Was Cantor Surprised?

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:10.4169/amer.math.monthly.118.03.198 198 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118 Cantor himself initially believed that a higher-dimensional figure would have a larger cardinality than a lower-dimensional one. Even after he had found the argument that demonstrated that cardinality did not respect dimensions: that one-, two-, three-, even n-dimensional sets all had the same cardinality, he said, “I see it, but I don’t believe it.” [2, p. 179] Did Cantor’s comment suggest that he found it hard to believe his own theorem even after he had proved it? Byers was by no means the first to say so. Many mathematicians thinking about the experience of doing mathematics have found Cantor’s phrase useful. In his preface to the original (1937) publication of the Cantor-Dedekind correspondence, J. Cavaillès already called attention to the phrase: . . . these astonishing discoveries—astonishing first of all to the author himself: “I see it but I don’t believe it at all,”1 he writes in 1877 to Dedekind about one of them—, these radically new notions . . . [14, p. 3, my translation] Notice, however, that Cavaillès is still focused on the description of the result as “surprising” rather than on the issue of Cantor’s psychology. It was probably Jacques Hadamard who first connected the phrase to the question of how mathematicians think, and so in particular to what Cantor was thinking. In his famous Essay on the Psychology of Invention in the Mathematical Field, first published in 1945 (only eight years after [14]), Hadamard is arguing about Newton’s ideas: . . . if, strictly speaking, there could remain a doubt as to Newton’s example, others are completely beyond doubt. For instance, it is certain that Georg Cantor could not have foreseen a result of which he himself says “I see it, but I do not believe it.” [10, pp. 61–62]. Alas, when it comes to history, few things are “certain.” 2. THE MAIN CHARACTERS. Our story plays out in the correspondence between Richard Dedekind and Georg Cantor during the 1870s. It will be important to know something about each of them. Richard Dedekind was born in Brunswick on October 6, 1831, and died in the same town, now part of Germany, on February 12, 1916. He studied at the University of Göttingen, where he was a contemporary and friend of Bernhard Riemann and where he heard Gauss lecture shortly before the old man’s death. After Gauss died, Lejeune Dirichlet came to Göttingen and became Dedekind’s friend and mentor. Dedekind was a very creative mathematician, but he was not particularly ambitious. He taught in Göttingen and in Zurich for a while, but in 1862 he returned to his home town. There he taught at the local Polytechnikum, a provincial technical university. He lived with his brother and sister and seemed uninterested in offers to move to more prestigious institutions. See [1] for more on Dedekind’s life and work. Our story will begin in 1872. The first version of Dedekind’s ideal theory had appeared as Supplement X to Dirichlet’s Lectures in Number Theory (based on actual lectures by Dirichlet but entirely written by Dedekind). Also just published was one of his best known works, “Stetigkeit und Irrationalzahlen” (“Continuity and Irrational Numbers”; see [7]; an English translation is included in [5]). This was his account of how to construct the real numbers as “cuts.” He had worked out the idea in 1858, but published it only 14 years later. 1Cavaillès misquotes Cantor’s phrase as “je le vois mais je ne le crois point.” March 2011] WAS CANTOR SURPRISED? 199 Georg Cantor was born in St. Petersburg, Russia, on March 3, 1845. He died in Halle, Germany, on January 6, 1918. 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 1872. 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. 843–878]. 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 I was exceptionally pleased to receive your answer to my last letter. I put my question to you because I had wondered about it already several years ago, and was never certain whether the difficulty I found was subjective or whether it was inherent in the subject. Since you write that you too are unable to answer it, I may assume the latter.—In addition, I should like to add that I have never seriously occupied myself with it, because it has no special practical interest for me. And I entirely agree with you when you say that for this reason it does not deserve much effort. But it would be good if it could be answered; e.g., if it could be answered with no, then one would have a new proof of Liouville’s theorem that there are transcendental numbers. Cantor first concedes that perhaps it is not that interesting, then immediately points out an application that was sure to interest Dedekind! In fact, Dedekind’s notes indicate that it worked: “But the opinion I expressed that the first question did not deserve too much effort was conclusively refuted by Cantor’s proof of the existence of transcendental numbers.” [8, p. 848] These two letters are fairly typical of the epistolary relationship between the two men: Cantor is deferential but is continually coming up with new ideas, new questions, new proofs; Dedekind’s role is to judge the value of the ideas and the correctness of the proofs. The very next letter, from December 7, 1873, contains Cantor’s first proof of the uncountability of the real numbers. (It was not the “diagonal” argument; see [4] or [9] for the details.) 