Incompleteness Page 8
Russell’s and Whitehead’s proffered challenge to logicians to come up with a less ad hoc solution to block the formation of paradoxical sets was what lured Wittgenstein away from aeronautical engineering. This problem that had stumped the great Lord Russell was obviously something worth thinking about. Wittgenstein went to Cambridge, where Russell was the most prominent philosopher on staff, and immediately made himself known to the distinguished philosopher, mathematician, political activist, and aristocrat.12
At first Russell was a bit wary before the strange intensity of the newcomer: “My ferocious German [sic] came and argued at me after my lecture,” Russell wrote to his current lover, Ottoline Morrell, the aristocratic wife of the Liberal MP Phillip Morrell. During their affair Russell wrote to her on average three times a day, so there is a lot of useful documentation on which to draw from this period of his life. If only adultery regularly yielded such scholarly riches. “He is armour-plated against all assaults of reasoning. It is really rather a waste of time talking with him.” But within a short span of time (while Wittgenstein was still an undergraduate) the “ferocious” convictions of the Austrian had a devastating effect on Russell’s confidence in his own logical powers:
We were both cross from the heat—I showed him a crucial part of what I had been writing. He said it was all wrong, not realizing the difficulties—that he had tried my view and knew it couldn’t work. I couldn’t understand his objection—in fact he was very inarticulate—but I feel in my bones that he must be right, and that he has seen something I have missed. If I could see it too I shouldn’t mind, but, as it is, it is worrying and has rather destroyed the pleasure in my writing—I can only go on with what I see, and yet I feel it is probably all wrong, and that Wittgenstein will think me a dishonest scoundrel for going on with it.
The force of Wittgenstein’s personality and his reforming attitude toward philosophy, the holy severity of the mission of disabusing his contemporaries of their presumptions (which had a great deal to do with his Viennese sense of the decadent exhaustion of old traditions13), transformed Anglo-American philosophy. Like Russell, the Cambridge philosophers and students of philosophy who came to surround Wittgenstein seemed not to have to understand him to know in their “bones that he must be right.” His evident brilliance, oracularly (if inarticulately) dispensed against the backdrop of a fierce and formidably austere personality, made for a powerfully convincing display. Wittgenstein quite often gave way to lamentations that his Cambridge colleagues and students did not understand him.
Partly it was the Viennese aspect in his thinking that eluded them. It was not just in his determination to seize hold of a methodology for sweeping away the decay of the old ways and making the entire field new again that he was a Viennese at heart. Wittgenstein’s tormentedly dramatic way of pursuing his field, the cult of genius that he propagated, was also highly Viennese. He had read in his youth, and always retained a high respect for, the strange Viennese writer Otto Weininger (1880–1903), “a quintessentially Viennese figure” who had argued that the only way for a man to justify his life (for a woman there is no way) is by acquiring and cultivating genius. Weininger had chosen to shoot himself to death in the very house in which Beethoven, the genius he revered above all others, had died. Wittgenstein himself was suicidal for nine years (his three older brothers committed suicide, also a quintessentially Viennese act), until he came to Cambridge and was pronounced a genius by Russell.
Back in Vienna, Wittgenstein, in absentia, was also producing a profound effect. His first published work, Tractatus Logico-Philosophicus, partly written in the trenches of the First World War, had singularly impressed Schlick’s group. As stylistically arresting as its creator, this work achieved in its austere elegance a sort of poetry.14 The traditional tool of the philosopher—the argument—is dispensed with; each assertion is put forth, as Russell once remarked, “as if it were a Czar’s ukase.” The poet’s obscurity of meaning is preserved despite (by means of?) the formal precision of its elaborate numbering system, which hierarchically arranges its assertions: so that, say, proposition 3.411 (In geometry and logic alike a place is a possibility: something can exist in it) is an elaboration of proposition 3.41 (The propositional sign with logical co-ordinates—that is the logical place) which is an elaboration of 3.4 (A proposition determines a place in logical space). The numbering system is borrowed from the mathematician Peano, who had used it in axiomatizing arithmetic, and it is the numbering system that Russell and Whitehead had also employed in Principia Mathematica.
The Cambridge philosopher G. E. Moore suggested the title, modeled on Spinoza’s Tractatus Theologico-Politicus. Bertrand Russell wrote the introduction that finally, after much difficulty, secured the author a publisher. Wittgenstein detested the introduction, especially after it was translated into German: “All the refinement of your English style,” he wrote Russell, “was, obviously, lost in the translation and what remained was superficiality and misunderstanding.” Russell’s and Wittgenstein’s former intimacy cooled considerably over the following years. “He had the pride of Lucifer,” was one of Russell’s later summations of Wittgenstein’s character.
It was Kurt Reidemeister, a geometer associated with the Circle, who in 1924 or 1925, at Schlick’s and Hahn’s request, studied the Tractatus and suggested that the group read it together.
