Incompleteness Read online

  Just as in Cambridge, Wittgenstein’s effect on the logical positivists, particularly Schlick and Waissmann, almost defies explanation. Schlick’s wife recalled her husband leaving the house to go to see Wittgenstein for the first time as if he were setting off on a religious pilgrimage. “He returned in an ecstatic state, saying little, and I felt I should not ask questions.”

  Feigl, in later life, reported, “Schlick adored him and so did Waismann, who, like others of Wittgenstein’s disciples, even came to imitate his gestures and manner of speech. Schlick ascribed to Wittgenstein profound philosophical insights that in my opinion were in fact formulated much more clearly in Schlick’s own early work.”

  The note of asperity in Feigl’s tone is worthy of comment, since “Feigl had always had an unusual ability to get along with everyone,” an ability borne out in his memoirist article, where almost everyone he had ever crossed paths with is described as having only the most amiable personality, the acutest of abilities. Feigl’s obliquely expressed distaste for Wittgenstein appears to be traceable back to Feigl’s “limitless admiration for Carnap,” a systematically precise and conscientious thinker. After banishing Carnap from his presence, Wittgenstein told Feigl, “If he doesn’t smell it, I can’t help him. He just has got no nose!” When Feigl’s admiration for Carnap became clear, he, too, was banished.

  Wittgenstein’s exasperation with his disciples even in his native Vienna, his insistence that although he might sound like a positivist he decidedly was not one, revolves around the meaning of the closing proposition of his Tractatus, numbered simply 7, the severely fulminating (so like a prophet of old): Wovon man nicht sprechen kann, darüber muss man schweigen, or: Of what we cannot speak we must remain silent. Schlick’s Circle interpreted Wittgenstein as saying in his concluding statement, as well as throughout the book, that the misuse of the conditions of language not only (tautologically) ends in nonsense, but also that outside the bounds of the sayable there is nothing at all; whereas for Wittgenstein there really was “that whereof we cannot speak.” The ethical or—what amounts to the same thing for him—the mystical is that whereof we cannot speak. The ethical, the mystical, is both real and inexpressible. He believed that he had explained all that can be said in the Tractatus, but as he told one potential publisher (who ultimately passed) what he had not said in the Tractatus—because it could not be said—was more important than what he had said:

  I once wanted to give a few words in the foreword which now actually are not in it, which, however, I’ll write to you now because they might be a key for you: I wanted to write that my work consists of two parts: of the one which is here, and of everything I have not written. And precisely this second part is the important one. For the Ethical is delimited from within, as it were, by my book; and I’m convinced that strictly speaking it can ONLY be delimited in this way. In brief, I think: All of that which many are babbling today, I have defined in my book by remaining silent about it.

  He took himself to have demonstrated how little one has actually said after one has finished saying all that can be said.

  The question is whether the requisite silence, imposed in proposition 7, hides nothing at all or rather all of the most important things. The positivists certainly interpreted Wittgenstein to be saying the former, which is almost certainly one of the reasons why he dismissed them as not understanding him in the least.

  Ironically, the Vienna Circle, united by their core distaste for mystery, were embracing a thinker committed to mystery, at least in so far as questions of ethics, aesthetics, metaphysics, the meaning of life—all the matters they had banished from the realm of reasonable consideration—were concerned. The “unsayable” is, for Wittgenstein, as the “unknowable” was for the traditional empiricists, a measure of our limits. By taking the measure of all that we can say, delimiting it, in his words from within, he is taking the measure of all that we can’t say, indicating it without expressing it, since expression is in principle impossible.

  Of course, insofar as the sayable was concerned, he truly was propounding a doctrine compatible with positivism, banishing mystery: The meaning of a nontautologous proposition is its method of verification; and so far as mathematical truth goes, Wittgenstein did present a view compatible with the positivists’ own, dissolving the seeming mystery of its aprioricity and certainty into the rules of syntax.

