find bookkeeping sentences like "We first remark ...","We must now rule out the case ...";these suggest the struc ture of a boolean expression.It will be quite a large expression, perhaps taking an entire page. If written in the usual unformatted fashion of proofs in current algebra texts, it will be quite unreadable. The same problem oc curs with computer programs, which can be thousands of pages long;to make them readable they must be care fully formatted, with indentation to indicate structure. We will have to do likewise with proofs. A formal proof is a boolean calculation using boolean algebra; when we learn to use it well, it will enable us
to go farther, faster, more succinctly, and with much greater certainty.But there is a great resistance in the math ematical community to formal proof, especially from those who are most expert at informal proof. They com plain that formal proof loses meaning, replacing it with symbol manipulation. The current state of boolean al gebra, not as an object of study but as a tool for use, is very much the same as number algebra was five centuries ago.
Boolean Calculation
Given an expression, it is often useful to find an equivalent but simpler expression. For example, in number al gebra
xx(z+1) - yX(z-1) - zX(x-y) (xxz + xX1) - (yxz - yX1) - (zxx- zxy) xxz + x - yxz + y - zxx + zxy x + y + (xXz - xxz) + (yXz - yxz) x+ y
distribute unity and double negation symmetry and associativity zero and identity
We might sometimes want to find an equivalent expression that isn't simpler;to remove the directionality I'll say "calculation" rather than "simplification". We can use operators other than
= down the left side of the calcula
tion; we can even use a mixture of operators, as long as there is transitivity. For example, the calculation (for real
x)
2:
xx(x + 2) x2 + 2 Xx x2 + 2 Xx + 1 - 1 (x + 1)2 - 1
distribute add and subtract
1
factor a square is nonnegative
-1
tells us
xX (x + 2) 2: -1 Boolean calculation is similar. For example,
-
-
-
-
And so
{:=:
(a==>b) V (b==>a) -,a V b V --,b V a a V -,a V b V --,b ru t e V tru e
replace implications
V is symmetric excluded middle, twice
V is idempotent
tru e
(a==>b) V (b==>a) has been simplified to 3n· n + n2 = n�' instance 0 + Q2 = Q3 arithmetic
ru t e
,
which is to say it has been proven. Here is another example.
ru t e And so
(3n· n + n2 = n3)
<=
ru t e
,
and so
3n· n + n2 = n3
is proven.
VOLUME 26, NUMBER 2, 2004
5
Solving simultaneous equations can also be done as a boolean calculation. For example,
x + xxy + y = 5 1\ x - xxy + y = 1 x - xxy + y + 2xxxy = 5 1\ x - xxy + y = 1 1 + 2 Xx Xy = 5 1\ x - xxy + y = 1 2 XxXy = 4 1\ x - xxy + y = 1 xxy = 2 1\ x - xxy + y = 1 xxy = 2 1\ x - 2 + y = 1 xxy = 2 1\ x + y = 3 x=1 1\ y=2 V x=2 1\ y=1 x=1 1\ y=2
-
¢=
subtract and add
2 XxXy in first equation
use second equation to simplify first
use first equation to simplify second
These examples show that simplifying, proving, and solving are all the same: they are all just calculation. When an expression is too long to fit on one line, it must be nicely formatted for easy reading, and when a hint is too long to fit on the remainder of a line, it can be written on as many lines as it takes, but we do not consider formatting further here.One point worth mentioning is that subcalculations (if boolean, they are called subproofs or lemmas) can save copying unchanged parts of a calculation through many lines. These subcalcu lations can be done in another place and referenced, or they can be done in-place, nicely formatted, to provide a structured calculation (structured proof). By far the best way to handle subcalculations is provided by win dow inference systems
[21 ] , [2], which open a new window for each subcalculation, keep track of its sense (di
rection), and make its context available. For example, in solving the simultaneous equations, we used the sec ond equation to simplify the first, and then the first to simplify the second.
In this brief introduction to boolean calculation, I have not taken the time to present all the rules.For a com
plete presentation, the reader is referred to
[14) .
see
[7) . A [10) . For further discussion of calculational proofs
A research monograph that uses calculational proof is
textbook on discrete math that uses calculational proof is
[9] , [17) .
Traditional Terminology
Formal logic has developed a complicated terminology that its students are forced to learn. There are terms which are said to have values. There are formulas, also known as propositions or sentences, which are said not to have values, but instead to be true or false. Operators (+,-) join terms, while connectives (1\,
V) join for tru e or false , but that's different from being true or false. It is difficult to fmd a definition of predicate, but it seems that a boolean term like x=y stops being a boolean term and mysteriously starts being a predicate when we admit the possibility of using quantifiers (3, V). Does x+y stop being a number term if we admit the possibility of using summation and product (I, IT)? There are at least three different equal signs: = for terms, and <=> and = for formulas and predicates, with one of mulas. Some terms are boolean, and they have the value
them carrying an implicit universal quantification. We can even find a peculiar mixture in some textbooks, such as the following:
a+ b = a V a+ b = b a and b are boolean variables, + is a boolean operator (disjunction), a+ b is a boolean term (having tru e or false ), a+b = a and a+ b = b are formulas (so they are true or false), and finally V is a
Here, value
logical connective. Fortunately, in the past few decades there has been a noticeable shift toward erasing the distinction between being true or false and having the value
tru e or false . It is a shift toward the calculational style of proof.But
we have a long way to go yet, as I find whenever I ask my beginning students to prove something of the form
af!Jb where
$ is pronounced "exclusive or". They cannot even start, because they expect something that looks
(af!Jb)=tru e or a�b , they are (a�b)=tru e confuses them again because it seems to have too many verbs. If I ask them to prove something of the form aVb , they take an unwittingly con structivist interpretation, and suppose that I want them to prove a or prove b , because that is what "do a or b means in English.The same lack of understanding can be found in many introductory programming texts, grammatically like a sentence. If I change it to either of the equivalent forms happy because they can read it as a sentence with a verb. But
"
where boolean expressions are not taught in their generality but as comparisons because comparisons have verbs. We find
whileflag= tru e do something but not the equivalent, simpler, more efficient
whileflag do something
6
THE MATHEMATICAL INTELUGENCER
because flag isn't the right part of speech to follow
. Our dependence on natural language for the un
while
derstanding of boolean expressions is a serious impediment. Traditional Notations Arithmetic notations are reasonably standard throughout the world. The expression
738 +45= 783 is recognized and understood by schoolchildren almost everywhere. But there are no standard boolean nota tions. Even the two boolean constants have no standard symbols. Symbols in use include
true false
t
tt
T
1
0
1=1
f
ff
F
0
1
1=2
Quite often the boolean constants are written as 1 and
0
, with
+
for disjunction, juxtaposition for conjunc
tion, and perhaps - for negation. With this notation, here are some laws.
x(y +z) = xy + xz (x+y)(x +z) x + yz x + -x = 1 0 x( -x) =
=
The first law above coincides with number algebra, but the next three clash with number algebra. The near-uni versal reaction of algebraists to notational criticisms is: it doesn't matter which symbols are used; just introduce them, and get on with it. But to apply an algebra, one must recognize the patterns, matching laws to the ex pression at hand. The laws have to be familiar. It takes an extra moment to think which algebra I am using as I apply a law. The logician R. L. Goodstein
(8] chose to use 0 and + as a variable and x
down a little more. A big change, like using
1
the other way around, which slows me
as an operator, would slow me down a lot.
I think it matters even to algebraists, because they too have to recognize patterns. To a larger public, the reuse of arithmetic symbols with different meanings is an insurmountable obstacle. And when we mix arithmetic and boolean operators in one expression, as we often do, it is impossible to disambiguate. The most common notations for the two boolean constants found in programming languages and in pro gramming textbooks seem to be
true
and
false
. I have two objections to these symbols. The first is that they
are English-based and clumsy. Number algebra could never have advanced to its present state if we had to write out words for numbers.
seven three eight + four five = seven eight three is just too clumsy, and so is
true 1\ false
V true == true
Clumsiness may seem minor, but it can be the difference between success and failure in a calculus. My second, and more serious, objection is that the words
tru e
and
false
confuse the algebra with an appli
cation. One of the primary applications of boolean algebra is to formalize reasoning, to determine the truth or fal
sity of some statements from the truth or falsity of others. In that application, we use one of the boolean con
stants to represent truth, and the other to represent falsity. So for that application, it seems reasonable to call them
true
and
false
.
The algebra arose from that application, and it is so much identified with it that many
people cannot separate them; they think the boolean values really are
true
and
false
. But of course boolean
expressions are useful for describing anything that comes in two kinds. We apply boolean algebra to circuits in which there are two voltages. We sometimes say that there are Os and 1s in a computer's memory, or that there are
trues
and
falses.
Of course that's nonsense; there are neither Os and 1s nor
trues
and
falses
in there;
there are low and high voltages. We need symbols that can represent truth values and voltages equally well. Boolean expressions have other applications, and the notations we choose should be equally appropriate for all of them. Computer programs are written to make computers work in some desired way. Before writing a pro gram, a programmer should know which ways are desirable and which are not. That divides computer behavior into two kinds, and we can use boolean expressions to represent them. A boolean expression used this way is called a
specification.
We can specify anything, not just computer behavior, using boolean expressions. For ex
ample, if you would like to buy a table, then tables are of two kinds: those you find desirable and are willing to buy, and those you find undesirable and are not willing to buy. So you can use a boolean expression as a table specification. Acceptable and unacceptable human behavior is specified by laws, and boolean expressions have been proposed as a better way than legal language for writing laws (1]. They can be used to calculate the at tractions and repulsions among a set of magnets.
VOLUME 26, NUMBER 2, 2004
7
For symbols that are independent of the application, I propose the lattice symbols T and l_ , pronounced "top" and "bottom". Since boolean algebra is the mother of all lattices, I think it is appropriate, not a misuse of those symbols. They can equally well be used for true and false statements, for high and low voltages (power and ground), for satisfactory and unsatisfactory tables, for innocent and guilty behavior, or any other opposites. For disjunction, the symbol V is fairly standard, coming from the Latin word "vel" for "or". For conjunction,
the symbol is less standard, the two most common choices being & and 1\ . We are even less settled on a sym bol for implication. Symbols in use include
The usual explanation says it means "if then", followed by a discussion about the meaning of "if then". Appar ently, people find it difficult to understand an implication whose antecedent is false ; for example, "If my mother had been a man, I'd be the king of France." [19]. Such an implication is called "counter-factual". Some people are uneasy with the idea that false implies anything, so some researchers in Artificial Intelligence have pro posed a new defmition of implication. The following truth table shows both the old and new definitions. a
old
new
b
a�b
a�b
true
false
false
false
false
false
true
true
true
true
true
true
false
true
unknown unknown
where unknown is a third boolean value. When the antecedent is false , the result of the new kind of impli cation is unknown
. This is argued to be more intuitive. I believe this proposal betrays a serious misunder
standing of logic. When people make statements, they are saying that each statement is true. Even if the state
ment is "if a then
" and a is known to be false, nonetheless we are being told that "if a then " is true. It is the consequent that is unknown. And that is represented perfectly by the old implication: there are two rows in which a is false and a�b is tru e ; on one of these rows, is true , and on the other b is false .
b b
b
b
Debate about implication has been going on for a long time; 22 centuries ago, Callimachus, the librarian at Alexandria, said, "Even the crows on the roof croak about what implications are sound."[3],[18]. In case you think that confusion is past, or just for beginners, consider the explanation of implication in Contemporary Logic
Design, 1994 [16]: As an example, let's look at the following logic statement: IF the garage door is open AND the car is running THEN the car can be backed out of the garage It states that the conditions-the garage is open and the car is running-must be true before the car can be backed out. If either or both are false, then the car cannot be backed out. Even a Berkeley computer science and electrical engineering professor can get implication wrong.
Implication is best presented as an ordering. If we are calling the boolean values "top" and "bottom", we can say
"lower than or equal to" for implication. It is easy, even for primary school students, to accept that l_ is lower
than or equal to T , and that l_ is lower than or equal to l_ . With this new pronunciation and explanation, three
other neglected boolean operators become familiar and usable; they are "higher than or equal to", "lower than", and "higher than". For lack of a name and symbol, the last two operators have been treated like shameful secrets,
and shunned. If we are still calling the boolean values "true" and "false", then we shall have to call implication "falser than or equal to". As we get into boolean expressions that use other types, ordering remains a good expla
nation: x<4 is falser than or equal to x<6 , as a sampling of evaluations illustrates (try x=3, 5, 7). I have tried using the standard words "stronger" and "weaker", saying x<4 is stronger than x<6 ; but I find that some of my students have an ethical fixation that prevents them from accepting that falsity is stronger than truth.
That implication is the boolean ordering, with T and l_ at the extremes, is not appreciated by all who use
boolean algebra. In the specification language Z [24], boolean expressions are used as specifications. Specifica tion
A
refines specification B if all behavior satisfying
A
also satisfies B . Although increasing satisfaction
is exactly the implication ordering, the designers of Z defined a different ordering for refinement where T is
not satisfied by all computations, only by terminating computations, and
l_
is satisfied by some computations,
namely nonterminating computations. They chose to embed a new lattice within boolean algebra, rather than to use the lattice that it provides.
8
THE MATHEMATICAL INTELLIGENCER
Implication has often been defined as a "secondary" operator in terms of the "primary" operators negation and disjunction: (a�b)==-.aVb Proofs about implications proceed by getting rid of them in favor of the more familiar negation and disjunction, as we did earlier in an example. This avoids the informal explanation, but it makes an unsupportable distinction between "primary" and "secondary" operators, and hides the fact that it is an ordering. When we learn that im plication is an ordering, proofs about implications become shorter and easier. If we present implication as an ordering, as I prefer, then we face the problem of how to use this ordering in the formalization of natural-language reasoning. To what extent does the algebraic operator "lower than or equal to" correspond to the English word "implication"? Philosophers and linguists can help, or indeed dominate in this difficult and important area. But we shouldn't let the complexities of this application of boolean algebra complicate the algebra, any more than we let the complexities of the banking industry complicate the definition of arithmetic. Symmetry and Duality
In choosing infix symbols, there is a simple principle that really enhances our ability to calculate: we should choose symmetric symbols for symmetric operators, and asymmetric symbols for asymmetric operators, and choose the reverse of an asymmetric symbol for the reverse operator. The benefit is that a lot of laws become visual: we can write an expression backwards and get an equivalent expression. For example, x equivalent to
+ y < z is
z > y + x . By this principle, the arithmetic symbols + X < > = are well chosen but - and
=!= are not. The boolean symbols 1\ V � ¢::: == EB are well chosen, but =I= is not.
Duality can be put to use, just like symmetry, if we use vertically symmetric symbols for self-dual operators, and vertically asymmetric symbols for non-self-dual operators with the vertical reverse for their duals. The vi
(T /\- _1_) V _1_ is the negation of 1\ T if you allow me to use the vertically symmetric symbol - for negation, which is self-dual. There
sual laws are: to negate an expression, tum it upside down. For example, (_1_ V- T)
are two points that require attention when using this rule. One is that parentheses may need to be added to main
tain the precedence; but if we give dual operators the same precedence, there's no problem. The other point is that
variables cannot be flipped, so we negate them instead (since flipping is equivalent to negation). The well-known ex
ample is deMorgan's law: to negate a V b , tum it upside down and negate the variables to get -a 1\ -b . By this principle, the symbols
T _1_ - 1\ V are well chosen, but �
¢::: == =I= EB are not. By choosing better symbols we
can let the symbols do some of the work of calculation, moving it to the level of visual processing. From Booleans to Numbers
Some boolean expressions are laws: they have value
T no matter what values are assigned to the variables.
Some boolean expressions are unsatisfiable: they have value _1_ no matter what values are assigned to the vari ables. The remaining boolean expressions are in between, and "solving" means fmding an assignment of values
for the variables for which the boolean expression has value T . (Solving is not just for equations but for any
kind of boolean expression.) A lot of mathematics is concerned with solving. And in particular, number algebra has developed by the desire to solve. To caricature the development, we choose an unsatisfiable boolean ex pression and say, "What a pity that it has no solutions. Let's give it one.". This has resulted in an increasing se quence of domains, from naturals to integers to rationals to reals to complex numbers. The boolean expression
x +1 = 0 is unsatisfiable in the natural numbers, but we give it a solution and thereby invent the integers. Sim ilarly we choose to give solutions to xX2 = 1 , x2 = 2 , x2 = -1 , and thereby progress to larger domains. This progression is both historical and pedagogical. At the same time as we gain solutions, we lose laws, since the laws and unsatisfiable expressions are each other's negations. For example, when we gain a solution to x2
2 , we lose the law x2 =!= 2 .
=
As the domain of an operation or function grows, we do not change its symbol; addition is still denoted +
as we go from naturals to complex numbers. I will not argue whether the naturals are a subset of the complex numbers or just isomorphic to a subset; for me the question has no meaning. But I do argue that it is important
1 and complex 1 because they behave the same way, and for natural + + because they behave the same way on their common domain. To be more precise, all boolean
to use the same notation for natural and complex
expressions over the naturals retain the same solutions over the complex numbers, and all laws of complex arith metic that can be interpreted over the naturals are laws of natural arithmetic. The reason we must use the same symbols is so that we do not have to relearn all the solutions and laws as we enlarge or shrink the domain. And indeed, it is standard mathematical practice to use the same symbols. For exactly the same good reasons that we have a unified treatment of number algebras, we must now unify boolean and number algebras. The question whether boolean is a different type from number is no more rele-
VOLUME 26, NUMBER 2, 2004
9
vant than the question whether natural and integer are different types. What's important is that solutions and laws are learned once, in a unified system, not twice in conflicting systems. And that matters both to primary school students who must struggle to learn what will be useful to them, and to professional mathematicians who must solve and apply laws. Historically, number algebra did not grow from boolean algebra; but pedagogically it can do so. As already argued, the use of 0 1 + X for J.. T V 1\ doesn't work To find an association between booleans and num bers that works for unification, we must use a number system extended with an infinite number. Such a system is useful for many purposes; for example, it is used in [ 13] to prove things about the execution time of programs (some execution times are infinite). For a list of axioms of this arithmetic, please see [13],[14]. The association that works is as follows. boolean
number
T
infinity
bottom
j_
minus infinity
negation
--,
top
negation
(\
disjunction
v
minimum
implication
�
order
conjunction
maximum
equivalence
equality Ell
exclusive or
inequality
With this association, all number laws employing only these operators correspond to boolean laws. For example, boolean law T
number law
= --,_]_
x=- -x xjoc=co oc
a VT=T a /\_i=_i
-00
X t-oo= X j-oo=X xtoo=x
-x
a V _i=a
aAT=a
a�T
a V (b A c) =(aVb)A (aVe)
aA (b V
=
c) =(al\b)
xj(ytz)= (xjy)t(xjz) d (y jZ) = (dy) i(d Z) X j Y =-(-d-y) dy =-(-xj-y)
V (a/\c)
a V b =--,(--,al\--,b)
aA b =--,(--,aV--,b)
There are boolean laws that do not correspond to number laws, just as there are integer laws that are not real laws. That's another way of saying that there are unsatisfiable boolean expressions that correspond to satisfi able number expressions. We will use this for our unified development. Unified Algebra
Here is my proposal for the symbols of a unified algebra. unified top bottom negation conjunction disjunction "nand"
T
j_ (\
v !:;.
minus infinity negation minimum maximum negation of minimum
"nor"
'V
negation of maximum order
implication
"'
reverse implication
2:
reverse order
strict implication
<
strict order
strict reverse implication
>
strict reverse order
'*'
inequality
equality
equivalence exclusive or
10
infinity
THE MATHEMATICAL INTELLIGENCER
The symbols
- ::::; 2:: < > = are world-wide standards, used by school children in all countries, so I dare not
suggest any change to them. The symbol =F for inequality is the next best known, but I have dared to stand up
the slash so that all symmetric operators have symmetric symbols and all asymmetric operators have asymmet
ric symbols. (Although it was not a consideration,
=I=
also looks more like EB
.
) The "nand" symbol is a com
bination of the "not" and "and" symbols, and similarly for "nor". But I am worried that 1\
and V
choices because they point the wrong way to be minimum and maximum; it might be better to use for conjunction and disjunction, and
t
and
t
t
are poor and
j
for "nand" and "nor". One suggestion: note that V is wide at
the top, and 1\ is narrow at the top. Another suggestion: note that V holds water, and 1\ doesn't.
Duality has been sacrificed to standards; the pair ::::; < are duals, so they ought to be vertical reflections of each other; similarly the pair 2:: > , and also
=
=I=
; addition and subtraction are self-dual, and happily
+
and - are vertically symmetric; multiplication is not self-dual, but X is unfortunately vertically symmetric. Having unified the symbols, I suppose we should also unify the terminology. I vote for the number terminol ogy in the right column, except that I prefer to call
T and
l. "top" and "bottom".
The association between booleans and numbers suggested here allows the greatest number of boolean laws to be generalized to all numbers. For example, if
a , b , and c are boolean, it is usual to define if a then b
else c as follows: (if If
a then b else c) = (a 1\ b) V (-a 1\ c)
a remains boolean but b and c are numbers, the if-expression on the left is still sensible (the Algol if), and
furthermore it is still equal to the expression on the right. This generalization requires the particular association between booleans and numbers suggested here. The next examples, written in boolean notations, are the laws
(a 1\ b =>c) (a V b =>c)
= =
(a =>c) V (b =>c) (a =>c) 1\ (b =>c)
A common error is to use conjunction twice, or disjunction twice. The boolean reading " a and b implies c
a implies c or b implies c " sounds no more reasonable than " a and b implies c if and a implies c and b implies c ". In unified notation,
if and only if only if
(a 1\ b :::s c) = (a:::sc) V (b::Sc) (a V b :::s c) = (a::Sc) 1\ (b::Sc) a and b is less than or equal to c when at least one of a or b is c , and the maximum of a and b is less than or equal to c when both a and b are less than or equal to c . They are laws for all numbers, not just the booleans. The arithmetic expression x - y varies directly with x and inversely with y . Thus if we increase x , we increase x - y , and if we decrease y we increase x - y . We calculate: it is more obvious that the minimum of
less than or equal to
:::s :::s
x- y (x+1) - y (x+1) - (y-1)
x to x+1 and so increase the whole expression decrease y to y-1 and so increase the whole expression increase
x 2:: y varies directly with x and inversely with y (no matter whether x y are numbers and 2:: is number comparison, or x and y are boolean and 2:: is reverse implication, or x and y are a mixture of number and boolean). We calculate as follows:
Similarly the boolean expression and
:::s :::s
x 2: y (x+1) 2:: y (x+1) 2:: (y-1)
increase decrease
x to x+1 and so increase the whole expression y to y-1 and so increase the whole expression
It is exactly the same calculation. By unifying number algebra with boolean algebra we carry our ability to cal culate over from numbers to booleans. Unified Development
Suppose we start with boolean algebra in the unified notation, with the terminology "top", "bottom", "minimum", "maximum", "less than", and so on. Now we say: what a pity that new solution is denoted 0
.
x=-x has no solution; let's give it one. The
While gaining a solution to some boolean expressions, we lose some laws such as
the law of the excluded middle
x V -x .
Now we have an algebra of three values:
T , l. , 0 . In one application they can be used to represent "yes", 27
"no", and "maybe"; in another they can be used to represent "large", "small", and "medium". This algebra has one-operand operators, one of which is - , defined as
VOLUME 26, NUMBER 2, 2004
11
T
X
In
j_
0
j_
-x
T
0
has 19683 two-operand operators, four of which are: TT
TO
T_l_
OT
00
Qj_
_l_T
j_Q
j_j_
x:s;y
T
j_
j_
T
T
j_
T
T
T
xEBy
j_
T
0
T
j_
0
j_
T
xy
x=y
x<
T
T
j_
j_
0
Whether :5 or
j_
«
j_ T
T 0
0
j_ 0
j_ 0
j_ 0
T 0
or another operator represents implication in the presence of uncertainty can be debated,
but the algebra is not affected by the debate. The operator EEl is modular (or circular) addition, and the other operators of modular arithmetic can be given similarly. We might continue our development with a four-valued algebra and five-valued algebra, but at this point I rec ommend filling in the space between T and 0 , and between 0 and .l , with all the integers. And then on to the rationals, the reals, and the complex numbers as usual. The argument in favor of this unification of boolean algebra and number algebra is just as strong as the argument in favor of using the same notations for the different number algebras. But the latter is familiar, and so it seems right, while the former is unfamiliar, and for that reason alone it may seem wrong. tntimately, the benefits will outweigh the unfamiliarity. For example, the data structure lmown as AND-OR trees and the algorithm that uses them become the same as the data structure and algorithm lmown as minimax methods; they should not have to be learned twice. A different unification of boolean algebra and number algebra that aims at the same goal (using the same cal culations for booleans and numbers), but emphasizes traditional modular arithmetic along the way, can be found in [5), a provocative work of grand scope. From Informal to Formal
Many mathematical notations began their lives as abbreviations for some words. For example, duced in [20) to mean "is equal to":
was intro-
And to avoide the tediouse repetition of these woordes "is equalle to" I will lette as I doe often in woorke bse, a paire of paralleles or Gemowe [twin] lines of one lengthe, thus: = , because noe 2 thynges, can be moare equalle. Later, = became associated with some algebraic properties, namely reflexivity, symmetry, transitivity, and sub stitutivity. Today, it is defmed by those properties, not as an abbreviation for some words. Someone might say that Alice and Bob are equal tennis players because they have played each other 10 times, and each has won 5 matches. They might similarly say that Bob and Carol are equal tennis players because they too have played each other 10 times, and each has won 5 matches. But this kind of equality is not transitive. As it happens, Alice and Carol are unequal tennis players: they have played each other 10 times, and Alice has won 8 matches. Because of the lack of transitivity, no mathematician today would use = for tennis equality. In the notation commonly used for small sets, such as (1, 3, 7} , the comma was introduced as just punctu ation, not as a mathematical operator. As soon as the notation is introduced, we must say that the order in which elements are written is irrelevant so that (1, 2}=(2, 1} ; the way to say that formally is A,B=B,A (comma is commutative). We must also say that repetitions of elements are irrelevant so that (3, 3}=(3} ; the way to say that formally is A,A =A (comma is idempotent). And we should say that comma is associative A,(B,C)=(A,B),C so that parentheses are unnecessary. Evidently the comma can be seen as a mathematical operator with alge braic properties that aggregates elements into a structure that is simpler, more primitive, than sets; let us call them bunches. Even the curly braces can be seen as an operator that applies to a bunch and makes a set; its in verse - applies to a set and makes a bunch: -(1,2}=1,2 . When a child first learns about sets, there is often an initial hurdle: that a set with one element is not the same as the element. It would be easier to present a set as packaging: a package with an apple in it is obviously not the same as the apple. Just as (1} and 1 differ, so (1,2} and 1,2 differ. Bunch theory tells us about aggrega tion; set theory tells us about packaging. The two are independent. Apart from being cute, are bunches useful? The subject of functional programming has suffered from an in ability to express nondeterminism conveniently. To say something about a value, but not pin it down completely, one can express the set of possible values. Unfortunately, sets do not reduce properly to the deterministic case; in this context it is again a problem that a set containing one element is not equal to the element. What is wanted 12
THE MATHEMATICAL INTELLIGENCER
is bunches. One can always regard a bunch as a "nondeterministic value". Bunches can also be used as a "type theory" with the advantage that it is unnecessary to duplicate the operators of the value space at the type level because the two are unified. And finally, the easiest way to present sets is via bunches. For details see [13],[14]. Formalization of the lowly comma leads to a beautiful and useful algebra. We have just seen two examples of formalization, one from the past and one from the future. Now here's an example of a formalization gone astray: functions defined as sets of ordered pairs. This way of defining func tions is part of the very interesting demonstration that all of mathematics can be based on sets. The demon stration requires us to make a set-model of functions, and numbers, and everything else. For example, the nat ural numbers can be equated to sets, with no inconsistency, as follows:
0
=
n +1
0 =
(the empty set)
nU{n}
So, for example, 3
= {0, {0}, {0, {0}}} . Few people would say that 3 really is the set {0, {0}, {0, { 0 } } } ; the
set-model of natural numbers was constructed by John von Neumann just to serve this one demonstration. Num bers are best formalized, not by building a set-model, but by an algebra showing how they participate in arith metic operations. Similarly, functions are best formalized by showing the laws of application and function com position (in general, set union and intersection are not useful ways of combining functions). But the set-model of functions has somehow taken root in the current mathematical culture; many people (and textbooks) say that a function really is a set of ordered pairs. A useful formalization is not one that answers the question "what is it?", but one that answers the question "how do we use it?". I write a function, or local scope, according to the following example:
(n: nat � n+ 1) This is essentially a "lambda-expression" [6], although Church did not use angle brackets and arrows. He borrowed a "hat" notation from Whitehead and Russell, but moved the hat down in front; the most similar available character
in the typesetter's tray was A ; thus the lambda calculus was born [22]. Following van de Snepscheut [23], I use an
gle brackets to delimit the scope of the variable. I use an arrow to facilitate the unification of ftmctions with ftmc tion spaces, which I do not discuss in this paper (see [14]). Next, I want to get rid of the idea that all possible vari ables (infinitely many of them) already "exist", and that the ftmction notation "binds" a variable, and any variable that
is not bound remains "free". I prefer the programmer's terminology of "local" and "nonlocal" variables. Variables
do not automatically "exist"; they are introduced (rather than bound) with a limited scope by the ftmction notation. Two notations that have not yet made the transition from informal beginning to formal, calculational tool are the quantifiers
V
and
3
. For most mathematicians today they remain abbreviations for the words "for all" and
"there exists", and their meaning is just whatever can be understood from those words. The word "all" sounds clear and unambiguous, but there is debate as to whether so-called "undefined" range elements, or other "non standard" elements, are included. Existence is even more contentious, as can be seen from the debate between classical and constructive mathematicians. Only a formal definition, equivalent to an automated theorem prover, is clear and unambiguous. Only a formal definition gives us calculation. Quantifiers
There are several notations that introduce a local (bound, dummy) variable. For example,
Vx:D· Px
{ fx
I
xED}
The introduction o f the local variable and its domain are exactly the job o f the function notation, s o all expres sions requiring a local variable can be uniformly expressed as an operator applied to a function. If the body of a function is a number expression, then we can apply
+(n: nat �
+
to obtain the sum of the function results. For example,
112n)
There is no syntactic ambiguity caused by this use of
+
,
so no need to employ another symbol
I for addi
tion. We can apply any associative symmetric operator, such as
X(n: nat � 1/2n) /\(n: nat � n>5) V(n: nat � n>5) The minimum operator ing
= and
1\
replaces "for all", and the maximum operator
V
replaces "there exists". By apply
=F to functions we obtain the two independent parity operators. Set comprehension and integrals
can be treated this same way.