And so the positivists began a joint study of the Tractatus, proposition by proposition, their Thursday-evening meetings now dedicated to Wittgenstein. They read it through not once, but twice, the endeavor taking the better part of a year. (These weekly readings of a portion of the Tractatus are reminiscent of the Jewish tradition of weekly readings from the Torah. It happens that 9 out of the 14 original members of the Circle were Jewish by birth, though of course not by conviction—all theistic utterances being regarded as paradigmatically meaningless.)
The Viennese positivists interpreted the cryptic Tractatus as offering precisely the new, purifying foundations they sought. Proposition 4.003, for example, could not summarize more perfectly their fundamental conviction:
Most of the propositions and questions to be found in philosophical works are not false but nonsensical. Consequently we cannot give any answer to questions of this kind, but can only point out that they are nonsensical. Most of the propositions and questions of philosophers arise from our failure to understand the logic of our language. . . . And it is not surprising that the deepest problems are in fact not problems at all.
They ascribed to Wittgenstein their own verificationist criterion of meaningfulness, viz. that the meaning of a proposition is identical with the method for verifying it; or, alternatively, that the meaning of a sentence can be reduced to the specification of the experiences that would make the proposition known to be true. They read a vindication of their own positivism in such propositions as 6.53, exhorting one to “say nothing except what can be said, that is, propositions of natural science,” and 4.11, “the totality of all true propositions is the whole of natural science (or the whole corpus of the natural sciences).”
They also believed that Wittgenstein had accounted for the truths of mathematics and logic, reducing them to tautologies, devoid of any descriptive content. Proposition 4.461 states that “propositions show what they say: tautologies and contradictions show that they say nothing. A tautology has no truth-conditions, since it is unconditionally true; and a contradiction is true on no condition.” There might be terms that refer to items in the world contained in such tautologies as, say, “Socrates is either mortal or he is not.” But those referring words are irrelevant to the truth of the tautology. It is the meaning of the purely logical constants—or and not—that determine the truth of the tautology, and, “4.0132 My fundamental idea is that the ‘logical constants’ are not representatives; that there can be no representatives of the logic of facts.” All logic is ultimately tautological: “6.1262 Proof in logic is merely a mechanical expedient to facilitate the recognition of tautologies in
complicated cases.” Because all logic is tautological, it says nothing: “5.43 But in fact all the propositions of logic say the same thing, to wit nothing.”
“6.125 Hence there can never be surprises in logic.”
(Gödel, of course, was poised to deliver the greatest surprise in the history of logic, one which, in the logician Jaakko Hintikka’s words, is “stranger than others by orders of magnitude.” So the reader might already suspect, at this early point in the discussion, that Wittgenstein’s views on the philosophy of logic, leading as they do to proposition 6.125, would put him starkly at odds with Gödel’s result. As we shall see, Wittgenstein never accepted that Gödel had proved what he provably did prove. This, too, might strike the reader as verging on the paradoxical.)
Wittgenstein’s discussion in the Tractatus of mathematics, as opposed to logic, is brief. Mathematics, he says, is a method of logic (6.2 and again 6.234) and so, presumably, all that he has said of logic applies to mathematics. Mathematics, he says (6.2), also says nothing, has no descriptive content, though because it is expressed in equations it seems to:
6.2323 An equation merely marks the point of view from which I consider the two expressions; it marks their equivalence in meaning.
6.2341 It is the essential characteristic of mathematical method that it employs equations. For it is because of this method that every proposition of mathematics must go without saying.
Mathematical propositions, just like the tautologies of logic, do not represent any facts because they are, in a certain sense, merely grammatical. “6.233 The question whether intuition is needed for the solution of mathematical problems must be given by the answer that in this case language itself provides the necessary intuition.” (Proposition 6.233 also puts him starkly at odds with Gödel’s result, as we will see.) By language itself, Wittgenstein means syntax, the rules that stipulate that which can be said. Mathematics, like logic, is syntactic. Meanings are irrelevant to the determination of truth, even the meanings of the logical constants and the mathematical “=,” for their “meanings” are nothing over and above the grammatical rules that stipulate how we use them:
3.33 In logical syntax the meaning of a sign should never play a role. It must be possible to establish logical syntax without mentioning meaning of a sign: only the description of expressions may be presupposed.
Interestingly, from this proposition alone, Wittgenstein claims to demonstrate the fundamental error of the Theory of Types: “3.331 From this observation we turn to Russell’s ‘theory of types’. It can be seen that Russell must be wrong, because he had to mention the meaning of signs when establishing the rules for them.” The next two propositions, 3.332 and 3.333, “dispose of Russell’s paradox.” Thus Wittgenstein, at least, was satisfied, at least while writing the Tractatus, that he had solved the problem that originally lured him into philosophy of logic.