  Though Wittgenstein raged at the positivists’ insistence on fitting him to the procrustean bed of their precision, they mostly responded with adoration, at least in the early days, while Gödel was still attending the Thursday-evening sessions. Olga Taussky-Todd, a mathematician who was Gödel’s age and who spent some time in the Vienna Circle, writes, “Wittgenstein was the idol of this group. I can testify to this. An argument could be settled by citing his Tractatus.” The visiting A. J. Ayer, who would make good use of his three-months stay in Vienna, wrote back to England, to his friend Isaiah Berlin, in February of 1933, “Wittgenstein is a deity to them all.” Bertrand Russell, whom they also respected as an empiricist in good standing, “was merely a forerunner to Christ (Wittgenstein).”

  Even the most sober-minded of the positivists, Rudolf Carnap, admits, in his autobiographical notes in the Schilpp volume in his honor, to a measure of near-religious awe:

  When he started to formulate his view on some specific philosophical problem we often felt the internal struggle that occurred to him at that very moment, a struggle by which he tried to penetrate from darkness to light under an intense and painful strain, which was even visible on his most expressive face. When finally, sometimes after a prolonged arduous effort, his answer came forth, his statement stood before us like a newly created piece of art or a divine inspiration. . . . The impression he made on us was as if insight came to him as through a divine inspiration, so that we could not help feeling that any sober rational comment or analysis of it would be a profanation.

  It was into this oddly Wittgenstein-enchanted Circle (all the odder for being a circle of positivists, the sworn enemies of cognitive bewitchment) that Kurt Gödel was to enter as a student, a reticent observer quietly taking in the opinions around him . . . and drawing his own conclusions.

  Gödel in the Vienna Circle: The Silent Dissenter

  Regardless of his profound, private disagreements with the positivists, Gödel’s association with the Circle led him into the most gregarious few years of his introverted life. He was meeting on a regular basis, not just on alternate Thursday evenings in the bare-bones room where the full Vienna Circle convened but also at late-night sessions in cafés in the garrulous city, men—and the occasional woman—who shared his interest in, if not his intuitions on, foundational issues.

  The ever-amiable Feigl reports:

  On the personal side, I should mention that Gödel, together with another student member of the Circle, Marcel Natkin (originally from Lodz, Poland) and myself became close friends. We met frequently for walks through the parks of Vienna, and of course, in cafés had endless discussions about logical, mathematical, and epistemological and philosophy-of-science issues—sometimes deep into the hours of the night.

  Karl Menger, in a course on dimension theory he was teaching, had a student by the name of Kurt Gödel, “a slim, unusually quiet young man. I do not recall speaking with him at that time.” It was through the regular meetings of the Circle that Menger began the acquaintanceship with Gödel that, though tried, would persist until the end of Gödel’s life.

  Almost all who were present at those meetings described Gödel in similar terms, as clearly brilliant (though one wonders if this clarity of brilliance emerged in hindsight, after the implications of the announcement of 1930 had fully penetrated) but always quiet, holding his own counsel. Gödel was, according to Feigl, “a very unassuming, diligent worker, but his was clearly the mind of a genius of the very first order.” “I never heard Gödel speak in these meetings or participate in the discussions; but he evinced interest by slight motions in the head indicatin
g agreement, skepticism, or disagreement,” said Menger, who also reports:

  After one session in which Schlick, Hahn, Neurath and Waismann had talked about language, but in which neither Gödel nor I had spoken a word, I said on the way home: “Today we have once again out-Wittgensteined these Wittgensteinians: we kept silent.” “The more I think about language,” Gödel replied, “the more it amazes me that people ever understand each other.”

  What did Kurt Gödel think of these “Wittgensteinians?” We know, of course, even though they did not, that he disagreed profoundly with the logical positivists, most specifically on their interpretation of mathematical truth, but far more generally as well. A man whose soul had been blasted by the Platonic vision of truth would not be sympathetic to denunciations of metaphysics. He would not accept a theory of meaning that branded as “meaningless” all descriptive statements that are in principle not empirically verifiable. The essence of mathematical Platonism is the claim that mathematics, though not empirical, is nonetheless descriptive. Gödel was a backbencher among the positivists. Though he shared their commitment to precision, as well as their interest in the philosophical relevance of the logical advances of Frege, Russell, and Whitehead, he could not have been more at odds with their metaconvictions. He told Hao Wang many years later (21 November 1971) that the positivists were fundamentally in error in thinking that all meaningful thought could be reduced to sense perceptions:

  Some reductionism is correct, [but one should] reduce to (other) concepts and truths, not to sense perceptions. . . . Platonic ideas are what things are to be reduced to.