VOLUME 26, NUMBER 2, 2004
13
If function
2 Jx
f
has domain
D
, then
f=(x : D
�
fx )
, so quantifications traditionally written
'tfx : D· Px
x:D
which we have just learned to write as
1\(x : D
+ (x : D � fx )
�
Px )
can be written even more succinctly as 1\P
+f
Using juxtaposition for composition, deMorgan's laws
-.(Vx : D· Px)
=
(3x : D· -.Px)
-.(3x : D· Px)
=
('tfx : D· -.Px)
become -1\P
=
-VP = 1\ -P
V-P
or even more succinctly (-/\)
=
(-V)
(V-)
=
(/\-)
The Specialization and Generalization laws say that if
('tfx : D· Px)
=::}
Py
y
is an element of
D ,
Py =::} (3x : D· Px)
They now become 1\P::::;
Py
Py :SVP
which say that the minimum item is less than or equal to any item, and any item is less than or equal to the max imum item. These laws hold for all numbers, not just for the booleans. Given function f , all function values jx are at least y if and only if the minimum function value fx is at least y . Traditionally, that's a universal quantification equated to a minimum. In unified algebra, it is just fac toring. Leaving the non-null domain of f implicit, we write
1\(x � fx � y ) 1\(x � fx ) � Y 1\f � y
factor out
�y
If we go in the other direction, "unfactoring" is called "distribution". And it works whether fx and y are num bers and � is the number ordering, or fx and y are booleans and � is reverse implication. It's no differ ent from the factoring/distribution law that says the minimum value of (fx - y) equals (the minimum value of
(fx) - y . 1\(x �fx - y ) = 1\(x fx ) Y 1\f- y �
factor out
-y
-
=
If we factor from the other side of the
1\(x � y - fx ) y - V(x �fx ) y - Vj
factor out
y-
factor out
y�
sign, we have to change minimum to maximum:
And similarly
1\(x y �fx ) y � V(x �fx ) y � Vj �
Once again, it works for numbers and booleans equally well. Unified algebra gives us many other factoring/ distribution laws just like these (see [14]). The goal is to create an algebra that's easy to learn and easy to use. That goal is not always consistent with traditional mathematical terminology and symbology. Readers are cautioned against matching the algebra di rectly with their own familiar terms and symbols. Although I have been using the words "minimum" and "maxi14
THE MATHEMATICAL INTELLIGENCER
mum" for 1\ and V , the words "greatest lower bound" and "least upper bound", or "infimum" and "supremum", may be more traditional in some contexts. For example,
1\(n: nat
_,.,.
1/n) = 0
Even more caution must be used with the words "all" and "exists". Intuition about existence in mathematics (like intuition about anything else) depends on what you have learned. We tend to believe that what we have learned is true. But mathematical truth is constructed, and we must be open to the possibility of constructing it differ ently. Unlearning can be more difficult than learning. Quantifier Examples
Is (3x·Px) => (Vy·Qy) equivalent to Vx· Vy· (Px => Qy) ? Even experienced logicians don't find it obvious. To see whether they are equivalent, those who reason informally say things like "suppose some x has property P ", and "suppose all y have property Q ".They are led into case analyses by treating V and 3 as abbrevi ations for "for all" and "there exists" (as they originally were). Of the very few who reason formally, most don't know many laws; perhaps they start by getting rid of the implications in favor of negation and disjunction, then use deMorgan's laws. Let me rewrite the questionable equivalences in the new notations.
(VP :S 1\Q) = 1\(x _,.,. 1\(y
_,.,.
Px :S Qy))
We might read the left side as saying that the maximum P is less than or equal to the minimum Q , and we might read the right side as saying that all P are less than or equal to all Q . Informal readings can be mis leading, and we should never attach our understanding to an informal reading, but sometimes we can get inspi ration from it. In this case, the reading sounds reasonable enough to suggest we might prove it, and not just for booleans, but for all numbers. Leaving the non-null domains implicit, here's the proof:
1\(.x: _,.,. 1\(y _,.,. Px :S Qy)) 1\(x _,.,. Px :S 1\Q) VP :S 1\Q
factor out Px:S factor out :SI\Q
Let L be a nonempty list (a function whose domain is an initial segment of the naturals). +L is its sum, and VL is its maximum; let #L be its length. We can say that the average item in the list is less than or equal to the maximum item as follows.
:S
+LI#L :S VL (+LI#L > 1) :S (VL > 1) (+L > #L) :S V(i _,.,. Li > 1)
now apply >1 to both sides of the inequality multiply by #L ; distribute >1
leaving the domain implicit. The bottom line is the "pigeon-hole principle"; it says that if the total number of things is greater than the number of places to put them, then some place has more than one thing in it. Notice what has happened: we read V as "maximum" on the top line, and as "some" on the bottom line; we read :S as "less than or equal to" on the top line, and as "if then" on the bottom line. Here is a further illustration of the benefits of unified algebra. Let f be a function from the naturals to the reals. If f is nondecreasing, then fO is its minimum. Traditionally, this might be written (leaving the domain implicit) as
(Vn· fn :S f(n+1))
=>
(JO = MIN ( fn I O:Sn
Rewriting this in the new notation, and weakening it to say that fO is less than or equal to the minimum, we get
1\(n
_,.,.
fn :S f(n+ 1)) :S (JO :S 1\f)
Now we apply the portation law, which says that for boolean a and any b and c ,
(a :S (b :S c)) = (al\b :S c) to obtain
fO 1\ 1\(n
_,.,.
fn :S f(n+ 1)) :S 1\f
If f happens to have a boolean range, this is induction, more traditionally written
JO 1\ (Vnjn
=> f(n+ 1)) =>
(Vn· fn)
Thus we see induction as a special case of a more general law saying that the first item in a nondecreasing se quence is its minimum.
VOLUME 26, NUMBER 2, 2004
15
Probability
The seminal work [4] by Boole on boolean algebra refers to both logic and probability. The standard theory of
0 to an event that cannot happen, 112 to an event that is equally likely to happen or not hap 1 to an event that is certain to happen. In a set of events in which exactly one event must happen, the probabilities sum to 1 . The integral of a probability distribution must be 1 .
probability assigns pen, and
Perhaps there is another way to develop probability theory based on unified algebra. Perhaps an event that
cannot happen has probability .l , an event that is equally likely to happen or not happen has probability
0 ,
and an event that is certain to happen has probability T . In a set of events in which exactly one event must happen, the average probability is
0 . The integral of a probability distribution must be 0 . Perhaps the new
probability space is related to the logarithm of the old space; essentially, probabilities are replaced by informa tion content. My hope is that the complicated formulas for distributions in the standard theory can be simplified by transforming the space of probabilities. Metalogic
In the study of logic, at or near the beginning, logicians present the symbol
1- to represent theoremhood. I ask
you to put yourself in the place of a beginning student. This symbol is applied to a boolean expression just like the boolean operators; but we know all the boolean operators and this isn't one of them. It sometimes has a left operand as well as a right operand, and then the explanation makes it seem just like implication. To say that it is a "meta-operator" just labels it, and doesn't explain it. Saying that it applies to the form, rather than the mean ing, is confusing too, since the entire point of the algebra is to enable us to work with the form and ignore the meaning. The distinction between metanotations and the object notations is not easily seen. To make things worse, there are different levels of meta-operators. Proof rules are sometimes presented us ing a horizontal bar, which is yet another level of implication. Consider, for example, the Modus Ponens proof rule, which uses all three kinds of implication:
A 1- x, B 1- x� y A, B 1- y Rewriting comma as conjunction, and turnstile and bar as implication, we get a tautology:
Rewriting any proof rule this way gives a tautology (if 1- has nothing to its left, use
T
). Rewriting any tau
tology whose main connective is implication gives a valid proof rule. It is hard to see the difference between the
meta-operators and the object-level operators because there is no formal difference! The proof rules are used to explain how to use the boolean expressions; natural language is used to explain how to use the proof rules. For beginners (and others) it would be better to skip the meta-notations altogether and just use natural language to explain how to use the boolean expressions. At a more advanced level, when we want a formalism to study formalisms, we will need an operator that ap plies to the form of an expression. For that purpose, we do not need any new kind or level of operator. Rather, we need to do exactly what Gtidel did when he encoded expressions, but we can use a better encoding. We need to do exactly what programmers do: distinguish program from data. One person's program may be a compiler writer's data, but when it is data, it is a character string. The character string for the expression
a
V
-a
. We apply
pression represented by string
"
a
V
-a
"
can be used as a code
1- to character strings so that 1-s is a theorem when the boolean ex
s is a theorem.
We have a name, "theorem", for a boolean expression that can be simplified to
T
, and an operator,
1- ,
whose purpose is to identify theorems. Strangely, logicians have not introduced a name, say "antitheorem", for
a boolean expression that can be simplified to .l , and no operator such as -1
, whose purpose is to identify
antitheorems. Perhaps that's because "antitheorem" just means "negation of a theorem" in those logics having negation and an appropriate proof rule. But we bother to name both booleans, even though one is just the nega tion of the other.
1- and Is to be a theorem if
I propose that logicians can improve metalogic by taking another lesson from programming. Instead of
-1 , we need only one operator to serve both purposes. It is called an interpreter. I want
and only if
s represents a theorem, and an antitheorem if and only if s represents an antitheorem. It is related
to 1- and -1 by the two implications
1-s
:s
Is
:s
- -Is
In fact, if we have defined 1- and -1 , those implications define
I . But I want I to replace 1- and
-1
, so
I shall instead defme it by showing how it applies to every form of boolean expression. Here is the beginning of its definition.
16
THE MATHEMATICAL INTELLIGENCER
I" T "
= T I"j_" = j_ I(" - "s) = -Is I(s"/\"t) = Is 1\ It I(s"V"t) = Is V It And so on. In a vague sense I acts as the inverse of quotation marks; it "unquotes" its operand. That is what an interpreter does: it turns passive data into active program. It is a familiar fact to programmers that we can write an interpreter for a language in that same language, and that is just what we are doing here. Interpreting (unquoting) is exactly what logicians call Tarskian semantics. In summary, an interpreter is a better version of 1- , and strings make metalevel operators unnecessary.
Using I , the famous Gi:idel incompleteness proof is just 3 lines. Suppose that every boolean expression is
either a theorem or an antitheorem (a complete logic), and define
Q by
Q = " - IQ" Then
IQ I " - IQ" - IQ
replace
Q with its equal
I unquotes
which proves a boolean expression equal to its negation, showing the logic to be inconsistent. A logic in which we can define an interpreter, and in which we can replace an expression with its equal, must be inconsistent or incomplete. We choose consistency, and we choose to allow the replacement of an expression with its equal, so we are forced to give up the ability to define a complete interpreter; in particular, I cannot unquote "-IQ". For further details of this version of Gi:idel's incompleteness theorem, see [ 1 2 ] , [23] . You cannot learn a programming language by reading an interpreter for it written in that same language. And you cannot learn logic, or a logic, by reading an interpreter for it written in logic. Not only is it inscrutable to a novice, but also it may be subject to more than one interpretation. Logic is better presented as algebra [ 1 1 ]. We don't present number algebra with the aid of a metaoperator that applies to number expressions and results in their values, and we should not present boolean algebra that way. I think boolean algebra should be presented with a little natural language and a lot of laws, because laws don't use any metanotations. Terms of Honor
My final comment concerns mathematical terminology intended to honor mathematicians. In some parts of math ematics it is standard: Lie algebra, Stone algebra, Cartesian product, Jordan decomposition, Cayley transform, Hilbert space, Banach space, Hausdorff space, Borel measure, Lebesgue integration, Fredholm index, Wedder burn's Theorem, and so on. It is well known that the person so honored is sometimes the wrong person; often it is only one of many who equally deserve to have their names attached to the idea. I suspect that sometimes the intention is not so much to honor a person as to use the person's prestige to lend respectability to an idea. Even when the intention is to honor, the effect is to obscure and make the mathematics forbidding and inac cessible. It may be argued that this is good, keeping the uninitiated from thinking they understand when they don't, but I reject that argument as elitist. I know what nand and nor are, but I forget which is the Scheffer stroke and which the Peirce arrow. To say that an operator is symmetric or commutative is much more descriptive and understandable than calling it Abelian. DeMorgan's laws would be better named duality laws. We who are used to the terms forget what a barrier they pose to beginners. The term "boolean algebra" honors George Boole. (It is popularly thought that the word "algebra" honors someone, but it comes from an arabic word meaning "the reintegration and reunion of broken parts". In any case, the word is now standard, known by people everywhere.) The best way to honor George Boole is to make the algebra that he created [ 4] a well known and well used tool, and to do that we might have to remove his name from it, and give it a more descriptive and accessible name, like "binary algebra".
Conclusions
Logic has been well studied and is now well understood, but it is not well used. Programmers learn that logic is a foundation of programming, but they don't often use it to program. Mathematicians study about logic, but they don't often use it in their proofs. Logic is a tool, like a knife. People have looked at it from every angle; they've described how it works at great length; now it's time to pick it up and use it. To use logic well, one must learn it early, and practice a lot. Fancy versions of logic, such as modal logic and metalogic, can be left to university study, but there is a simple basic algebra that can be taught early and used widely.
VOLUME 26, NUMBER 2, 2004
17
Number algebra is used by scientists and engineers everywhere. It is used by economists and architects. It is taught first to 6-year-olds, without a metanotation, very concretely as addition and subtraction of numbers. Then variables and equations are introduced, and always the applications are emphasized. As a result of that early and long education, scientists and engineers and mathematicians are comfortable with it. Boolean algebra can be equally useful if it is taught the same way. At present, it is not in a good state for presentation to a wide audi ence. We need to simplify the terminology, get rid of the metanotations, adopt the view that proof is calculation, choose some good symbols, detach it from its dominant application in which the boolean values represent true and false statements, and explain it as algebra. There is a small advantage to choosing uniquely boolean symbols: we can give them a precedence after the arithmetic operators, which reduces the need for parentheses. On the other hand, there is a large advantage to uniting boolean and number symbols in the way I have suggested: the laws and solutions are familiar and can be interpreted either as booleans or numbers. In addition, by placing booleans in the same context as numbers, we move quickly away from debates about the meanings of operators. The fact that the booleans can be em bedded in the extended integers just as smoothly as the integers are embedded in the rationals seems a com pelling reason to do so. Quantifiers can be simplified, made uniform, and generalized by treating them as operators on functions. We should stop speaking about "existence", and speak instead about the maximum of a function. Similarly, we should stop speaking about "all", and speak instead about the minimum of a function. We should stop trying to say what functions and other mathematical ideas are, and say instead how to write them and use them. An interpreter serves the same purpose as the metalevel theoremhood operator with the added advantage that it gives antitheoremhood as well as theoremhood. And by applying it to strings, we avoid having to introduce a separate metalevel of operators. Metalogic is an advanced topic, not a good introduction to boolean algebra for those who are new to the subject. This paper has not presented a detailed proposal for a change to our primary and secondary mathematics cur riculum, but it has presented the case for making a change, and several suggestions. The main suggestion is to unify boolean algebra with number algebra so that we can begin with the simplest algebra and move smoothly to the more complicated algebras, all using the same notations and in the same calculational framework. Acknowledgments
Theo Norvell provided references [16] and [19]. Benet Devereux provided references [3] and [18]. Rutger M. Dijk stra, Wim Hesselink, and Jim Grundy corrected some errors and caused me to improve some explanations.
REFERENCES
[0] Unfortunately, 500-year-old algebra texts are hard to find. This is not a quotation, but my own creation. I think it is representative of the work of the time, such as [20] . [1 ] L. E. Allen: "Symbolic Logic: a Razor-Edged Tool for Drafting and Interpreting Legal Documents", Yale Law Journal v.66 p.833-879, 1 957. [2] R. Back, J. Grundy, J . von Wright: "Structured Calculational Proof", Formal Aspects of Computing v.9 n. 5 p.469-483, 1 997.
[3] J.M. Bochenski: A History of Formal Logic second edition, trans lated and edited by lvo Thomas, Chelsea Publishing, New York, 1 970. [4] G. Boole: An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities,
MacMillan, 1 854, reprinted by Dover, 1 973. [5] R. Boute: "Binary Algebra and Functional Predicate Calculus: a Practical Approach", University of Ghent, Belgium, 1 999. [6] A. Church: "The Calculi of Lambda-Conversion", Annals of Math ematical Studies v. 6, Princeton University Press, 1 941 .
[7] E.W. Dijkstra, C.S.Scholten: Predicate Calculus and Program Se mantics, Springer-Verlag, 1 990.
18
THE MATHEMATICAL INTELLIGENCER
[8] R.L. Goodstein: Development of Mathematical Logic, Springer-Ver lag, 1 97 1 . [9] D. Gries: "Improving the Curriculum through the teaching of Cal culation and Discrimination", Communications of the ACM v.34 n.3 p.45-55, 1 991 March. [1 0] D. Gries, F.B. Schneider: A Logical Approach to Discrete Math , Springer-Verlag, 1 993. [1 1 ] P. Halmos, S. Givant: Logic as Algebra , Mathematical Association of America, 1 998. [1 2] E.C.R. Hehner: "Beautifying Godel", chapter 1 8 in Beauty is our Business, a birthday tribute to Edsger Dijkstra, Springer-Verlag,
1 990. [1 3] E.C.R. Hehner: A Practical Theory of Programming, Springer-Ver lag 1 993. The second edition, 2004, is at www.cs.toronto.edu/ �hehner/aPToP [1 4] E.C.R. Hehner: "Unified Algebra", www.cs.toronto.edu/�hehner/ UA.pdf. [1 5] I . N . Herstein: Topics in Algebra p.323, Blaisdell , 1 964. [1 6] R.H. Katz: Contemporary Logic Design, Benjamin Cummings, 1 994, p . 1 0.
( 1 7] L. Lamport: "How to Write a Proof" , American Mathematical
A U T H O R
Monthly v. 1 02 n.7 p.600-608, 1 995 Aug.
[ 1 8] J. Lukasiewicz: "On the H istory of the Logic of Propositions", Se lected Works , citing Sextus Empiricus in Adversus Mathematicos ,
circa 200. [1 9] C. Navarre: quoted in Barbara W. Tuchman: A Distant Mirror: the Calamitous Fourteenth Century, Knopf, 1 978.
(20] R. Recorde: The Whetstone of Witte, London 1 557, reprinted by Da Capo Press, Amsterdam, 1 969. (2 1 ] P.J. Robinson, J. Staples: "Formalizing a Hierarchical Structure of Practical Mathematical Reasoning", Journal of Logic and Compu
ERIC C . R. HEHNER
tation 3(1 ):47-61 , February 1 993.
Department of Computer Science
[22] J.B. Rosser: " Highlights of the History of the Lambda-Calculus",
University of Toronto
Toronto M5S 3G4, Canada
Annals of the History of Computing v.6 n.4 p.337-349, 1 984 Oc
e-mail:
[email protected]
tober. (23] J.L.A. van de Snepscheut: What Computing is All About, Springer Verlag, 1 993.
[24] J.M. Spivey: The Z Notation: a Reference Manual, Prentice-Hall,
Eric Hehner has spent most of his professional life at the Uni
versity of Toronto, first as a graduate student in Physics, then as
1 989.
graduate student and faculty (full
professor since 1 983)
in
Computer Science. To vary the scenery, he has taken visiting appointments at Xerox Palo Alto, at Oxford, at Grenoble, at
University of British Columbia, and elsewh ere . His research, publications, and edit ing have been mainly
in the field of for
mal programming methods .
He is the father of two children, and symmetrically the child
of two parents. He plays fiddle, piano, and five guitars (but not all simultaneously).
clg£b.@Scientific WorkPlace· Mathematical Word Processing
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Failing Phoenix: Tauber, Helly, and Viennese Life Insurance Karl Sigmund
This column is afornmfor discussion of mathematical communities throughout the world, and through all time. Our definition of "mathematical
Marj o r i e S e n ec h a l , E d itor
T
heir names are familiar to most mathematicians: Alfred Tauber and Eduard Helly have theorems, con stants, and properties named after them, and their work contributed sub stantially to the development of analy sis in the early part of the twenti eth century. Yet both failed to find an adequate academic position. They worked as actuaries for the Viennese life insurance company Phoenix, as did their Hungarian-born colleagues Stefan Vajda and Eugene Lucas, and Z. W. Birnbaum from Poland-a constellation worthy of a first-class university de partement. The crash of the Phoenix in 1936, one of the major disasters in the history of the first Austrian republic, and the Anschluss to Hitler's Germany in 1938 dispersed them. Eventually, all found shelter in England or the United States, with the tragic exception of Tauber, who died in Theresienstadt. This article traces their fates, which crossed in an insurance company that failed to provide security.
community" is the broadest. We include "schools" of mathematics, circles of correspondence, mathematical societies, student organizations, and informal communities of cardinality greater than one. What we say about the communities is just as unrestricted. We welcome contributions from mathematicians of all kinds and in all places, and also from scientists, historians, anthropologists, and others.
Please send all submissions to the Mathematical Communities Editor, Marjorie Senechal, Department
of Mathematics, Smith College, Northampton, MA 01 063 USA e-mail:
[email protected]
Reluctant Actuary
"May I add in my favor that it should be acknowledged in some way that I have given the same course [an intro duction to actuarial mathematics] year after year, to an extent which made it almost impossible to lecture on other
I
sity in Vienna and had obtained the coveted title of Dozent at the age of twenty-five, two years after his doc torate. This "habilitation" gave him the right to lecture on any mathematical topic of his choosing, but it implied nei ther an appointement nor a salary (be yond a nominal fee). Tauber's father was a Jewish lumber-merchant who had migrated from Slovakia to the cap ital of the Austro-Hungarian empire. "Have you ever seen a poor lumber merchant?" is a quip from Johann Ne stroy, Vienna's foremost comedian at that time, and seems to imply that Tauber's family was reasonably well off. But the young Dozent Tauber did not have to live from his father's pock ets: one year after his habilitation, he became chief mathematician of the huge Phoenix life insurance firm. The job must have left Tauber with some time for research: he published extensively on complex analysis and potential theory. At the age of thirty, he discovered a gem, the forerunner of all "Tauberian Theorems": A neces sary and sufficient condition for the convergence of the series I-an is n that both lim s--? 1 - I-ans exists and lim N- 1 I.nan = 0. The necessity of the two conditions had been shown by Niels Abel and by Leopold Kronecker, respectively. Neither condition is suffi
fields." When Alfred Tauber wrote this,
cient. Tauber proved, in a succinct note
in 1907, to the Dean of the Philosophi cal Faculty of the University of Vienna, the monotonous chore of his introduc tory course had lasted-"year after year"-for more than a decade. Barely over forty, Tauber was already thor oughly frustrated in his academic ca reer. He had at long last been ap pointed professor extraordinary for actuarial science, but this seemed hardly appropriate for a mathemati cian of his renown. And the wretched introductory course was to plague him for thirty more years. It had started so well. Alfred Tauber, born in Bratislava on November 5, 1866, had attended school and univer-
in the Monatshefte fur Mathematik und Physik, that their conjunction is. Within ten years, mathematicians of the rank of J. E. Littlewood and Ed mund Landau understood the value of this approach for the investigation of series, integrals, and summability methods. This launched a flurry of in vestigations into what G. H. Hardy and Littlewood called, in 19 13, Tauberian theorems. In 1930, Norbert Wiener published, under this title, a hundred page treatise in the Annals of Mathe matics. A glimpse into the Mathemat ical Reviews will confmn that the topic is still going strong. Indeed, it is diffi cult to delimit this field in a few words:
© 2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 2, 2004
21
Phoenix. The largest life insurance company in the Austro-Hungarian Empire, its branches extended over all of Central Europe. The spectacular crash in 1 936 robbed most Austrians of their sense of security, not to mention their insurance policies.
it is a programme rather than a collec tion of results. This makes it look as if the Univer sity of Vienna had committed an inex cusable oversight in relegating Tauber to actuarial maths. But full professorships were scarce at the Mathematical Semi nar. When Leopold Gegenbauer and Franz Mertens were appointed, in 1893 and 1894, Tauber was still in his twen ties, too young to be a serious candidate. The next vacancy, in 1904, was filled by Wilhelm Wirtinger, obviously an impec cable choice. By 1913, when Mertens re tired, Tauber was too old, and Philipp Furtwangler got the chair. For years Tauber had, as if in protest, published no mathematics, but only tables for cal culating pension schemes, and short commentaries on insurance laws. He loathed his introductory course of lec tures, and seems to have made no at tempt to hide his disgust. Eye-witnesses gave devastating accounts of his perfor mance: Tauber usually started the les son by pressing a sheet of paper filled with formulas into the hands of a luck less student, and asking him to copy them on the blackboard. He would then leave the lecture room, and return, half an hour later to mumble quickly through the list of formulas. The student was al lowed to keep the sheet of paper. 22
THE MATHEMATICAL INTELLIGENCER
Phoenix to Ashes
Tauber's childless marriage was by then over. His health was frail, his tem per querulous, and his mood despon dent. His personal file at the University archives is replete with complaints and disputes. He kept running feuds with the janissary who overheated the lec ture room, with a ministry official who had offended him, and with his luckier rival Wilhelm Wirtinger, who got all the good students and could lecture on whatever he fancied. Wirtinger, for his part, was obviously not bent on easing tensions: he persistently attempted to reform Tauber's course in actuarial mathematics, and argued successfully at faculty meetings against a motion to reward Tauber with the title of a full professor. (Conferring such a title was a symbolic gesture without any effect on salary and duties.) Finally, in 1919, after Austria had been dismembered, Wirtinger gave up his opposition (ex plaining that by now, it did not matter anyway), and Tauber received the cov eted distinction. This belated sign of recognition seems to have afforded him a new lease on life. He started pub lishing mathematics again, mostly on numerical solutions of linear differen tial equations and on potential theory. At sixty-six, Tauber had to take his
retirement. His pension as emeritus ex traordinary professor was pitifully low, and he was obliged, for financial rea sons, to keep giving his introductory course on actuarial mathematics, that bane of his life, lecturing at the Univer sity during the winter term, and at the Technical University during spring. He also kept working on mortality tables, and was widely viewed as a foremost expert on life insurance-he had kept up a Consultancy job with the Phoenix. In 1936, the Phoenix crashed. It had weathered the disasters of the First World War, the collapse of the Austro Hungarian empire, a catastrophic in flation, and the world-wide economic crisis. Austrians viewed it as the epit ome of solidity. Its crash was a hard blow, especially for the civil servants, who were traditionally underpaid but looked for compensation in the pros pect of a safe retirement. (To quote Nestroy again: "Civil servants get nix, but this is secure.") Now it turned out that the accounts had been based on fraud, and the investments were lost. Eight directors were arrested. A huge swamp of corruption came to light, tainting some of the pillars of society, including a former head of the govern ment, one General Vaugoin, a member of the administrative committee of the Phoenix.
Alfred Tauber. This may be the only surviving photograph of Tauber. He had few friends and no family, and the few belongings he was allowed to take with him to Theresienstadt vanished without a trace.
tained the selection theorem for func
The political tensions racking the young, unloved Austrian Republic had
Klein, Hermann Minkowski, and Carl Runge. A letter from Richard Courant
tions of bounded variation and, as an
led, in the early thirties, to the estab
to his fmancee provides a snapshot: it describes in detail how a question by
application, the criterion for weak-star convergence in C[ a,b 1 (re-discovered
Helly stopped Richard, then a nervous first-year student about to give his first seminar lecture, dead in his tracks. A confused discussion followed, with Hilbert and Minkowski joining the fray and some in the audience smiling in dulgently. In the end, the lecture had to be postponed for a week. By then, it was understood that Helly's question had been based on a misunderstanding. It is not clear why Helly, who was obviously earmarked for an academic career, published only two research pa pers in the next seven years. In retro spect, these contributions were seen, together with contemporary papers by David Hilbert (the famous "Mitteilun gen"), Erhard Schmidt, and Frigyes Riesz, as important steps in the devel opment of an as-yet-unchristened "functional analysis." Helly's first pub lication, in particular, a thirty-page treatise "On linear functional opera tors," was truly pathbreaking. To a large extent, it was motivated by the characterization of bounded linear op erators on C[a,b], which Riesz had dis covered in 1906. Helly's paper con-
by Hubert Bray some ten years later). It had versions of Banach-Steinhaus and the uniform boundedness princi ple for functionals on C[ a,b1. In each case it exhibited the general, func tional-analytic kernel of the results, us ing an Abstandsbestimmung which
lishment of an embattled clerico-fas cist regime which had outlawed both the Left and the Nazis. The ban of op position papers gave free rein to the wildest rumors. But it turned out in court that it was no mere rumor that the Phoenix company had systemati cally funded not only the private army of the regime but also the illegal Nazi party, the Jewish National Fund, an in fluential weekly, and a state official who had recently blown his brains out. The scandal shook Austria to the core. Alfred Tauber realised that he would have to keep lecturing, now that his consultancy fees had dried up: the small income from the course was es sential to cushion his retirement. The active members of the former Phoenix mathematical department were harder hit. The head of the de partment, Alfred Berger (who had suc ceeded Tauber both at the university and at the firm), was indicted, but eventually acquitted of all blame. The other mathematicians found them selves on the street; most took to giv ing private lessons. For Eduard Helly, this was nothing new. He had spent several years before the First World War giving private tu ition, and even had published four booklets with solutions to the class room exercises contained in Sup pantschitsch, the almost legendary textbook used for mathematics in the schools of pre-war Austria. Some felt that he was overqualified for such an enterprise.
and the constant M be given. There ex ists a continuous linear functional U on C[a,b 1 with norm bounded by M and satisfying U(fi) = ci for all i iff n
I I J.Lici l i� 1
:5
n
M I I f.Ldi ll i� 1
holds for all real numbers f.Li and all n. This is an extension theorem for functionals defined on the subset ! f1 > f2, . . . } of C[a,b 1. Its proof, in fact, con tains the main ingredient of the Hahn Banach theorem: it is shown that if the inequality holds for a given n (and all f.Li) and if fn + 1 is an additional contin uous function on [a,b), then it is possi ble to find Cn + 1 such that n+1
I I f.LiCi I i�1
:5
n+1
M il I f.Lifi I I i�1
holds for all choices of f.Li· Both Hans Hahn and Stefan Banach were later to use this idea in their proofs of the Hahn-Banach theorem, without quot ing Helly. It would take decades before Helly got due credit for his achieve ments. By 1913 Helly's prospects looked good. He had secured a position as a
Preparing a Course
Indeed, Eduard Helly, born June 1, 1884, in Vienna, had done brilliantly as a student there under Ludwig Boltz mann, Franz Mertens, Wirtinger, and the young Dozent Hans Hahn. In 1907, he wrote his Ph.D. thesis on one of the hottest topics of that time: "Contribu tions to the theory of Fredholm's inte gral equation." Wirtinger was impressed and organised a stipend, from the Fund for the Training of Future University Teachers, which allowed Helly to spend his postdoc years where it mattered: in Gottingen, with David Hilbert, Felix
was nothing else but a norm. As Harro Heuser wrote later, "in a curious way the abstract norm existed before the abstract normed space." In the same paper, Helly extended Riesz's solution of the moment prob lem to prove the following theorem (in modem terminology): Let the func tions fi E C[a,b), the real numbers ci ,
Eduard Helly. There exist many photographs of Helly, but not one that shows him smiling. Yet despite his pensive, occasionally melancholic look, he is consistently described as a very warm and friendly person with a large circle of friends.
school teacher (just as Frigyes Riesz had done a few years earlier), he was highly visible in the Viennese Mathe matical Society, and he gave courses at the Viennese Volksbildungsheim-a kind of worker's university, quite ad vanced for its time. In addition, he spent a lot of his free time at the Vien nese Mathematical Seminar, which had
VOLUME 26, NUMBER 2, 2004
23
Seeking knowledge, eighteen-year-old Eduard enrolled at the University of Vienna, the Technical University, and (later) the University of Got tingen. Students had to keep a booklet to register the "testates" of their professors. Helly's book contains the signatures of Boltzmann, Wirtinger, Mertens, Hahn, and Hilbert.
just moved into an imposing new build ing not far from his flat. Helly had also just discovered a ba sic theorem of convex geometry. In deed, while handling the above system of inequalities, he had used a simple statement: if a family of compact in tervals has the property that aU pair wise intersections are non-empty, then the intersection of aU intervals is non empty. He now hit upon a splendid gen eralisation: if a family of compact con vex sets in R"' has the property that any n + 1 of them have non-empty inter section, then the intersection of aU the sets is non-empty. Helly gave a lecture on this result, which would lead later to a luxurious growth of "Helly-type" the24
THE MATHEMATICAL INTELLIGENCER
orems, but he did not publish it-there seemed no hurry, and he wanted first to work out some generalisations. Moreover, Helly was in love: a young mathematics student, Elise (Liese!) Bloch, who had the good sense to work on functions of bounded variation she tried to extend its definition to sev eral variables, no trivial task-was clearly sensitive to his attentions. Ed uard had dedicated a handwritten ver sion of his first paper to Fraulein Bloch as early as 191 1 , but it would be ten years before they would marry. The Far End of the Line
In 1914, when the war broke out, Helly volunteered on the spot, like many
young intellectuals. His military train ing took place close to Vienna, and he was even able, for a time, to keep lec turing at the Volksbildungsheim. But in 1915, his unit was sent to the Russian front, where things were getting out of hand. In September of that year he was shot through the lung, and captured. The shot had dislocated his heart, but Helly somehow survived the wound, and a series of Russian hospitals, in Kiev, Kursk, and Voronezh. In 1916, he was deemed fit enough for transporta tion to Siberia; his camp, Berezovka, was close to Tobolsk. By 1917, the war with Russia ground to an end, and POWs were supposed to be returned to their home countries. But there was a
"Fur Elise." In 1 91 1 , Eduard Helly dedicated two theorems to his beloved, Fraulein Bloch. It would take ten years before the two were married, and even longer before the theo rems were fully appreciated.
