by E. T. Bell
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.
A striking and important instance of the “existence” way of looking at the theory of sets is afforded by what is known as Zermelo’s postulate (stated in 1904). “For every set M whose elements are sets P (that is, M is a set of sets, or a class of classes), the sets P being non-empty and non-overlapping (no two contain things in common), there exists at least one set N which contains precisely one element from each of the sets P which constitute M.” Comparison of this with the previously stated definition of a set (or class) will show that the “we” men would not consider the postulate self-evident if the set M consisted, say, of an infinity of non-overlapping line segments. Yet the postulate seems reasonable enough. Attempts to prove it have failed. It is of considerable importance in all questions relating to continuity.
A word as to how this postulate came to be introduced into mathematics will suggest another of the unsolved problems of Cantor’s theory. A set of distinct, countable things, like all the bricks in a certain wall, can easily be ordered; we need only count them off 1, 2, 3, . . . in any of dozens of different ways that will suggest themselves. But how would we go about ordering all the points on a straight line? They cannot be counted off 1, 2, 3, . . . . The task appears hopeless when we consider that between any two points of the line “we can find,” or “there exists” another point of the line. If every time we counted two adjacent bricks another sprang into being between them in the wall our counting would become slightly confused. Nevertheless the points on a straight line do appear to have some sort of order; we can say whether one point is to the right or the left of another, and so on. Attempts to order the points of a line have not succeeded. Zermelo proposed his postulate as a means for making the attempt easier, but it itself is not universally accepted as a reasonable assumption or as one which it is safe to use.
Cantor’s theory contains a great deal more about the actual infinite and the “arithmetic” of transfinite (infinite) numbers than what has been indicated here. But as the theory is still in the controversial stage, we may leave it with the statement of a last riddle. Does there “exist,” or can we “construct,” an infinite set which is not similar (technical sense of one-to-one matching) either to the set of all the positive rational integers or to the set of all points of a line? The answer is unknown.
Cantor died in a mental hospital in Halle on January 6, 1918, at the age of seventy three. Honors and recognition were his at the last, and even the old bitterness against Kronecker was forgotten. It was no doubt a satisfaction to Cantor to recall that he and Kronecker had become at least superficially reconciled some years before Kronecker’s death in 1891. Could Cantor have lived till today he might have taken a just pride in the movement toward more rigorous thinking in all mathematics for which his own efforts to found analysis (and the infinite) on a sound basis were largely responsible.
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Looking back over the long struggle to make the concepts of real number, continuity, limit, and infinity precise and consistently usable in mathematics, we see that Zeno and Eudoxus were not so far in time from Weierstrass, Dedekind, and Cantor as the twenty four or twenty five centuries which separate modern Germany from ancient Greece might seem to imply. There is no doubt that we have a clearer conception of the nature of the difficulties involved than our predecessors had, because we see the same unsolved problems cropping up in new guises and in fields the ancients never dreamed of, but to say that we have disposed of those hoary old difficulties is a gross mis-statement of fact. Nevertheless the net score records a greater gain than any which our predecessors could rightfully claim. We are going deeper than they ever imagined necessary, and we are discovering that some of the “laws”—for instance those of Aristotelian logic—which they accepted in their reasoning are better replaced by others—pure conventions—in our attempts to correlate our experiences. As has already been said, Cantor’s revolutionary work gave our present activity its initial impulse. But it was soon discovered—twenty one years before Cantor’s death—that his revolution was either too revolutionary or not revolutionary enough. The latter now appears to be the case.
The first shot in the counter-revolution was fired in 1897 by the Italian mathematician Burali-Forti who produced a flagrant contradiction by reasoning of the type used by Cantor in his theory of infinite sets. This particular paradox was only the first of several, and as it would require lengthy explanations to make it intelligible, we shall state instead Russell’s of 1908.
We have already mentioned Frege, who gave the “class of all classes similar to a given class” definition of the cardinal number of the given class. Frege had spent years trying to put the mathematics of numbers on a sound logical basis. His life work is his Grundgesetze der Arithmetik (The Fundamental Laws of Arithmetic), of which the first volume was published in 1893, the second in 1903. In this work the concept of sets is used. There is also a considerable use of more or less sarcastic invective against previous writers on the foundations of arithmetic for their manifest blunders and manifold stupidities. The second volume closes with the following acknowledgment.
“A scientist can hardly encounter anything more undesirable than to have the foundation collapse just as the work is finished. I was put in this position by a letter from Mr. Bertrand Russell when the work was almost through the press.”
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Russell had sent Frege his ingenious paradox of “the set of all sets which are not members of themselves.” Is this set a member of itself? Either answer can be puzzled out with a little thought to be wrong. Yet Frege had freely used “sets of all sets.”
