# Was Cantor Surprised?

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1 Was Cantor Surprised? Fernando Q. Gouvêa Abstract. We look at the circumstances and context of Cantor s famous remark, I see it, but I don t believe it. We argue that, rather than denoting astonishment at his result, the remark pointed to Cantor s worry about the correctness of his proof. Mathematicians love to tell each other stories. We tell them to our students too, and they eventually pass them on. One of our favorites, and one that I heard as an undergraduate, is the story that Cantor was so surprised when he discovered one of his theorems that he said I see it, but I don t believe it! The suggestion was that sometimes we might have a proof, and therefore know that something is true, but nevertheless still find it hard to believe. That sentence can be found in Cantor s extended correspondence with Dedekind about ideas that he was just beginning to explore. This article argues that what Cantor meant to convey was not really surprise, or at least not the kind of surprise that is usually suggested. Rather, he was expressing a very different, if equally familiar, emotion. In order to make this clear, we will look at Cantor s sentence in the context of the correspondence as a whole. Exercises in myth-busting are often unsuccessful. As Joel Brouwer says in his poem A Library in Alexandria,... And so history gets written to prove the legend is ridiculous. But soon the legend replaces the history because the legend is more interesting. Our only hope, then, lies in arguing not only that the standard story is false, but also that the real story is more interesting. 1. THE SURPRISE. The result that supposedly surprised Cantor was the fact that sets of different dimension could have the same cardinality. Specifically, Cantor showed (of course, not yet using this language) that there was a bijection between the interval I = [0, 1] and the n-fold product I n = I I I. There is no doubt, of course, that this result is surprising, i.e., that it is counterintuitive. In fact Cantor said so explicitly, pointing out that he had expected something different. But the story has grown in the telling, and in particular Cantor s phrase about seeing but not believing has been read as expressing what we usually mean when we see something happen and exclaim Unbelievable! What we mean is not that we actually do not believe, but that we find what we know has happened to be hard to believe because it is so unusual, unexpected, surprising. In other words, the idea is that Cantor felt that the result was hard to believe even though he had a proof. His phrase has been read as suggesting that mathematical proof may engender rational certainty while still not creating intuitive certainty. The story was then co-opted to demonstrate that mathematicians often discover things that they did not expect or prove things that they did not actually want to prove. For example, here is William Byers in How Mathematicians Think: doi: /amer.math.monthly c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118

3 Georg Cantor was born in St. Petersburg, Russia, on March 3, He died in Halle, Germany, on January 6, He studied at the University of Berlin, where the mathematics department, led by Karl Weierstrass, Ernst Eduard Kummer, and Leopold Kronecker, might well have been the best in the world. His doctoral thesis was on the number theory of quadratic forms. In 1869, Cantor moved to the University of Halle and shifted his interests to the study of the convergence of trigonometric series. Very much under Weierstrass s influence, he too introduced a way to construct the real numbers, using what he called fundamental series. (We call them Cauchy sequences. ) His paper on this construction also appeared in Cantor s lifelong dream seems to have been to return to Berlin as a professor, but it never happened. He rose through the ranks in Halle, becoming a full professor in 1879 and staying there until his death. See [13] for a short account of Cantor s life. The standard account of Cantor s mathematical work is [4]. Cantor is best known, of course, for the creation of set theory, and in particular for his theory of transfinite cardinals and ordinals. When our story begins, this was mostly still in the future. In fact, the birth of several of these ideas can be observed in the correspondence with Dedekind. This correspondence was first published in [14]; we quote it from the English translation by William Ewald in [8, pp ]. 3. ALLOW ME TO PUT A QUESTION TO YOU. Dedekind and Cantor met in Switzerland when they were both on vacation there. Cantor had sent Dedekind a copy of the paper containing his construction of the real numbers. Dedekind responded, of course, by sending Cantor a copy of his booklet. And so begins the story. Cantor was 27 years old and very much a beginner, while Dedekind was 41 and at the height of his powers; this accounts for the tone of deference in Cantor s side of the correspondence. Cantor s first letter acknowledged receipt of [7] and says that my conception [of the real numbers] agrees entirely with yours, the only difference being in the actual construction. But on November 29, 1873, Cantor moves on to new ideas: Allow me to put a question to you. It has a certain theoretical interest for me, but I cannot answer it myself; perhaps you can, and would be so good as to write me about it. It is as follows. Take the totality of all positive whole-numbered individuals n and denote it by (n). And imagine, say, the totality of all positive real numerical quantities x and designate it by (x). The question is simply, Can (n) be correlated to (x) in such a way that to each individual of the one totality there corresponds one and only one of the other? At first glance one says to oneself no, it is not possible, for (n) consists of discrete parts while (x) forms a continuum. But nothing is gained by this objection, and although I incline to the view that (n) and (x) permit no one-to-one correlation, I cannot find the explanation which I seek; perhaps it is very easy. In the next few lines, Cantor points out that the question is not as dumb as it looks, since the totality ( ) p q of all positive rational numbers can be put in one-to-one correspondence with the integers. We do not have Dedekind s side of the correspondence, but his notes indicate that he responded indicating that (1) he could not answer the question either, (2) he could show that the set of all algebraic numbers is countable, and (3) that he didn t think the question was all that interesting. Cantor responded on December 2: 200 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118