1)) .
fi!Ltl 03� �-�
9� aJt o( eo
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"Siberiaks" would meet, years later, and reminisce about the Siberian clar ity of their seclusion. Camp authorities did not greatly worry about escape at tempts; where might escapees turn? Su pervision, while erratic, was usually light. In summer, posses of POWs would roam far and wide as lumberers: in win ter, most camps produced their own brand of cultural life, with makeshift universities, printing presses, and mu sical performances. Eduard Helly found some fellow-inmates interested in mathematical research. One, Paul Elbogen, wrote an "Introduction to ax iomatic methods" which was pub lished, ten years later, by his bereft mother. In the foreword, Helly wrote about "the terrible physical and psy chical pressure" under which the book let had been written. He himself re sumed the lecturing which he had been obliged to give up at the worker's uni versity, and found an exceedingly tal-
1 '1 -1 -1 .
snag: the Russian revolution had caused a fierce civil war, and indepen dent units of the Whites, the Reds, Eng lish and Japanese intervention troops, a free-roaming Czech unit, and sundry semi-military gangs led by desperate warlords ranged up and down the for mer Czarist empire and fought for chunks of the Trans-Siberian railway. In 1918, the prisoners finally left their camp, but only to end up in another camp, even further away, in Nikolsk Ussurisk, the last stop on the line be fore Vladivostok. Siberian internment camps, from well before Dostoevski to well after Solzhenitsin, were wellsprings of intellectual heroism. The fierce climate and the utter remoteness led to a solidarity among the inmates which often endured after their release: such old
Pop science. The municipality of Vienna strongly supported movements for spreading knowl edge to the masses. Mathematicians Eduard Helly, Hans Hahn, and Heinrich Tietze, physi cists Erwin Schrodinger and Friedrich Kohlrausch, and philosopher Karl Popper lectured to non-academic audiences. Helly would later use the skills acquired at this "workers' univer sity" to teach calculus to fellow prisoners of war, then to US Army officer candidates.
VOLUME 26, NUMBER 2, 2004
25
.-.
-
---
A day in the life of Lt. Helly (POW}. Despite the terrible hardships of Siberian camps, some of the young men were later to view their years in trans-Baikalean remoteness as a formative experience of almost other-worldly quality.
ented young Hungarian inmate named Tibor Rad6, a student destined to go far. Incidentally, Rad6 went far in more than one sense; in 1920 he escaped north and walked home (a hike of sev eral thousand miles through the Arc tic), and in 1930-by then a professor at Ohio State University-he immor talised his name by solving the Plateau problem. In the late summer of 1920 things fi nally got moving for Helly. After stopovers in Manchuria, Japan, and Egypt (where he spent another few months in a British internment camp), he returned to Vienna in December 1920, ending a five-year odyssey. His Penelope was waiting for him: Elise Bloch, after finishing her PhD thesis under Wirtinger's supervision in 19 15, had found a job as a school teacher. 26
THE MATHEMATICAL INTELLIGENCER
They would not lose any more time; they would marry as soon as Eduard got this habilitation.
cepted by the Monatshefte, although there was no money to go to press. Functional analysis had come of age after the war, with three almost simul taneously appearing memoirs by Ba nach, Wiener, and Hahn. A few weeks after Helly's return, Hahn was recalled from Bonn as the third full professor at the Mathematical Seminar. Furtwan gler and Wirtinger were glad to see someone from the younger generation in charge, and Hahn took command with boundless energy. He immedi ately recognised the value of Helly's manuscript and wrote a glowing re port. Within a few months, Helly be came Dozent and got married. But his hope for an appointement at the Uni versity failed to materialise. Only one position was vacant, that of an extra ordinary professor for geometry. Hahn paved the way, not for Helly, but for a young German, Kurt Reidemeister. Elise Helly, much later, hinted at an anti-Semitic factor behind the decision. Anti-Semitism was indeed rampant in Austria, and particularly at the Univer sity. The current head of state, prelate Ignaz Seipel, went on record with his opinion that the percentage of Jewish students was too high and would have to be limited. (He coined the word Notwehrantisemitismus-Notwehr means "self-defense".) However, Hahn himself was Jewish. Some might have felt that the appointement of another Jewish professor would be undiplo matic, but Hahn was unlikely to be moved by such timorous considera tions. In any case, Reidemeister proved to be a brilliant choice, and functional
Intersections
The Mathematical Seminar had not changed much. Rooms were unheated, ofcourse, in that freezing winter, and stu dents looked undernourished. Wirtinger had gone a bit deafer, Furtwangler was lame, and cranky old Tauber lived like a recluse. Helly, for that matter, had also lost some of his health and hair, but none of his enthusiasm for mathe matics. At the age of thirty-seven, he still had only two publications to his credit, but he had not been idle in Siberia, and he submitted a superb new manuscript in long-hand, "On systems of linear equations with infinitely many unknowns." It was immediately ac-
I· '
'
" . '
¥'
:a.l
' '
--
To my Mama. Stranded in Siberia for five years, Helly could communicate with his mother only very rarely. An intrepid Countess Kinski, the "Angel of Siberia," managed to visit the camps and distribute some help-in Helly's case, a loan of forty rubles, to be paid back by Eduard's mother.
Elise Helly (1 892-1992), nee Bloch, in a nearly professional glamour shot by her husband. Miss Bloch got her PhD in mathematics un der the guidance of Wilhelm Wirtinger. In the early years of twentieth-century Europe, it be came almost fashionable for girls with brains to study mathematics {the wife of the writer Thomas Mann being
probably the best
known example). During the First World War, they dominated the PhD lists.
met success. Inflation was overcome, and Vienna knew its share of the golden twenties. Working hours were long for Helly, but he found time to finish a note pre senting a proof of his theorem on in tersections of convex sets, which he had discovered ten years before. In the meantime, an Austrian and a Hungar ian, Johann Radon and Denes Konig, had each published his own proof of Helly's theorem, duly quoting him. Helly still was persuaded that the convexity assumption was too narrow, and he hunted for a topological extension. Topology was flourishing in Vienna during those years, and Helly had con tacts with Leopold Vietoris, Adolf Hur witz, and Karl Menger. He kept fre quenting the Mathematical Seminar. On Monday evenings he met his friends in the famous Cafe Central, where
analysis, as everyone agreed, was well covered by Hahn himself. Helly had to look for another job and luckily found one right away, as a bank clerk of the Bodencreditanstalt, a highly respected bank dealing mostly with landed property. Prospects looked bright enough for Liesel Helly, who was frail of health, to give up her teacher's job. A promise of prosperity hung in the air. Within a few years, the tough fi nancial measures of chancellor Seipel
Charming country. Eduard Helly was a highly gifted and competent photographer, and a pas sionate hiker. He sold this picture to the Austrian Tourist Office. In his later years of exile, the innocent text must have appealed to Helly's sense of humour.
VOLUME 26, NUMBER 2 , 2004
27
lively discussions and dead smoke filled the air. He was a charming, gre garious man, with a vast range of in terests-an avid chess player, a gifted photographer, and a passionate hiker, despite his dislocated heart. Although not a member of the Vienna Circle, he knew and associated with many of its members, and in particular two mem bers-at-large who often returned to their home town, Richard von Mises and Philipp Frank Von Mises headed the large Institute for Applied Mathe matics in Berlin, and Frank held a chair of physics in Prague. Helly must have thought of looking for a professorship abroad, but perhaps he was too timid to apply, or just too happy to be back in his native town. The Hellys kept an open house; Liesel enjoyed their con-
•
.. . . .. . .. .
• • . . " .
!
. I ea• •
Crash Course
Thus Helly resigned himself to staying in the Bodencreditanstalt. But in Oc tober 1929, within weeks of Wall Street's Black Friday, this bank, one of the largest in Austria, long patron ised by the imperial family and the landed gentcy, had to declare insolvency. Other Austrian banks were forced to
. · :,; - -:;
. . •.• .
•
was to be the fifth of Reily's mathemat ical publications. It came too late to help him get the job as Reidemeister's suc cessor, when the latter left for Konigs berg. In the meantime, a young shooting star named Karl Menger had written a dozen papers on dimension theocy, and Hahn preferred to appoint twenty-five year-old Menger rather than forty-one year-old Helly.
tacts with mathematicians. She lec tured at the Volksbildungsheim and translated some monographs. Helly met his topological mentor in Walther Mayer (of Mayer-Vietoris fame), a Dozent in mathematics who stood out, among the run of Viennese coffee-house intellectuals, for owning a coffee-house himself. (Albert Einstein would also look for Mayer's expertise: in 1932, he arranged to have him as his assistant in Berlin, and soon afterwards in Prince ton.) Eventually, Helly found how to ex tend his theorem beyond convex sets: if there exists, in R", a family of cells with the property that the intersection of any n + 1 of them is a cell, then the inter section of the whole family is a cell. (Helly defines a cell as a compact non empty set with all Betti numbers 0.) This
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: llow -.u ... "ip... J:.t, .
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Letter of condolence. In the twenties, young Kurt Godel had attended
Privatdozent, and the right to teach at the university. But on August 2,
lectures by Helly on algebraic curves and non-Euclidean geometry.
1938, the passport was extended; on August 8, he received an immi
The Godels had reached Princeton in spring 1940, traveling the long
gration visa authorised by the American vice-consul; and on Septem
way-via Siberia. Elisabeth Helly kept this letter in an album for her
ber 1 5th he, his wife Elise, and their eight-year-old son Walter em
son Walter, with the note; "Now all is well . . . and Papa dies."
barked in Cuxhaven for New York.
28
THE MATHEMATICAL INTELLIGENCER
shoulder the Bodencreditanstalt's obligations-which helped to delay the crash for two years--but its employees found themselves on the street. For Helly, it could not have come at a worse time: his wife was pregnant. But he managed to find another job, as actuary at the Phoenix. He did not be come the firm's chief mathematician; Alfred Berger held that job, which had been Tauber's years ago. Next in se niority came Ernst Fanta (1878-1939), an elderly dozent in actuarial science. Younger than Helly were two Hungar ian mathematicians, Eugene Lukacs (1906-1987) from Budapest, a former student of Walther Mayer, and Stefan Vajda (1901-1995) from Szombathely. These two young actuaries born into land-locked Hungary little dreamed that in later life they would head the theoretical divisions of the two largest sea powers of the world. Visit ing Phoenix was also the Polish Zyg mund W. Birnbaum (1903-2000), who came from the Banach-Steinhaus school of functional analysis. The younger men were soon to ap preciate Helly's encyclopaedic knowl edge, his talent, and his modest, friendly ways. Birnbaum described the congenial atmosphere: "Helly was a de lightful man, cheerful in the face of ad versity, with a gentle sense of humour. . . . Whenever a non-routine question came up, the difference between Helly and the other two [of Z. W.'s superiors] became apparent. Helly gave the prob lem a mathematical formulation and obtained a solution which could be used over and over again in similar cases; the other two worked the prob lem numerically in each case, grinding it out on their hand-operated desk cal culators. Incidentally, even the manner in which Helly handled the desk cal culator was ingenious." Helly took ob viously great interest in the job. His last research paper, published in 1934, was on actuarial mathematics. Additionally, Privatdozent Helly kept lecturing at the Mathematical Seminar, for a nominal fee. His topics were highly diverse, and his style was uni versally admired: "Helly's lectures were wonderful" (Edmund Hlawka), "works of art, spiced by his never-fail-
ing enthusiasm" (Elise Stein), "per fectly prepared, delivered in a free, al most dashing style" (Karl Strubecker). When in 1934 Hahn died unexpectedly, Helly and Menger were seen as the nat ural candidates for the vacant position. But the ministry decided not to fill the vacancy. In addition to the fiscal bot tleneck, pique may have been a motive: indeed, Hans Hahn, a forceful speaker for progressive causes, was certainly not well liked by the clerico-fascist regime. In the following year, Wirtinger retired, and his chair was filled by Karl Mayrhofer, an industrious assistant professor but certainly not of Helly's or Menger's calibre. Menger left in frustration for the United States. Helly remained with Phoenix Life Insurance, and saw it go down the next year, when the Phoenix scandal burst. By then, he must have felt dogged by bad luck He and Liesel had to give up their flat and move into her mother's house. After a few months, he managed to find another actuarial job. But it was not to last for long. In March 1938, the Nazis took over. Walter Rudin, a seventeen-year-old Vi ennese youth at the time of the An-
D•·
i
schluss, later wrote: "In a way we had it better than the German Jews. In Ger many, the screws were tightened grad ually. . . . For this reason many German Jews hesitated until it was too late. In Austria, it became clear within a few days that there was no alternative to leaving the country." The group of mathematicians from Phoenix dispersed in all directions. Sixty-year-old Ernst Fanta died a lonely death in 1939 in a shabby hotel room in Sao Paolo. Z. W. Birnbaum, who had returned to Poland before the Phoenix collapse, managed to find a job as a foreign correspondent in the U.S. He later embarked on a distin guished career as professor of statis tics at the University of Washington. Eugene Lukacs reached the United States in February 1939. For a number of years he taught Latin and mathe matics at several colleges. After the war, he turned to probability theory, and became in 1953 head statistician at the Office of Naval Research. Later, as professor at the Catholic University in Washington, D.C., he wrote a mono graph on characteristic functions which was to dominate the field for many years. Stefan Vajda, with the help
UNIVOJOF. ALFRE�.TAUBER TELiFO�42 4 15 t,.Y•�·
"Unfortunately impossible." A letter from Alfred Tauber to the Rektor of the Technical Uni versity explaining that he could not deliver the requested proof of Aryan descent. By May 1 939, he had been obliged to give up his flat and move to the Jewish quarter. All Jews were required to use a new middle name, Israel or Sara.
VOLUME 26, NUMBER 2, 2004
29
of his ex-Viennese friend Karl Popper, emigrated to England in 1939. When war broke out, he was interned for a spell, but eventually became (as the successor of John Todd) head of the mathematical departement of the British Navy, and he wrote the first textbook on linear programming. In 1965, he became professor in Birmingham, and later at the University of Sussex.
eight-year-old son, Walter, for New York. Before they could leave, they had to pay the extortionist Reichsjluchts teuer, a tax required by the Nazis from whoever intended to "desert" the Third Reich. In 1939, immigrant mathematicians were not exactly a scarce commodity
in the United States. They had been ar riving for years. The Rockefeller Foun dation had long since turned into a rescue squad, with Stephen Duggan, Harold Hardy, and Oswald Veblen per forming miracles of selfless help, as sisted by early arrivals from Germany such as Emmy Noether, Hermann
A Lesson for Professors
The Nazis lost no time in "purifying" the University of Vienna (their term was Siiuberung ). The student body had already been subverted, to a large ex tent, before the Anschluss. From 1931 onward, the National Socialist student movement had been the strongest group in the Austrian Students' Union. It had been officially disbanded in July 1933, but Viennese authorities proved helpless to suppress the movement. Among the professors, Nazi sympa thies were much less pronounced. Right after the Anschluss, the newly commissioned rector of the University addressed this issue squarely in his first speech to the assembled faculty. He added ominously that "all this has changed now. The object-lesson which the professorial group had had occa sion to appreciate during the FUhrer's stay in Vienna, will not fail to have its effect." The German Act for the Restitution of a Professional Civil Service was im mediately implemented. All staff mem bers of the University had to swear a public oath of allegiance: "I swear to be loyal and obedient to the FUhrer of the German Reich and Volk, Adolf Hitler. . . . " Of course only true and proved Aryans were allowed to deliver the oath. At the Philosophical Faculty of the University of Vienna, 14 of the 45 full professors, 11 of the 22 extra ordinary professors, and 56 of 159 do cents were summarily dismissed. Helly was one of them. He also lost his ac tuarial job. Even tutoring was out of question. Elise had relatives in the United States, and after frantic months of ut ter insecurity and bureaucratic hassle, the Hellys managed to get an immigra tion visa and to leave Austria with their
30
THE MATHEMATICAL INTELLIGENCER
"Not yet sworn in." On June 1 938, the university produced a list of staff members who had failed to take the oath to Hitler. Not everyone on the list was Jewish. The mathematician Karl Strubecker, for instance, just ahead of Tauber on the list, had already been sworn in at the Technical University. But for many, this list was the beginning of an ordeal leading finally to another list-the Theresienstadt Book of the Dead.
"ARBEIT MACHT FREt."
"Work liberates" was the cynical motto on the gate of Theresienstadt. This was not an extermination camp, like
Auschwitz, but the forced labour killed many of its inmates. In a grotesque propaganda film, Nazis tried to pass off Theresienstadt as a com fortable place for retirement.
Weyl, Richard Courant, Harald Bohr, and John von Neumann. But after five years of this, their glowing letters of recommendation were becoming less and less successful. The Hellys, for a time, had to resume their private coaching of apprehensive school kids. Once, Helly was invited to give a sem inar talk at the Institute for Advanced Study in Princeton, where his old friend Walther Mayer was still working as Einstein's assistant. The lecture, de livered to an audience of luminaries, was a splendid career opportunity, but according to an eye-witness, Helly per formed less than brilliantly on that oc casion. In the end, the combined patron age of Einstein and Weyl landed him and his wife teaching jobs at the Col lege of Paterson, in New Jersey. The couple worked hard, and one year later were promoted to Monmouth Ju nior College. They kept at it, and at long last, in early 1943, the break through came: Eduard and Elisabeth were appointed instructors for the Signal Corps Training Program. The position was modest, but it was at the brand-new Illinois Institute of Tech nology, in Chicago: after many years, the Hellys had resumed contact with university life, and with research mathematicians. In September 1943, Helly was pro moted to visiting lecturer in mathe matics for the Army Specialised Train ing Program at liT. This was, at the age
of fifty-nine-a long-awaited profes sorship! But Helly was not destined to enjoy an academic career. His dislo cated heart had started to give him trouble. He survived a first attack, but on the morning of November 28, 1943, after a pleasant evening with Ernst Hellinger, Herbert Busemann, and other colleages, he failed to wake up. He had survived Tauber by one year. The "proof of Aryan descent" had been made mandatory even for those who, like Tauber, had long been emeritus. As Tauber wrote to His Magnificence, the Rektor, it was impossible for him to give such a declaration. He thus found himself on the long list of the univer sity's staff who had "not yet" sworn their allegiance to the FUhrer. There was to be no more lecturing for Tauber. In fact, he was not even allowed to en ter the university building, or even to use its library. He was an ailing outcast at the age of seventy-two, and he had desperately few contacts abroad. Most mathematicians would have been as tonished to learn that the discoverer of the first Tauberian theorem was still alive. But Tauber, who had been obliged to sell most of his possessions and to move into reduced quarters, knew that he would have to leave town. The Totenbuch
Adolf Eichmann had taken up his work in Vienna. It consisted, at first, in "organising" emigration to Palestine, blackmailing the tormented Jewish
community. When war broke out, this was no longer feasible. Ominous mass deportations started leaving "for the East." People whispered that the trans ports were heading for Theresienstadt. The Nazis launched the rumour that this was a safe haven, a place of re tirement for elderly and prominent Jews, looking not in the least like a concentration camp. Goebbels's pro paganda ministry even produced a film, The FUhrer presents the Jews with a town, purporting to show, in docu mentary style, that Theresienstadt had all the trimmings of a charming shtetl, with sidewalk cafes and promenade concerts. But no one sent to There sienstadt had ever returned to confirm these stories. Meanwhile, the Gestapo dragnet tightened remorselessly. Tauber remembered a former stu dent, Emil Schonbaum, who had se cured a position as civil servant in Quito, Ecuador. Correspondence to and from Ecuador was somewhat ir regular, but the two men devised a glorious plan to establish mortality rates for the whole of Latin America, and in November 1941, a few weeks before Pearl Harbour, Schonbaum was able to invite Tauber: he would take care of the visa and the travel costs. Tauber wrote to Mrs. Michal up, the mother of another former student: "I want to try to find a job as substi tute teacher in Quito, where the uni versity advertises for European appli cants, and as a retired university
VOLUME 26. NUMBER 2. 2004
31
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"Book of the dead." On hundreds of pages, the "Totenbuch Theresienstadt" (published long afterward but drawn from Nazi records) lacon ically lists its inmates. Alfred Tauber's birthday is wrongly given as 8.1 1 .1 886 (instead of 1866). Another Viennese mathematician on the list is George Pick (1 859-1942), the discoverer of the eponymous theorem on the area of polygons with integer-valued vertices, who had stud ied in Vienna and worked as assistant of the physicist Emst Mach before becoming, in 1892, professor in Prague.
32
THE MATHEMATICAL INTELLIGENCER
professor I may have some chances, despite my advanced age." Brave words, but it was too late. At the se cret Wannsee conference, high-rank ing Nazi officials had decided to speed up the "final solution." On June 28, 1942, Mrs. Michalup, coming to bring a few victuals to the old professor, who was now living in a shabby side street in the Jewish dis trict, caught a barely perceptible signal by the concierge. Instead of climbing the narrow stairs to deliver her fat and flour, Mrs. Michalup passed on, as in conspicuously as possible. Two weeks later, she returned to see the concierge, and learnt that two Gestapo men had, at the time, been in the building to es cort Tauber to the nearby railway sta tion. The police file duly kept up with the change of address: Nach There sienstadt abgemeldet. The frail old man did not survive a month: he died on July 26, 1942, as recorded in a bleak and poignant list, the Totenbuch There sienstadt.
ter Helly, New York, as well as to the archives of the University and the Technical University of Vienna. REFERENCES
Einhorn. R. (1 985), Vertreter der Mathematik und Geometrie an den Wiener Hochschulen
1 900- 1940, PhD Thesis, Technical Univer
sity of Vienna. Binder,
C.
(1 984)
"Alfred Tauber (1 866-
1 942) - Ein i:isterreichischer Mathernatiker," Jahrbuch
Oberblicke
Mathematik,
Bibli KARL SIGMUND
ographisches lnstitut, pp. 1 51 - 1 66.
Un1Vers1tat Wien
Jahresbericht der DMV 82, 1 8-51 .
Strudlhofgasse 4
Hochstadt, H. ( 1 980), "Eduard Helly, father of
1 090 Vienna
the Hahn-Banach Theorem," Mathematical
Austria
Jntelligencer 2 , 1 23-1 25.
Heuser, H. (1 995). "Commentary on Hans Hahn's contributions in functional analysis," in Col lected Works of Hans Hahn (eds. K. Sigmund
and L. Schmetterer), Springer-Verlag, Vienna.
Pin I, M. and A. Dick (1 97 4), "Kollegen in einer dunklen 1 66-204.
Zeit, "
Jahresbericht
Nachtrag
und
DMV
75,
Berichtigung
Jahresbericht DMV 77 (1 976), 1 6 1 -1 64.
ropa " - Wien
This article could not have been writ ten without the help of Christa Binder, Hans Ploss, Josef Teichmann, and Inge Troch. I am grateful to Professor Wal-
lnslltut fur Mathematik
Butzer, P.L. et a!. (1 980), "Eduard Helly,"
Sigmund, K. (2001 ), "KOhler Abschied von Eu
Acknowledgements
A U T H O R
1 938 und der Exodus der
Mathematik. Catalogue for an exhibition of
the Austrian Mathematical Society.
Rudin, W. (1 996) The way I remember it, Amer
Karl Sigmund is a mathematician by professio n, historian by marriage, and Viennese by coffee-house allegiance.
All of this has emerged in his several past lntelligencer contributions. He
says he has always been fasc1nated by non-equilibrum situations: in er godic theory (hyperbolic systems), in biomathematics (heteroclin1c anrac tors). 1n game theory (evolutionary dy namics), and in history of mathemat
ics (Vienna 1n the 'th i rties) .
ican Mathematical Society, Providence, Rl.
Pure mathematics is a branch of applied mathematics.
-Joe Keller
VOLUME 26, NUMBER 2, 2004
33
Mathematical Tourism in Siberia George Lugosi
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the caje 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 to this column 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 Tourist Editor, Dirk Huylebrouck, Aartshertogstraat 42,
8400 Oostende, Belgium e-mail:
[email protected]
34
I
n Western Siberia, in the lower basins of the rivers Ob and Irtysh, live the Khants (or Rants, Hanti, or Ostyaks). Although they are but a small group of about 20,000 to 23,000 people, they are the third largest group of the North. The region is not very densely popu lated, as the territory is unforgiving ter rain, inundated by snow and ice in win tertime and by water in summertime. In this harsh local climate, the Khants' log-houses, covered by birch bark and raised on stilts, provide suffi cient shelter. Clay ovens surround them, and for centuries, bread was baked and fish smoked in the same tra ditional way. However, since the 1960s, land devastation by ruthless oil or in frastructure development and the con sequent pollution has caused an eco nomic and social crisis. It severely threatens the Khant culture. It is timely to study what is left of their culture, and in particular the aspects of inter est to mathematicians. Their folk art is remarkable. Pat terns on coats and dresses are of particular interest (Fig. 3). The rein deer-skin fur is trimmed in part with reindeer and in part with hare fur, cut to an even length. The names of the decorative band patterns are "swan's legs" and "hare's ears." In several areas (especially in Eastern and South Eu rope), coat decoration determines whether the garment belongs to a girl or to a woman, but Khant folk art does not make such a distinction, and the topcoats of girls of marriageable age and married women hardly differ. We can describe certain patterns in shirts like the one in Figure 3 in terms of permutation. Indeed, the cross stitch patterns on the shirts can be summarized in a matrix M = (m ij) ij � l ,
. .
.
, n
with values 0 (white) or 1 (black). To the left of each row, a number indicates how many ones there are in the corre sponding row, and above each column,
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
another number specifies the number of ones in the corresponding column. In these patterns the mij element will be 1 when the sum of the numbers to the left of the row i and above the col umn j is greater than n. For instance, if n 4, and if 1 , 2, 3, 4 designates the desired number of ones in the respective rows, and 4, 3, 2, 1 the desired number in the columns, then m 1 , 1 1 + 4 (>n), m 1 ,2 = 1 + 3 (=n), m l, 3 = 1 + 2 (:Sn) . . . . This yields the configuration =
=
In the same way, the indication 3, 2, 1, 4 left of the rows and 4, 3, 1 , 2 above the columns, produces
4
3
1
2
3 2 1 4 To generate all of the configurations possible using the same column se quence, a diagram is made as follows. Starting with the row sequence 1, 2, 3, 4, the permutation of rows provides 24 4! possibilities of Figure 4. Per muting the column sequence, another 24 possibilities result in 24 tables, each containing 24 configurations. A block with all possibilities will thus contain 576 = 4! X 4! different configurations. In every table, some configurations are =
Figure 1. Map showing the Khant region.
Figure 2. A. Y. Filtchenko's photograph of a traditional Khant log-house (1 998).
Figure 3. A shirt with rich cross-stitched patterns. The arrows indi cate patterns explained in the text (Photo: Markku Haverinen).
VOLUME 26. NUMBER 2, 2004
35
September 29, 2003, the museum or ganized a special exposition titled: "Sibe
ria Ufe on the Taiga and Tundra" (see
http://www.nba.fi/NEWS/LEHDISTO/ 2002/siperialehdistokuvat.htm). The mu seum is housed in a former indoor ten nis complex, at the "Etelilinen Rauta tiekatu
8 I Salomonkatu 15" in Helsinki.
The multiplex has been renovated to house the museum as well as the Helsinki City
Art Museum, Finnkino's
cinema centre, and a number of shops, restaurants, and cafes.
R&D&I- Research, Develop, Invent 2 Union Street Kew 3 1 0 1 Victoria, Australia e-mail: g .
[email protected]
Figure 4.
Figure 6.
Figure 5.
the horizontal mirror images of each
these patterns in different regions of
other, whereas in each block the con
Siberia (see Figure
figurations of some tables are vertical
plex example), or elsewhere in the
7 for a more com
mirror images. The pattern of the con
world, is an interesting ethno-mathe
nections, valid inside the tables and in
matical subject. The mathematical set
side the blocks, is shown in Figure 5.
up here may help to classify different
6 is a magnified image of Fig
stitched patterns and to compare con
ure 3; it uses repeatedly the configura
figurations made by cultures from dif
tion 1, 3, 2, 4 (rows) and 3, 1, 4, 2
ferent regions or from different eras.
Figure
n = 4, and the configu
A museum that attaches great im
ration 1, 4, 2, 5, 3 (rows) and 5, 1, 3, 2,
portance to these fur coats with their
mirror images.
Helsinki
(columns) for
4 (columns) for n = 5, along with their To investigate the occurrence of
36
THE MATHEMATICAL INTELLIGENCER
intriguing geometrical patterns is the Museum
of Cultures.
Re
cently, between May 14, 2002, and
Figure 7. More cross-stitched Khant pat terns.
1Mfijii§jr@ih££h§4£i 11 i.J i§.id ..
Autologlyphs
M i chael K l e b e r a n d Ravi Vak i l , Ed itors
For solutions, see page 411.