Many ways were proposed for evading or eliminating the contradictions which began exploding like a barrage in and over the Frege-Dedekind-Cantor theory of the real numbers, continuity, and the infinite. Frege, Cantor, and Dedekind quit the field, beaten and disheartened. Russell proposed his “vicious circle principle” as a remedy: “Whatever involves all of a collection must not be one of the collection”; later he put forth his “axiom of reducibility,” which, as it is now practically abandoned, need not be described. For a time these restoratives were brilliantly effective (except in the opinion of the German mathematicians, who never swallowed them). Gradually, as the critical examination of all mathematical reasoning gained headway, physic was thrown to the dogs and a concerted effort was begun to find out what really ailed the patient in his irrational and real number system before administering further nostrums.
The present effort to understand our difficulties originated in the work of David Hilbert (1862-) of Göttingen in 1899 and in that of L. E. J. Brouwer (1881-) of Amsterdam in 1912. Both of these men and their numerous followers have the common purpose of putting mathematical reasoning on a sound basis, although in several respects their methods and philosophi
es are violently opposed. It seems unlikely that both can be as wholly right as each appears to believe he is.
Hilbert returned to Greece for the beginning of his philosophy of mathematics. Resuming the Pythagorean program of a rigidly and fully stated set of postulates from which a mathematical argument must proceed by strict deductive reasoning, Hilbert made the program of the postulational development of mathematics more precise than it had been with the Greeks, and in 1899 issued the first edition of his classic on the foundations of geometry. One demand which Hilbert made, and which the Greeks do not seem to have thought of, was that the proposed postulates for geometry shall be proved to be self-consistent (free of internal, concealed contradictions). To produce such a proof for geometry it is shown that any contradiction in the geometry developed from the postulates would imply a contradiction in arithmetic. The problem is thus shoved back to proving the consistency of arithmetic, and there it remains today.
Thus we are back once more asking the sphinx to tell us what a number is. Both Dedekind and Frege fled to the infinite—Dedekind with his infinite classes defining irrationals, Frege with his class of all classes similar to a given class defining a cardinal number—to interpret the numbers that puzzled Pythagoras. Hilbert, too, would seek the answer in the infinite which, he believes, is necessary for an understanding of the finite. He is quite emphatic in his belief that Cantorism will ultimately be redeemed from the purgatory in which it now tosses. “This [Gan tor’s theory] seems to me the most admirable fruit of the mathematical mind and indeed one of the highest achievements of man’s intellectual processes.” But he admits that the paradoxes of Burali-Forti, Russell, and others are not resolved. However, his faith surmounts all doubts: “No one shall expel us from the paradise which Cantor has created for us.”
But at this moment of exaltation Brouwer appears with something that looks suspiciously like a flaming sword in his strong right hand. The chase is on: Dedekind, in the role of Adam, and Cantor disguised as Eve at his side, are already eyeing the gate apprehensively under the stern regard of the uncompromising Dutchman. The postulational method for securing freedom from contradiction proposed by Hilbert will, says Brouwer, accomplish its end—produce no contradictions, but “nothing of mathematical value will be attained in this manner; a false theory which is not stopped by a contradiction is none the less false, just as a criminal policy unchecked by a reprimanding court is none the less criminal.”
The root of Brouwer’s objection to the “criminal policy” of his opponents is something new—at least in mathematics. He objects to an unrestricted use of Aristotelian logic, particularly in dealing with infinite sets, and he maintains that such logic is bound to produce contradictions when applied to sets which cannot be definitely constructed in Kronecker’s sense (a rule of procedure must be given whereby the things in the set can be produced). The law of “excluded middle” (a thing must have a certain property or must not have that property, as for example in the assertion that a number is prime or is not prime) is legitimately usable only when applied to finite sets. Aristotle devised his logic as a body of working rules for finite sets, basing his method on human experience of finite sets, and there is no reason whatever for supposing that a logic which is adequate for the finite will continue to produce consistent (not contradictory) results when applied to the infinite. This seems reasonable enough when we recall that the very definition of an infinite set emphasizes that a part of an infinite set may contain precisely as many things as the whole set (as we have illustrated many times), a situation which never happens for a finite set when “part” means some, but not all (as it does in the definition of an infinite set).
Here we have what some consider the root of the trouble in Cantor’s theory of the actual infinite. For the definition of a set (as stated some time back), by which all things having a certain quality are “united” to form a “set” (or “class”), is not suitable as a basis for the theory of sets, in that the definition either is not constructive (in Kronecker’s sense) or assumes a constructibility which no mortal can produce. Brouwer claims that the use of the law of excluded middle in such a situation is at best merely a heuristic guide to propositions which may be true, but which are not necessarily so, even when they have been deduced by a rigid application of Aristotelian logic, and he says that numerous false theories (including Cantor’s) have been erected on this rotten foundation during the past half century.
