Some more decades-smoldering resentment was allowed to escape—this time actually sent off—when his old acquaintance from the Vienna days, Karl Menger, pointed out to him some passages in Wittgenstein’s posthumously published Remarks on the Foundations of Mathematics, in which Gödel is mentioned. Writes Menger:
In the early 1970’s I began writing a book on my recollections of the Schlick Circle. For the sake of completeness, I looked for ideas about Gödel published by Wittgenstein. A few were in the latter’s book Remarks on the Foundations of Mathematics, which appeared in 1956. Aside from noncommittal remarks in Part 5, the Appendix I of Part I . . . contains a discussion of the problem—without, however, any adequate appreciation of Gödel’s work. In fact, Wittgenstein goes so far astray as to say that the only use of undecidability proofs is for “logische Kunststücke” [little logical articifices or conjuring tricks].
After Menger pointed out the passages to Gödel, Gödel responded to Menger:
As far as my theorems about undecidable propositions are concerned, it is indeed clear from the passage that you cite that Wittgenstein did not understand it (or that he pretended not to understand it). He interprets it as a kind of logical paradox, while in fact it is just the opposite, namely a mathematical theorem within an absolutely uncontroversial part of mathematics (finitary number theory or combinatorics). Incidentally, the whole passage you cite seems nonsense to me. See e.g. the “superstitious fear of mathematicians” of contradiction.20
These decidedly irked responses to Wittgenstein come after Wittgenstein’s own reaction to Gödel’s famous incompleteness results; and the nature of Wittgenstein’s reaction was such as to prompt umbrage even if none had existed before. Wittgenstein never came to accept that Gödel had, through strict mathematics, achieved a result with metamathematical implications. That there could be a mathematical result with metamathematical implications went against Wittgenstein’s conception of language, knowledge, philosophy, everything. The reticent logician’s historically audacious ambitions were, according to the Wittgensteinian point of view, unrealizable in principle. No wonder the anything-but-reticent philosopher would dismiss Gödel’s theorems with the belittling phrase “logische Kunststücke,” a dismissal that many mathematians have found, to this day, galling in the extreme, as apparently Gödel himself did. (Nary a mathematician I have spoken with has a good word to say about Wittgenstein. One particularly incensed mathematician I know characterized Wittgenstein’s famous proposition 7: Whereof we cannot speak we must remain silent as “accomplishing the difficult task of being at once portentous and vacuous.” The logician Georg Kreisel, who as a student worked with Wittgenstein and later knew Gödel, wrote: “Wittgenstein’s views on mathematical logic are not worth much, because he knew very little and what he knew was confined to the Frege-Russell line of goods.” Kreisel also decried the transformative influence Wittgenstein had had on students, including himself.)
Yet at a deeper level than even the foundations of mathematics, there was more affinity between the early Wittgenstein’s views and Gödel’s result than would be apparent from the Vienna Circle’s (mis)understanding of Wittgenstein. Wittgenstein really was no positivist, as he insistently protested; and his proposition 7 of the Tractatus amounts to a version of his own incompleteness thesis. Of course, to fully appreciate both the substance of the disagreement and affinity between Wittgenstein and Gödel, it is necessary to understand what Gödel actually did. So we will come back to the thorny relationship between Wittgenstein and Gödel later, after the proofs of the incompleteness theorems are presented.
In any case, though Wittgenstein may have loomed large in the circle of men among whom Gödel found himself in his fervently formative years, how seriously the young logician and confirmed Platonist ever took Wittgenstein is, in the end, unknowable. Beyond Wittgenstein towered the figure of the most influential mathematician of the day, David Hilbert, a figure Gödel could not possibly dismiss as mathematically inadequate. Like Wittgenstein’s, Hilbert’s views on the nature of mathematics could not have been more incompatible with the mathematical result that the young Gödel was soon to spring on an unsuspecting world.
1 It was also, interestingly, the location of the Augustinian monastery where Gregor Mendel (1822–1884) performed his highly tedious and important experiments with pea plants, resulting in his discovery of the laws of dominance and recessiveness in heredity.