6 that interval, but rather all with the exception of certain y. However, it assumes each of the corresponding values y only once, and that seems to me to be the essential point. For now I can bring y into a one-to-one relation with another quantity t that assumes all the values 0 and 1. I am delighted that you have found no other objections. I shall shortly write to you at greater length about this matter. This is a remarkable response. It suggests that Cantor was very confident that his result was true. This confidence was due to the fact that Cantor was already thinking in terms of what later became known as cardinality. Specifically, he expects that the existence of a one-to-one mapping from one set A to another set B implies that the size of A is in some sense less than or equal to that of B. Cantor s proof shows that the points of the square can be put into bijection with a subset of the interval. Since the interval can clearly be put into bijection with a subset of the square, this strongly suggests that both sets of points are the same size, or, as Cantor would have said it, have the same power. All we need is a proof that the powers are linearly ordered in a way that is compatible with inclusions. That the cardinals are indeed ordered in this way is known today as the Schroeder- Bernstein theorem. The postcard shows that Cantor already knew that the Schroeder- Bernstein theorem should be true. In fact, he seems to implicitly promise a proof of that very theorem. He was not able to find such a proof, however, then or (as far as I know) ever. His fuller response, sent two days later on June 25, contained instead a completely different, and much more complicated, proof of the original theorem. I sent you a postcard the day before yesterday, in which I acknowledged the gap you discovered in my proof, and at the same time remarked that I am able to fill it. But I cannot repress a certain regret that the subject demands more complicated treatment. However, this probably lies in the nature of the subject, and I must console myself; perhaps it will later turn out that the missing portion of that proof can be settled more simply than is at present in my power. But since I am at the moment concerned above all to persuade you of the correctness of my theorem... I allow myself to present another proof of it, which I found even earlier than the other. Notice that what Cantor is trying to do here is to convince Dedekind that his theorem is true by presenting him a correct proof. 2 There is no indication that Cantor had any doubts about the correctness of the result itself. In fact, as we will see, he says so himself. Let s give a brief account of Cantor s proof; to avoid circumlocutions, we will express most of it in modern terms. Cantor began by noting that every real number x between 0 and 1 can be expressed as a continued fraction x = a b + 1 c + 2 Cantor claimed he had found this proof before the other. I find this hard to believe. In fact, the proof looks very much like the result of trying to fix the problem in the first proof by replacing (nonunique) decimal expansions with (unique) continued fraction expansions. March 2011] WAS CANTOR SURPRISED? 203

