Men of Mathematics (109 page)

The year 1874 which saw the appearance of Cantor's first revolutionary paper on the theory of sets was also that of his marriage, at the age of twenty nine, to Vally Guttmann. Two sons and four daughters were born of this marriage. None of the children inherited their father's mathematical ability.

On their honeymoon at Interlaken the young couple saw a lot of Dedekind, perhaps the one first-rate mathematician of the time who made a serious and sympathetic attempt to understand Cantor's subversive doctrine.

Himself somewhat of a
persona non grata
to the leading German overlords of mathematics in the last quarter of the nineteenth century, the profoundly original Dedekind was in a position to sympathize with the scientifically disreputable Cantor. It is sometimes imagined by outsiders that originality is always assured of a cordial welcome in science. The history of mathematics contradicts this happy fantasy: the way of the transgressor in a well established science is likely to be as hard as it is in any other field of human conservatism, even when the transgressor is admitted to have found something valuable by overstepping the narrow bounds of bigoted orthodoxy.

Both Dedekind and Cantor got what they might have expected had they paused to consider before striking out in new directions. Dedekind spent his entire working life in mediocre positions; the claim—now that Dedekind's work is recognized as one of the most important contributions to mathematics that Germany has ever made—that Dedekind
preferred
to stay in obscure holes while men who were in no sense his intellectual superiors shone like tin plates in the glory of public and academic esteem, strikes observers who are themselves “Aryans” but not Germans as highly diluted eyewash.

The ideal of German scholarship in the nineteenth century was the lofty one of a thoroughly coordinated “safety first,” and perhaps rightly it showed an extreme Gaussian caution toward radical originality—the new thing might conceivably be not quite right. After all an honestly edited encyclopaedia is in general a more reliable source of information about the soaring habits of skylarks than a poem, say Shelley's, on the same topic.

In such an atmosphere of cloying alleged fact, Cantor's theory of the infinite—one of the most disturbingly original contributions to mathematics in the past 2500 years—felt about as much freedom as a skylark trying to soar up through an atmosphere of cold glue. Even if the theory was totally wrong—and there are some who believe it cannot be salvaged in any shape resembling the thing Cantor thought he had launched—it deserved something better than the brickbats which were hurled at it chiefly because it was new and unbaptized in the holy name of orthodox mathematics.

*  *  *

The pathbreaking paper of 1874 undertook to establish a totally unexpected and highly paradoxical property of the set of
all
algebraic numbers. Although such numbers have been frequently described in preceding chapters, we shall state once more what they are, in order to bring out clearly the nature of the astounding fact which Cantor proved—in saying “proved” we deliberately ignore for the present all doubts as to the soundness of the reasoning used by Cantor.

If
r
satisfies an algebraic equation of degree
n
with rational integer (common whole number) coefficients, and if
r
satisfies no such equation of degree less than
n,
then
r
is an algebraic number of degree
n.

This can be generalized. For it is easy to prove that any root of an equation of the type

c
₀
x
ⁿ
+
c
₁
x
^n–1
+ . . . +
c
_n–1
x
+
c
_n
= 0,

in which the
c's
are any given
algebraic
numbers (as defined above), is itself an algebraic number. For example, according to this theorem, all roots of

are algebraic numbers, since the coefficients are. (The first coefficient satisfies
x
²
−2x
+ 10 = 0, the second,
x
²
- 4x−
421 =0, the third, x
³
−90 = 0, of the respective degrees 2, 2, 3.)

Imagine (if you can) the set of
all
algebraic numbers.
Among
these will be
all
the positive rational integers 1, 2, 3, . . ., since any one of them, say
n,
satisfies an algebraic equation,
x—n
= 0, in which the coefficients (l, and
—n)
are rational integers. But
in addition to these
the set of
all
algebraic numbers will include
all
roots of
all
quadratic equations with rational integer coefficients, and
all
roots of
all
cubic equations with rational integer coefficients, and so on, indefinitely. Is it not
intuitively evident
that the set of
all
algebraic numbers will contain
infinitely more
members than its
sub-set
of the rational integers 1, 2, 3, . . .? It might indeed be so, but it happens to be false.

Cantor proved that the set of all rational integers 1, 2, 3, . . . contains precisely as many members as the “infinitely more inclusive” set of
all
algebraic numbers.

A proof of this paradoxical statement cannot be given here, but the kind of device—that of “one-to-one correspondence”—upon which the proof is based can easily be made intelligible. This should induce in the philosophical mind an understanding of what a
cardinal number
is. Before describing this simple but somewhat elusive concept it will be helpful to glance at an expression of opinion on this and other definitions of Cantor's theory which emphasizes a distinction between the attitudes of some mathematicians and many philosophers toward all questions regarding “number” or “magnitude.”