Henry Segerman and Paui-Oiivier Dehaye
This column is a place for those bits of contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
G O +2 3 8lJ OODVOD
Please send all submissions to the Mathematical Entertainments Editor,
� �� � OO � ffi CW
Ravi Vakil, Stanford University,
Department of Mathematics, Bldg. 380, Stanford , CA 94305-2 1 25, USA e-mail:
[email protected]
© 2004 SPRINGER-VERLAG NEW YORK. LLC. VOLUME 26, NUMBER 2. 2004
37
Il 11 � I 1 11 I
11 11 11 11
THE MATHEMATICAL INTELLIGENCER
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VOLUME 26, NUMBER 2, 2004
39
ip.i$�•ffli•i§
[email protected]?Ji
Kolmogorov and Aleksandrov in Sevan Monastery, Armenia, 1 919 Vahe G . Gurzadyan
Does your hometown have any mathematical tourist attractions such as statues, plaques, graves, the cafe where thefamous conjecture was made, the desk where the famous initials are scratched, birthplaces, houses, or memorials? Have you encountered a mathematical sight on your travels?
L
D i rk H uy l e b ro u c k , Ed itor
ake Sevan is one of the most beau
and princess Mariam. It includes two
tiful spots in Armenia. It has always
churches, the Arakelotz (the Church of
attracted poets and painters. Maxim
the Apostles) and Astvatsatsin (the
Gorky called it "a piece of sky which
Church
came down among the mountains. " In
panoramic view on the lake from the monastery. In the 1930s, the water flow from the lake via the river Hrazdan was
33, two mathematicians just starting
increased for irrigation and hydroelec
their life-long friendship, visited Sevan.
tric power purposes. This lowered the
For about a month they enjoyed the
level of the lake, and transformed the
beauties of the lake, living in a cell in
island into a peninsula. However, in
the monastery on the island of Sevan.
1929 the island still existed, and Kol
They visited other sites of Armenia, but
mogorov colorfully described the life
still actively continued their research.
on that isolated spot:
Both mathematicians mentioned Sevan in their memoirs, many decades later:
At that time, Sevan was in its full glory. We bathed several times a day in its cool limpid waters. We also did a lot of work; in particular, I worked on "Aleksandrov-Hopf' (the type writer was always with me) (Alek sandrov [ 1]). Naturally we were immediately at tracted by the rocky islet, which with the decrease in the level of the Lake Se van has now became a peninsula, and we wanted to settle there. This proved to be a simple matter. The cells of the monastery were empty and we occu pied one of them (Kolmogorov [2}). Sevan provides an unusual combi
column a picture, a description of its
titude lake, 1900 m above sea level, 75
may follow in your tracks.
km in length and 55 km in breadth. The
water is fresh and transparent, and the people of the region drink it abun dantly. A number of rivers flow into the lake and only one, the Hrazdan, flows from it. The lake is famous for its trout and beautiful peninsula, which was once an island. During the Middle Ages various battles occurred
over that
Please send all submissions to
rocky island, where Armenians often
Mathematical Tourist Editor,
sheltered from their enemies.
Dirk Huylebrouck, Aartshertogstraat 42,
The monastery on Sevan island was
8400 Oostende, Belgium
founded in 874 AD by the Armenian
e-mail:
[email protected]
king Ashot, of the Bagratuni dynasty,
40
There is a
N. Kol
nation of natural conditions, a high-al
a map or directions so that others
of the Virgin).
mogorov, 26, and Pavel S. Aleksandrov,
the summer of 1929, Andrei
If so, we invite you to submit to this mathematical significance, and either
j
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
The permanent population of the is land consisted then of the archiman drite of the monastery (who had a fairly big house), his wife (who looked after some cows), the head of the me teorological station with his small family, and finally the "Captain, " who did indeed command "the Sevan fleet" consisting of one motor boat and a few of the unusual pleasure boats. His picturesque figure has occasion ally been described in literature (for example, by Marietta Shaginyan). Every day the archimandrite opened the lower church (there were two abandoned temples on the top of the hill), lit candles and, in complete soli tude, recited the service. Obviously the head of the meteorological station car ried out his duties. At times, the Cap tain brought honored guests, such as Sar'yan or the then President of the All-Union Central Executive Commit tee of Armenia, but he was also ready to support humble tourists. On the island, we both set to work. With our manuscripts, a typewriter, and a folding table, we sought out the secluded bays. In the intervals be tween our studies, we bathed a lot. To study, I took refuge in the shade, while Aleksandrov lay for hours in full sun light wearing only dark glasses and a white panama. He kept this habit of working completely naked under the burning sun well into his old age.
to the monastery of Hayravank under the guidance of our captain). The Hayravank monastery with its IX-century church is situated on a high rock facing Lake Sevan. According his student Hrant Maranjian, Kolmogorov recalled tiny details about that stay at Sevan's island during his visit in Ar menia in 1973.
For Aleksandrov the day was ap proaching on which he had arranged to journey to Gagra, and we set off to gether for Yerevan (where we stayed for some days in a student hostel). The Pavel Aleksandrov and Andrei Kolmogorov (photo Uspekhi Mat. Nauk, 1 986). temperature was 40°C, the sky was a hazy blue, and only after sunset did there unexpectedly appear the peak of On Sevan, Aleksandrov worked on "On analytic methods in the theory of Ararat, suspended in this blue sky. We visited Echmiadzin (where we de various chapters of his joint mono probability. " Given its position, Lake Sevan cided not to visit Katholikos as we did graph with Hop!, "Topology" [3}, and he helped me to write the German text mostly enjoyed sunny weather, but not have the right clothes). From Ech of my article on the theory of inte sometimes clouds com'ing from the miadzin, we walked to Alagez (spend grals. Besides writing this paper, I East fiUed up from the mountains, ing the first night by the lake, where was busy with ideas about the ana dropped down to the water and then, physicists working on cosmic showers lytic description of Markov processes on contact with it, va,nished. We stayed very kindly put us up). After spending with continuous time, the end prod there for about 20 days without leav one night (stiU without suits, wearing uct of which later became the memo'ir ing the pla,ce (apart from excursions only shorts), we climbed the south
The Sevan monastery where Kolmogorov and Aleksan-
General view of the Sevan peninsula, in 1 929 still an island, where the monastery
drov lived in 1 929 (picture by the author, August 2003).
is situated.
VOLUME 26, NUMBER 2, 2004
41
The summit of Mount Aragatz, climbed by Kolmogorov (photo H. Badalian).
Alagez summit, which did not present any complications (4000m). From the top there opened up a view of the rocky northern summit (41 00m), separated from the south by a huge ridge of snow, at the very bottom of which could be seen a small lake, its shores frozen and covered with snow. Of course, Aleksandrov wanted to climb down there and bathe, but I preferred climbing the northern summit. At the time, reaching the top of Mt. Aragatz (Alagez) involved a 30 km walk over mountainous terrain, the roads not having been built until the 1940s. Aragatz has a spectacular 3km-wide crater with a glacier and 4 summits surrounding it. The northern summit is not only the highest but also the most difficult one for climb ing; its easiest route is classified "lB" in the mountaineering category of dif ficulty. The cosmic ray station of Yerevan Physics Institute is situated 3200 m above sea level. The nearby beautiful lake, mentioned by Kolmogorov, is an
42
THE MATHEMATICAL INTELLIGENCER
artificial reservoir that was con structed in the Second Millennium BC, as archaeological studies have found [5]. The surface of the lake is free of ice only 2 to 3 months a year, and Alek-
sandrov's desire to bathe in the icy wa ter is as characteristic as Kolmogorov's climbing of the most difficult northern summit of Aragatz. After the dissolution of the Soviet
The cosmic ray station, 3200 m, where Kolmogorov and Aleksandrov stayed the night, before climbing the Aragatz mountain the next day. Many scientists who went mountaineering later visited the station. The photo shows (from right to left), L. D. Faddeev (St. Petersburg), Alexan der Migdal (Princeton), R. Kallosh (Stanford), Arcady Migdal (Landau Institute, Moscow), A. M. Polyakov (Princeton), V. G. Gurzadyan (Yerevan Physics Institute), and A. G. Sedrakian (Yerevan Physics Institute) descending the Aragatz summits in 1 983. Walking behind was an other member of the group, A. Linde, now a famous cosmologist {his hat is visible in the back).
Union, the former communist con straints on religion have been re moved in the Republic of Armenia. The activity of the Sevan monastery is much increased, not only because of the larger number of priests but also because a seminary has been opened on the peninsula. The monastery and the peninsula are now attractive tourist areas. Few of the tourists, however, can guess that
the gloomy cells, with walls completely blackened by many centuries of candle smoke, once accommodated two lead ing mathematicians of the twentieth century.
[3] P. Alexandroff, H. Hopf, Topologie, Berlin, 1 935. [4] P. Cuneo, Architettura Armena , De Luca Edittore, Rome, 1 988. [5] A. Kalantar, Armenia: From the Stone Age to the Middle Ages, Civilisations du Proche
Orient, Neuchatei-Paris, 1 994.
REFERENCES
[1 ] P. S. Aleksandrov, Russian Math. Surveys 35, 3 1 5 , 1 980.
Vahe G. Gurzadyan
[2] A. N . Kolmogorov, Russian Math. Surveys
Yerevan Physics Institute, Armenia
4 1 ' 225, 1 986.
e-mail:
[email protected]
Mathematics a nd Cultu re
Mathematics and Culture mmer, Univer iry of Rome 'La
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VOLUME 26. NUMBER 2. 2004
43
Solutions to Autologlyphs
(pp. n-39)
Henry Segerman and Paui-Oiivier Dehaye
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THE MATHEMAT1CAI.. INTEU.JGENCER
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N. J. WILDBERGER
Foundations of Fi n i t e Grou p Theory for a (Future) Com puter ood morning, Daisy. In 202 7, your Development Committee approached me with a curious proposal. You, a dynamic AI system, were soon to be launched, and to demonstrate your capability to discover new mathematics with a minimum of instruction, they wanted a beginner 's course in elementary group theory (meant for a computer, of course). You were going to absorb the ma terial and immediately start doing research. They claim that in capacity to comprehend written Eng lish, and in familiarity with basic mathematics, you are like a good second-year undergraduate. But your ability to as similate and compute is astounding. You trounced Deep Blue ten minutes after learning the rules for chess. GhengisV, one of the main gurus on the project, ex plained what was required. "This beauty is a whiz. Give her the main definitions, VO specifications, and general directions-then we'll let 'er rip. Getting her to do human mathematics is not so much the point, since GAP and MAGMA already do that very well. Mathematics by a computerfor a computer is the goal here. "That's what we got with classical logic in one of her training sessions, where she devoured, almost instanta neously, all the exercises we had found, and started churn ing out new stuff." "What do you mean by I/0 specifications? And why don't you give her some group theory books?" I asked. "Computers can't easily work with abstract objects un less definitions are tied down with conventions for input and output," he replied. "For example, the term 'graphics
file' is vague, whereas 'gif file' or 'jpeg file' is precise. It's a concept mathematicians should take more seriously. "You define a group to be a particular kind of multipli cation table. So a computer will, by default, expect groups to be input and output as multiplication tables. When sub sequently groups are discussed in terms of generators and relations, or physical symmetries, or abstract operators, the computer gets confused. She looks for software to convert these forms into the one agreed upon in the definition, be cause she wants to work with objects, not just talk about them. "To put it another way, when you say 'Let G be a group,' you are presumably initiating a thought experiment. But the subsequent development of that experiment will be, or ought to be, dependent on the form you have in mind for that unknown group. In the absence of a convention for this form, what does the phrase mean? "Please make sure your basic definitions are completely concrete, that is, understandable by a computer. But don't worry about programming her, or organizing the subject in some kind of computer style. "In fact, forget about the way we AI systems work Daisy programs herself. Arranges her internal structures in a
© 2004 SPRINGER-VERLAG NEW YORK, LLC, VOLUME 26, NUMBER 2, 2004
45
spooky way which even we don't understand, and is smarter than you can probably even imagine. Just explain to her the most important ideas, as you see them, and let her discover the rest. "And keep the following in mind: An hour after you teach her group theory, she'll be teaching you group theory." Overview
So I took the job, and the result is this object-oriented course. May I begin with a brief overview, even though it won't make a lot of sense till you have absorbed the terms and perused the literature? (It may help human readers at least.) The usual introduction to group theory starts out with the main definitions and some examples, typically the sym metric groups sn> groups arising from symmetries of familiar two- and three dimensional objects, and groups of matrices. It then goes on to a discussion of subgroups, Lagrange's the orem, homomorphisms, and quotient groups. This is followed by material on conjugacy classes, actions, gener ators and relations, and solvability. The course often ends with the Sylow theorems. In a second course, or perhaps a related course in lin ear algebra, there is a treatment of the classification of fi nite abelian groups and duality, and the extension of these ideas to the non-commutative setting, involving character theory and representations. The latter subject is quite im portant in applications. This course is unorthodox. The order of topics is dra matically different from the above; there are very few ex amples, and the treatment of many areas is non-standard, even heretical. Some of the usual material is not included. The order of topics is different because I want to ad dress right away the two problems which I feel are at the heart of the subject. These are: "How do you tell if two given groups are isomorphic?" and "How do you find the character table of a given group?" I want you thinking about these issues as soon as possible. There are very few examples because you are in a much better position to generate them than I am. Most of the groups I know are of a particular geometric or algebraic kind, and they are really rather unrepresentative. This re flects the human inability to process large amounts of in formation unless it is presented to us geometrically (that is, visually). Not having such a limitation, you should find it relatively easy to generate explicitly millions, even bil lions, of groups, and subsequently perform all manner of empirical investigations on them. The treatment is non-standard because I am influenced by two theories which shed a lot of light on group theory. One is the theory of multisets, as expounded in [W1]. The other is the theory of hypergroups, which developed in the 1970s and provides a unified arena in which to investigate
objects which have been studied under a plethora of dif ferent names. These include generalized translation opera tors, pseudo-groups, Heeke algebras, hypercomplex sys tems, paragroups, superselection sectors, Bose-Mesner algebras, Racah-Wigner algebras, centralizer algebras, table algebras, association schemes, and fusion rules of confor mal field theories. For an introduction and explicit refer ences, see [BI], [BH], [W3], and [OW]. I will show that with the right attitude toward multisets, hypergroup-theoretic ideas can be introduced into group theory in a direct and elementary way. This allows us to get to the heart of the subject even in a first course, with out resorting to group algebras. Finally, a word or two about programs. To me, a pro gram is a finite collection of explicit instructions that takes an input of a specified kind and yields an output of a specified kind. A "pro gram" that does not always halt is like a "theorem" which is not always true. Neither interests us. Ob taining bounds on running times of programs in terms of sizes of inputs is a valuable goal. I will suggest that you develop particular programs, usu ally to count something or other. These Exercises and Prob lems are necessarily somewhat open-ended. The problems are very difficult, at least for me.
I am i nfl uenced by two
theories wh ich shed a l ot of l ig ht on g roup theory:
multisets and hypergroups .
46
THE MATHEMATICAL INTELLIGENCER
Definitions, Specifications, and Examples
Let G be a finite set of specified mathematical objects. That means that G is given to us, either explicitly, as in the or dered set G 1 { 1, 2, 3, 4 } or the (unordered) set G2 = { 0_6_\7 } , or indirectly via a program. Clearly both of us could write a program to list the elements of =
G3 = {n ln
=
1, 2, · · · , 10 100 } ;
but I would have no idea how to proceed with
G4
=
{n E [ 5,
6,
· · · , 10 10 } l 3k, l E [ 5, 6, · · · }, n5k2 - nl8 k7 =
+
l3
+
1 },
so to me the latter has only been described, not specified. A multiplication program on G is a program that in puts ordered pairs {x,y) from G and outputs a single ele ment from G, denoted either by x · y or by xy. A multipli cation program on G is associative if (xy)z x(yz) for all x, y, z E G, is commutative if xy yx for all x, y E G, and has an identity if there exists a distinguished element e of G such that ex = xe = x for all x E G. If a multiplica tive program on G has an identity e, then it also has in verses if for each element x E G there exists an element y E G with xy yx e. If G has an associative multiplication program, then if an identity exists, it is unique, and if in addition inverses exist, they are also unique. An associative multiplication program on G which has an identity and inverses is called a group. We say simply that "G is a group." If the multi=
=
=
=
plication program of a group G is commutative, then G is a commutative group. The order of a group G is the num ber of elements of G, denoted IGI. To specify a group G, we should specify both the un derlying set G and the multiplication program on G. This may be done directly, by listing in some fashion all the el ements of G and giving an explicit table for the multipli cation program, or indirectly, by providing programs that will output the elements of G and the results of multipli cations. Example 1. Let G = { 1, 2, 3, 4 ) , with multiplication pro gram given by the following table, where by convention the element in row i and column j is the group element i . j. 2
i ·i 2
2
4
4
2
3
3
3
4
4
3
4
3
4
2
2
3
This is an example of a commutative group which has been directly specified. .
G {n 1 n = 1, 2, 1 0 1 00 ) , with mul tiplication program given by the formula
Example 2 . Let
=
i j ·
=
·
·
·
,
i + j - 1 (mod 1 01 00) .
This is another example of a commutative group. It has been indirectly specified by a program. Example 3. Let S be the set ofpermutations of { 1 , 2,
, n n), with a typical permutation of the form u = n], and the [u(l )u(2) u(n)] , the identity e = [ 12 multiplication defined by composition ·
·
·
·
·
·
·
·
·
(u1 · uz)(k) = u1 (uz(k)). This definition hovers between a description and a spec ification, since a modicum ofprogramming skill, but not much more, is required to list all n! elements and the mul tiplication program explicitly for any given n.
1
2
I I
1
1
2
2
2
3
4
5
6
7
8
9
10
11
4
3
12
10
11
9
8
6
7
8
6
3
3
3
4
5
5
9
12
10
6
4 6
4
3
6
11
8
8
12
10
10
12
12
7
9
11
7
9
11
5
2
2
7
10
9
5
8
7
11
8
5
6
4
10
11
12 6
7
9
5
8
9
4
11
7
6
11
7
5
9 3
10
8
5
12
10
11
6
12
12
4
3 8
9
7
10
8
4
9
2
4
2
4
6
7
4
8
10
2
3
10
3
9
9
6
7
12 2
9
5
3
5
3 12
·
·
·
·
·
·
=
-m < 0.
For elements x, y of a group G, define the element yxy- 1 to be the conjugate of x by y. A subset C of G of the form { yxy - 1 ly E G ) is called a conjugacy class. The conjugacy classes form a partition of G. A non-empty subset H <;;; G with the property that xy E H and x- 1 E H for all x, y E H is a subgroup of G. For fi nite groups, which is all we deal with here, the first of these conditions implies the second. A subgroup H which is a union of conjugacy classes is a normal subgroup. For ex ample, the group of Example 1 is a normal subgroup of �. A group G which contains no normal subgroups except for { e} and G itself is simple. A group G is cyclic if there is an element x in G such that every element has the form xn for some integer n. In such a case x is called a generator for G. Cyclic groups are necessarily commutative. A subset S of a group G gen erates G if every element of G is a product of elements of S. A group G is ap-group for some prime p if IG is a power of p. Having established all this terminology, it is time for our first exercise.
Write a program which determines if a spec ified multiplication program is a group, and subse quently checks if this group is commutative.
Exercise 1
Theorem 1
6
8
3
x0 = e xn = xx x (n times) if n > 0 xn = x- 1x- 1 x- 1 (m times) if n
8
12
2
4
For each x in a group G we denote its inverse by x- 1 . Associativity further implies that for any integer n, xn is well defined, where
5
12
10
11
5
7
11
11
2
12
10
5
4
2
3
7
[ 1234] , [2 143], [3412], [4321 ] , [2314] , [3124] , [2431], [4132], [324 1 ] , [42 13], [ 1342] , [ 1423].
Implicit in the this exercise are some interesting issues. How many checks are required to verify associativity in an arbitrary multiplication program? More generally, how much information is there in the multiplication program of a group? How much information is required to determine commutativity? The following result, an extension of Cay ley's theorem, may help with such questions. It is surely known, but perhaps not well-known.
Example 4. Let G = { 1, 2, 3 , 4, 5, 6, 7, 8, 9, 10, 1 1 , 12 } with multiplication table i ·j
This is a directly specified non-commutative group. It is usually referred to in the literature as A4, the set of even permutations of a 4-set. The particular multiplication table above comes from listing these permutations in the following order.
2
6
11
An n X n multiplication table with an iden tity on a set G is a group if and only if its rows form a subgroup of the group of permutations of G. Suppose, by possibly relabeling the elements, that G { 1 , 2, , n j. Then the group of permutations of G is 811• Let ui denote the i-th row of the multiplication table, so that ui lj] = k if and only if i · j k. Suppose that G is a group. Since i j = i · k implies that j k, each row ui contains each element of G exactly once, Proof. =
·
·
·
=
·
=
VOLUME 26, NUMBER 2, 2004
47
so is an element of Sn. We thus write u; [ j ] u;(j ) as the effect of the bijection u; on the elementj. For two rows u; and O"j we compute the product in Sn as follows: =
(u; O"j)(k) ·
=
u;(uj(k))
=
u;(j
i
=
for arbitrary k, which shows that the product of two rows is a row. It fol lows that the rows form a subgroup of Sn. Conversely, suppose that the rows of the multiplica tion table for G form a sub group of Sn and that 1 is the identity. We first check as sociativity. For fixed i and j and variable k,
·
0
k) (j
0
k) = (i
0
j)
0
k = O";.j(k)
One other i m portant and
natural p roblem is that of
classification : how do we
systematical ly organ ize the zoo of g rou ps?
k) = u;(uj(k)) = (u; O"j)(k) = O"; x j(k) = (i X j) k for some element i X j of G. To show that i X j = i j, let k = 1. To check inverses, for every i the row u; contains 1, so there is some elementj such that i j = 1. There is then also an element k such that j k 1, and so 1 = (j (i J)) k (j i) (j k) = j i. Thus i has inversej. • i
(j
·
·
·
·
·
·
·
·
·
·
=
·
·
·
=
·
There are some common subgroups of a group G that occur frequently. The center of G is the normal subgroup defined by
Z
=
(x E G l xy = yx Vy E G).
The commutator subgroup of G is the smallest subgroup containing all commutators of the form xyx- 1y- 1 . Others can be found in standard texts. Exercise 2 Write programs that determine the center, the commutator subgroup, and other important sub groups of a group G. Let G and H be groups with identity elements ec and eH, respectively. The product G X H is a group with identity el ement [ec, eH] and multiplication [x1 ,Yd [x2,Y2 l = [x1x2, Y1Y2l· Isomorphism, Classification, and Invariants
Two groups G and H are isomorphic if there exists a bi jection u : G � H such that
u(xy) = u(x)u(y) u(x- 1 ) u(x)- 1 =
for all x, y E G, in which case we write G = H. The bijec tion u will typically be a program. Here is the first-and perhaps most fundamental-prob lem in the subject. Problem 1 Write a program which determines if two given groups G and H are isomorphic.
Classical mathematics generally manages to finesse this issue, by somehow assuming that we can theoretically iden48
THE MATHEMATICAL INTELLIGENCER
tify isomorphic objects without prescribing how. It also gets into logical difficulties, rarely acknowledged, by dis cussion of "isomorphism classes." There is a reasonably simple program that accomplishes the task-just run through all the possible bijections be tween two groups by brute force and check if any is an iso morphism. There are n! possibly different multipli cation programs on a group G of order n, so this is hopelessly inefficient in general. Therefore an important goal is to develop tools for distinguishing groups. Somewhat informally, an invariant is a program that inputs a group and outputs a "simpler," or at least "smaller," mathematical object, in such a way that iso morphic groups yield identical, or isomorphic, outputs. There are three families of invariants which arise in a first course: the order invariants, the central invariants, and the subgroup invariants. These will be among our primary tools for studying group theory. Please keep in mind that these are generally going to be programs. To make some initial contact with the isomorphism problem, let's suppose that G = ( 1,2, . . . , n). The multipli cation program of G yields an n X n array, or matrix, whose element in row i and column j is the group element i j. We can equivalently encode this data in the list ·
L
=
[1
·
1, 1 2, . . . , 1 ·
·
n, 2
·
1, 2 · 2, . . . 2 n · 1,
n, . . . , n 2, . . . n n] .
·
·
·
The objects in L are the numbers 1, 2, . . . , n, each with multiplicity n. Permutations of the labels for the elements of G will yield generally different lists. Of all such lists, there is exactly one which is minimal in the lexicographi cal ordering of lists of natural numbers, we call it the min imal list of G. This is a canonical invariant of a group G which "solves" the isomorphism problem: two groups are isomorphic if and only if they have the same minimal list. However, finding it seems impractical. One other important and natural problem is that of classification: how do we systematically organize the zoo of groups? Clearly the problem of identifying isomorphic groups needs to be addressed first, for without it classi fication is too unwieldy. A simple-minded approach is to list all the groups G of a given order by arranging their minimal lists in increasing (or decreasing) lexicographi cal order.
Write a program that inputs n and outputs, in some systematic fashion (for example, by the order de scribed above), a complete list, up to isomorphism, of groups G of order n. Problem 3 Find a multiplication programfor each sim ple group described in the Atlas of Finite Simple Groups [CCNPW]. Problem 2
Multisets from Groups
The next definitions are highly non-standard (be warned!) but are indispensable in this treatment of the subject. Our purpose is develop a language for efficiently compressing the information content of a multiplication table into sig nificantly smaller tables, thus yielding invariants. There will be two basic ways of doing this: central analysis-the study of conjugacy classes and their algebraic structure, and sub group analysis-the study of the relationship between a group G and a subgroup H. First we need just a quick review of positive rational multisets (see [W1]). A multiset is an unordered collection of mathematical objects, with repetitions allowed, such as A = [1_1_2) or B = [3_3]. Note the convention of square brackets, and the use of space or underscores, not com mas, to separate elements. Square brackets with commas denote lists, as in computer science. Sets and ordered sets are special cases of multisets and lists, and can be denoted by either square or curly brackets. Thus C [4_5) = {4_5 ) is a set. We say A is a 3-multiset, since A = 3, whereas B is a 2multiset. Besides the (usual) operations of U and n, we may also add two multisets, by simply combining all ele ments of both. This allows us to give meaning to expres sions like =
act somewhat like "generalized group elements." In special cases the multiplication structure of appropriately chosen 1-multisets forms a useful invariant of a group, which is of ten a key to further study. An obvious way of creating a 1-multiset from a group G is to take a (non-empty) positive multiset S � G and form the associated 1-multiset
s=
A = [.ra j a E A) is the left translate of A by .r, and A{x) = [a.r j a E A) {.r)
is the right translate of A by .r. Example 5.
let A =
= A + A = 4[1) + 2[2) 3A + 2B = 6[1) + 3[2) + 2[3).
t
A X B = [[a,bJ ia E A, b E B). [ [ 1 ,3) _ [ 1 ,3) _ [ 1,3) _ [ 1 ,3) _ [2,3) _ [2,3)).
We say A is from B, denoted by A � B, if for any object x =>
For the group G = A4 defined in Example 4, and B = [6_10_ 1 1 ) . Then A - 1 = [2_3) = A B- 1 [5_9_12) AB = [6_6_7_7_10_ 1 1 ) BA = [6_7_7_10_ 1 1_ 1 1 )
Order Invariants
The main order invariants are the order of G, namely iG', and the order multiset of G defined by
j
ord(G) = [ord(.r) .r
In the example,
mA(x) > 0
[2_3)
=
More generally, we allow rational combinations like :3_A + B, which is a 1-multiset. We may also form the direct rod uct A X B by the rule
A XB =
lSf ·
This will be a convention: general multisets denoted by cap itals, and the associated 1-multisets denoted by the corre sponding small letter. Special cases of the product of multisets deserve men tion. For x E G and A a multiset from G,
2A
±
s
mB(x) > 0
where mA(x) is the multiplicity of the object x in the mul tiset A. If A and B are multisets from a group G, in other words if A � G and B � G, then define the multisets
A- 1 = [a- 1 I E A) AB = [ ab I [ a,b] E A X B) . a.
In this latter definition of the product AB (not to be con fused with the direct product A X B defined above), we must consider all possible products ab with repetitions included. Note that this generally yields a multiset even if A and B are subsets of G, so that AB has here a dif ferent meaning from that used in almost all texts in the subject! We see that if A and B are multisets from a group G, then
VlEI = f,4 , jBf . It follows that the product of any two 1-multisets from a group G is itself a 1-multiset. That means that 1-multisets
E G)
where ord(.r) is the least nonzero natural number n such that xn = e. Clearly if two groups are isomorphic then their orders and order multisets must agree. Exercise 3 Write a program which outputs the order mult·iset of a group G. There are secondary order invariants which one may
define, which I have not seen studied. Given x and y in a group G, an ordered pair of integers [n,m] such that xn = y"' may be called a period pair for [.r,y] . Period pairs for [.r,y] form an integral lattice and include the vec tors [ord(.x:),O] and [0, ord(y)] . One may associate to this some invariants, the simplest of which is the area a(x,y) of a fundamental domain. Define the area multiset of G by
a(G) = [a(x,y) i x,y E G) . Similarly, for a triple of elements .r, y, z we can consider period triples [n,m,l] satisfying xn = ym = z1, an associ
ated volume multiset, and so on. For any element x in a group G, the subgroup {.x:" I k = 1, 2, ord(.r) ) is called the cyclic subgroup generated by x. Any subgroup of a cyclic group is also cyclic. The set of cyclic subgroups of a group G is a poset under inclusion, ·
·
·
,
VOLUME 26, NUMBER 2, 2004
49
which we call the cyclic subgroup poset of G. Clearly the period lattices for x and y are closely related to the cyclic subgroups generated by x and y. We must be attuned to the possibility of counting some thing at all times in this subject. To this effect, multisets play an important organizational role. For another exam ple, if the number of elements that commute with an ele ment x is c(x), then we may study the multiset
[c(x) I x E
G].
You will be able to define and study many more such or der invariants as you proceed with your investigations. Conjugacy Classes and the Class Hypergroup
Now we come to central analysis on a group G. Let C; be a conjugacy class of G. Then the associated 1-class is the 1-multiset - C;
C; -
lCJ .
A multiset S from G with the property that {x) S {x- 1 ) = S for all x E G will be called G-invariant, or central. De noting the conjugacy classes by C0 = { e), C1 , · · · , Cr, any central multiset is a linear combination of them in the mul tiset sense, so any central 1-multiset s is a combination n
s = I r;c; ;�o where r; 2: 0 and �?� o r; = 1 .
Since the product o f central multisets is central, for each
i and j there are positive rational numbers nt summing to 1 such that
n
c;ci = I n t ck. k�O
The coefficient nt is the probability that the product xy is in ck if X is in C; and y is in Cj. The multiplication c; ci = c;ci of 1-classes is associa tive, has identity c0, and has a partial notion of inverse: for every c; there is a unique c� = c;* with the property that c;c7 contains a non-zero multiple of c0; ci is the conjugacy class of inverse elements of c;. We require that the map c; ----> ci be an involution. This defines a (finite) hypergroup. In terms of the coefficients nri and the map i ----> i*, the definition amounts to the following conditions.
erally much smaller than G, especially if G is highly non commutative. The monster group M, largest of the sporadic simple groups, has roughly 8 X 1053 elements. Its class hy pergroup K(M) has 194 elements, and so its multiplication program contains at most 1943 rational numbers. Furthermore, K(G) is always commutative, meaning that c;cj = CjCi for all i,j. Thus classical tools of harmonic analy sis, such as characters, the Fourier transform, Plancherel the orems, Parseval identities, and so on are available. Commu tative hypergroups provide us with a powerful extension of abelian harmonic analysis with applications far beyond group theory; to physics, number theory, combinatorics, and func tional analysis (see [BH], [W2], and the references therein).