7 where the partial quotients a, b, c,..., etc. are all positive integers. This representation is infinite if and only if x is irrational, and in that case the representation is unique. So one can argue just as before, interleaving the two continued fractions for x and y, to establish a bijection between the set of pairs (x, y) such that both x and y are irrational and the set of irrational points in [0, 1]. The result is a bijection because the inverse mapping, splitting out two continued fraction expansions from a given one, will certainly produce two infinite expansions. That being done, it remains to be shown that the set of irrational numbers between 0 and 1 can be put into bijection with the interval [0, 1]. This is the hard part of the proof. Cantor proceeded as follows. First he chose an enumeration of the rationals {r k } and an increasing sequence of irrationals {η k } in [0, 1] converging to 1. He then looked at the bijection from [0, 1] to [0, 1] that is the identity on [0, 1] except for mapping r k η k, η k r k. This gives a bijection between irrationals in [0, 1] and [0, 1] minus the sequence {η k } and reduces the problem to proving that [0, 1] can be put into bijection with [0, 1] {η k }. At this point Cantor claims that it is now enough to successively apply the following theorem: A number y that can assume all the values of the interval (0... 1) with the solitary exception of the value 0 can be correlated one-to-one with a number x that takes on all values of the interval (0... 1) without exception. In other words, he claimed that there was a bijection between the half-open interval (0, 1] and the closed interval [0, 1], and that successive application of this fact would finish the proof. In the actual application he would need the intervals to be open on the right, so, as we will see, he chose a bijection that mapped 1 to itself. Cantor did not say exactly what kind of successive application he had in mind, but what he says in a later letter suggests it was this: we have the interval [0, 1] minus the sequence of the η k. We want to put back in the η k, one at a time. So we leave the interval [0, η 1 ) alone, and look at (η 1, η 2 ). Applying the lemma, we construct a bijection between that and [η 1, η 2 ). Then we do the same for (η 2, η 3 ) and so on. Putting together these bijections produces the bijection we want. Finally, it remained to prove the lemma, that is, to construct the bijection from [0, 1] to (0, 1]. Modern mathematicians would probably do this by choosing a sequence x n in (0, 1), mapping 0 to x 1 and then every x n to x n+1. This Hilbert hotel idea was still some time in the future, however, even for Cantor. Instead, Cantor chose a bijection that could be represented visually, and simply drew its graph. He asked Dedekind to consider the following peculiar curve, which we have redrawn in Figure 1 based on the photograph reproduced in [4, p. 63]. Such a picture requires some explanation, and Cantor provided it. The domain has been divided by a geometric progression, so b = 1/2, b 1 = 3/4, and so on; a = (0, 1/2), a = (1/2, 3/4), etc. The point C is (1, 1). The points d = (1/2, 1/2), d = (3/4, 3/4), etc. give the corresponding subdivision of the main diagonal. The curve consists of infinitely many parallel line segments ab, a b, a b and of the point c. The endpoints b, b, b,... are not regarded as belonging to the curve. The stipulation that the segments are open at their lower endpoints means that 0 is not in the image. This proves the lemma, and therefore the proof is finished. 204 c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118

8 a a d a d C a d b b b a d b O b b 1 b 2 b 3 b 4 P Figure 1. Cantor s function from [0, 1] to (0, 1]. Cantor did not even add that last comment. As soon as he had explained his curve, he moved on to make extensive comments on the theorem and its implications. He turns on its head the objection that various mathematicians made to his question, namely that it was obvious from geometric considerations that the number of variables is invariant: For several years I have followed with interest the efforts that have been made, building on Gauss, Riemann, Helmholtz, and others, towards the clarification of all questions concerning the ultimate foundations of geometry. It struck me that all the important investigations in this field proceed from an unproven presupposition which does not appear to me self-evident, but rather to need a justification. I mean the presupposition that a ρ-fold extended continuous manifold needs ρ independent real coordinates for the determination of its elements, and that for a given manifold this number of coordinates can neither be increased nor decreased. This presupposition became my view as well, and I was almost convinced of its correctness. The only difference between my standpoint and all the others was that I regarded that presupposition as a theorem which stood in great need of a proof; and I refined my standpoint into a question that I presented to several colleagues, in particular at the Gauss Jubilee in Göttingen. The question was the following: Can a continuous structure of ρ dimensions, where ρ > 1, be related oneto-one with a continuous structure of one dimension so that to each point of the former there corresponds one and only one point of the latter? Most of those to whom I presented this question were extremely puzzled that I should ask it, for it is quite self-evident that the determination of a point in an extension of ρ dimensions always needs ρ independent coordinates. But whoever penetrated the sense of the question had to acknowledge that a proof was needed to show why the question should be answered with the self-evident no. As I say, I myself was one of those who held it for the most likely that the March 2011] WAS CANTOR SURPRISED? 205