“A mathematician never defines magnitudes in themselves, as a philosopher would be tempted to do; he defines their equality, their sum and their product, and these definitions determine, or rather constitute, all the mathematical properties of magnitudes. In a yet more abstract and more formal manner he
lays down
symbols and at the same time
prescribes
the rules according to which they must be combined; these rules suffice to characterize these symbols and to give them a mathematical value. Briefly, he creates mathematical entities by means of arbitrary conventions, in the same way that the several chessmen are defined by the conventions which govern their moves and the relations between them.”
^II
Not all schools of mathe
matical thought would subscribe to these opinions, but they suggest at least one “philosophy” responsible for the following
definition
of cardinal numbers.

Note that the initial stage in the definition is the description of “same cardinal number,” in the spirit of Couturat's opening remarks; “cardinal number” then arises phoenix-like from the ashes of its “sameness.” It is all a matter of
relations
between concepts not explicitly defined.

Two sets are said to have
the same cardinal number
when all the things in the sets can be
paired off
one-to-one. After the pairing there are to be no unpaired things in either set.

Some examples will clarify this esoteric definition. It is one of those trivially obvious and fecund nothings which are so profound that they are overlooked for thousands of years. The sets
(x, y, z), (a, b, c)
have
the same cardinal number
(we shall not commit the blunder of saying “Of course! Each contains
three letters”) because
we can
pair off
the things
x, y, z
in the first set with those,
a, b, c
in the second as follows,
x
with
a, y
with
b, z
with
c,
and having done so, find that none remain unpaired in either set. Obviously there are other ways for effecting the pairing. Again, in a Christian community practising technical monogamy, if twenty married couples sit down together to dinner, the set of husbands will have the same cardinal number as the set of wives.

As another instance of this “obvious” sameness, we recall Galileo's example of the set of all squares of positive integers and the set of all positive integers:

1
²
, 2
²
, 3
²
, 4
²
, . . . ,
n
²
, . . .

1, 2, 3, 4, . . ., n, . . .

The “paradoxical” distinction between this and the preceding examples is apparent. If all the wives retire to the drawing room, leaving their spouses to sip port and tell stories, there will be precisely twenty human beings sitting at the table, just half as many as there were before. But if all the squares desert the natural numbers, there are just as many left as there were before. Dislike it or not as we may
(we should not, if we are rational animals), the crude miracle stares us in the face that
a part of a set may have the same cardinal number as the entire set.
If anyone dislikes the “pairing” definition of “same cardinal number,” he may be challenged to produce a comelier. Intuition (male, female, or mathematical) has been greatly overrated. Intuition is the root of all superstition.

Notice at this stage that a difficulty of the first magnitude has been glossed.
What is a set, or a class?
“That,” in the words of Hamlet, is “the question.” We shall return to it, but we shall not answer it. Whoever succceeds in answering that innocent question to the entire satisfaction of Cantor's critics will quite likely dispose of the more serious objections against his ingenious theory of the infinite and at the same time establish mathematical analysis on a non-emotional basis. To see that the difficulty is not trivial, try to imagine the set of
all
positive rational integers 1, 2, S, . . . , and ask yourself whether, with Cantor, you can hold this totality—which is a “class”—in your mind as a definite object of thought, as easily apprehended as the class x,
y, z
of three letters. Cantor requires us to do just this thing in order to reach the
transfinite
numbers which he created.

Proceeding now to the definition of “cardinal number,” we introduce a convenient technical term: two sets or classes whose members can be paired off one-to-one (as in the examples given previously) are said to be
similar. How many
things are there in the set (or class)
x, y,
_f
z?
Obviously three. But what is “three”? An answer is contained in the following definition: “The
number
of things in a given class is the
class
of all classes that are similar to the given class.”

This definition gains nothing from attempted explanation; it must be grasped as it is. It was proposed in 1879 by Gottlob Frege, and again (independently) by Bertrand Russell in 1901. One advantage which it has over other definitions of “cardinal number of a class” is its applicability to both finite and infinite classes. Those who believe the definition too mystical for mathematics can avoid it by following Couturat's advice and not attempting to
define
“cardinal number.” However, that way also leads to difficulties.

Cantor's spectacular result that the class of all algebraic numbers is similar (in the technical sense defined above) to its sub-class of all the positive rational integers was but the first of many wholly unexpected properties of infinite classes. Granting for the moment that
his reasoning in reaching these properties is sound, or, if not unobjectionable in the form in which Cantor left it, that it can be made rigorous, we must admit its power.

Consider for example the “existence” of transcendental numbers. In an earlier chapter we saw what a tremendous effort it cost Hermite to prove the transcendence of
a particular
number of this kind. Even today there is no general method known whereby the transcendence of any number which we suspect is transcendental can be proved; each new type requires the invention of special and ingenious methods. It is suspected, for example, that the number (it is a constant, although it looks as if it might be a variable from its definition) which is defined as the limit of