Let G = A4, with conjugacy classes Co = { [ 1234] ) c1 = { [2143]_[3412]_[432 1 ] ) c2 = { [2314]_[4132]_[3241]_[ 1423] ) c3 = { [3124]_[2431 ]_[42 13]_[1342] ). The multiplication program for the class hypergroup K(A4) = { co, c 1 , c2 , c3) is given by the table Example 6.
c1 · c1
Co
Co
Co
c1 c,
c,
c,
l eo + l c ,
c2
CJ
c2
c2
c2
c3
l eo + l c ,
CJ
c3
3
3
c2
c3
c2
c3
l eo + l c ,
CJ
4
4
4
4
c2
We can see that ci = c; for all i (in such a case the hy pergroup is called Hermitian). So determination of the 1-class structure constants nt is one of the first and most important tasks to be performed for any group G. Here is my favourite group-theory problem.
Problem 4. Find a good formula for the 1-class struc ture constants for the symmetric group Sn.
·
�f=o n�j = 1 Vi,j nt 2: 0 \:fi,j,k m l l m \:fi,j,k,m 3 . "'-l=O n;J n lk - .:-1� 0 n il nJk 4. n�; = nro = 8 ;k \:fi,k 5. n � > 0 � j = i* Vi,j 6. nt = nJ:i \:fi,j,k The set K(G) = { co,cb , c,) 1.
2.
"""
_
,.-n
*
·
·
·
with the multiplication and involution defined above is the class hypergroup of G. Why would we consider this seemingly complicated ob ject? There are two good reasons. First of all, K(G) is gen50
THE MATHEMATICAL INTELLIGENCER
Character Tables
Let's now establish some further central invariants, espe cially the character table of a group. This latter forms the foundation for harmonic analysis on non-abelian groups, with applications to chemistry, physics, and other branches of mathematics, as well as to further development of the subject. This presentation has its origins in the work of Frobenius [F] and Kawada [K]. Suppose that K(G)
= {co, c 1 ,
·
·
·
,
c,)
is the class hypergroup of a group G, with structure equa tions n
c;ci = I ntck. k�O This can be viewed as a system of (n + 1)2 rational equa tions in the variables c;, with the property that the right
hand side of each equation is a linear expression, and the left-hand side is a quadratic product.
Perhaps thefundamental computational problem in the subject is the determination of all complex-valued solutions to such a system. That is, we want to replace the variables ci with complex numbers so that the equations still hold. Almost surely you already possess algorithms (well devel oped in MAPLE and MATHEMATICA) that will input a sys tem of quadratic/linear equations, and in certain cases, find all solutions. Let's outline the basic idea. First define a character of the commutative hypergroup above to be a complex-valued function x which satisfies n ci c1 = ) x( )x( I ntxCck) 'Vi,j. k�O
There is always at least one character, the constant func tion
'Vi ;
xo(ci) = 1
x (co) = 1. Example 7. There are exactly jour characters of the hy pergroup K(A4), given by the rows of the following table C (here a- = e27Ti13)
X1
X2
Co
- 1 /3
0
0
(T2
X3
(T
It turns out that there are always exactly (n + 1) char acters of a fmite commutative hypergroup K = {co, c1, · · · , Cn }. We can find them as follows: For each element ci E K, let Mi be the matrix of multi plication by ci with respect to the ordered basis { c0, c1, , cnl · The matrices Mi satisfy exactly the same multiplicative relations as the elements ci, and in particular they mutu ally commute. Furthermore, they are linearly independent, for if � i ziMi = 0, then multiplying � i ZiCi by cJ and con sidering the resulting coefficients of c0 yields z1 = 0. The matrices Mi can be simultaneously diagonalized by some invertible matrix Q. That is, 1 QMiQ- = Di •
·
·
for some diagonal matrices Di. The diagonal entries of the matrices Di are then the characters of K, that is, we define the hypergroup character table C(i , J) = Xi (c1) = Dj(i,i). In particular, there are exactly as many characters as con jugacy classes. It is worth pointing out that in general only one of the matrices Mi is required. Suppose that M1 happens to have distinct eigenvalues. If a1 is one of these, then replacing c1 with a1 in the n - 2 equations for c1c2, · · · , c1cn yields a , Cn which has a unique system of linear equations in c2 , solution up to a scalar. Substituting into the equation for c1 yields an absolutely unique solution, so this gives us one character, or row of the character table. As aj runs through the eigenvalues of M1, the rows so obtained constitute the full character table. ·
·
·
Example 8.
For K(A4), the matrices Mi are
[� � � �l [� � ! �] ] [! Mo
0 0 1 0
0 0 0 1
114 3/4 0 0
Mz
and simultaneous diagonalization ofthese yields the table of Example 7.
and for any character x we must have
Xo
So the other equations, not involving c1, can be viewed as providing additional information to cover the case of eigenvalues of M1 with multiplicity greater than one.
The hypergroup character table C contains much useful information. Its columns furnish us with a model of the hy pergroup K, in the sense that pointwise multiplication of columns obeys the same structure equations as does hy pergroup multiplication, while the rows, being the charac ters, can also be multiplied pointwise to obtain what turns out (in the case of the class hypergroup of a group) to be a hypergroup structure in complete duality with K, called the character hypergroup and denoted K(G"). In the particular case when G is itself a commutative group of order n, so that G = K(G), this procedure shows that both G and G" = K(G") are isomorphic to a subgroup of the n-fold product en. In fact it is not hard to see that in this case all the entries of the character table must be n th roots of unity, so G and G " are subgroups of products of cyclic groups. (From this one may deduce, with some more work (!), that both G and G" are themselves iso morphic to products of cyclic groups.) Weights and Orthogonality Relations
The character table for a finite commutative hypergroup exhibits some remarkable orthogonality. To describe this, first define the weight w(c;) of an element Ci of a hyper group K to be
which in the case of K(G) is just the size of the associated conjugacy class Ci. Define the weight w(K) of a hypergroup K = { co, c1, · · · , cn l to be n w(K) I w(ci) , =
i�O
so that in our case w(K(G)) ! G j . For any hypergroup K the rows of the character table, 1 viewed as elements in en + , are orthogonal with respect to the usual Hermitian inner product modified by the weights w(ci) , in other words by the inner product =
VOLUME 26, NUMBER 2, 2004
51
while the columns are similarly orthogonal with respect to the dual weights w(xi) of K(G"). The weight list
[
WK(G) = w(co) , · · · , w(cn)
J
Write multiplication programs for the class hypergroups and character hypergroups of all simple groups. Problem 6.
and the dual weight list WK(G A) = [w (xo) , · · · , w(xn) l
are implicit in the character table, but the orthogonality re lations suggest that it is useful to identify them explicitly as important invariants. Calculating the following five central invariants of a group G is thus a primary task
Write programs which determine the class hypergroup, the hypergroup character table, the charac ter hypergroup, the weight list, and the dual weight list of a given group G. Problem 5.
Example 9. You may check that the character hyper group K(A4) has multiplication table XI " XJ
Xo
X1
X2
Xa
xo
Xo
X1
X2
X3
X1
X1
X1
X1
X2
X2
+ �X1 + _1_X2 + _1_X3 _1_Xo 9 3 9 9 X1
X3
xo
X3
X3
X1
Xo
X2
and that the weight list is WKc.A4) weight list is wKc.A4) = [ 1 ,9, 1 , 1 ] .
=
while the dual
[ 1,3,4,4]
By multiplying each row Xi of the hypergroup character table of the class hypergroup K(G) by the square root of the dual weight w(xi), we obtain the group character table C'. In other words, we define Xi(Cj)
=
Xi(cj)
VwCXJ
and let C'(i,J) Xi(Cj) · The group character table has the advantage of often having many integral entries, but it has the disadvantage of obscuring the symmetry between the class and character hypergroups. Its definition is usually left to a second course in group theory and traditionally in volves the deeper notion of a representation of a group. One of the advantages of the approach sketched here is the introduction of character tables without representation theory, which was actually Frobenius's original point of view! Recall that the passage from the multiplication table to the character table as I have outlined it involves only counting and solving a quadratic/linear system of equations, or equivalently diagonalizing some matrices. The weight list WK(G) is the list of sizes of the conjugacy classes, while the dual weight list WK(G A) has also the in terpretation as the list of squares of the dimensions of the irreducible representations (which are not defined here). =
Example 10.
Here is the group character table C' ofA4.
X,(CJ )
Co
c1
c2
Ca
3
-1
0
0
Xa
x1
x2
1
1
u
u2
x3
52
1
THE MATHEMATICAL INTELLIGENCER
The next problem is in principle straightforward, since the character tables for simple groups are clearly laid out in the Atlas [ CCNPW].
1
u2 u
It is a consequence of the classification of simple groups that a simple group is determined by its group character table, and so by its hypergroup character table. The following fundamental problem probably requires some new insight for its resolution. (The pattern of cir cles on a simple group, introduced later, might help in this regard.) Problem 7. Write a program that constructs a simple group from its group or hypergroup character table. Remark 2. The dual of an arbitrary commutative hy pergroup is not always a hypergroup, as negative coeffi cients can occur. Nevertheless, by enlarging our view to include so-called signed hypergroups, there is a complete duality theory generalizing Pontryagin duality forfinite commutative groups. This fact was discovered in the slightly different context of C-algebras by Kawada [K]; see [W2] and [OW] for a modern treatment. Cosets and Subgroup Analysis
Let's now introduce subgroup analysis. Let H be a sub group of G. A multiset S from G with the property that { y } S = S (respectively, S{y } S) for all y E H will be called H left-invariant (respectively, H right-invariant). A mul tiset S from G which is both H left-invariant and H right invariant will be called H hi-invariant. The subgroup H is itself H hi-invariant. For any x E G, the sets {x) H and H {xj are H right-invariant and H left-in variant, respectively, and are called left cosets and right cosets of H, respectively. The set of all left cosets of H forms a partition of G, as does the set of all right cosets of H. These two partitions of G are equal precisely when H is a normal subgroup of G. Because l lxJHI IH(xJ I = IHI, we conclude =
=
IHI divides IGI for any subgroup H of a group G. For any x E G the multiset H{x}H [h1xh2 l h 1 . h2 E H] is a double coset of H. It is H hi-invariant and is an IHI2Theorem 2 (Lagrange's Theorem).
=
set. In contrast to left and right cosets, a double coset is generally not a set. Note carefully that at this point our notation diverges from the standard usage of HxH as a
set.
Suppose that the distinct left cosets of H are H = H0, H1, . . . , Hz. Define the associated left 1-cosets by the rule
with h be
H· = hj 1�1
=
h0. Now defme the GIH =
right quotient
{ho, h1, · · ·
, hzJ.
of G by H to
Note that G/H is not defined to be a set of cosets, but rather a set of 1-cosets! Because H is a subgroup, we have
h2 = h h- 1 = h. Furthermore, the product of H right-invariant 1-multisets is also an H right-invariant 1-multiset, so there exist ratio nal numbers Pt such that l
hihj = I Pt hk. k�O This is a product on G/H which shares some of the prop
erties of a hypergroup; for example, it is associative and for fixed i andj the right quotient structure constants Pt are probabilities. However in general ho is a right identity, that is, hiho = hi for all i, but not necessarily a left identity. There are inverses but they are not unique, and the product is gen erally not commutative. Nevertheless, the structure constants Pt are computable invariants of the quotient G!H. So any
right quotient G/H carries an algebraic structure.
The situation is entirely symmetrical when we interchange lefts and rights in the above discussion and consider right cosets of H in G and the associated left-quotient H\G. In the special case when H is a normal subgroup of G, the left-invariant and right-invariant 1-cosets coincide, and the above product structures coincide and indeed form a group called the quotient group and denoted by either G/H or H\G. In general the connection between the left quotients and right-quotients is given by the inverse map ping x � x- 1 , which transforms left cosets to right cosets and vice-versa. So the right-quotient and left-quotient struc ture constants are essentially the same. In fact there is a simpler and more fundamental invari ant associated to a subgroup H of a group G which is a hy pergroup. A 1-multiset of G of the form all
H {x } H a= � will be called a double 1-coset of H in G. The double 1cosets of H in G are either identical or disjoint, and we let
ffiGIH = {ao, a 1
arJ denote the set of all double 1-cosets of H in G, with a0 = ho = H!ll:ij. Then there are rational numbers q�1, which again ,
·
·
·
,
have an interpretation as probabilities, such that
The multiplication table is a3
a1 · a1
ao
a1
a2
ao
ao
a,
a2
a3
a,
a,
ao
a2
a3
a2
a2
a2
a3
l ao + l a ,
a3
a3
a3
l ao + l a, 2
2
2
a2
2
This is obviously commutative, so (G,K) is a Gelfand pair. The character table is ao
x;lail xo
a1
a2
-1
X1 X2
0 a a2
X3
a3
0
r? a
The similarity with the group character table in this case is largely accidental. Given a group G and a subgroup H, (write a program to) find the structure constants p�1 and q t above and determine whether (G,H) is a Gelfand pair. Exercise 8.
Circles: Translates of Conjugacy Classes
A translate of a conjugacy class in G will be called a cir cle (this is not a usual definition). More particularly, a translate of the i-th conjugacy class ci will be called an i-circle. Since {x}Ci = Ci{x) for any x E G and any conju gacy class Ci, it is unnecessary to specify left or right trans late. We will say that the circle {x)Ci has color i and cen ter x, both of which may fail to be unique. Example 12.
Let
G = s3 = { [ 123] , [213] , [132] , [321] , [231] , [312] } with conjugacy classes Co = { [ 123] } c1 = { [213]_[132]_[321]} Cz = { [231]_[312] } . Then of course every element is a 0-circle trivially; fur thermore,
{[123] } c1 = { [231J J c1 = { [312]} c1 = c1 and ! [213]} c1 = { [ 132] } c1 = { [321]} c1 { [123]_[231]_[312]}, so there are only two !-circles and their centers are not unique. There are six distinct 2 -circles, each with a unique center, namely =
r
aiaJ = I q�J ak k�O which makes ffiGJH into a bona fide hypergroup which we call the double coset hypergroup of H in G. In general ffiGIH is not commutative; when it is, (G,H) is known as a Gelfand pair.
Let G = A4 and K = { 1_2} . Then K\GIK = {ao_a1-a2-a3) where ao = 21 [1_2] a 1 = 21 [3_4] az = ± [5_8_9_12] a3 = ± [6_7_1 0_1 1]
Example 1 1 .
( [123] } Cz = {[231]_[312]} { [231]} Cz = {[123]_[312]} { [312] } Cz = {[123]_[231]} ( [213] } c2 = ( [132]_[321] ) { [321]} Cz = { [213]_[ 132] } {[132] } c2 = ( [213]_[321]} . Example 13. For G = A4 the class structure equations reveal that a circle may have more than one color: C2 and VOLUME 26. NUMBER 2, 2004
53
C3 are translates of each other, so each is both a 2 -circle
and a 3-circle.
Restricting to simple groups, the situation is much more regular than the above examples might lead us to suspect. Theorem 3. Let G be a finite non-commutative simple group. Then every circle in G has a unique color and a unique center. Every point x E G lies on precisely lci l i-circles, and so lies on a total of I G I circles.
Let Ci,Ci be two distinct conjugacy classes of G. We first show that no translate of Ci can also be a trans late of Ci. If Hi (z E G I (z}Ci = Gil then Hi is a subgroup of G and is normal. It cannot be all of G unless G is trivial, in which case there are not two distinct conjugacy classes. Thus Hi = (e) since G is simple. Now suppose that (x}Ci = (y)Cj for some x,y E G, or equivalently that (z}Ci = Ci for some z. Conjugating by w E G, we see that (wzw- 1 }Ci = Ci. But then (z- 1wzw- 1 }Ci = Ci and so by the previous re mark z- 1wzw- 1 = e. Since w is arbitrary, z is in the cen ter of G and so z = e. But this is impossible because Ci and cj are distinct. For the second claim, note that the statement is true for x = e, for the centers of the i-circles on which e lies are ex actly the elements of the class c; of inverses of the ele ments of Ci, which has I Gil elements. Since the family of i circles is invariant under translation, every point x E G lies on precisely I Gi l i-circles, and so on a total of I GI circles. • Returning to general groups, the circles of G are related to the structure constants n �i of the class hypergroup K(G) = ( co, C1 , , Cn } . Theorem 4. For any conjugacy classes C;, Ci, Ck of G and z E Ci, Proof.
=
•
•
•
I (z} ci n ckl = lci l n�i· Proof. Suppose that the product CiCi contains Ck a total of N �i times, necessarily an integer. Then since ci = C/ I Ci I , Nfi l ck l _ - nkij· I ci I I cj I Of course I (z} cj n ck I = I (xzx- 1} cj n Ck l for any x, so that counting in different ways yields l ci l l lz} Cj n ck l = N�j l ck l· • Combining, we get the required equality. In particular we see that ICj I n�j is always a positive in teger, which in tum allows further deductions, such as the following. Corollary 1 . If Ci and Ci are conjugacy classes of G whose orders are relatively prime, then CiCi is a multi
could recover the circle pattern of a simple group from the character table, we could hope to recover G from its action, on the left and right, as symmetries of the circle pattern. Relations and Isomorphism Theorems
So far we have been focusing on individual groups, their conjugacy classes, and their subgroups. Now let us con sider possible relations between two or more groups. The following treatment is more general than usual, but I leave most of the proofs to you. If G and H are groups, then a subgroup R of the prod uct group G X H will here be called a group relation be tween G and H. Note that the order of G and H matters; interchanging the order of all ordered pairs in R yields the transpose group relation RT between H and G. A homomorphism from a group G to a group H is a map if! : G � H satisfying for all x, y E G
cp(xy) = cp(x) cp(y). Perhaps you have already been pre-programmed to think of homomorphisms, not relations, as the natural objects of study in algebra. In that case, note that, given a homomor phism cp from G to H, we may define a group relation R be tween G and H by the rule R = ( [x, cp(x) J i x E G). If R is a group relation between G and H, then define the image and the transpose image of R, respectively, by im(R) imT(R)
=
=
(y E H I [x,y] E R for some x E G) (x E G I [x,y] E R for some y E H).
These are subgroups of H and G, respectively. If both imT(R) = G and im(R) = H then we define R to be full. The next problem is deep, and generalizes aspects of repre sentation theory.
Write a program that inputs an ordered pair ofgroups and finds all full group relations between them.
Problem 9.
If R is a group relation between G and H, then define the kernel and the transpose kernel of R, respectively, by ker(R) = (x E G I [x,eH] E R} kerT(R) = (y E H I [ea,Yl E R},
where eH and ea are the identities in H and G, respectively. More generally, for any x E G and y E H define RY = (x E G I [x,y] E R} Rx = (y E H I [x,y] E R}
ple of a single conjugacy class. Proof. In this case n�i is both an integer and a probabil
so that
The pattern of circles on a group is an in teresting geometrical object, closely connected to the sub ject of association schemes (see for example fBI]). If we
Rea· Then Rx is both a left coset and a right coset of kerT(R) for all x E imT(R), and RY is both a left coset and a right
ity for each k, and so must be either 0 or 1 .
Remark 3.
54
THE MATHEMATICAL INTELLIGENCER
•
ker(R) kerT(R)
= Ren
=
coset of ker(R) for all y E im(R), so that ker(R) and kerT(R) are normal subgroups of imT(R) and im(R), respectively.
A U T H O R
r�
If R is a group relation between groups G
Theorem 5.
and H, then
.
imT(R)Iker(R) = im(R)IkerT(R).
j\
Given a group relation R between groups G and H, we may associate to each x E imT(R) the 1-multiset r(x) = rx = Rxl I R:r I in H. In that case, the homomorphism equa tion applies, showing that a group relation can be viewed as a (partially defined) homomorphism that is allowed to have 1-multisets as values. If R is a group relation between G and H satisfying imT(R) = G and kerT(R) eg, then R determines a homo morphism q; from G to H by the rule
..
•
:
·
N . .J. WILDBERGER
UNSW
Sydney, NSW 2052 Australia
e-mail:
[email protected] A native Canadian, N. J. Wildberger has settled 1n the alto gether sunnier environment of Australia, where he enjoys bush
walking, sw1mm1ng, and plaYJng GO. He studied at the Uni
In this case the theorem reduces to
versity of Toronto and Yale, and then taught at Stanford and Toronto before heading down under.
G!ker(R) = im(R). Then R and RT both determine homomorphisms if and only if R determines an isomorphism. Consider a useful example. Suppose G is a group with normal subgroups K and L, which have associated 1-mul tisets k and l, respectively. Define a group relation R be tween the groups G/K and GIL by the rule that [ la} k, lb}l] E R if and only if I a} k n I b} l is non-empty. Let's first check that this makes sense. Because K and L are normal, la} k = k laJ and lb} l = l lbJ for any a, b E G. Thus if X 1 E lai } k n lbi } l and X2 E la2 } k n lb2}l, then X1X2 E lai} k la2 J k = la1a2} kk la 1a2 } k. Similarly X1X2 E lb 1b2 } l; thus R is indeed a group relation. Now kl lk because both are G-invariant, and so (kl) 2 kl. This implies that kl is the associated 1-multiset to a nor mal subgroup M (which is usually denoted by KL in the lit erature) containing both K and L. It is not hard to see that ker(R) = MIK and kerT(R) MIL, so the above theorem yields the equation =
=
=
GIK = GIL . MIK MIL In the special case when K is a subgroup of L (and so a normal subgroup of L), we get M = L and the following iso morphism theorem.
Acknowledgments
I thank Peter Donovan, Hendrik Grundling, and David Har vey for useful comments and discussions. REFERENCES
[BI] E. Bannai and T. Ito, Algebraic combinatorics /-Association Schemes, Benjamin and Cummings, Menlo Park, 1 984.
[BH] W. R . Bloom and H. Heyer, Harmonic analysis of probability mea sures on hypergroups , de Gruyter Studies in Mathematics, Vol. 20,
Walter de Gruyter, Berlin, 1 995. [CCNPW] J. H. Conway, R. T. Curtis, S. P. Norton, R. A Parker and R. A Wilson, Atlas of Finite Groups, Clarendon Press, Oxford, 1 985. [F] G. Frobenius, Ober Gruppencharacktere, Gesammelte Abhandlun gen, Vo l . I l l , pp. 1 -37, Springer-Verlag (1 968), Berlin. [K] Y. Kawada, Ober den Dualitatssatz der Charaktere nichtcommuta tiver Gruppen , Proc. Phys. Math. Soc. Japan 24(3) (1 942), 97-1 09.
[OW] N. Obata and N. J . Wildberger, Generalized hypergroups and or thogonal polynomials , Nagoya Math. J . 1 42 (1 996), 67-93.
[W1 ] N . J . Wildberger, A new look at multisets, preprint, 2003.
Theorem 6.
[W2] N. J. Wildberger, Duality and Entropy for finite commutative hy
GIK = GIL . LIK
Show that this theorem extends to general (non-normal) subgroups by developing a theory of quo tients for the "hypergroup-like" objects GIH.
Exercise 10.
.
School of Mathematics
=
[x,q;(x)] E R.
�
.1. .
r(xy) = r(x)r(y)
=
.
. .
pergroups and fusion rule algebras, J . London Math. Soc. 56(2)
(1 997), 275-291 . [W3] N. J . Wildberger, Finite commutative hypergroups and applica tions from group theory to conformal field theory, In Applications of
Hypergroups and Related Measure Algebras, Proc. Seattle 1 993 Conference, Contemporary Math. 1 83 (1 995), 41 3-434.
VOLUME 26, NUMBER 2 , 2004
55
Mathemati c a l ly Bent
Col i n Ad a m s , Ed itor
This Theorem Is Big Colin Adams The proof i s i n the pudding.
Opening a copy of The
Mathematical
Intelligencer you may ask yourself uneasily, "lthat is this anyway-a mathematical journal, or what?" Or you may ask, "lthere am /?" Or even "ltho am !?" This sense of disorienta tion is at its most acute when you open to Colin Adams's column. Relax. Breathe regularly. It's mathematical, it's a humor column, and it may even be harmless.
Column editor's address: Colin Adams, Department of Mathematics, Bronfman Science Center, Williams College, Williamstown, MA 01 267 USA e-mail :
[email protected]
56
H had to call you, because I have a ey, Barry. It's me, Sid. Listen, I just
theorem like no other theorem you ever saw. You are going to go apeship over this. This one is going to be big. And not just a little big. We are talking another Atiyah-Patodi-Singer Index Theorem. Think Mostow Rigidity or Reine-Borel. This is mega blockbuster material. I'm shopping it around, and let me tell you, there's a lot of interest. But I wanted you to have a shot at it. I know you guys haven't had a big hit in a while, not since Riemann-Roch. So I know you're hungry. This one has the potential to be a blockbuster. Get ready, because here it is. The Mandelbrot Thurston-Wiles Pi-Orbifold Syzygy The orem. How about that, huh? You don't get more star power than Mandelbrot, Thurston, and Wiles. These are box office heavies. And if you're picking a real number to put in the title of a theorem, you can't get hot ter than 7T. And, hell, we have a word in there whose only vowel is "y" AND it appears three times. This word is so end-of-the alphabet-heavy, it screams brilliance. Now you are probably wondering how I got these three big names to agree to this project. And the honest truth is that I don't yet have them all on board. But this is where you come in. The only reason they are hesitating is the need for a sponsoring institution, a place of your caliber. They aren't go ing to sign on with some community college with a 4-4 teaching load. No, they need to know that you're going to provide the resources necessary for this theorem to get the attention it de-
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
serves. Let me tell you what I have in mind. I want you to get invitations for all three to speak at the Institute for Ad vanced Study, the Fields Institute, and the International Congress. I'm betting you can make that happen. That's why I called you. And I'm thinking there should be some awards from some wealthy Middle Eastern countries. Or maybe Japan. You may have to pull some strings, but I'm betting you can do that. I'm thinking big gold ingots, with a picture of Gauss stamped on them, or maybe a big belt, like they give out at the World Wrestling Federation. On the PR front, we have lots of ideas. We're thinking of spicing things up a bit, having a shoving match be tween Thurston and Wiles at the AMS meetings. All faked, of course; but imagine the PR potential. Stars not get ting along. Disagreement over the credit for the corollaries. This is hot stuff. We'll get the front page of the tabloids. And of course, there will be the love interest. Get this. Thurston, dyed-in the-wool topologist, right? He falls for analytic number theory. See? So that's what he contributes to the theorem. But then, there is some lemma trouble. ANT's pretty rigid stuff. Thurston's frustrated. Thinking maybe he's made a mistake. Misses his old sweetheart topology. She was supple, forgiving. But in the end, the lemmas work out, and love transcends all. I'm telling you, there won't be a dry eye in the house. You're probably wondering what the theorem will say. Me, too. Quite frankly, I haven't got a clue. But I haven't brought the writers on board yet. I'm thinking maybe it will have to do with the structure of singularities. Yeah, that has a ring to it. But there won't be three or four singularities, there will be 7T of them. Yeah, an irra tional number of them. We'll general ize the very definition of cardinality of sets. And we'll have a sequence. No, a lot of sequences. Wait, wait, make
those spectral sequences. Bockstein this. The co-authors don't care. They spectral sequences. And they are con prove it anyway. Wow, this is good. But verging to other spectral sequences. of course, we're still in the early de Grothendieck spectral sequences. And velopment stages. We'll need Mandel the indices on the sequences are them brot, Thurston, and Wiles to fill in some selves sequences. And there's a tower of the details. of these sequences reaching way up Oh, and the theorem will have im into the sky. plications. Oh yes, will it ever have im And there will be some incomplete plications. It will certainly imply the ness. Say a Cauchy sequence that Riemann Hypothesis at the very least. doesn't converge. Or a statement that Maybe Poincare, and Goldbach, too. is true but unprovable. Yeah, but get And it will show the airlines how to dis-
tribute their planes to the airports to maximize their profit. Maybe then the President could thank the three au thors in a special televised address to the nation, with a whole lot of happy pilots standing behind him. Oh, this is good. Anyway, let me know what you think We have to move on this fast, be fore someone else does it. Who knows, if we play this theorem right, maybe Ron Howard will make it into a movie. Sorry for the long message. Call me.
MIS either Confirms nor Denies Allegations Concerning Alan Turing Andrew INine d ·p culation t hat 115 may
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to both modem comput r t elm logy rutd . How v r, critks of seer t gow m m nt ag ncit> ha\·e found it u pici ou t hat lr Turing also was involv d in r ru·ch on method t o distinguish human b ings fr m non human, artificial rt>ason ing machines. In his artid om pul ing fa ·hinery and I nt el li gence Turing advallc d t he
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gramme of countemwasure. to substitution of artificial ertain associates of tr Tming have su gge te
s d ind(>ed, o f e:xllerimental surgery at tempting the first hum an cyborg, t hat is, a hybrid of or gani m and computing machine. They ugge t that hi '"Turing t t" may have been de ·igned for u on hinlS lf, \\it h the aim of allowing hi research team to discern th human or rutificial nature of h is re pon. e . Th pre,iou ly top secret documents rei as d today confurn that Tming underwent a serie of tate-dir ct d proc dur · nl'ar the end of his life. It had b .n report d in th ' past that Turing was atTe ted in 1 952 for gros in d cency, and after c onviction under the British Ob c ni ty Act was subjN:ted to treatments designed to wcure" his ho mo ·e.·uality, and that thi had led to h is suicid in 1954. Many people, in clud ing fonner 115 and , 116 ag nts, have long regarded this as a mere �cover tory." "Which tory i more piau ible'?" ask on fonner ag nt 110\ work ·
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VOLUME 26, NUMBER 2, 2004
57
l'i¥1(9·{·1.1
Dav i d E.
Rowe ,
Editor
I
There is hardly any doubt that for physics special relativity theory is of much greater consequence than the general theory. The reverse situation prevails with respect to mathematics: there special relativity theory had com paratively little, general relativity the ory very considerable, influence, above aU upon the development of a general scheme for differential geometry. -Hermann Weyl, ''Relativity as a Stimulus to Mathematical Research, " pp. 536-537
The Mathematicians' Happy Hunting Ground: Einstein's N General Theory of Relativity David E. Rowe
o one was more familiar with the impact of Einstein's general the ory of relativity (GRT) on mathemat ics and mathematicians than Hermann Weyl, who threw himself headlong into this new field shortly after the appear ance of Einstein's classic paper [Ein stein 1916]. In the summer semester of 1917 Weyl taught a three-hour course at the ETH in Zurich on "Raum, Zeit, Materie." The idea to publish a book based on these lectures came from Ein stein's close friend, Michele Besso [CPAE, vol. 8B, 663]. One year later the first edition of Weyl's classic Raum Zeit-Materie [Weyl 1918] was already in print. Einstein praised it to the skies:
I am reading with genuine delight the page proofs of your book, which I am receiving piece by piece. It is like a symphonic masterpiece. Every word has its relation to the whole, and the design of the work is grand. What a magnificent method the infinitesimal parallel displacement of vectors is for deriving the Riemann tensor! How naturally it all comes out. And now you have even given birth to the child I absolutely could not muster: the con struction of the Maxwell equations out of the g11/s! (CPAE, val. 8B, 669-670}.