11 Against this, I declare (despite your theorem, or rather in consequence of reflections that it stimulated) my conviction or my faith (I have not yet had time even to make an attempt at a proof) that the dimension-number of a continuous manifold remains its first and most important invariant, and I must defend all previous writers on the subject... For all authors have clearly made the tacit, completely natural presupposition that in a new determination of the points of a continuous manifold by new coordinates, these coordinates should also (in general) be continuous functions of the old coordinates... Dedekind pointed out that, in order to establish his correspondence, Cantor had been compelled to admit a frightful, dizzying discontinuity in the correspondence, which dissolves everything to atoms, so that every continuously connected part of one domain appears in its image as thoroughly decomposed and discontinuous. He then set out a new conjecture that spawned a whole research program:... for the time being I believe the following theorem: If it is possible to establish a reciprocal, one-to-one, and complete correspondence between the points of a continuous manifold A of a dimensions and the points of a continuous manifold B of b dimensions, then this correspondence itself, if a and b are unequal, is necessarily utterly discontinuous. In his next letter, Cantor claimed that this was indeed his point: where Riemann and others had casually spoken of a space that requires n coordinates as if that number was known to be invariant, he felt that this invariance required proof. Far from wishing to turn my result against the article of faith of the theory of manifolds, I rather wish to use it to secure its theorems, he wrote. The required theorem turned out to be true, indeed, but proving it took much longer than either Cantor or Dedekind could have guessed: it was finally proved by Brouwer in The long and convoluted story of that proof can be found in [3], [11], and [12]. Finally, one should point out that it was only some three months later that Cantor found what most modern mathematicians consider the obvious way to prove that there is a bijection between the interval minus a countable set and the whole interval. In a letter dated October 23, 1877, he took an enumeration φ ν of the rationals and let η ν = 2/2 ν. Then he constructed a map from [0, 1] sending η ν to η 2ν 1, φ ν to η 2ν, and every other point h to itself, thus getting a bijection between [0, 1] and the irrational numbers between 0 and MATHEMATICS AS CONVERSATION. Is the real story more interesting than the story of Cantor s surprise? Perhaps it is, since it highlights the social dynamic that underlies mathematical work. It does not render the theorem any less surprising, but shifts the focus from the result itself to its proof. The record of the extended mathematical conversation between Cantor and Dedekind reminds us of the importance of such interaction in the development of mathematics. A mathematical proof is, after all, a kind of challenge thrown at an idealized opponent, a skeptical adversary that is reluctant to be convinced. Often, this adversary is actually a colleague or collaborator, the first reader and first critic. A proof is not a proof until some reader, preferably a competent one, says it is. Until then we may see, but we should not believe. REFERENCES 1. K.-R. Biermann, Dedekind, in Dictionary of Scientific Biography, C. C. Gillispie, ed., Scribners, New York, c THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118

12 2. W. Byers, How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics, Princeton University Press, Princeton, J. W. Dauben, The invariance of dimension: Problems in the early development of set theory and topology, Historia Math. 2 (1975) doi: / (75)90066-x 4., Georg Cantor: His Mathematics and Philosophy of the Infinite, Princeton University Press, Princeton, R. Dedekind, Essays in the Theory of Numbers (trans. W. W. Beman), Dover, Mineola, NY, , Gesammelte Mathematische Werke, R. Fricke, E. Noether, and O. Ore, eds., Chelsea, New York, , Stetigkeit und Irrationalzahlen, 1872, in Gesammelte Mathematische Werke, vol. 3, item L, R. Fricke, E. Noether, and O. Ore, eds., Chelsea, New York, W. Ewald, From Kant to Hilbert: A Source Book in the Foundations of Mathematics, Oxford University Press, Oxford, R. Gray, Georg Cantor and transcendental numbers, Amer. Math. Monthly 101 (1994) doi: / J. Hadamard, An Essay on the Psychology of Invention in the Mathematical Field, Princeton University Press, Princeton, D. M. Johnson, The problem of the invariance of dimension in the growth of modern topology I, Arch. Hist. Exact Sci. 20 (1979) doi: /bf , The problem of the invariance of dimension in the growth of modern topology II, Arch. Hist. Exact Sci. 25 (1981) doi: /bf H. Meschkowski. Cantor, in Dictionary of Scientific Biography, C. C. Gillispie, ed., Scribners, New York, E. Noether und J. Cavaillès, Briefwechsel Cantor Dedekind, Hermann, Paris, FERNANDO Q. GOUVÊA is Carter Professor of Mathematics at Colby College in Waterville, ME. He is the author, with William P. Berlinghoff, of Math through the Ages: A Gentle History for Teachers and Others. This article was born when he was writing the chapter in that book called Beyond Counting. So it s Bill s fault. Department of Mathematics and Statistics, Colby College, Waterville, ME March 2011] WAS CANTOR SURPRISED? 209

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