Men of Mathematics (110 page)

as
n
tends to infinity, is transcendental, but we cannot prove that it is. What is required is to show that this constant is not a root of
any
algebraic equation with rational integer coefficients.

All this suggests the question “How many transcendental numbers are there?” Are they
more
numerous than the integers, or the rationals, or the algebraic numbers as a whole, or are they
less
numerous? Since (by Cantor's theorem) the integers, the rationals, and
all
algebraic numbers are equally numerous, the question amounts to this: can the transcendental numbers be counted off 1, 2, 3, . . .? Is the class of all transcendental numbers
similar
to the class of all positive rational integers? The answer is no; the transcendentals are
infinitely more numerous than the integers.

Here we begin to get into the controversial aspects of the theory of sets. The conclusion just stated was like a challenge to a man of Kronecker's temperament. Discussing Lindemann's proof that π is transcendental (see Chapter 24), Kronecker asked, “Of what use is your beautiful investigation regarding
π?
Why study such problems, since irrational [and hence transcendental] numbers do not exist?” We can imagine the effect on such a skepticism of Cantor's proof that the transcendentals are infinitely more numerous than the integers 1, 2, 3, . . . which, according to Kronecker, are the noblest work of God and the
only
numbers that
do
“exist.”

Even a summary of Cantor's proof is out of the question here, but
something of the kind of reasoning he used can be seen from the following simple considerations. If a class is similar (in the above technical sense) to the class of all positive rational integers, the class is said to be
denumerable.
The things in a denumerable class can be counted off 1, 2, 3, . . .; the things in a non-denumerable class can
not
be counted off 1, 2, 3, . . . : there will be more things in a non-denumerable class than in a denumerable class. Do non-denumerable classes exist? Cantor proved that they do. In fact the class of all points on any line-segment, no matter how small the segment is (provided it is more than a single point), is non-denumerable.

From this we see a hint of why the transcendentals are non-denumerable. In the chapter on Gauss we saw that any root of any algebraic equation is representable by a point on the plane of Cartesian geometry. All these roots constitute the set of all algebraic numbers, which Cantor proved to be denumerable. But if the points on a mere line-segment are non-denumerable, it follows that
all
the points on the Cartesian plane are likewise non-denumerable. The algebraic numbers are spotted over the plane like stars against a black sky; the dense blackness is the firmament of the transcendentals.

The most remarkable thing about Cantor's proof is that it provides no means whereby a single one of the transcendentals can be constructed. To Kronecker any such proof was sheer nonsense. Much milder instances of “existence proofs” roused his wrath. One of these in particular is of interest as it prophesied Brouwer's objection to the full use of classical (Aristotelian) logic in reasoning about infinite sets.

A polynomial
ax
ⁿ
+ bx
ⁿ
–
¹
+ . . . + l, in which the coefficients
a, b, . . . l
are rational numbers is said to be
irreducible
if it cannot be factored into a product of two polynomials both of which have rational number coefficients. Now, it is a meaningful statement to most human beings to assert, as Aristotle would, that a given polynomial either
is
irreducible or
is not
irreducible.

Not so for Kronecker. Until some definite process, capable of being carried out in a
finite
number of nontentative steps, is provided whereby we can settle the reducibility of any given polynomial, we have no logical right, according to Kronecker, to use the concept of irreducibility in our mathematical proofs. To do otherwise, according to him, is to court inconsistencies in our conclusions and, at best, the use of “irreducibility” without the process described, can give us only
a Scotch verdict of “not proven.” All such
non-constructive
reasoning is—according to Kronecker—illegitimate.

*  *  *

As Cantor's reasoning in his theory of infinite classes is largely non-constructive, Kronecker regarded it as a dangerous type of mathematical insanity. Seeing mathematics headed for the madhouse under Cantor's leadership, and being passionately devoted to what he considered the truth of mathematics, Kronecker attacked “the positive theory of infinity” and its hypersensitive author vigorously and viciously with every weapon that came to his hand, and the tragic outcome was that not the theory of sets went to the asylum, but Cantor. Kronecker's attack broke the creator of the theory.

In the spring of 1884, in his fortieth year, Cantor experienced the first of those complete breakdowns which were to recur with varying intensity throughout the rest of his long life and drive him from society to the shelter of a mental clinic. His explosive temper aggravated his difficulty. Profound fits of depression humbled himself in his own eyes and he came to doubt the soundness of his work. During one lucid interval he begged the authorities at Halle to transfer him from his professorship of mathematics to a chair of philosophy. Some of his best work on the positive theory of the infinite was done in the intervals between one attack and the next. On recovering from a seizure he noticed that his mind became extraordinarily clear.

Kronecker perhaps has been blamed too severely for Cantor's tragedy; his attack was but one of many contributing causes. Lack of recognition embittered the man who believed he had taken the first—and last—steps toward a rational theory of the infinite and he brooded himself into melancholia and irrationality. Kronecker however does appear to have been largely responsible for Cantor's failure to obtain the position he craved in Berlin. It is usually considered not quite sporting for one scientist to deliver a savage attack on the work of a contemporary to his students. The disagreement can be handled objectively in scientific papers. Kronecker laid himself out in 1891 to criticize Cantor's work to his students at Berlin, and it became obvious that there was no room for both under one roof. As Kronecker was already in possession, Cantor resigned himself to staying out in the cold.