Send submissions to David E. Rowe, Fachbereich 1 7 - Mathematik, Johannes Gutenberg University, 055099 Mainz, Germany.
58
Weyl, for his part, always held Ein stein's theory of general relativity in the highest esteem, and Raum-Zeit Materie did much to promote Ein stein's fame as the "New Copernicus." Thirty years later, when Weyl reassessed
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER·VERLAG NEW YORK. LLC
its impact in "Relativity as a Stimulus to Mathematical Research" [Weyl 1949], he spoke far more soberly, but still with decided enthusiasm. That lecture and the passage from it cited above came on an auspicious occasion. On March 19, 1949, more than three hundred scientists-including such eminent figures as J. R. Oppenheimer, Eugene Wigner, I. I. Rabi, and H. P. Robertson-gathered in Princeton to celebrate Albert Einstein's seventieth birthday, which fell five days earlier. The celebrant's young assistant, John Kemeny, later recalled the excitement in Princeton during the days leading up to this stellar event, held as a tribute to Einstein's scientific achievements. "People fought over tickets like mad," Kemeny remembered. "I had nothing to do with the tickets, but people some how thought that being Einstein's as sistant I had some pull, and more big shots came to me begging for an extra ticket. They were absolutely dying to get in, and Einstein just had no sense at all about what absolute reverence there was for him" [Sayen 1985, 227] . Einstein in Berlin and GoHingen
For Einstein, Weyl's address must have brought back some fond memories of the enthusiastic response his general theory of relativity received from lead ing mathematicians, especially after November 1915 when he succeeded in finding an elegant set of generally co variant gravitational field equations:
R11v
=
-K(
TJLV -
i ) g}Lvr
(*)
Most physicists, by contrast, found Einstein's whole approach to gravita tion a daring speculation, at best. Max von Laue, who later became one of the strongest advocates of GRT in Ger many, was originally dismayed even by Einstein's original theory based on the equivalence principle alone. Laue re lated his misgivings to Einstein in a let ter dated 27 December 1911: "I have now carefully studied your paper on gravitation and have also lectured
about it in our colloquium [Arnold Sommerfeld's colloquium in Munich). I do not believe in this theory because I cannot concede the full equivalence of your systems K and K'. After all, a body causing the gravitational field must be present for the gravitational field in system K, but not for the accelerated system K ' " [CPAE, vol. 5, 384] . Laue's criticism came before Ein stein had wandered into the thickets of the Ricci calculus, an adventure made possible through his collaboration with the Zurich mathematician Marcel Gross mann [Pais 1982, 208-227]. After Ein stein left the ETH in Zurich to become a member of the Prussian Academy in Berlin, he found virtually no sympa thy in his new surroundings for the Einstein-Grossmann approach, often called the Entwurf theory. Indeed, the only encouragement he received came from an Italian mathematician who happened to be the world's leading au thority on Ricci's absolute differential calculus, Tullio Levi-Civita. As Einstein mentioned to his friend Heinrich Zang ger in April 1915, corresponding with Levi-Civita gave him great pleasure: The theory of gravitation will not find its way into my colleagues' heads for a long time yet, no doubt. Only one, Levi Civita in Padua, has probably grasped the main point completely, because he is familiar with the mathematics used. But he is seeking to tamper with one of the most important proofs in an in cessant exchange of correspondence. Corresponding with him is unusually interesting; it is currently my favorite pastime [CPAE, val. 8A, 1 77-1 18}.
Meanwhile Berlin's two leading the oreticians, Laue and Max Planck, re mained skeptical. Einstein's fortunes in Germany began to change, however, when he visited Gottingen to deliver six lectures on general relativity during the summer of 1915. Afterward, Arnold Sommerfeld and David Hilbert both be gan to take a keen interest in Einstein's daring approach to gravitation and in ertia. The three of them corresponded 1 Einstein identified the term
nostrifizieren
fairly regularly throughout the autumn, by which time Hilbert had developed a strategy for combining Einstein's gravitational theory and Mie's electro magnetic theory of matter within the framework of a generally covariant mathematical formalism [Rowe 2001]. Einstein was less than enthusiastic but saw that he had to abandon the re stricted covariance of the Einstein Grossmann theory in order to make progress. This led to his four famous notes on general relativity from No vember 1915, the last of which con tained the equations (*). Writing to Sommerfeld three days later, he de scribed his dramatic struggle: "you must not be cross with me that I am answering your kind and interesting letter only today. But in the last month I had one of the most stimulating and exhausting times of my life, and indeed one of the most successful as well" [CPAE, vol. 8A, 206]. His unhappiness with Hilbert at this time can be seen most clearly from another letter to Zangger in which Einstein wrote:
The theory is beautiful beyond com parison. However, only one colleague has really understood it, and he is seeking to "partake" [nostrifizieren]l in it . . . in a clever way. In my personal experience I have hardly come to know the wretchedness of mankind better than as a result of this theory and everything connected to it. But it does not bother me [CPAE, val. 8A, 205}. It bothered him enough, however, that he wrote to Hilbert suggesting that they forget about this momentary distur bance to their friendship [CPAE, vol. 8A, 222]. Afterward they remained on excellent terms, in large part because of their mutual dedication to the ideal of an international scientific commu nity free from political pressures.
physical problems of greatest interest. He applied these methods in deriving first results for the three classic tests of GRT: the perihelion motion of Mer cury, gravitational redshift, and the bending of light rays in the sun's grav itational field. All of these findings faced challenges from the scientific community during the decade follow ing Einstein's breakthrough in 1915, but leading proponents of GRT found a series of improvements that helped strengthen Einstein's case. The first important result was con veyed to Einstein in December 1915 by the astronomer Karl Schwarzschild, who was then stationed at the Russian front. Einstein replied in late Decem ber and then in a longer letter dated 9 January 1916, which began: "I exam ined your paper with great interest. I would not have expected that the ex act solution to the problem could be formulated so simply. The mathemati cal treatment of the subject appeals to me greatly" [CPAE, vol. 8A, Nr. 181]. Schwarzschild gave the first exact solution of Einstein's equation (*) in the vacuum case, where the right-hand side is zero, by showing how to calcu late the exterior gravitational field of a static massive body that was spheri cally symmetric, like the sun, but treated as a mass point [Schwarzschild 1916). The resulting metric thus gave a precise means of calculating the tiny deviations from Newton's theory that were of such crucial importance for the success of general relativity. An even more useful derivation of this result was obtained by Johannes Droste, a student of H. A Lorentz, in May 1916, the very month in which Schwarzschild died after contracting a terminal skin disease in Russia. Jean Eisenstaedt has pointed out in [Eisen staedt 1989, 216] that the famous Schwarzschild solution found in all standard textbooks on general relativity
The Schwarzschild Solution
The Einstein equations (*) are notori ously difficult to solve, a circumstance that led Einstein to devise methods of approximation for handling the special
( �)
- 1 -
+
dt2
+
( � rl 1 -
r(drf!- + sin2
e
dr
dcf)
(**)
with the name Max Abraham, who was famous in Gbttingen for his back-stabbing behavior. But, if Einstein learned of this ex
pression from Abraham, it was, in fact, a familiar one in Gbttingen mathematical circles, where visitors were often warned that good ideas had a way of getting "nos trified" once they got into the local atmosphere. See [Reid 1 976, 1 2 1 ] .
VOLUME 26. NUMBER 2, 2004
59
was actually first presented in modem form in Droste's dissertation [Droste 1916]. In Droste coordinates, one sees immediately that the metric breaks down at r 0 and r 2M. Converting from geometrized units, the condition r 2M becomes =
=
=
r
=
2 M G c2
=
3
( l!_ )
km,
Ms
where Ms is the solar mass. Clearly, for any conventional astronomical object, the Schwarzschild radius r lies well in side the body, so that the vacuum equa tions no longer apply. Indeed, for sev eral decades all leading experts agreed that the Schwarzschild radius had no physical relevance whatsoever (see [Eisenstaedt 1989]). The kind of cata strophic gravitational collapse that leads to black holes was simply un thinkable in those days. Arthur Stanley Eddington was em phatic about this point in Space, Time and Gravitation, where he described what would happen if an observer were to approach the surface r 2M with a measuring rod. "We can go on shifting the measuring rod through its own length time after time," he wrote, "but dr is zero; that is to say we do not re duce r. There is a magic circle which no measurement can bring us inside" [Ed dington 1920, 98]. Hermann Weyl, who derived the Schwarzschild solution in Droste coordinates in Raum-Zeit-Ma terie, noted that the gravitational radius of the earth's mass was a mere 5 mm. David Hilbert claimed that the Schwarz schild solution contained two singular ities, the obvious one at r 0 and the more mysterious one located on the two-sphere with radius r 2M. Follow ing the lead of Ludwig Flamm, Hilbert calculated the trajectories of non-radial light rays that pass near this imaginary sphere, concluding that those which ap proach it can never penetrate its surface because their velocity will approach zero before arrival (see Figure 1). Hilbert presented these findings in an unpublished lecture course offered in Gottingen [Hilbert 19 16-17], but Max von Laue borrowed a copy of the course notes when he was writing Die Relativitatstheorie [Laue 1921], the first textbook on general relativity in the German language. Laue repeated Hil=
=
=
60
THE MATHEMATICAL INTELLIGENCER
-"'--�--------------
--
-------- -
---·---- · -A_,_
Figure 1 . Hilbert's visualization in [Hilbert 1 916-17] of the trajectories of light rays in the neigh borhood of a strong gravitational field. Note how the rays fail to penetrate the "magic circle" (Eddington's term) determined by the Schwarzschild radius.
bert's pronouncement regarding the singular nature of the two-sphere with r 2M, and he even replicated Hil bert's diagram showing how light rays fail to penetrate the sphere (Figure 2). Like the Gottingen mathematician, he was convinced of the essential nature of this singular region, which "cannot be eliminated by using any other coor dinates; it is essential to the nature of the thing" [Laue 192 1 , 215]. Physically, this seemed patently obvious, since "every mass m . . . has a radius greater than . in fact we do not know as yet any counter-example even in the nucleus of atoms." (For a modem treat ment of the trajectories associated with the Schwarzschild solution, see [Wald 1984, 136-148].) Johannes Droste had actually pre sented solutions to the Einstein vac uum equations for a mass point in three different coordinate systems, and he was well aware that the r coordinate had no direct physical significance [Eisenstaedt 1989, 216-2 17]. Einstein, too, was completely clear about this cir cumstance. His friend, Paul Painleve, had been troubled by the plethora of different representations of the Schwarzschild solution, and passed his misgivings on to Einstein. Noting that one could obtain an infmite number of different formulas for the metric, he suggested this "gave a clear indication of the hazardous character of such pre dictions," concluding that "it is pure imagination to claim that such conse quences can be derived from the ds2" =
2C::
..
[Eisenstaedt 1989, 222]. Einstein's re sponse illuminated the mathematical and physical issues that bothered Painleve and others:
When in the ds2 of the static solution with central symmetry you introduce any junction of r instead of r, you do not obtain a new solution because the quantity r in itself has no physical meaning . . . only conclusions reached after the elimination of coordinates may pretend to an objective signifi cance. Furthermore, the metrical in terpretation of the quantity ds is not ''pure imagination" but rather the in ner-most core of the theory itself (quoted in [Eisenstaedt 1989, 223}). Einstein's original treatment of the Schwarzchild problem was based on a specialized class of coordinate sys tems. He also postulated that the met ric tensor be spherically symmetric, independent of time, stationary, and asymptotically flat at spatial infmity. These conditions hold, of course, for the Schwarzschild-Droste metric (**) , and Hilbert showed that the last con dition was not required to deduce this solution. Then, in 1923, G. D. Birkhoff proved a far stronger result: Birkhoffs Theorem: Any sphericaUy symmetric solution of Einstein's vac uumjield equations is necessarily sta tic and has a Schwarzschild metric. In Relativity and Modern Physics [Birk hoff 1923], the Harvard mathematician
F
!b
sideration of systems of geodesics, but
E
another vast terrain was opened by the
Blj
study of general path spaces, a specialty
0
C
of the Princeton geometers. Weyl de scribed this research activity in the fol
3 }'3
2a
����----�
I
3 l'3 2
--
Ct
Figure 2. Max von Laue's diagram of the same phenomena in [Laue 1 921] was clearly taken from [Hilbert 1916-17].
showed that the assumption of spheri
One ofthose who followed Einstein's
cal symmetry alone was sufficient to en
lectures closely was the differential
sure the existence of a coordinate sys
geometer
tem in which the solution
Princeton's senior mathematician. In
(**) holds for
Luther
Pfahler
Eisenhart,
the vacuum case. The Schwarzschild so
fact, Eisenhart instigated the invitation
lution thus bears a strong analogy with
that led Einstein to visit Princeton that
the Coloumb field associated with a sta
spring, although he originally hoped to
tic charged particle in electrodynamics:
coax him into coming as a guest lecturer
in both cases the fields are unique, show
1920-21 (Eisenhart to Einstein, 20 October 1920, [CPAE, vol. 7, 231]). Einstein declined this offer, but in February 1920 Kurt Blu
ing that a monopole cannot emit radia tion. (For a modem treatment of Birk hoffs Theorem, see [Hawking and Ellis
1973, Appendix B].)
during the winter semester of
menfeld
persuaded the now-famous
physicist to undertake a trip to the United States to support the Zionist
Relativity and Differential Geometry
movement and raise funds for Hebrew
Einstein had met Birkhoff on his very
University [CPAE, vol.
first visit trip to the United States in
1921,
a whirlwind tour that ended with a stop
7, 231: note 48),
a circumstance that led him to recon sider Eisenhart's invitation.
in Princeton. There, in May, Einstein de
Einstein surely recalled Eisenhart's
livered five Stafford Little Lectures on
avid interest in general relativity when
the theory of relativity. The first two of
he heard Hermann Weyl's lecture on
these were of an introductory nature,
"Relativity as a Stimulus to Mathemati
whereas the latter three entered into the
cal Research." Weyl noted that a major
technicalities of the theory. Later that
part of this stimulation began with in
year, Einstein worked on a revision of
vestigations of affinely connected man
his
last three talks and these were pub
ifolds, which generalized Riemannian
lished in booklet form under the title The
manifolds but still admitted infinitesimal
1922],
parallel transport of vectors. Since co
Tran
variant differentiation and the whole ap
scripts of his first two talks, on the other
paratus of tensor analysis carried over
hand, were only recently published in
to this more general setting, it was not
7 of the Collected Papers of Al bert Einstein [CPAE, vol. 7, App. C ] .
necessary to assume the existence of a
Meaning of Relativity [Einstein one of his best known works.
volume
cused on the conformal structure of Rie mannian spaces, which led to the con
metric tensor. One area of research fo-
lowing words:
. . . here is clearly rich food for math ematical research and ample oppor tunityforgeneralizations. Thus schools of differential geometers sprang up in the wake of general relativity. Here in Princeton Eisenhart and Veblen took the lead, Schouten in Holland. In France, E. Cartan's fertile geometric imagination disclosed many new as pects of the subject. Some of their out standing pupils are Tracy Thomas and J. M. Thomas in Princeton, van Dantzig in Holland and Shiing Shen Chern of the Paris school. A lone wolf in Zurich, Hermann Weyl, also busied himself in this field; unfortunately he was aU too prone to mix up his math ematics with physical and philosophi cal speculations [Weyl 1949, 538}. The work of Elie Cartan and Weyl eventually opened up the whole vast theory of fiber spaces of differential manifolds.
Cartan's
investigations
dealt with geometries possessing tran sitive transformation groups that act on their tangent spaces, an approach that drew much inspiration from Lie theory and Klein's Erlangen Program (see [Cartan
1974]). Weyl developed an
axiomatic treatment of maps of tan gent spaces by utilizing parallelism along smooth curves. The latter idea was first developed by Levi-Civita in
1917 for Riemannian spaces, but it was soon generalized by Weyl to manifolds with an affme connection. Using these constructs, Weyl sought to develop a purely
infinitesimal
geometry
that
could serve as a basis for a unified field theory for gravity and electromagnet ism [Scholz
2001].
Eisenhart took his inspiration from the generalization to general affine spaces of geodesics in Riemannian geometry, namely the paths satisfying the differential equations
d2xi dt2
--
+ f �.
]k
d:x} dx" dt dt
-
-
0.
(***)
VOLUME 26, NUMBER 2, 2004
61
In Riemannian geometry one uses the In an article in Science (December space caused by the Sun's gravitational metric tensor to define the Christoffel 21, 1923) the Princeton geometer came field. Eisenhart's article appeared in symbols rjk from which one then to Einstein's defense after Philipp the wake of a similar defense of Ein proves that the solutions of (***) are Lenard republished portions of an stein's priority published by David geodesics, the shortest paths joining older theory of gravitation presented Hilbert and Max Born in the widely any two points within an appropriate by Georg Soldner. Lenard claimed that read Frankfurter Zeitung (see the ac neighborhood of the manifold. In an Soldner had derived precisely the same companying box). Hermann Weyl also affine geometry, the rjk are defmed in result for the deflection of light in the countered the criticisms of anti-rela dependent of a metric, and the solu vicinity of the sun as that found by Ein tivists, but this only hardened the tions of (***) determine a congruence stein in 1911. Einstein later revised his views of experimental physicists like of paths with special properties. The figure to 1. 75" of arc in 1915, twice the "Lenard and Co.," who regarded gen geometry of paths in affine spaces can original value, due to the curvature of eral relativity as a theory devoid of also be regarded as a generalization of methods introduced by Riemann in his famous Habilitationsschrift of 1854, Soldner und Einstein in particular his systems of normal co ordinates. Eisenhart worked directly Von Prof. Dr. Hilbert und Prof. Dr. Born Gottingen with objects of the manifold, in con Der Aufsatz von L. Baumgardt veranlaBt uns zu folgender sachlicher trast to the methods of Cartan and Berichtigung: Weyl, which dealt with the associated tangent spaces. Unfortunately, the for 1. Die von Soldner 1801 abgeleitete Formel fur die Lichtablenkung stimmt mer approach leads to complications, mit der iiberein, die Einstein im Jahre 1911 als Resultat einer vorHiufigen Uber because the intrinsic geometric objects legung tiber den EinfluB der Schwere auf das Licht veroffentlicht hat. Diese associated with paths do not satisfy the Arbeit von 1911 entha.It aber noch nicht diejenigen Grundgesetze der Physik, transformation law for generalized die man als, "allgemeine Relativitatstheorie" bezeichnet. Christoffel symbols, a central feature 2. Die richtigen Grundgleichungen dieser allgemeinen Relativitatstheo in the affine theories of Weyl and Car rie wurden erst Ende 1915 von Einstein gefunden. Dabei ergab sich, daB tan. Eisenhart summarized his contri der von Einstein vorher 1911 vermutete Wert, der mit der Soldnerschen butions to the geometry of paths as Angabe iibereinstimmt, vom Standpunkt der Relativitatstheorie falsch ist; well as those of his students in Non vielmehr stellte sich die Lichtablenkung doppelt so groB heraus. DaB Ein Riemannian Geometry [ �isenhart stein bei seinem Suchen nach den richtigen Gesetzen der Schwere zunachst 1927], whereas the related theory of durch vorlaufige Abschatzungen zu dem falschen Wert gelangt war, ist le differential invariants was sketched by icht begreiflich; das Genie sieht lange, ehe ein Gedanke begrifflich bis in Oswald Veblen in "Invariants of Qua alle Einzelheiten geklart und formuliert ist, die wichtigsten Zusammen dratic Differential Forms" [Veblen hange voraus und schatzt die experimentell kontrollierbaren Wirkungen 1927, Chapter VI]. ihrer GroBe nach zunachst mit rohen Mitteln ab. 3. Der endgtiltige Wert der Lichtablenkung von Einstein folgt vollig ein Einstein's Enemies: deutig aus der allgemeinen Relativitatstheorie. Es ist ein lrrtum zu glauben, The Anti-Relativists daB Soldner eine Folge der Relativitatstheorie richtig vorausgesehen habe; Not surprisingly, these mathematical vielmehr ist es umgekehrt: wenn die von Soldner auf Grund der Newton developments were largely ignored by schen Theorie berechnete GroBe der Lichtablenkung (die mit Einsteins physicists, few of whom had sufficient vorlaufiger Schatzung von 191 1 zufa.Ilig tibereinstimmt) durch die Beobach knowledge of differential geometry to tungen als richtig erwiesen wiirde, so ware damit eine vollgiiltige Wider make any sense of them. Opponents of legung der Einsteinschen Relativitatstheorie erzielt. relativity, on the other hand, had long Die englischen Astronomen, die bei der Sonnenfinstemis des Jahres claimed that Einstein's theory was of 1920 die Messungen der Lichtablenkung ausgefiihrt haben, sind der Mein purely mathematical interest, pointing ung, daB nicht der auf der Newtonschen Attraktionstheorie fuBende Wert to Minkowski's four-space formalism von Soldner, sondem der von der Einsteinschen Relativitatstheorie vo to make their case. Aside from Weyl, rausgesagte Wert tatsachlich gilt. Neue Untersuchungen werden bei der most prominent mathematicians chose Sonnenfinstemis des Jahres 1922 stattfinden. to avoid engaging in polemics with vi tuperative anti-relativists. Still, just two years after Einstein's visit to Text of an article by Hilbert and Born in the Frankfurter Zeitung defending the originality of Princeton in 1921, the soft-spoken Einstein's prediction for the deflection of light in the sun's gravitational field. The authors em Eisenhart became embroiled in con phasize that Einstein's original prediction from 1 91 1 , which agreed with the value earlier ob troversies surrounding Einstein's the tained by Georg Soldner, was incorrect, and that Einstein's revised figure from 1 91 5 repre ory, which by then had spilled over sented the true relativistic result. Strangely, Hilbert and Born state that the British confirmed from Germany into the United States. Einstein's result during the solar eclipse of 1 920; the eclipse took place in May 1 91 9. 62
THE MATHEMATICAL INTELLIGENCER
12
March
ZS, 1922
Reuterdahl vs. Einstein : Nailing a Fallacy Former Insists Latter's· Theory Wrong and Questions Its Originality S THE Einstein enthusiasm is waning, we may well ask what thoughts and influences have 1 caused this slowing down of . the meteorical rise ·and advance of Einstein in giving the study of relativity its tremendous and unexpectedly sudden impetus. The History of Relativity shows us that in 1889 Rudolf Mewes, in an article entitied, "Das Wesen der Materie und des Naturerkennens/' gave a brief outline of his theory of relativity. Mewes held that the es sence of matter consists in its action as manifest ip continuous transformation in a.ccordance with cau sality. He found in his basic concept .a sow-ce of the relativity of space, time and matter. For him all reality depends upon the combination of space and time. From 1892 to 1894 M ewes made a cardul analysis of the derivation of Webn's Ltlw from the Do[ltler Principle. This resulted in the tvafuation by M�wcs of the transformation formula which later made · Lo rentz famous through its use by Einstein. Mewes clearly antedates 4rmor, Lorentz and Einstein in this development. In addition, Mewes grounds his system in sound philosophy. Arvid Reuterdahl presented a brief · outline of his Space-Time Potential and Theory of Interdependence · at the inaug'ural meeting of the American Electrochem ical Society, held ih · Philadelphi2., April 5, 1902, und
A
What the Records Show
HENRI ZIEGLER, in September, 1902, laid the • fou n dation o f a cosmic the�ry in a lecture entitl�d "Die Universelle Weltformcl und 1hre Bedeutung fur dte wahre Erkenntniss aller Dinge.''. · Ziegler's cosmology is based upon the fundamental conception that the world is a unitary structure generated from the uni· versal trinity of space, time and · force, with the in clusion of an Absolute Principle, the constancy of the velocity of light. In .. Einstein 'and the New Science," Arvid Reuter... dahl gives us the history of relativity, and · give• i1 most co.ncise chro!lologl�l �cc-?unt of the au�ors and stientssts precedmg Emstem- m the study of- the theory of relativity-Out also at -t he same time gives a serious blow to both the originality and the validity of "Einstein's Theory of Relativity.� . In this little book of 26 pages of concentrated. facts, we · find that Reu terdahl commends; lauds, encour.�ges and acknowl �ork.s of the many edges with the .greatest free om 1c:ientific. cqntr ibutors, and m an article 10 THE Df:AR aoaN IND.P&NDIENT· o£ April 30, 1921, entitled, .. Kincrtia vs. Ein•tein," gloriously dcfcnds ·'·the;- g�cat . work o f "Kincrtia " although w e · know· � that ReutcrdahJ:. and � "K.inertia. '- .. do not -entirely · agrcei_.irr·r.tbeir. · ideas.' · '·-· But it shows Reuterdahl's unselfishpess �d· fairness·, in-con· traposition to . E instein's ·��.Qfl, .a�:kuo�g� mcnts to the most important ·o[� his .vredcccssors. · Reuterdahl has, in the past, been wilfully · ignored. Attempts have been made to intimjdate him and'·.his work has been practically suppressed by the machina tions of hb jealous enemies. U.lldaunted by these 00- · stacles, Reuterdaht•s i aith - in the : ul.tirnate victory of truth :r�:mains unshaken. Now the. tune has · come for som«J� who knows the facts to· pustflt to : the �world the truth concerning ReUterd.ahl�th� man who� dared to challenge Einstein, the all"!!cd"'SUJJUman 'ol:.sciencc. Back in 1913, Reuterdahl . discussed With me · m .J(an sas City his .. Space- Time P9tential, a concept ion of Gravitation and Electricity/' and' had his manuscript. typewritten in my office. I spoke about the subject on RlY I�cture tours covering other matters. and with the result that in December, 1914, Reuterdahl was invited ' to tecture at the Kansas State College of Agriculture and the University of Kansas, which lectures . , were given c;orly in 1915.
J
�
the
By E. LEE HEIDENREICH A RVID REUTERDAHL, d ean of the � dep artment of engineering �nd archi tecture at the College of S t. Thomas, St. Paul, Minne,;ota, sinc e 1918, as far back as 1896, while a student at · Brown Uni versity, Providence, Rhode I s l a n d, con c ei v ed the basic outline of h is work, "S c i e n t i fi c
Theism Ver sus Material ism, tbe Space T i m e Poten tial." He has lectured exten sively on en gineering, aci entific. nutbe matical and p h i l o s o p h ical subjects, and bas pub-. li ghed a num ber of books on scientific subj ects. W h e n the t r u m p e t s of E i n s t e i n beg an t o sound ARVID REUfBRDAHL the blasts · of glory for Rei- . ativity and its discover er, the American scientists wer e swept off their feet- be. cause they .were unfamiliar with . these strange· doctrines, but Dr. Reuterdahl bad been . thinking in "Space·Time" terms since :1896; in fact, he was the first to employ. the ,hyphen between the two words, space and time, and his Space� · Time Potential antedates Einstein's Space Time ContioUum.. .,rofeaao' r Rc�terdahl waa the first in tl>e United states to de. no·un�.e- Einsteinism .as a- fallacy. Dean Reuterdabl ·was., born February lS,'!\1876, at Karlstad, Sweden, the son of .Christ ina. Reuterdabl. :Jn 1882 the- family came to the United rs�ates, eventually locating in Providence. The son�s early education c onsist ed in private instruction given by his. father, a former officer in th e Swedish army. and an es c cllent mathematician and philosopher.
J�·-�
between 41space" and "time" is a mere coincic:ilmce. In a statement given to the New York World (J une 12, 1921 ) , he again disclaims all knowledge of Reuterdahl. This disclaimer is irreconcilable with his own previous statement to the Afteuposten. The question naturally "Why was Einstein s o anxious to efface the arises ; name of Reuterdahl in connection w ith the Theory of Re-lativity? Does the five years' sojourn in Europe of Reuterdahl's manuscript answer th is question ?
to an infinitesimal portion of the great frame are the same as hold for a part conceived as great as the imagination may permit in conformity with the frame. The Einsteinean uJorld r�ference frame is Gaussian and four-dimr:nsional in type, the co-ordinate syste'm being .an attemvt at the amalgamation of space and time. Any reference section arbitrarify cut out of.."the fram� ·is a veritable co-ordinate fabric. Such section has been called a "R•fer•>�ce-Mollusk" by Einstein. •We may conceive that a large reference section or '"Mollusk" shrivels or contracts into a small section or vice ver sa. During such diminutions or augmentati()ns the laws of the relatKmal factors remain the same, and the re sulting section is either a reduced or - increased replica of the original. The changes are similar to the · effect produced by looking at an original,. first ·with a reducing glass, and then with a magnifying glass. All the parts of the co-ordinate fabric will remain in the same· rela · tive juxtaposition irrespective of the type of th�· ·gtass through which it is observed. In �uch a system force regarded as an independent entity does not exist.;. Action at a distance no longer becomes a problem because the world frame is a uni tary interacting system. Einstein holds that a further consequence of his worldsystem is the disappearance of the mass M (which pertains to a particular point) from the law of motion. He reduces the motion of a particle under the so-called inftuen.:e of gravitation to a mere "Following" of a geodetic line in his four-dimensional world� Nowhere has Einstein stated his own case as clearly as it has been- resented. in the above an�Jysi�. w�ich· is p due lo Reuterdahl. It IS even doubtful If E10stem has grasped the significance of his own vague and stumbling attemp ts at formulating this great world synthesis. His p itiable inability to express the essentials of the system m a clear and intelligent mann�r can readily be ac counted for on the supposition which grow! upon us that Einstein has taken his entire thought of this syn thesis from the work of Reuterdahl. Not grasping dearlr the intent of Reuterdahl's Space-Time Potential, Emstein camouflaged it in a dis torted manner into his own Space- Time Continuum by transferring the conception from Euclidean to non Euclidean. Rcutcrdahl's Space-Time Potential is a syn thesis,· not only of the same intent as Einstein's, but of a vastly more comprehensive order. Reut�rdahl's com plete cosmology proceeds direct from physical facts, wh_ereas Einstem presents us with a skeleton, incapable of harmonious functioninJ�. btcause - it is constructed fro� the fibers of incons1stent mathematical specula tion, which predict coming events with the hope that they will conform-and if they do not, they arc quietly thrown out of court. Rcutcrdahl has distinctly stated, that force docs not exist in the physical universe . as . all independent and distinct entity, and action-at-a �
rdinate curves which constitute ,the fabric g{ 6ts reference-mollusk>, Reutcrdahl also · numbers hi� .�o . tential loci and shows carefully, that relative condtU'oh's existing- : between any two dements of a se<:tion, also pertain without change betwr:en the similarly numbered elements in any other section, irrespective of its s 1ze, and whether located in the rr:alm of microcosm or that of macrocosm. His Spa�-Time Potential is a chart or d iagra m d�llning the motion of every activity center in 'the entire interacting systc:m. We rtaliz:e the similarity of the two systenu, t.b e one complete and the other an incomplete instanta.nr:o�s st4i,tic view of the dynamics in the: former. Reuterdahl's priority is undisputed. the relative dates being nailed down chronologicaJiy. If the Space· Time Continuum is not a camouflaged edition o f Space-Time Potential, it cutainly looks like another toinci.denct. Einstein's version suffers sorely when compared with the earlier synthesis of Reuterdahl. Einstein abandons the classical mechanics, while Reuterdahl adheres strictly to its experimentally demonstrated principles. Einstein resorts to non-Euclidean geometry for the fibr:rs of his Space�Time Continuum, whereas Reuterdahl con tents himself with observed physical data.