2 Aristotle is commonly acknowledged as the father of logic. His work in logic is laid out in the Prior Analytics, which is part of the posthumous consortium known as the Organon. The philosopher had the seminal insight that in a deductive logical argument, some words are logically relevant while others are not. The irrelevant words can be dispensed with by making them variables. So, for example, in the stock syllogism: if all men are mortal, and Socrates is a man, then Socrates is mortal, the words “men” and “mortal,” and “Socrates” are disposable. This particular syllogism is just an instantiation of the more general syllogism-scheme: if all X’s are Y, and i is an X, then i is a Y. The move toward denoting logically irrelevant words with variables was a move toward generality and thus toward the science of logic. Aristotle, however, generalized too much, asserting that all deductive reasoning is syllogistic. The modern developments of the nineteenth century, most especially those of the German Gottlob Frege (1848–1925), revealed the greater variety of deductive arguments.
3 “K.-K.” stands for Kaiserliche-Königliche (imperial-royal) and referred to all pertaining to the Austrian crownlands. Kaiserliche und königliche was applied to that which was jointly administered by Austria and Hungary, and königliche alone referred to that pertaining to Hungary. This system of imperial abbreviation seems ready-made for satirizing, and so it was, magisterially, by Robert Musil in his novel, Man Without Qualities.
4 The Secession was founded in 1897 by artists dissenting from the policies of the Viennese artistic establishment. An exhibition hall for the Secessionists was opened in 1898. Ludwig Wittgenstein’s father, the enormously wealthy steel magnate, Karl Wittgenstein, was one of the three “benefactors” whose names are inscribed on the plaque inside the doors. The other two names belong to famous artists of the day: Rudolf von Alt and Theodor Hörmann.
5 Jean Cocteau wrote in 1926 that “The worst tragedy for a poet is to be admired through being misunderstood.” For a logician, especially one with Gödel’s delicate psychology, the tragedy is perhaps even greater.
6 The student, Johann (or Hans) Nelböck, had already twice been committed to a psychiatric ward for threatening Schlick. He had constructed a delusion in which the influential philosopher was responsible not only for Nelböck’s romantic problems but also for his difficulties in finding employment. He shot Schlick on the center staircase of the university’s main building (a brass inscription still marks the spot) as the philosopher was hurrying to deliver a lecture to his class. Interestingly, Vienna’s Nazis—by 1930 a sizable presence—applauded the psychotic murderer as winning a victory in their own battle for demographic hygiene, with Schlick, a German Protestant, coruscated in the dailies as a godless Jew. Of course, it’s true that he was godless, but he wasn’t the slightest bit Jewish. In fact, he derived from minor Prussian nobility.
7 Boltzmann had succeeded in deducing the laws of thermodynamics from a statistical analysis of the behavior of a great many molecules. His work was under-appreciated because of the dominant Mach’s positivistic rejection of the reality of molecules. Boltzmann committed suicide, perhaps partly out of professional despair.
8 The Prussian mathematician Christian Goldbach (1690–1764) had conjectured that every even number greater than 2 is the sum of two prime numbers (i.e., a number which is only divisible by 1 or by itself). So, for example 4 = 2 + 2, 6 = 3 + 3, 8 = 5 + 3, and so on. Goldbach’s conjecture has been confirmed for every even number that has ever been checked; however, no proof has of yet been discovered for the universal conclusion that every even number greater than 2 is th
e sum of two primes. The fact that Goldbach’s conjecture remains unproved means (at least according to the Platonist) that lurking out there beyond the point where mathematicians have checked there might be a counterexample: an even number that isn’t the sum of two primes. Then again (according to the mathematical Platonist), there may not be a counterexample: every even number may be the sum of two primes, without there being a formal way of proving that this is so. A Platonist asserts that there either is or isn’t a counterexample, irrespective of our having a proof one way or the other.
9 It was a highly musical family. The philosopher’s brother, Paul, was a concert pianist who lost his right arm in the First World War. He managed to gain such proficiency with his left hand that he was able to continue his career. Ravel’s famous Concerto for the Left Hand was written for Paul Wittgenstein.