  In the unsent letter to the sociologist Grandjean, after his opening volley in which he pounced on the sociologist’s ingratiating suggestion that his work had been “a facet of the intellectual atmosphere of the early twentieth century,” he went on immediately to say:

  It is true that my interest in the foundations of mathematics was aroused by the “Vienna Circle,” but the philosophical consequences of my results, as well as the heuristic principles leading to them, are anything but positivistic or empiricistic. See what I say in Hao Wang’s recent book “From Mathematics to Philosophy” in the passages cited in the Preface. See also my paper “What is Cantor’s Continuum Problem?” in “Philosophy of Mathematics” edited by Benacerraf and Putnam in 1964; in particular pp. 262–265 and pp. 270–272.18

  I was a conceptual and mathematical realist since about 1925. I never held the view that mathematics is syntax of language. Rather this view, understood in any reasonable sense, can be disproved by my results.

  There we have the matter, stated in an unposted letter. Gödel had become a mathematical realist in 1925, had attended the Vienna Circle’s meetings between 1926 and 1928, and by 1928 had begun to work on the proof for the first incompleteness theorem which he interpreted as disproving a central tenet of the Vienna Circle, the very tenet that had caused them to append “logical” to the Machian viewpoint of positivism. He had used mathematical logic, beloved of the positivists, to wreak havoc on the positivist antimetaphysical position. Yet here he was, in 1974, still having to explain, in missives that he never mailed, that he was not a positivist, that the intended import of his celebrated theorems had been, in fact, to prove the positivists wrong. The positivists had endorsed the Sophist’s man-measurement of truth. Gödel had sought to vindicate the Sophist’s implacable antagonist, Plato.

  Gödel, unlike his friend Einstein, did not have a well-developed sense of the ironic, which is, all things considered, a shame.

  Gödel and Wittgenstein

  It is hard to imagine two more disparate personalities. Wittgenstein and Gödel were both geniuses, both tortured geniuses, in fact. But how they presented that tormented genius to the world could not have been more starkly different.

  Wittgenstein had decided views on the nature and duties and privileges of genius. He had once spoken to Russell about Beethoven:

  [A] friend described going to Beethoven’s door and hearing him “cursing, howling, and singing” over his new fugue; after a whole hour Beethoven at last came to the door, looking as if he had been fighting the devil, and having eaten nothing for 36 hours because his cook and parlour-maid had been driven away from his rage. That’s the sort of man to be.

  And Wittgenstein was that sort of man, acting out the high drama of genius, so that Russell, when he was still enthralled, described him to Lady Ottoline as “. . . perhaps the most perfect example I have ever known of genius as traditionally conceived, passionate, profound, intense, and dominating.”

  He was the sort of genius to attract disciples so fanatical they took to wearing their shirts unbuttoned at the top as he did and aping his tics and mannerisms, such as clapping their hands to their foreheads when struck by a philosophical insight or its lack. They may have disagreed with each other on the correct interpretation of Wittgenstein, but agreed that the correct interpretation, if only it could be attained, would almost of necessity be true. (This conviction persists still in significant pockets of Anglo-American philosophy.) Though he more pronounced than argued, still the pronouncements present themselves, singly and in configuration with one another, with the logical austerity craved by rigor-seeking thinkers.

  This austerity attached to his person as well, as if the purity of formal logic had been embodied in the man, its standards of absolute truth imposed on human behavior. An anecdote chosen almost at random (there are so many), this one told by Fania Pascal who had known him in Cambridge in the 1930s, bears this out:

  I had my tonsils out and was in the Evelyn Nursing Home feeling sorry for myself. Wittgenstein called. I croaked: I feel just like a dog that has been run over. He was disgusted: “You don’t know what a dog that has been run over feels like.”