Arvid Reuterdahl, author of a booklet Einstein and the New Science, used Henry Ford's Dearborn Independent to polemicize against relativ ity theory.
physical content. (For details, see the editorial "Einstein's Encounters with German Anti-Relativists" in [CPAE, vol. 7, 101-1 13].)
These were episodes Einstein surely had no reason to think about in 1949, now that Lenard was dead, and with him the whole Aryan physics move-
ment too. During the early 1920s, in fact, Einstein had mainly been content to let others battle the anti-relativists. The one time, in August 1920, that he
VOLUME 26, NUMBER 2 . 2004
63
did take pen in hand to attack his op ponents in the pages of the
Tageblatt
Berliner
[CPAE, vol. 7, 344-349], his
words caused such a stir that he seri ously contemplated leaving Berlin for quieter quarters. The streets of the Prussian capital were, indeed, amuck with chaos, but even in the quiet back woods of Minnesota an American anti relativist was keeping a close eye on the latest events. Dr. Aivid Reuterdahl, who had come to the United States from Sweden as a young lad, was President of Ramsay In stitute of Technology in St. Paul. He was also a regular contributor to Henry Ford's newspaper,
pendent,
The Dea-rborn Inde
remembered today for its
blatantly anti-Semitic editorials. Reu terdahl wrote a column called "Inter national Science Briefs," in which he regularly inveighed against the Ein steinians as corruptors of scientific truth. Even Eisenhart was taken to task for his defense of Einstein's theory in the face of Soldner's ancient derivation of light deflection
sans
a principle of
relativity. A column titled "The Einstein Film and the Debacle of Relativity" caught the eye of G. D. Birkhoff, who sent a copy on to Eisenhart asking, "Did you see this article!" In it Reuterdahl unleashes a flurry of invective against the Einsteinians, singling out Eisenhart for purportedly claiming that both of Einstein's deriva tions-the 1915 value based on GRT and the 19 1 1 derived from the equiva lence principle alone-yielded the cor rect value for the deflection of a light ray in the neighborhood of the sun, de
Karl Schwarzschild in academic attire.
spite the fact that the latter was twice the former. "Both are right according
Clearly, Reuterdahl's voice carried
equals one.
more than a little hysteria of its own, not
posed to portray this four-dimensional
In the presence of the mighty 'invari
to mention an apparently abysmal un
space, depict nothing but conceptual
ants' and 'covariants' of modern rela
derstanding of the fundamental ideas of
tivity such mere trivial inconsistencies
Einstein's theory of gravitation.
to the Einsteinians:-two
It was
real. . . . Mathematical expressions, sup
speculations having no counterpart in a real physical world."
must be ignored as negligible 'Ein
all just some mathematical hocus-pocus
steinian effects' " [Reuterdahl 1924].
to him. He called a popular 1922
film ex
stance was the common currency of
Continuing in this vein, Reuterdahl
plaining relativity "balderdash" and "an
nearly all anti-relativists from this time.
claims that Eisenhart gave an hysteri
insult to common sense"; Max von Laue
They came out in droves during the early
This
dogmatic
anti-mathematical
cal argument to defend Einstein from
criticized it, too, but of course for dif
the specter of Soldner by asserting that
ferent reasons. According to Reuter
tions approvingly. In fact, Reuterdahl
the velocity of light decreases in the
dahl, relativists ask us "to admit that we
corresponded with a number of leading
vicinity of the sun, thereby contradict
live in a four-dimensional space without
figures within the anti-relativists' net
ing the fundamental tenet of (special)
knowing it. We admit that we are not
work, including Ernst Gehrcke,
relativity which asserts the constancy
aware of this four-dimensional space,
most tenacious of the German anti-rel
of the speed of light.
but we refuse to accept [it] as physically
ativists. Thus it comes as no surprise
64
THE MATHEMATICAL INTELLIGENCER
1920s and cited each other's publica
the
Schwarzschild (left) with fellow officers dur ing the Great War.
that Reuterdahl praises Gehrcke for having "exposed the Einsteinian use of Gerber's earlier formula of 1898," an al lusion to Gehrcke's contention that
Einstein had plagiarized
an
earlier pa
per written by a Gymnasium instructor named Paul Gerber. The anti-relativists attached great significance to the fact that Gerber had come up with a fommla that Einstein later derived in accounting for the per ihelion motion of Mercury. Immediately after the appearance of [Einstein 1916] in
Annalen der Physik,
Gehrcke pub
lished a "rebuttal" in the same journal, where he charged that Einstein had surely taken the formula from Gerber without citing him [Gehrcke 1916]. Both Gehrcke and Lenard tried to make this plagiarism charge stick, but Max von Laue easily showed that Gerber's whole approach
to
gravitation was faulty,
Einstein showing off his favorite tensor.
VOLUME 26, NUMBER 2. 2004
65
whatever the status of his formula (Laue 1917]. Nevertheless, Reuterdahl was still trumpeting this news seven years later. Few readers of The Dearborn In dependent could have suspected that Gehrcke had turned up nothing more than a curiosity in the physics literature. In case some might have wondered about Gehrcke's scientific credentials, Reuterdahl led them to believe he was one of the giants of German physics by claiming that Gehrcke had succeeded Hermann von Helmholtz as Director of the Physikalisch-Technische Reich sanstalt in Charlottenburg. In fact, he was only one of several scientists work ing on the PTR's staff; not its director. Reuterdahl and Gehrcke shared a zealous passion not only to refute the basic tenets of Einstein's theory but also to reveal that its creator and his followers had perpetrated a massive fraud on the scientific world and the general public. Thus, Reuterdahl was proud to claim that he "was the first in the United States" to call attention to Georg Soldner's ancient result on light deflection from 1801. Indeed, he was convinced-thanks to the revelations of the works by Gerber and Soldner that the Einsteinian relativists were now more vulnerable than ever, and once "forced into the open, the com plete debacle of Einsteinianism is in evitable in the near future." This was a strange vision coming from an American writing in 1924, but hardly an isolated opinion. In Ger many, it became part of the dogma of the German physics movement whose hero was Philipp Lenard. By the time Einstein joined the faculty at Prince ton's new Institute for Advanced Study in 1933, only the political hostilities re mained as a residue of the anti-relativ ity movement from the early 1920s. Once the Nazis gained power, the orig inal aim of refuting relativity on scien tific grounds gave way to racial argu ments. Lenard and Johannes Stark no longer claimed that relativity theory and quantum mechanics were wrong scientifically. They protested instead that these theories represented an alien spirit and hence posed a deadly threat to "true" German physics, which had no room for empty mathematical abstractions. 66
THE MATHEMATICAL INTELLIGENCER
REFERENCES
Space-Time, Cambridge: Cambridge Univer
[Birkhoff 1 923] George D. Birkhoff, Relativity
sity Press, 1 973.
and Modern Physics, Cambridge, Mass. :
[Hilbert 1 9 1 6-1 91 7] David Hilbert, "Die Grund
Harvard University Press, 1 923. [Cartan 1 97 4] E lie Cartan, Notice sur les
lagen der Physik I I , " Vorlesung, Winterse
travaux scientifiques, Paris: Gauthier-Villars,
Mathernatisches lnstitut, Universitat Gottin
1 974.
rnester 1 9 1 6-1 7, ausgearbeitet von R. Bar, gen.
[Droste 1 9 1 6] Johannes Droste, "The Field of
[Laue 1 9 1 7] Max von Laue, "Die Fortpflan
a Single Centre in Einstein's Theory of Grav
zungsgeschwindigkeit der Gravitation. Be
itation and the Motion of a Particle in that
rnerkungen zur gleichnahmigen Abhandlung
Field," Proceedings of the Section of Sci
von P. Gerber, Annalen der Physik, 52 (1 9 1 7),
ences,
Konink/ijke Akadernie van
Weten
2 1 4-21 6.
schappen teArnsertdarn, 1 9 (1 9 1 6): 1 97-2 1 5.
[Laue 1 92 1 ] Max von Laue, Die Relativitatsthe
[Eddington 1 920] Arthur Stanley Eddington,
orie, Band 2 . Die allgemeine Relativitatsthe
Space, Time and Gravitation , Cambridge:
orie und Einsteins Lehre von der Schwerkraft.
Cambridge University Press, 1 920.
Braunschweig: Vieweg, 1 92 1 .
[CPAE, vol. 5] Collected Papers of Albert Ein stein (CPAE), Vol. 5; The Swiss Years: Cor
respondence, 1 902- 1 9 1 4 , Martin J . Klein et a!., eds . , Princeton, NJ: Princeton University
Press, 1 993.
[Pais 1 9S2] Abraham Pais, 'Subtle is the Lord. ': The Science and the Life of Albert Einstein ,
Oxford: Clarendon Press, 1 9S2. [Reid 1 976] Constance Reid, Courant in G6t tingen and New York, New York: Springer
[CPAE, vol. 7] Collected Papers of Albert Ein
Verlag, 1 976.
stein (CPAE) , Vol. 7: The Berlin Years: Writ
[Reuterdahl 1 924] Arvid Reuterdahl, "The Ein
Princeton, NJ: Princeton University Press,
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ings, 1 9 1 8- 1 92 1 , Michel Janssen et a/., eds.
2002.
stein Film and the Debacle of Relativity." The 1 5.
[CPAE, vol. SA] Collected Papers of Albert Ein
[Rowe 200 1 ] David E. Rowe, "Einstein Meets
stein (CPAE) , Vol. SA: The Berlin Years: Cor
Hilbert: At the Crossroads of Physics and
Robert Schul
Mathematics," Physics in Perspective, 3
respondence,
1 9 1 4- 1 9 1 7,
mann et a!., eds. Princeton, NJ: Princeton University Press, 1 99S.
(200 1 ): 379-424. [Sayen 1 9S5] Jamie Sayen, Einstein in Amer
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[Scholz 200 1 ] Erhard Scholz, "Weyls lnfinitesi
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[Einstein 1 91 6] Albert Einstein, "Die Grundlage
1 91 7-1 925,"
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(Einstein 1 922] Albert Einstein, The Meaning of Relativity. Four Lectures Delivered at Prince
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Hermann
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[Schwarzschild 1 91 6] Karl Schwarzschild, " U ber das Gravitationsfeld eines Massen punktes nach der Einsteinschen Theorie," K6niglich Preu}Sische Akademie der Wis
[Eisenhart 1 927] Luther P. Eisenhart, Non-Rie mannian Geometry, Princeton, NJ: Princeton
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[Eisenstaedt 1 9S9] Jean Eisenstaedt, "The Early Interpretation of the Schwarzschild So
Quadratic Differential Forms," Cambridge: Cambridge University Press, 1 927.
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[Wald 1 9S4] Robert M . Wald, General Relativ
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Howard and J. Stachel, Basel: Birkhauser, 1 9S9, pp. 2 1 3-233. [Gehrcke 1 91 6] Ernst Gehrcke, "Zur Kritik und Geschichte der neueren Gravitationstheo rien."
1 9S4. [Weyl 1 9 1 S] Hermann Weyl, Raum-Zeit-Ma
(1 9 1 6):
[Weyl 1 949] Hermann Weyl, "Relativity Theory
[Hawking and Ellis 1 973] S. W. Hawking and
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der
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1 1 9-1 24. G. F. R. Ellis, The Large Scale Structure of
as a Stimulus in Mathematical Research," Society 93 (1 949), 535-541 .
I.Hftiit§rr@ihfll#fi%§4@11,j,i§,id
Implicit Proof Kevin Wald
M i chael Kleber a n d Ravi Vak i l , E d itors
24. Equal to its complex conjugate
10. The triangle man
25. The Greatest, in a ring
1 1 . Without speaking
26. With veneration and wonder
16. Concluding section (var.)
28. Regard (two wds).
1 7. Cosmetics company
3 1 . Gun rights group
22. Simulated
32.
Second line of the proof
37. "Who
__
23. Giuliano da
contagious mathematics that travel from person to person in the community, because they are so elegant, suprising, or appealing that one has an urge to pass them on. Contributions are most welcome.
39. To a Nightingale, or f"
-
40.
Third line of the proof
45.
__
f=
5
, two, three, . . .
46. Convert cones to frusta, say 4 7.
Fourth line of the proof
wds.) 28. The
__
's Paradox
29. Paid for by yours truly (two wds.) 30. Of the ears 33. Like a Platonic solid (abbr.)
54. Old nuclear power agency (abbr.)
34. Symbol for the scalar product
55. Off-road vehicle, for short
35.
56. Reaction to a mouse
36. Originally called
57.
Fifth line of the proof
0
41. Amount of size
E
62. Georgia or Estonia, once (abbr.)
42. Under a hex (var.)
63. Converted from [) to ()?
43. Fixed
64. Not remote (two wds.)
44. Important numerical datum about
a UV-block
67. Successfully apply game theory? 68.
__
bene (mark well)
69. Presenters of arguments 74. Top
X
right?
x be
76. 78.
4 7. Substance obeying
PV
=
nRT
48. Mandatory (abbr.) 49. Section of a hospital (abbr.) 50. Source of caffeine
. . .
51. One whose age E [ 13, 20)
77. Knock
T
(15th-16th
27. Discussed in a publication (two
38. Ingest
This column is a placefor those bits of
__
c. Florentine architect)
? 2460 1 ! " (two wds.)
Sixth line of the proof
52. Iron-containing component of blood
81. Maze solver, commonly he method of infinite descent gives
82. 1 .5-tUITI maneuver
a very simple proof of the irra
83. Any of
V+F
-
2
+ 2g items
tionality of the golden ratio. Indeed, it
84. Intracity train systems
could probably be written out in one
85. The Untouchable Eliot
line; however, in this crossword, it
86. Action
53.
__
out (obtained with difficulty)
58. Al-Khwarizmi, Ramanujan, and billions of others 59. Famous London ringers (two wds.) 60. Narcotic
takes six.
61. Give utterance to Across
Down
1. Vivacity
1. Prolate speroidal aircraft
5. The Beehive State
2. Allude (to)
9. %/10,000 (abbr). 12. Letterman rival Jay 13. Descartes or Thorn 14. Water for Descartes or Thorn 15.
First line of the proof
18. More water for Descartes or Thorn
3. Beginning section 4. Reaction to a "proof from the book" 5. Member of a teaching order of nuns 6. Location of an Arizona State campus
19. Young dog
7. Prefix for lemma or basis
20. Sweet in a symphony
8. Contained
2 1 . Projections perpendicular to the
9. Removed the two-dimensional
plane of symmetry
68
boundary of
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
64. Enthusiastic (two wds.) 65. Gaussian 66. Ermines 70. Agnesi and Noether, to Bourbaki 71. Wear away 72.
lf(x) : x E domain)
73. � 75. Catcher of counterfeiters 79. It's used in bifurcating trees 80. Vegas bet that wins with proba bility 9/19
e-mail: [email protected] 56A Cedar Street, Somerville, MA 021 43, USA
a 5 13
6
7
8
9
10
11
14
a
b
b - a = a(a/b ) 78
79
'
81
82
84
85
VOLUME 26, NUMBER 2 , 2004
69
la§lji§i.t;j
O s m o Peko n e n , E d itor
I
1 089 and All That by David Acheson NEW YORK, OXFORD UNIVERSITY PRESS, 2002. 1 78 pp. US $1 9.95, ISBN 0-19-851623-1
REVIEWED BY JOHN MIGHTON
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
Column Editor: Osmo Pekonen, Agora Centre, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: [email protected]
70
T
he title of Acheson's book harkens back to a classic of British humour, published in 1930 by W. C. Sellar and R. J. Yeatman, titled 1 066 and AU That. What Sellar and Yeatman once did for the teaching of history, Acheson has now done for mathematics: in what other book of popular mathematics can you find a tour of algebra based on a magic trick from the I-SPY Manual for 1956, or an explanation of the mathe matics behind pendulums based on a device (invented by the author) that out-does the Indian rope trick? (And where else can you find a one-word ex planation of the secret of life, which, according to the author's elderly biol ogy teacher, was "chlorophyll"?) I was surprised to see how much substantial mathematics David Ache son has squeezed into this well-crafted little book. In a single chapter of seven pages, for instance, he introduces enough calculus to allow the reader a glimpse, in subsequent chapters, of the mathematics behind soap bubbles, simple harmonic motion, and even Euler's stunning formula linking e with the trigonometric functions and the imaginary numbers. Acheson demon strates a number of important mathe matical principles with well-chosen examples: he uses a surprising topolog ical transformation to show how easy it is to jump to the wrong conclusion in mathematics, and he uses the Konigs berg Bridge Problem to illustrate the power of proof by contradiction. With a wealth of intriguing exam ples and insights, richly illustrated with mathematical diagrams and cartoons, 1 089 and AU That captures the "ele ments of mystery and surprise" that
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
run through much of mathematics. The book will spark the imaginations of people who were convinced by their teachers that mathematics is boring and inaccessible. (As well as the imag inations of those who have yet to be convinced: my ten-year-old daughter read the book with my guidance and loved it.) Even mathematicians will find fresh perspectives on old themes in this playful and inventive book
The Fields Institute 222 College Street Toronto, Ontario, Canada M5T 3J1 e-mail: [email protected]
Mathematical Constants by Steven R. Finch NEW YORK: CAMBRIDGE UNIVERSITY PRESS, 2003 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, #94 US $95.00, 602 pp., ISBN 0-521 -81 805-2
REVIEWED BY JET WIMP
l A �ile adjudicating a thesis at the WW University of Uppsala last year, I was a guest at the home of a congenial middle-aged couple who were book dealers. Every room in their palatial home was a library. Bookshelves cov ered the walls. Books sprouted from end tables, from desks, from the car peted floors. When I retired the first evening, I had my pick of dozens of put me-to-sleep volumes, neatly stacked on my bedside table. I chose one, a pho tographic essay of period furniture in Swedish country homes at the tum of the century. I found it to be a puissant competitor for any pharmaceutical sleep inducer. The next morning, I commented on the profusion of books and inquired about the inevitable difficulty of fmding one. My hostess confided to me that it was a mistake to look for a book "It
la§lji§i.t;j
O s m o Peko n e n , E d itor
I
1 089 and All That by David Acheson NEW YORK, OXFORD UNIVERSITY PRESS, 2002. 1 78 pp. US $1 9.95, ISBN 0-19-851623-1
REVIEWED BY JOHN MIGHTON
Feel like writing a review for The Mathematical Intelligencer? You are welcome to submit an unsolicited review of a book of your choice; or, if you would welcome being assigned a book to review, please write us, telling us your expertise and your predilections.
Column Editor: Osmo Pekonen, Agora Centre, University of Jyvaskyla, Jyvaskyla, 40351 Finland e-mail: [email protected]
70
T
he title of Acheson's book harkens back to a classic of British humour, published in 1930 by W. C. Sellar and R. J. Yeatman, titled 1 066 and AU That. What Sellar and Yeatman once did for the teaching of history, Acheson has now done for mathematics: in what other book of popular mathematics can you find a tour of algebra based on a magic trick from the I-SPY Manual for 1956, or an explanation of the mathe matics behind pendulums based on a device (invented by the author) that out-does the Indian rope trick? (And where else can you find a one-word ex planation of the secret of life, which, according to the author's elderly biol ogy teacher, was "chlorophyll"?) I was surprised to see how much substantial mathematics David Ache son has squeezed into this well-crafted little book. In a single chapter of seven pages, for instance, he introduces enough calculus to allow the reader a glimpse, in subsequent chapters, of the mathematics behind soap bubbles, simple harmonic motion, and even Euler's stunning formula linking e with the trigonometric functions and the imaginary numbers. Acheson demon strates a number of important mathe matical principles with well-chosen examples: he uses a surprising topolog ical transformation to show how easy it is to jump to the wrong conclusion in mathematics, and he uses the Konigs berg Bridge Problem to illustrate the power of proof by contradiction. With a wealth of intriguing exam ples and insights, richly illustrated with mathematical diagrams and cartoons, 1 089 and AU That captures the "ele ments of mystery and surprise" that
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER-VERLAG NEW YORK, LLC
run through much
of mathematics. The book will spark the imaginations of people who were convinced by their teachers that mathematics is boring and inaccessible. (As well as the imag inations of those who have yet to be convinced: my ten-year-old daughter read the book with my guidance and loved it.) Even mathematicians will find fresh perspectives on old themes in this playful and inventive book The Fields Institute 222 College Street Toronto, Ontario, Canada M5T 3J1 e-mail: [email protected]
Mathematical Constants by Steven R. Finch NEW YORK: CAMBRIDGE UNIVERSITY PRESS, 2003 ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS, #94 US $95.00, 602 pp., ISBN 0-521 -81 805-2
REVIEWED BY JET WIMP
l A �ile adjudicating a thesis at the WW University of Uppsala last year, I was a guest at the home of a congenial middle-aged couple who were book dealers. Every room in their palatial home was a library. Bookshelves cov ered the walls. Books sprouted from end tables, from desks, from the car peted floors. When I retired the first evening, I had my pick of dozens of put me-to-sleep volumes, neatly stacked on my bedside table. I chose one, a pho tographic essay of period furniture in Swedish country homes at the tum of the century. I found it to be a puissant competitor for any pharmaceutical sleep inducer. The next morning, I commented on the profusion of books and inquired about the inevitable difficulty of fmding one. My hostess confided to me that it was a mistake to look for a book "It
would drive you crazy," she said. In stead of searching for a book, she would just allow her eye to drift over a proxi mate collection of volumes until it lit on something interesting. "And that," she explained," is what I tell myself I was ordained to read at that moment." For me it takes almost superhuman self-discipline not to keep on looking for that one volume in my personal li brary of many thousands of books. What has saved me, on occasion, is to know exactly where the ilber book is the compendium, handbook, enchirid ion, whatever you want to call it-the book that summarizes the contents of many other volumes, the book that makes that desired book irrelevant. In any bookstore, a book with the title, Handbook ofX, whether X is Chamber Music, Mathematical Functions, even Subtle-Energy Therapies or Arc and Oxyacetylene Welding, is certain to grab my attention, as indeed does the present book, a handbook of con stants. "Constants! How banal!" the reader may think. After all, aren't we all conditioned to denigrate the mere constant? "Up to a constant," we've heard, or "except for a constant," as though these were objects that didn't deserve our attention. But constants are the alphabet of mathematics, and their properties have obsessed mathe maticians since the early Greeks. They still do, and books about constants, such as Pi and the AGM, by Jonathan M. and Peter Borwein, now in paper back, are often mathematical best-sell ers. Although I have misgivings about it, I think the present book is truly un usual and abundantly exciting, whether as a desert island read or as a tool for breaking the logjam of mathematical researcher's block: open it to almost any page and you'll find an open problem, some of which I will mention in this re view. Early on, the author points out that it is not even known whether Euler's con stant y .5772156649015328606065120 . . . is rational, let alone transcendental, an attribute of uncertainty it shares with Catalan's constant G, =
G= =
lo
( - l)n (2n + 1 )2
.9159655941772 190150546035 . . .
Inexpert as the exposition is, one cannot fault the author's assiduity. Each section, however brief it may be, contains an exhaustive bibliography. For instance, the 12-page section on the Hardy-Littlewood constants has a bibliography of 77 items, and the bib liography for the eight-page section on Apery's constant, -;(3), has 1 3 1 . The author has clearly done his home work The author's failings are, I think, due equally to inexperience and to being overwhelmed by the subject matter. The essential stratagem of expositional writing, of course, is to make certain that what is clear to you will also be clear to your intended reader. Or per-
Constants are the alphabet of mathematics, and their p roperties have obsessed mathematicians since the early G reeks . haps a talent for such writing is innate. I doubt whether I have it. (Looking at old papers of mine, I am often horrified at their incomprehensibility, and when I receive queries about them, I am help less.) Hardy had the talent in spades. Knopp, Apostol, the Borweins, too. Sometimes, had the present author added a line or two, the murky would have become clear. For instance, in the section on Sierpiriski's constant (p. 122), the author is discussing r(n), "de fined to be the number of representa tions of the positive integer n as a sum of two squares, counting order and sign. For example, r(l) = 4." I shook my head at this, and consulted Hardy and Wright (An Introduction to the Theory of Numbers , p. 240), who give the formula 1 = (± 1)2 + 02 = 02 + (± 1)2, which explains everything.
r(1)
= 4,
Occasionally, the author forgets he hasn't defined something, e.g., p. 163: "a specific integer sequence defined by the greedy algorithm: x1 = 1 and Xi
=
max (j l:SJ:Sl- 1
+
1)(i - A.J)."
What is a greedy algorithm? (The term is used again shortly thereafter, for a different algorithm.) Often, I found, a trivial rewording could have banished puzzlement. Speaking about a certain constant A that will guarantee the positivity of a co sine sum (p. 242), he says, "Gasper . . . proved that A 4.5678018826 . . . and has minimal polynomial . . . " and my reaction was to ask myself why A should be expected to be the solution of a polynomial equation. I would have said, "Gasper . . . proved that A = 4.5678018826. . . . Further, A is a root of the following polynomial, and of no polynomial with integer coefficients of lower degree." In other words, good ex position doesn't arouse distracting re flections in the reader. We don't need to be told when a section ends, as the author often clumsily does, "This com pletes the story for cosine sums." There are occasional Goldwynisms, "Lower bounds on A are clearly of enormous interest to everybody con cerned" (p. 204), and there are apposi tional errors, "and then carry it back to the frequency domain, that is, return ing to where we were initially" (p. 204). It would have helped considerably had the author numbered his equations. However, if exposition and some times grammar get short shrift, the mathematics shines. I want to hurl great gobbets of the text at the reader. (What constitutes "fair use," anyway?) Nevertheless, I promise restraint. I'll single out some particularly vitalizing results after a precis of the contents. Chapter 1: Well-known Constants; Chapter 2: Constants Associated with Number Theory, Chapter 3: Constants Associated with Analytic Inequalities; Chapter 4: Constants Associated with the Approximation of Functions; Chap ter 5: Constants Associated with Enu merating Discrete Structures; Chapter 6: Constants Associated with Func tional Iteration; Chapter 7: Constants Associated with Complex Analysis; =
VOLUME 26, NUMBER 2, 2004
71
Chapter 8: Constants Associated with Geometry. There follows a 28-page table of constants, ordered by magni tude, starting with the Jeopardy-like answer to "what is the conjectured value of the Bruijn-Newman constant?" Everyone will like Chapter 1 . "There is a spigot algorithm for calculating 7T, " the author says. (What is a spigot al gorithm, and why does the author keep doing this?) "Far more important, how ever," he continues, "is the digit-ex traction algorithm discovered by Bai ley, Borwein & Plouffe . . . based on the formula . . . " and the author gives the splendid and by now well-known equa tion that allows one to display any given digit in the binary expansion of 7T. The series he actually gives is a gen eralization, whose specialization yields the desired equation
--- --- ) 1
1 8k + 6
8k + 5
°
Myself, I love y. Section 1.5 tells us many things about it. "A surprising re sult, due to de Ia Vallee Poussin . . . is 1 - y = I.1m
n---'>x
- k�� l { }, 1 L n
n k
-
,
where {x} denotes the fractional part of Surprising, indeed. Results like this don't come for free. Sacrifices are re quired. (I would not be surprised to learn of the mysterious disappearance of animals, even children, in the valley of chicks a century ago.) The above re sult has a probabilistic interpretation: if a large integer n is divided by each integer k = 1,2,3, . . . , n, then the av erage fraction by which the quotient nlk falls short of the succeeding inte ger is not 1/2, the value intuition sug gests, but y. Section 1 . 1 1 on Chaitin's constant is mind-blowing. This material is at the nexus of probability, computability, logic, and decidability, and it has its ba sis in Matiyasevic's proof that Hilbert's Tenth problem is unsolvable, i.e., the result that there exists no algorithm for determining whether a general Dio phantine equation has positive integer solutions. It is possible to construct a x.
72
THE MATHEMATICAL INTELLIGENCER
universal Diophantine equation with this property whose solutions DN de pend on the parameter N. Now define a real number A by its binary expan sion O.A 1A2A3 . . . where
AN =
{
�
1 f DN =I= 0 0 If DN 0 =
There is no algorithm for deciding for arbitrary N whether AN is 1 or 0, so A is a perfectly well-defined but incom putable real number. This assertion has
"There i s a spigot
ample can be found in Section 3.12 on Du Bois Reymond's constants. Define the mth Du Bois Reymond constant by
Much is known about the Cm. Watson proved that c2k is a polynomial of de gree k in e2 with rational coefficients. No such information is known about the c2k+ 1, but there is an expression as an infmite series for all Cm· Denote by �1 , �2, �3 . . . the positive solutions of x = tanx. Then
algorith m for calculating
1r"
disquieting philosophical implications. There are many open problems, some philosophical, some mathematical, in this area. Section 2. 13, on Mills's Constant, gives rise to perplexing thoughts, too. Mills showed that there exists a posi tive constant C such that I C3N I yields prime numbers for all integers N. ( I . I denotes the largest-integer func tion.) Actually, there are many Cs. It can be shown that one way of con structing one is to define q0 2 and let qn+ 1 be the least prime exceeding q� for each n 2': 0. Then such a C is given by
Constants arising in the approxima tion of functions-including the topics of Fourier series, approximation by generalized orthogonal polynomials, and best rational and polynomial ap proximation-exert a powerful fasci nation. I love Lebesgue constants, which originally arose in Fourier se ries, but can be defined for expansions in any set of orthogonal polynomials. If Sn is the nth partial sum of the Fourier series for a functionfwith ::::::; 1, then
I! I
=
It is easy to compute that C 1.3063778838. . . . However, this ap proach is unsatisfying. First, it requires the prime numbers themselves, and second, one would require C to many places to compute just a few primes. Mills's formula would be of use as a prime-generator only if a way could be found to generate an exact value of some C. Scattered through the book are se ductive kickshaws which could lead one to fritter away hours that could more productively be spent in grading final exams, or engaging in legitimate (at Drexel this means funded) re search. Many of these originally ap peared as problems in the American Mathematical Monthly. A typical ex=
and Ln is called the nth Lebesgue con stant. Lo = 1, L1 1 .43599 . . . , L2 = 1.642 18 . . . , L3 = 1.7783. . . . Is Ln monotone increasing? Szego showed it was, by demonstrating the series =
Ln =
16 _9 7r
x (2n+l)k
Ll y�l ? k�
1 4k2 - 1
1 2'(}· - 1 ·
Less easily decidable issues arise from Lebesgue constants occurring in more sophisticated contexts, the theory of orthogonal polynomials, for instance. I want to mention an example from the theory of best approximation. Let's restrict ourselves to functions bounded on [ - 1,1] with norm ! = sup IJCx) l .
ll ll - l:s "'>1 x
Let Hn denote the norm o f the error of the best polynomial approximation of degree n to the function lx /. Bernstein
showed that the following limit ex isted, lim nHn n--.x
= {3,
=
=
and that {3 < .286. He conjectured that {3 1/(2 y;) .2821, but many years later, Varga and Carpenter showed this cor\iecture to be incorrect by comput ing {3 to 50 decimals. A closed-form ex pression for {3 is still unavailable. Con trast this with the error, call it En, of the best approximation to f by the ra tio of two polynomials each of degree n. In 1992 Herbert Stahl proved that lim
n--.ac
en-Yn En
=
o(1). = Bv� � B = 2 VZb = 2.9904703993 . ln(Mn)
Here where b
=r 0
+
ln(l - ln(1
. .