However, he was not without some comfort. The sympathetic Mittag-Leffler not only published some of Cantor's work in his journal
(Acta Mathematica)
but comforted Cantor in his fight against Kronecker. In one year alone Mittag-Leffler received no less than fifty two letters from the suffering Cantor. Of those who believed in Cantor's theories, the genial Hermite was one of the most enthusiastic. His cordial acceptance of the new doctrine warmed Cantor's modest heart: “The praises which Hermite pours out to me in this letter . . . on the subject of the theory of sets are so high in my eyes, so unmerited, that I should not care to publish them lest I incur the reproach of being dazzled by them.”

*  *  *

With the opening of the new century Cantor's work gradually came to be accepted as a fundamental contribution to all mathematics and particularly to the foundations of analysis. But unfortunately for the theory itself the paradoxes and antinomies which still infect it began to appear simultaneously. These may in the end be the greatest contribution which Cantor's theory is destined to make to mathematics, for their unsuspected existence in the very rudiments of logical and mathematical reasoning about the infinite was the direct inspiration of the present critical movement in all deductive reasoning. Out of this we hope to derive a mathematics which is both richer and “truer”—freer from inconsistency—than the mathematics of the pre-Cantor era.

Cantor's most striking results were obtained in the theory of
non-denumerable
sets, the simplest example of which is the set of all points on a line-segment. Only one of the simplest of his conclusions can be stated here. Contrary to what intuition would predict, two unequal line-segments contain the
same number
of points. Remembering that two sets contain the same number of things if, and only if, the things in them can be paired off one-to-one, we easily see the reasonableness of Cantor's conclusion. Place the unequal segments
AB, CD
as in the figure. The line
OPQ
cuts
CD
in the point
P,
and
AB
in
Q; P
and
Q
are thus paired off. As
OPQ
rotates about
0,
the point
P
traverses
CD,
while
Q
simultaneously traverses
AB,
and each point of
CD
has one, and only one, “paired” point of
AB.

An even more unexpected result can be proved. Any line-segment, no matter how small, contains as many points as an infinite straight line. Further, the segment contains as many points as there are in an entire plane, or in the whole of three-dimensional space, or in the whole of space of
n
dimensions (where
n
is
any
integer greater than
zero) or, finally, in a space of a denumerably infinite number of dimensions.

In all this we have not yet attempted to define a
class
or a
set.
Possibly (as Russell held in 1912) it is not necessary to do so in order to have a clear conception of Cantor's theory or for that theory to be consistent with itself—which is enough to demand of any mathematical theory. Nevertheless present disputes seem to require that some clear, self-consistent definition be given. The following used to be thought satisfactory.

A set is characterized by three qualities: it contains all things to which a certain definite property (say redness, or volume, or taste) belongs; no thing not having this property belongs to the set; each thing in the set is recognizable as the same thing and as different from all other things in the set—briefly, each thing in the set has a permanently recognizable individuality. The set itself is to be grasped as a whole. This definition may be too drastic for use. Consider, for example, what happens to Cantor's set of all transcendental numbers under the third demand.

At this point we may glance back over the whole history of mathematics—or as much of it as is revealed by the treatises of the master mathematicians in their purely technical works—and note two modes of expression which recur constantly in nearly all mathematical exposition. The reader perhaps has been irritated by the repetitious use of phrases such as “we can find a whole number greater than 2,” or “we can choose a number less than
n
and greater than
n
−2.” The
choice of such phraseology is not merely stereotyped pedantry. There is a reason for its use, and careful writers mean exactly what they say when they assert that “we can find, etc” They mean that
they can do what they say.

In sharp distinction to this is the other phrase which is reiterated over and over again in mathematical writing: “There exists.” For example, some would say “there exists a whole number greater than 2,” or “there exists a number less than
n
and greater than
n—
2.” The use of such phraseology definitely commits its user to the creed which Kronecker held to be untenable,
unless,
of course, the “existence” is proved by a
construction.
The existence is not proved for the sets (as defined above) which appear in Cantor's theory.

These two ways of speaking divide mathematicians into two types: the “we can” men believe (possibly subconsciously) that mathematics is a purely human invention; the “there exists” men believe that mathematics has an extra-human “existence” of its own, and that “we” merely come upon the “eternal truths” of mathematics in our journey through life, in much the same way that a man taking a walk in a city comes across a number of streets with whose planning he had nothing whatever to do.

Theologians are “exist” men; cautious skeptics for the most part “we” men. “There exist an infinity of even numbers, or of primes,” say the advocates of extra-human “existence”; “produce them,” say Kronecker and the “we” men.

That the distinction is not trivial can be seen from a famous instance of it in the New Testament. Christ asserted that the Father “exists”; Philip demanded “Show us the Father and it sufficeth us.” Cantor's theory is almost wholly on the “existence” side. Is it possible that Cantor's passion for theology determined his allegiance? If so, we shall have to explain why Kronecker, also a connoisseur of Christian theology, was the rabid “we” man that he was. As in all such questions ammunition for either side can be filched from any pocket.