  Just as standards of mathematics have made nonmathematicians feel pitifully inadequate in comparison, so the mathematically severe behavioral standards Wittgenstein seemed to demand could easily make others feel fraudulent. Wittgenstein seemed, demonstrably, the real thing. His was genius turning itself inside out, with all the Sturm und Drang of supremely heightened consciousness exposed, and it was a sight before which a thinker, even a logical positivist, might be expected to swoon.

  In contrast to Wittgenstein’s dramatic display of genius—his Cambridge students recalled that you could see the suffering of his thinking—we have Gödel, indicating with slight motions of his head when he agrees, disagrees, is skeptical. His hermetically sealed genius allowing next to nothing of the Sturm und Drang of its heightened consciousness to show, Gödel breathed not a word of his fundamental dissent from the beliefs of the Vienna Circle until he had a rigorous mathematical proof to do all his talking for him, until he had mathematical theorems prolix enough to speak out his metaphysical convictions.

  It is intriguing to try to imagine the young Gödel, observing his Wittgenstein-bedazzled elders, perhaps more than a little aggrieved or disapproving—not only of the views but also of the very style of the genius so at odds with his own, genius making such a fuss about itself, demanding that others, too, participate in the fuss. One wonders (almost blasphemously) whether there might not have been a bit of human emotion spurring the silent dissenter on to find a conclusive refutation, to confront the “divine inspiration” of the philosopher with a higher authority: mathematics.

  Such a motivation, however collateral, is conjectural, of course, given the opacity of Gödel’s inner life. And we have Gödel’s word that Wittgenstein had not influenced his work in mathematical logic at all. In one of the two drafts of the unsent reply to the sociologist Grandjean’s questionnaire, in answer to the question, “Are there any influences to which you attribute special significance in the development of your philosophy?”, after Gödel had credited Professor Gomperz,19 he gratuitously remarked on whom, specifically, had not influenced his work: “Wittg[enstein]’s views on the phil[osophy] of math had no inf[luence] on my work nor did the interest of the Vienna Circle in that subj[ect] start with Wittgenst[ei
n] (but rather went back to Prof. Hans Hahn).”

  Of course influence, in a positive sense, is quite different from the sort of murkier incentive I am speculating about. And his adding the gratuitous disclaimer concerning Wittgenstein is telling, especially in so retentive a personality, of at least a retrospective resentment. The influence of the charismatic philosopher on the members of the Vienna Circle may have irked him, amused him (more doubtful), even helped to inspire him in the direction of his proof: we cannot really know. But the older Gödel did leave behind, in the written record, some few scant hints of exasperated pique toward Wittgenstein.

  For example, in 1971, mathematician Kenneth Blackwell pointed out to Gödel that there was a passage in Russell’s Autobiography in which Gödel is mentioned, with various inaccuracies (including his Jewish origins), as well as a rather facile and sarcastic reference to Gödel’s Platonism:

  Gödel turned out to be an unadulterated Platonist, and apparently believed that an eternal “not” was laid up in heaven, where virtuous logicians might hope to meet it hereafter.

  Gödel drafted a letter, which of course still lies unposted in the Nachlass, responding point by point to Russell’s inaccuracies on the topic of Gödel [including his alleged Jewishness: “I have to say first for the sake of truth that I am not a Jew (even though I don’t think this question is of any importance)”] and ending:

  Concerning my “unadulterated” Platonism, it is no more “unadulterated” than Russell’s own in 1921, when in the Introduction [to Mathematical Philosophy, first published in 1919, p. 169] he said “[Logic is concerned with the real world just as truly as zoology, though with its more abstract and general features].” At that time evidently Russell had met the “not” even in this world, but later on under the influence of Wittgenstein he chose to overlook it.

  Coming from Gödel, these are pointed words, which is of course why they still languish in a folder in Firestone Library.