'
- e-x))dx.
Discrete structu res (I
8.
This example, which shows that some times the complex is more amenable to analysis than the simple, is one of those results that aren't to be had for free. However, I know Herbert, and he is an honorable man, incapable of commit ting enormities for the sake of mathe matics. I believe the fauna of Berlin are safe. Discrete structures (I prefer the term "combinatorics") yields a plethora of in triguing constants. I mention an exam ple which the reader should find ap pealing, since probably most readers of The Intelligencer are drivers. Cars of unit length are parked at random on the street. If a new car overlaps a car, our convention allows it to slide off to the front or rear, whichever is closer. It is then parked if there is a space for it. If not, it is discarded. The mean lim iting density of the cars that can fit is
m=f
of all members of the group? Bill Goh and Eric Schmutz, through a redoubtable feat of asymptotic analysis, were able to find the asymptotic expression
p refer the term "combi natorics") yields a plethora of i ntri g u i ng constants . Stong later gave a simpler formula for b: _
b-
J
x
0
ln(x + eX -
1) 1 dx.
Combinatorics yields another pot pourri of constants, with applications to manifold areas of applied science. Typically these constants occur as the lead constant in the asymptotic ex pression for some important combina torial quantity. One example: denote by rn the number of constitutional iso mers of the alkane series (methane,
ethane, propane, butane, etc.) Then if
(2x + 1)exp[ -2(x +
{3(x)
m=
=
exp
e-x - 1) ] {3(x)dx,
( -2 L 1 -/-( dt) , X
and .8086525183 . . . This seems reasonable, but in real life it is proba bly lessened by the occurrence of those cyclopean SUVs. Group theory has contributed its share of constants. Given a permutation 1T of n symbols, defme its order, ec 7r), to be the least positive integer such that 7f"' identity. Obviously, 1 ::s ec7r) ::s n!. What is its asymptotic mean value, call it Mn, taken over the orders
=
m
K
= .3551817423 lim �rnn512
n--.x
. . . . one has
= .6563186958 . . .
The theory of trees is responsible for these results, and for many other re sults about chemical isomers. All of this material is fertile with un solved problems, which the author doles out generously. A polyomino of order n is a connected set of n adja cent squares, i.e., squares containing a common side. Two polyominoes are distinct only if they have different shapes or different orientations. Let A(n) denote the number of polyomi noes of order n.
We have
A(1)
=
=
= =
1, A(2) 2, A(3) 6, A(4) 19, A(5) 63, A(6) = 216, A(7) 760 . . . =
=
Almost nothing is known about A(n). No one has found a generating function for it, although Augean computations by various researchers have provided values up to A(47). It is suspected, but not known, that A(n) is not com putable in polynomial time. Though it is known that the limit
a=
lim [A(n) ] 11n n--.oo
exists, its value is not known. Various bounds exist. Slight changes in the model can pro duce problems that are more tractable. A polyomino is row-convex if every row consists of an unbroken line of squares. Denote by B(n) the number of possible row-convex polyominoes consisting of n squares.
=
B( I ) 1, B(2) = 2, B(3) 6, B(4) 19, B(5) 61, B(6) 196, B(7) 629 . . .
=
=
=
=
=
A generating function is known for [1] :
B(n) f(x) =
x(1 - x? 1 - 5x + 7x2 - 4x3
=
I B(n)xn,
n=O
and this tells us everything we need to know. It leads to an explicit represen tation for B(n), even to one in terms of radicals, and to the asymptotic formula
B(n) - ccin,
a = 3.205569431 · c=
. 1908155020
B(n) has very curious properties. B(n) grows very rapidly, too rapidly, one would think, to have a very robust di visibility profile. Not so. Let b(n) B(n) - 2. It can be shown that every positive integer divides an infinite number of the b(n). Whether A(n) above er\ioys similar properties is not known. I close with one last example from this bounteous book, one which ex hibits the resourcefulness, the impres sive ingenuity of the mathematical imagination in its fullest flowering, namely, the material on Conway's Con-
=
VOLUME 26, NUMBER 2, 2004
73
stant (Section 6. 12). We start with a string of digits, say, " 13." Describing this sequence as "one one, one three" leads to the derived sequence, " 1 1 13." This we describe as "three ones, one three" to obtain "31 13." The next num ber is "132 1 13," and so forth. What can be said about the length L(k) of the kth string? At first glance, this seems to be precisely the sort of problem one should flee from. Conway, however, did not. Defying all expectations, he proved that L(k) �CAk, k � oo, A 1.3035772690. . . .
page in the book is an adventure, and I am grateful that he could enlist all of us as his fellow explorers. REFERENCE
[1 ] Odlyzko, A M . , "Asymptotic Enumeration Methods," p. 1 1 04, Handbook of Combina torics, v.ll, ed. R. L. Graham, M. Gr6tschel,
and L. Lovasz, North-Holland, 1 995. Department of Mathematics Drexel University Philadelphia, PA 1 91 04 e-mail: [email protected]
=
An amazing feature of A is that it is a universal constant, in the sense that the above asymptotic estimate prevails (with the trivial exceptions of the initial strings "0" and "22") no matter what
I th i n k al l mathematicians shou l d own t h is book . one starts with. Doron Zeilberger and Shalosh B. Ekhad (about whom I have warned lnteUi gencer readers in the strongest possible terms), both figure in this saga, as does the motif of utilizing mathematical soft ware to prove theorems. There is a con nection with chemistry. A is the unique largest (in modulus) eigenvalue of the 92 X 92 transition matrix whose (i,J)th element is the number of atoms of ele ment j resulting from the decay of one atom of element i. Occasionally one en counters cynics who claim there is lit tle really conceptually new to be dis covered in mathematics, that future knowledge will be a mere elaboration of the known. They are deceived. I think all mathematicians should own this book. To say that it is a tri umph of substance over style is too dis missive, and even inaccurate. There are sections where the author conveys admirably his excitement over some unexpected and beautiful sequence of ideas. I pay the author the earnest com pliment of stating that nearly every his cimmerian cohort,
74
THE MATHEMATICAL INTELLIGENCER
Hypercomplex Iterations by Yumei Dang, Louis H. Kauffman and Daniel Sandin WORLD SCIENTIFIC, 2002 SERIES ON KNOTS AND EVERYTHING, VOL. 1 7 1 64 pp. US$33 ISBN 981 -02-3296-9
REVIEWED BY KEITH BRIGGS
T
he prototype complex discrete time dynamical system is the qua dratic polynomial z � z2 + c; z,c E C. Thanks to pioneering work in the 1980s by Douady, Hubbard, Peitgen, and oth ers, we now have a good understand ing of both the structure of orbits for fixed c, and the structure of parameter space, that is, the dependence of the dynamics on the constant c. The for mer is described by Julia sets, the boundaries of which separate bounded orbits from those which diverge to in finity; and the latter is described by the Mandelbrot set: a value of c is in the in terior of this set if the orbit starting at 0 converges to a cycle. The Mandelbrot set consists of components, each topo logically a circle, on which the period of the cycle is constant; components meet tangentially at bifurcation points. Furthermore, it is easily shown that any complex quadratic polynomial is affine conjugate to z2 + c for some c, so that we have essentially a complete understanding of this family of dynam ical systems. It is therefore natural to ask whether such a description is available for dy namics in other spaces, and the present
book is an attempt to produce a paral lel theory for quatemion polynomials. The quatemions IHl form a 4-dimen sional non-commutative division alge bra, in which the three imaginary units i, j, k satisfy i 2 j2 = k2 = ijk - 1. S o how to d o maps q � q 2 + c ; q,c E IHl behave? Unfortunately (or fortunately, depending on our point of view), we get nothing new here, since there is al ways a rotation of IHl which puts this into the form z � z2 + c for z,c E C. In other words, the dynamics take place on a 2-real-dimensional subspace, iso morphic to C, in IHI. Thus, we can say that the quatemion quadratic q2 + c is already fully understood. The authors note this fact on page 66, but then they go on to ignore this raison de ne pas etre. Additionally, this particular qua dratic is no longer a generic represen tative of the set of all quatemion qua dratics, so it is not clear why it should be the focus of study. This book suffers from several de fects: a large number of factual and typographical errors; a curious imbal ance in the level of rigor, giving rigor ous theorems (though sometimes with faulty proofs) that are not used, and re jecting rigor when it is really needed; as well as uncertainty as to the intended audience. The ostensible main aim is to produce estimates of distances in IHl to Julia sets in order to produce computer graphics of these. However, we have al ready seen that these distances could easily be found by rotating to C, so it is hard to see any justification for devot ing a whole book to this aim. Let us survey the book to note some specific examples of these defects. On page 20 we read that "The Mandelbrot set was discovered by Mandelbrot in 1980." In fact, it was discovered by Brooks and Matelski in 1978. The proof of Lemma 3. 1 makes use of the "fact" that cos(m - n)d O for integers m,n, m =I= n. The proof of Lemma 3.3 is by contradiction: it begins, "Suppose to the contrary that f is a constant" and concludes, "Then f(z) is a constant, a contradiction!" The exclamation mark is indeed ironic. We have circular def initions such as that of Julia set on page 44: "Let us denote by Ap(oo) the basin of attraction of the Julia set generated =
J51T
=
by the polynomial p(z). Then the Julia set of p(z) is defined as Jp = aAp(oo)." Even definitions that are not meaning less are confused, such as the immedi ately following definition of Mandel brot set: "M = {q E IHI I 0 E Kp forp(z) z2 + c) where p(z) = zn + c. " A long but irrelevant section on analyticity of quaternion functions based on the work of Sudbery is then followed by an in correct statement of the maximum modulus theorem of complex analysis, which puts a condition onf ' (z). Section 4.8 is taken word-for-word from a paper of Bedding and Briggs, though the cita tion in this section incorrectly gives the author as Briggs. In copying equation (4.27) from that paper, five typographi cal errors were introduced. In addition, in the sentence "Second, let us see whether anything more interesting is possible with quadratic quaternion functions," the crucial word "regular" has been omitted before "quadratic." Eventually we reach section 7.3, where the distance estimation results be gin to appear. Here the authors work with maps F : fRN � fRN for general N, without bothering to define what multi plication might mean in such a space. Section 7. 1 has defined a restricted form of multiplication on the unit sphere, but this again appears to be reducible to the complex case by rotation. The authors assume without proof that the set of points not attracted to infinity under it eration of F is compact. Everything de pends on this assumption; we are not told for which F this might be true, and it is not checked for any explicit F. Page 75 uses the chain rule for a de gree k polynomial incorrectly: "npn + 1 = k(Fn)D(Fn)." In any case, the "correct" rule npn + 1 k(Fn)k- 1D(Fn) is proba bly meaningless in view of the lack of a definition of multiplication already mentioned. More defects of this kind continue throughout the book Because of this, it is difficult to have confidence that the computer graphics shown are cor rect. Unfortunately, this book does not add to knowledge, but will confuse a naive reader and infuriate a knowl edgeable reader. My opinion differs from that of Peter Ha'issinsky in Math ematical Reviews: "This book is well =
=
written and self-contained. A CD-Rom comes with the book which provides striking animated illustrations of such fractals." The attached CD indeed con tains striking images, but these are es sentially meaningless when based on dubious mathematics. BTExact Adastral Park Polaris 1 34 Martlesham Heath, Suffolk IP5 3RE, UK e-mail: [email protected]
On Ouaternions and Octonions: Their Geometry, Arithmetic, and Symmetry By John H. Conway and Derek
A.
Smith
NATICK, MASSACHUSETTS: AK PETERS 2003 1 50 pp. US $29.00 ISBN 1 -56881 - 1 34-9
REVIEWED BY GEOFFREY DIXON
A
resonant spike above background noise in one parameter as another parameter is varied is a frequent indi cator in experimental science that one has found something significant. Reso nance is good. It justifies the expendi ture of our time and can lead to the kinds of rewards that prompted us to run the experiment in the first place. In a broad sense there is a reso nance in mathematics detectable in much the same way. Using as the input parameter the set of positive integers, and as the output some measure of the number and richness of mathematical structures associated with the inputs, one could easily argue that the integers with the highest "resonant" spikes above the background noise are 1, 2, 4, 8, and 24. Associated with the first four of these integers are the parallelizable spheres, the n-square identities, lat tices of particular beauty and elegance, and the composition algebras, R, C, H, and 0 (real numbers, complex num bers, quatemions, and octonions). There is a holographic aspect to much of this. Starting with the parallelizable spheres, for example, we can derive
the existence of the division algebras, and from there the n-square identities, and so forth. The whole in that case is implicit in a part. For all that, the eas iest bits to play with are the so-called division algebras, R, C, H, and 0. And by the way, I include 24 in the list be cause that is the dimension of the Leech lattice. These mathematical objects have from the beginning inspired a great deal of research in both mathematics and physics (and beyond: the most ef ficient way to build a 3-d engine for computer games and modeling is using the quatemions). During the latter part of the nineteenth century, throughout much of the twentieth and on into the twenty-first, there have been those, like myself, convinced that this notion of mathematical resonance is more than metaphor. It seems reasonable that it should be able to help us answer some of the big questions of physics. In particular, why is the universe struc tured as it is? The father and patron saint of divi sion algebra obsession is surely W. R. Hamilton, the discoverer in 1843 of the quaternion algebra. By all accounts he harbored few if any doubts that our 3dimensional geometry could be most naturally described in H, and perhaps even ascribed to the existence of H. Small wonder then that he was galled by, and spent much of his life in resis tance to, the ultimately dominant de scription of vectors in 3-space in terms of ordered triples, along with the dot and cross products, both of which are inherent in quatemion multiplication. Hamilton's friend John Graves was impressed by the discovery of H, and by its connection to 3-space. But Graves, more of a mathematician, first and foremost saw a potentially ex tendable sequence of algebras. Very soon after learning about the quater nions he found the next algebra in the sequence, the octonions. He little real ized at the time that it was the last of its sort, that the sequence of division algebras is finite. One can imagine that Hamilton might have been disap pointed at some level by the existence of 0 beyond his beloved H, lessening the likelihood that H was a mathemat-
VOLUME 26, NUMBER 2, 2004
75
ical pinnacle linked inexorably to our
ence. Together with what many theo
physical reality. Ultimately, however,
retical physicists feel is an insurmount
theory, which many feel has a higher
his obsession may prove not to have
able
unification
been wrong or misplaced, but simply
nonassociativity of
too limited.
obstacle
to
application-the
0-this failure en
space-time arises in the mysterious M potential
than
ordinary
String theory. What's more, this Jordan
In the arena of mathematical physics
hanced the perception in the majority
algebra also has spinors, which are
that these algebras were suspect, and
fermionic, a term in physics jargon re
Hamilton's efforts to promote H as a
that association with them could only
ferring to how they transform with re
description of 3-space over the more
hurt one's reputation. To
this day one
spect to Lorentz transformations. And
popular vector triplets tainted the for
still finds a hint of apology in most ar
the remarkable SOs triality transfor
mer idea. Being on the losing side of
ticles relating the octonions to physics.
mation, mixing the two 8-dimensional
such a battle leads to a general per
I myself was warned several times that
halves of these spinors with the 8 trans
ception that the legitimacy of the
work in this area was a potential career
verse space dimensions (the bosonic
loser's cause is questionable. Fortu
killer-and in fact it was.
part, in a similar piece of physics jar
nately this perception never entirely killed off curiosity about the
two
And yet the articles continue. For
gon), is a kind of supersymmetry. And
many it seems just too unlikely that al
then there are the links of this algebra
higher-dimensional division algebras
gebras linked to so much elegant math
to exceptional groups (the exceptional
among mathematicians and physicists.
ematics are not also linked to central
group Es in particular has been promi
To this day one
from the octonions. It all sounds just too
sti l l fi nds a h i nt of
alone who see the tendrils of division al
In the first half of the twentieth cen tury such prominent figures as Pascual Jordan, John von Neumann, and Eu gene Wigner attempted to apply the oc tonions to quantum mechanics. There is a description of quantum theory in terms of Jordan algebras, and of all the infmitely many of these algebras there is one that is truly exceptional, stand ing apart from the rest: the Jordan al gebra of 3
X
3 hermitian
matrices over
nent in String theory), all of which arise
idea that the tendrils will be there what
articles relating
ever direction is taken, so I threw them
all into the mix and ended up linking the
the octon ions
division algebras to the Standard Model of quarks and leptons, with its U(1)
to physics .
correlation of this algebra to any viable
gebra theory in their work My own work, for example, is founded on the
apology i n most
0. But they never found a satisfying
delicious. And it is not String theorists
SU(2)
X
X
SU(3) gauge symmetry.
After three decades of immersion in
quantum theory, and their efforts were
ideas in physics. Why would physics
mathematical physics, it dawned on me
soon overshadowed by developments
rest on less elegant structures? And
that the
in other areas, like quantum field the
then there are the hints from String
sundry theories naturally get excited
Although
a
adherents
of various
and
ory, and the construction of devices
theory.
10-dimensional
when the application of some mathe
that make very big booming noises.
Lorentzian space-time arises most fre
matical idea or structure leads to a
The need for bigger and better booms
quently in String theory, the dimen
sudden lessening of resistance to ad
dried up many esoteric research ideas,
sions 3, 4, 6, 10, and 26 have all been
vancement. However, it could very
and the quatemions and octonions lan
shown to be the only dimensions in
well be that this lessening of resistance
guished for a time.
which certain sets of conditions are
is more attributable to this mathemat
In the 1970s they arose again, most
met. The associated transverse sub
ical resonance than to any particular
prominently at Yale University. In that
spaces of these space-times have the
theory's merits, and that any theory,
familiar dimensions 1 , 2, 4, 8, and 24.
however absurd, will seem most right
decade I attended a talk at Harvard by someone from Yale about the relation
The Lorentzian metrics of the 3-, 4-, 6-,
ship of the octonions to the increasingly
and 10-dimensional spaces arise from
prominent gauge Lie group, SU(3). A
the determinants of 2
short time before I had found an elegant
matrices over R, C, H, and
X
and exciting when resonating in tune with this mathematics.
2 hermitian
Finally, to segue away from physics,
0. These
it was the String theorist Martin Ced
expression of all four of Maxwell's equa
collections of matrices are also Jordan
erwall who introduced the octonion x
tions as a single equation over the
algebras. But it is the last of these,
product ((Ax) (x*B), where x is a unit
quaternions, so I was ripe to be influ
linked to the more important 10-di
octonion, and x* its conjugate. I sub
enced by news of a related algebra of
mensional space-time, which has re
sequently generalized this to the xy
potentially
ceived the most attention.
product ((Ax)(y*B)), and showed that
greater
significance
to
physics. All this spawned in me an ob
But wait! The Jordan algebra of 3
X
0 has also
for general unit octonions x and y the algebra resulting from this new prod
session for these algebras that may have
3 hermitian matrices over
rivaled or exceeded Hamilton's. But the
been the focus of much attention in
uct is still the octonion algebra (with a
research at Yale ultimately failed to ful
physics. This contains not only a 10-di
new identity, yx*). Then, inspired by
mensional space-time, but an extra
Conway and Sloane's Sphere Packings, Lattices and Groups, I showed that for
fill the hopes of its participants, and at
least one was embittered by the experi-
76
THE MATHEMATICAL INTELLIGENCER
space dimension. And 1 1-dimensional
any given orthogonal octonion basis with the product of any pair of basis units being another basis unit (plus or minus)-the set of unit octonions x, or pairs of unit octonions x and y, for which the product of any pair of basis units is still a basis unit, generates, in the former case, a pair of 8-dimen sional laminated lattices (E8), and in the latter case a pair of 16-dimensional laminated lattices. Lattices of the for mer sort appear prominently in Con way and Smith's new book, On Quater nions and Octonions. For the pure mathematician the study of these algebras is a friendlier pursuit. Not required in general to make big booms, nor to link the results of their re search to a tantalizingly out-of-reach physical reality, they are free to let the mathematics lead them where it will. For most of us with any familiarity with these algebras, they are associ ated with continuum structures: Lie al gebras; Lie groups; Euclidean spaces; spheres; Jordan algebras, and so forth. In On Quaternions and Octonions there is a fair amount of attention paid to rotation groups on spaces of di mension 2,3,4, 7, and 8, the odd num bers arising from the dimensions of the imaginary parts of H and 0. But one of the authors of this new volume is co author of Sphere Packings, Lattices and Groups, devoted to discrete struc tures in all dimensions. Not surpris ingly much of this new volume is de voted to discrete structures (lattices of integers) in dimensions 1, 2, 4, and 8, and to finite groups associated with the division algebras. That one can even define notions of integer and prime integer in C, H, and 0 is a consequence of their being com position algebras. If x and y are in one of these algebras, and N(x) is the Eu clidean norm of x, then
N(xy)
=
N(x)N(y).
This property has endless conse quences and lies at the heart of why these algebras are so important. In di mensions 2, 4, and 8 it allows us to define integers, primes, and unique prime factorization, up to certain conditions. There are three parts to the book,
one for each of C, H, and 0. The first is the shortest, and most readers will find the material familiar, save perhaps the sections on Gaussian and Kleinian integers over C. Chapters 3 to 5 are devoted to the quaternions. The first two focus on 3-di mensional and 4-dimensional groups, both finite and Lie (the dimensions re ferring to the spaces upon which the groups act). The noncommutativity ofH allows us to rotate both the 3-dimen sional purely imaginary part ofH (uAu*, u a unit quaternion), and the full 4-di mensional algebra (uAv*, u and v unit quaternions). Chapter 5 introduces Lip schitz and Hurwitz integral quaternions,
"Seven Rig hts Can Make a Left . " primes, units, and factorization of inte gers. The latter system of integers is connected to the elegant D4 lattice. All of this is mostly precursor to the octonion chapters, 6 to 1 2 . These chapters begin with a l o o k at octonion multiplication, and at a proof of the Hurwitz theorem: R, C , H, and 0 are the only composi tion algebras. The importance of this theorem cannot be overstated. There is more than one way to gen eralize this sequence of algebras (Dickson's is one, and Dixon's an other), but the higher-dimensional algebras that result are-to con tinue the resonance metaphor a bit further-little more than back ground noise. There is no associ ated spike of interesting mathe matical structures and ideas. The octonion algebra is noncom mutative and nonassociative. Because of nonassociativity the set of left mul tiplication maps, Lx: A � xA , is not closed. That is, for x and y in 0, there is in general no z such that LxLy = Lz. To close the set one must include nested actions of the form, A � x(y ( . . . (zA) . . . ) ) . Curiously, despite non commutativity any left (or right) action can be expressed as a combination of right (or left) actions. In the quaternion
algebra this is certainly not the case: left actions and right actions are dis tinct and commute with each other. In the octonion case, left actions do not in general commute with right actions, and the algebra of all left actions is equal to the algebra of right actions. This is complicated further by two sided actions, and the whole is inti mately tied to triality. Chapters 6 to 8 beautifully discuss many aspects of these multiplicative complexities. My favorite section title is "Seven Rights Can Make a Left." An integer system on 0, linked to the E8 lattice as the Hurwitz integers on H are linked to D4, is defined in chapter 9, and elaborated on in chap ters 10 and 11. The notion of integral ity on 0 is more difficult than on C or H and certainly deserves the extra at tention. Just a note: the lattices E8 and D4 are associated with the Lie groups E8 and SOs, both of which crop up prominently from time to time in many unification models. On Quaternions and Octonions concludes in chapter 12 with an origi nal look at the octonion projective plane. Projective spaces of all dimen sions can be constructed over R, C, and H, but the nonassociativity of 0 prevents anything beyond the plane. Similarly, R, C, and H are associated with infinite sequences of classical Lie groups, while 0 is linked primarily to just the five exceptional groups. This projective plane, by the way, is very closely linked to that previously men tioned exceptional structure, the Jor dan algebra of 3 X 3 hermitian matri ces over 0. Although Conway and Smith's book is pure mathematics, this last chapter highlights the fact that when it comes to the division algebras, pure and applied and seldom far apart. REFERENCE
Conway, J. H . , and Sloane, N. J. A : Sphere Packings, Lattices and Groups. Springer-Ver
lag, 1 993. Department of Physics University of New Hampshire Durham, NH 03824
USA
e-mail: [email protected]
VOLUME 26, NUMBER 2, 2004
77
Stamping Through Mathematics by Robin J. Wilson SPRINGER-VERLAG, NEW YORK INC., 2001 1 26 pages US $29.95 ISBN 0-387-98949-8
REVIEWED BY VAGN LUNDSGAARD HANSEN
T
his cleverly designed book is based on the idea to tell the history of mathematics through motifs on stamps. The author, Robin Wilson, has a dis tinguished record in mathematics and has published extensively in graph the ory and combinatorics. Outside math ematics he holds a long-time passion for philately. He is well known to the readers of The Mathematical Intelli gencer for enlightening short stories about mathematicians, or mathemat ics, represented on stamps in the col umn "Stamp Comer," which he has edited for many years. It is commend able that Wilson by the publication of a book has shared his comprehensive knowledge about mathematics on stamps with a more general public. That the title of the book is witty is no surprise to those who have attended the author's entertaining lectures. The book is organized so that every right-hand page contains a number of stamps and the associated left-hand page tells an episode of mathematics illustrated by the stamps. The stamps are slightly enlarged and uncancelled to show each motif in complete detail, and the text is clear and informative. It is amazing how much mathematics Wilson has been able to extract from stamps. There are almost 400 different stamps in the book The reviewer counted 392, not counting the rare
78
THE MATHEMATICAL INTELLIGENCER
stamp on the back cover picturing the author himself, which still awaits for mal approval by the postal services. A stamp is a small piece of art, and the postal services in the various coun tries spend considerable time on preparing their graphics and texts. It is highly non-trivial for a person, an event, or a subject to get on a stamp. It therefore came as a surprise to this reviewer that enough mathematics-re lated topics have been selected for stamps to write a valuable short history of mathematics with stamps as the
A stamp is a smal l p iece of art . guide. Wilson has succeeded very well in this undertaking. He writes in the preface, "This is not a history of math ematics book in the conventional sense of the word. Several important mathe maticians or topics are omitted, due to the absence of suitable stamps featur ing them, whereas others may have as sumed undue prominence because of the abundance of attractive images. Where appropriate, I have felt free to let the stamps dictate the story." Wil son's book can be forgiven for not be ing the complete history of mathemat ics; in the opinion of the reviewer it is certainly a splendid first introduction. The great variety of topics touched upon is illustrated by some of the sec tion heads: Greek Geometry, Plato's Academy, Euclid and Archimedes, Early Islamic Mathematics, Go and Chess, Map-Making, The French Revo lution, The Geometry of Space, Mathe matical Education, and many more. In the section on The Nature of Light, I
missed a Danish stamp featuring the astronomer Ole R0mer, who discov ered that light has fmite speed. A sec tion on Mathematical Shapes is illus trated by stamps of different shapes, including a beautiful hexagonal stamp in the form of a honeycomb tessella tion and depicting various aspects of bee-keeping, issued by the Pitcairn Is lands in 1999. You can learn a lot just by studying such a stamp with an open mind, including geography! The oldest stamp I noticed in the book was issued by Colombia in 1869 and has the form of a scalene triangle. Another old stamp appears in the sec tion on Map-Making. It shows the world in Mercator projection and was issued by Canada in 1898. Among mathemati cians and mathematical physicists de picted on stamps, not surprisingly New ton and Einstein take the lead. Robin Wilson's book is attractively written, and even on a short train trip you can cover several episodes in mathematics. It should appeal to al most any kind of reader and it will be extremely useful for teachers and stu dents looking for a quick introduction to the history of mathematics. The book provides convincing evidence that mathematics can be found every where. It will be a good point of de parture for many interesting mathe matical conversations with philatelists and mathematicians alike. The design of the book is inviting and every bib liophile will find pleasure in paging through it. This magnificent book de serves to be widely read. Department of Mathematics Technical University of Denmark Building 303 DK-2800 Lyngby, Denmark e-mail: V.L. [email protected]
k1flrri .M$·h•i§i
Robin Wilson
The Philamath' s Alphabet- D
I
nite series and attempted to formalise
determine the distances between two
the idea of a limit so as to put the cal
places. It dates from around 300 AD.
culus on a firm basis. He was also the
1898), better known as Lewis Carroll,
describes the motion of a vibrating
was the author of
string. In his later years he wrote many
Alice's Adventures in Wonderland and other children's
of the mathematical and scientific arti
books. A mathematics lecturer at Christ
Encyclopedie,
Church, Oxford University, he wrote
which attempted to classify the knowl
books on symbolic logic and the alge
edge of the time.
bra of determinants, and he was the
cles for Denis Diderot's
D
'Alembert: Jean le Rond d'Alem bert ( 1 7 1 7-1 783) was a leading
Enlightenment figure. He stated the ra tio test for the convergence of an infi-
Dodgson: Charles Dodgson (1832-
first to obtain the wave equation that
Dedekind: Richard Dedekind (1831-
creator of many ingenious mathemati
1916) invented the concept of an ideal to
cal puzzles. A great enthusiast for the
explain why certain types of "number"
teaching of Euclid's
factorise in more than one way-for ex
dents, he produced several treatises
ample, 10
=2 X5=
(4 + v6)(4 - v6).
Elements to stu
(both serious and humorous) on it.
The stamp below shows the unique fac torisation of an ideal as a product of powers of ''prime ideals." He also intro duced the idea of a "Dedekind cut" in order to provide a rigorous defmition of a real number.
Descartes: D'Alembert
In
1637 Rene Descartes
(1596-- 1650) wrote Discours de la meth
ode, a philosophical treatise with a 100-page appendix, La geometrie, con taining fundamental contributions to an alytic geometry. Here he solved an an cient problem of Pappus on the path traced out by a point moving in a par
De Witt
ticular way, by naming two lengths x and
y and calculating all the other lengths in terms of them, thereby obtaining a conic as the required path. Thus Descartes in Dedekind
troduced algebraic methods into geom etry, but he did not initiate the "Carte sian coordinates" (with orthogonal axes) named after him.
De Witt: Johan de Witt (1625-1672) was a talented mathematician and po litical leader whose concern with Hol-
Distance-measuring
land's financial problems led to his writing a treatise on the calculation of life annuity payments, an early attempt Descartes
to apply probability theory to econom ics. His important Elementa curvarum linearum was one of the earliest ac counts of analytic geometry. He was
Please send all submissions to
murdered by an angry mob for politi
the Stamp Corner Editor,
cal reasons.
Robin Wilson, Faculty of Mathematics,
Distance-measuring
The Open University, Milton Keynes,
measuring instruments have survived
cart:
Various
MK7 6AA, England
from ancient China, including a distance
e-mail: [email protected]
measuring cart that one pulls along to
80
THE MATHEMATICAL INTELLIGENCER © 2004 SPRINGER� VERLAG NEW YORK, LLC
Dodgson
cart