Such a revolution in the rudiments of mathematical thinking does not go unchallenged. Brouwer’s radical move to the left is speeded by an outraged roar from the reactionary right. “What Weyl and Brouwer are doing [Brouwer is the leader, Weyl his companion in revolt] is mainly following in the steps of Kronecker,” according to Hilbert, the champion of the status quo. “They are trying to establish mathematics by jettisoning everything which does not suit them and setting up an embargo. The effect is to dismember our science and to run the risk of losing a large part of our most valuable possessions. Weyl and Brouwer condemn the general notions of irrational numbers, of functions—even of such functions as occur in the theory of numbers—Cantor’s transfinite numbers, etc., the theorem that an infinite set of positive integers has a least, and even the ’law of excluded middle,’ as for example the assertion: Either there is only a finite number of primes or there are infinitely many. These are examples of [to them] forbidden theorems and modes of reasoning. I believe that impotent as Kronecker was to abolish irrational numbers (Weyl and Brouwer do permit us to retain a torso), no less impotent will their efforts prove today. No! Brouwer’s program is not a revolution, but merely the repetition of a futile coup de main with old methods, but which was then undertaken with greater verve, yet failed utterly. Today the State [mathematics] is thoroughly armed and strengthened through the labors of Frege, Dedekind, and Cantor. The efforts of Brouwer and Weyl are foredoomed to futility.”
To which the other side replies by a shrug of the shoulders and goes ahead with its great and fundamentally new task of reestablishing mathematics (particularly the foundations of analysis) on a firmer basis than any laid down by the men of the past 2500 years from Pythagoras to Weierstrass.
What will mathematics be like a generation hence when—we hope—these difficulties will have been cleared up? Only a prophet or the seventh son of a prophet sticks his head into the noose of prediction. But if there is any continuity at all in the evolution of mathematics—and the majority of dispassionate observers believe that there is—we shall find that the mathematics which is to come will be broader, firmer, and richer in content than that which we or our predecessors have known.
Already the controversies of the past third of a century have added new fields—including totally new logics—to the vast domain of mathematics, and the new is being rapidly consolidated and coordinated with the old. If we may rashly venture a prediction, what is to come will be fresher, younger in every respect, and closer to human thought and human needs—freer of appeal for its justification to extra-human “existences”—than what is now being vigorously refashioned. The spirit of mathematics is eternal youth. As Cantor said, “The essence of mathematics resides in its freedom”; the present “revolution” is but another assertion of that freedom.
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Baffled and beaten back she works on still,
Weary and sick of soul she works the more,
Sustained by her indomitable will:
The hands shall fashion and the brain shall pore
And all her sorrow shall be turned to labour,
Till death the friend-foe piercing with his sabre
That mighty heart of hearts ends bitter war.
—JAMES THOMSON.
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I. Quoted from R. E. Moritz’ Memorabilia Mathematica, 1914. The original source is not accessible to me.
II. L. Couturat, Del ‘infini mathématique, Paris, 1896, p.49. With the caution that much of this work is now hopelessly out of date, it can be recommended for its clarity to the g
eneral reader. An account of the elements of Cantorism by a leading Polish expert which is within the comprehension of anyone with a grade-school education and a taste for abstract reasoning is the Leçons sur les nombres transfinis, by Waclaw Sierpinski, Paris, 1928. The preface by Borel supplies the necessary danger signal. The above extract from Couturat is of some historical interest in connection with Hilbert’s program. It anticipates by thirty years Hilbert’s statement of his formalist creed.
ABOUT THE AUTHOR
ERIC TEMPLE BELL was born in 1883 in Aberdeen, Scotland. His early education was obtained in England. Coming to the United States in 1902, he entered Stanford University and took his A.B. degree in 1904. In 1908 he was teaching fellow at the University of Washington, where he took his A.M. degree in 1909. In 1911 he entered Columbia University, where he took his Ph.D. degree in 1912. He returned to the University of Washington as instructor in mathematics and became full professor in 1921. During the summers of 1924-28 he taught at the University of Chicago, and in 1926 (first half) at Harvard University, when he was appointed Professor of Mathematics at the California Institute of Technology.
Dr. Bell was a former President of the Mathematical Association of America, a former Vice President of the American Mathematical Society and of the American Association for the Advancement of Science. He was on the editorial staffs of the Transactions of the American Mathematical Society, the American Journal of Mathematics, and the Journal of the Philosophy of Science. He belonged to The American Mathematical Society, the Mathematical Association of America, the Circolo Matematico di Palermo, the Calcutta Mathematical Society, Sigma Xi, and Phi Beta Kappa, and was a member of the National Academy of Sciences of the United States. He won the Bâcher Prize of the American Mathematical Society for his research work. His twelve published books include The Purple Sapphire (1924), Algebraic Arithmetic (1927), Debunking Science, and Queen of the Sciences (1931), Numerology (1933), and The Search for Truth (1934).