4. “THE SAME TRAIN OF THOUGHT . . . ” Cantor seemed to have a good sense for what question should come next. On January 5, 1874, he posed the problem of higher-dimensional sets: As for the question with which I have recently occupied myself, it occurs to me that the same train of thought also leads to the following question: Can a surface (say a square including its boundary) be one-to-one correlated to a line (say a straight line including its endpoints) so that to every point of the surface there corresponds a point of the line, and conversely to every point of the line there corresponds a point of the surface? It still seems to me at the moment that the answer to this question is very difficult—although here too one is so impelled to say no that one would like to hold the proof to be almost superfluous. Cantor’s letters indicate that he had been asking others about this as well, and that most considered the question just plain weird, because it was “obvious” that sets of different dimensions could not be correlated in this way. Dedekind, however, seems to have ignored this question, and the correspondence went on to other issues. On May 18, 1874, Cantor reminded Dedekind of the question, and seems to have received no answer. The next letter in the correspondence is from May, 1877. The correspondence seems to have been reignited by a misunderstanding of what Dedekind meant by “the essence of continuity” in [7]. On June 20, 1877, however, Cantor returns to the question of bijections between sets of different dimensions, and now proposes an answer: . . . I should like to know whether you consider an inference-procedure that I use to be arithmetically rigorous. The problem is to show that surfaces, bodies, indeed even continuous structures of ρ dimensions can be correlated one-to-one with continuous lines, i.e., March 2011] WAS CANTOR SURPRISED? 201 with structures of only one dimension—so that surfaces, bodies, indeed even continuous structures of ρ dimension have the same power as curves. This idea seems to conflict with the one that is especially prevalent among the representatives of modern geometry, who speak of simply infinite, doubly, triply, . . . , ρ-fold infinite structures. (Sometimes you even find the idea that the infinity of points of a surface or a body is obtained as it were by squaring or cubing the infinity of points of a line.) Significantly, Cantor’s formulation of the question had changed. Rather than asking whether there is a bijection, he posed the question of finding a bijection. This is, of course, because he believed he had found one. By this point, then, Cantor knows the right answer. It remains to give a proof that will convince others. He goes on to explain his idea for that proof, working with the ρ-fold product of the unit interval with itself, but for our purposes we can consider only the case ρ = 2. The proof Cantor proposed is essentially this: take a point (x, y) in [0, 1] × [0, 1], and write out the decimal expansions of x and y: (x, y) = (0.abcde . . . , 0.αβγ δ . . . ). Some real numbers have more than one decimal expansion. In that case, we always choose the expansion that ends in an infinite string of 9s. Cantor’s idea is to map (x, y) to the point z ∈ [0, 1] given by z = 0.aαbβcγ dδe . . . Since we can clearly recover x and y from the decimal expansion of z, this gives the desired correspondence. Dedekind immediately noticed that there was a problem. On June 22, 1877 (one cannot fail to be impressed with the speed of the German postal service!), he wrote back pointing out a slight problem “which you will perhaps solve without difficulty.” He had noticed that the function Cantor had defined, while clearly one-to-one, was not onto. (Of course, he did not use those words.) Specifically, he pointed out that such numbers as z = 0.120101010101 . . . did not correspond to any pair (x, y), because the only possible value for x is 0.100000 . . . , which is disallowed by Cantor’s choice of decimal expansion. He was not sure if this was a big problem, adding “I do not know if my objection goes to the essence of your idea, but I did not want to hold it back.” Of course, the problem Dedekind noticed is real. In fact, there are a great many real numbers not in the image, since we can replace the ones that separate the zeros with any sequence of digits. The image of Cantor’s map is considerably smaller than the whole interval. Cantor’s first response was a postcard sent the following day. (Can one envision him reading the letter at the post office and immediately dispatching a postcard back?) He acknowledged the error and suggested a solution: Alas, you are entirely correct in your objection; but happily it concerns only the proof, not the content. For I proved somewhat more than I had realized, in that I bring a system x1, x2, . . . , xρ of unrestricted real variables (that are≥ 0 and≤ 1) into one-to-one relationship with a variable y that does not assume all values of 202 c © THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 118 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 SchroederBernstein theorem. The postcard shows that Cantor already “knew” that the SchroederBernstein 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