Wittgenstein was later to reject many of the assertions of his Tractatus. In fact, the discontinuity in his thinking was judged so radical that he was bifurcated into “early” and “later” Wittgenstein. In place of the early monolithic logic of language, the later Wittgenstein speaks of many different “language-games,” each with its own rules. In the early Wittgenstein the interesting nonsense (so to speak), characteristic of philosophy, derives from the violation of the rules that govern the bounds of all meaningfulness; in the later Wittgenstein, the interesting nonsense is a result of confusing the rules of one language-game with those of another. (Consistent between the two Wittgensteins is the belief that all philosophical problems arise from confusion about syntax.) The homogeneous view of one language, with one set of rules, from which the positivists took solace, gave way to a postmodern-friendly pluralism of language-games. The later Wittgenstein came to place much more emphasis on the social aspects of rule-following. Rules are embodied in social forms of behavior (also appealing to the postmodern sensibility). Even the law of noncontradiction wasn’t to be regarded as an absolute:
We shall see contradiction in a quite different light if we look at its occurrences and its consequences as it were anthropologically—and when we look at it with a mathematician’s exasperation. That is to say, we shall look at it differently, if we try merely to describe how the contradiction influences language-games, and if we try to look at it from the point of the mathematical law-giver.
Wittgenstein’s attitude toward mathematical logic radically changed. The pluralistic theory of rule-following of the later Wittgenstein was meant to subvert monologicism: that there is but one logic and its name is Principia Mathematica. Whereas the early Wittgenstein had labored hard with Russell on problems of logic, the later Wittgenstein came to regard the entire field as a “curse” (while Russell, disheartened by his earlier labors with Wittgenstein—his inability to understand him—withdrew from the field and wrote bestsellers.15)
Still, even between the early and the later Wittgenstein there is agreement enough on many issues, including fundamental questions in the philosophy of mathematics. Wittgenstein’s view of rule-following changed, but he remained committed to the claim that the entire nature of mathematics enfolds from rule-following. All that happens in mathematics is a consequence of rule-following, which is why mathematical “intuitions” are figments of our obfuscation. If we saw clearly what we are doing when we do mathematics we would not resort to these figments.
Both the early and later Wittgenstein are in agreement, then, that there can be no genuine surprises in mathematics. When, therefore, a surprise on the order of Gödel’s result arrived, the thing had to be argued away.
Of What We Cannot Speak
Though Wittgenstein may have believed he had summarily disposed of Russell’s paradox, the very problem that had drawn him away from aeronautical engineering and into the world of philosophy of logic and language, the entire Tractatus constitutes a self-avowed paradox, as the philosopher himself freely admits. According to its own dictates, its very own propositions are meaningless. Wittgenstein forbade talking about a language within the language. The syntactical nature, whether of logic or of mathematics, cannot really, without violating the syntax of the language, be spoken about, but must rather be shown.
6.54 My propositions serve as elucidations in the following way: anyone who understands me eventually recognizes them as nonsensical, when he has used them—as steps—to climb up beyond them. (He must, so to speak, throw away the ladder after he had climbed up it.)
(This last metaphor, for which Wittgenstein is famous, was one that Wittgenstein borrowed from the drama critic/philosopher Fritz Mauthner, of whose Sprachkritik Wittgenstein tended to be rather critical in the Tractatus. 4.0031: “All philosophy is a ‘critique of language’ [though not in Mauthner’s sense”].)
Wittgenstein’s attitude toward the inherent contradiction of the Tractatus is perhaps more Zen than positivist. He deemed the contradiction unavoidable. Unlike the scientifically minded philosophers who took him as their inspiration, he was paradox-friendly. Paradox did not, for Wittgenstein, signify that something had gone deeply wrong in the processes of reason, setting off an alarm to send the search party out to find the mistaken hidden assumption. His insouciance in the face of paradox was an aspect of his thinking that it was all but impossible for the very un-Zenlike members of the Vienna Circle to understand.16
In his autobiography Carnap recalled how the Vienna Circle had struggled with Wittgenstein’s dictum concerning the question of whether “it is possible to speak about linguistic expressions.”17 Carnap asked Wittgenstein for elucidation on this point once too often and was summarily banished forever more from Wittgenstein’s presence.
Schlick and Waismann were permitted to meet with Wittgenstein in person on a regular basis, Waismann because he was writing a commentary on the Tractatus—although Wittgenstein eventually gave up on ever making Waismann understand him and the book was never completed. Waismann was, perhaps, of all the Wittgenstein-enchanted Circle, the positivist who suffered the deepest from philo
sophical infatuation. He changed his views every time Wittgenstein did, and, like some of the equally impressionable Cambridge students, began to mimic the philosopher’s behavioral tics. At each Thursday meeting of the Circle he would update the other members with the breaking news of the philosopher’s views, beginning with the disclaimer: “I shall relate to you the latest developments in Wittgenstein’s thinking but Wittgenstein rejects all responsibility for my formulations. Please note that.” Those members of the Circle whom Wittgenstein refused to meet were thus kept informed, through Schlick and Waissman, of the philosopher’s ideas, which were often quoted in their papers. Some Austrian philosophers expressed doubts of the very existence of this “Dr. Wittgenstein” to whom Schlick’s group so often made reference. Perhaps he was simply a figment of Schlick’s imagination, “a mythological character invented as a figurehead for the Circle.”