10 Here is why you can prove anything from a contradiction. The rule of inference known as modus ponens says that from a conditional proposition of the form if p, then q (where p and q are any random propositions) and from the proposition p, you can deduce q. Propositions of the form if p, then q are false if p is true and q is false, and true in all other cases. Therefore if p is a contradiction, then if p, then q will be true for whatever q one cares to choose. Therefore, from the assertion of a contradiction p anything at all can be made to follow.
11 Another paradox-breeding set is the set of all sets, known as the universal set. The mathematician Georg Cantor (1845–1918) had proved that the power set of a set (formed by taking all the subsets of the set) always has a higher cardinality than the set itself. But then consider the universal set. Clearly, no set could have a larger cardinality than the universal set. However, its power set would . . . a contradiction. This is known as Cantor’s paradox, and the rules of set theory must bar the formation of the universal set.
12 He was, at least after the death of his older brother, the third Earl of Russell. His grandfather, Lord John Russell, introduced the Reform Bill of 1832, and served as prime minister under Queen Victoria. Bertrand Russell was a political activist, in particular a pacifist. He was jailed twice: once, in 1918, for six months for an allegedly libelous article in a pacifist journal; and again in 1961, at the age of 89, for one week, in connection with his campaign for nuclear disarmament.
13 Wittgenstein was not apologetic, but perhaps even perversely proud, of how few of the historical philosophical greats he had ever studied. On the other hand, the frontisquotes for his two books “were taken from authors who could hardly have been more typically Viennese—Kürnberger for the Tractatus, Nestroy for the Investigations.”
14 That it was a sort of poetry was the damning praise of Frege: “The pleasure of reading your book can therefore no longer be aroused by the content which is already known, but only by the peculiar form given to it by the author. The book thereby becomes an artistic rather than a scientific achievement.” (From a letter from Frege to Wittgenstein, 16 September 1919, translated in Monk 1990, p. 174)
15 His History of Western Philosophy, published in 1945, a very comprehensive and readable account of precisely what its title promises, became a longtime bestseller for Simon and Schuster. He turned to such popularizations of philosophy after giving up the more technical work to which he had devoted himself before Wittgenstein entered his life.
16 Perhaps this is why he found conversation with them so fruitless that on the occasions that he agreed to meet with some of them, he often just turned himself to the wall and read aloud the poetry of Rabindranath Tagore, “an Indian poet much in vogue in Vienna at that time, whose poems express a mystical outlook diametrically opposed to that of the members of Schlick’s Circle.”
17 Carnap was to (prematurely) welcome Gödel’s incompleteness theorem as vindicating the Circle’s insistence on meaningful metalanguage, and thus as making operable the Circle’s positive program for eliminating all metaphysical elements. “By use of Gödel’s method,” he set about to demonstrate how “even the metalogic of the language could be arithmetized and formulated in the language itself.” But it was Gödel himself who soon took the wind out of Carnap’s sails, convincing him—or at any rate trying to—that the upshot of the result he had produced by means of his method was entirely at odds with the positivist program itself. “Although Gödel had not persuaded Carnap on this fundamental issue, he did move Carnap in a strongly Platonist direction in his definition of analyticity, the capstone of the syntax program.”
18 In the pages referenced Gödel forthrightly sets out his Platonist conviction, using the undecidability of Cantor’s continuum hypothesis (that there is no set that is both larger than the set of natural numbers and smaller than the set of real numbers), a mathematical result which he helped to prove, as provocation: “For if the meanings of the primitive terms of set theory . . . are accepted as sound, it follows that the set-theoretical concepts and theorems describe some well-determined reality, in which Cantor’s conjecture must be either true or false. Hence its undecidability from the axioms being assumed today can only mean that these axioms do not contain a complete description of that reality. Such a belief is by no means chimerical, since it is possible to point out ways in which the decision of a question, which is undecidable from the usual axioms, might nevertheless be obtained.”
19 In the alternative set of answers to the questionnaire, he answered the same question, “Are there any influences to which you attribute special significance in the development of your philosophy?” by including the “math[ematical lectures] by Phil. Furtwänger” as well as “phil[osophical] lectures (introductory) of Gomp[erz].”
20 Gödel is here citing a parenthetical remark in passage I, 17: “Die abergläubische Angst und Verehrung der Mathematiker vor dem Widerspruch: The superstitious fear and awe of mathematicians in the face of contradiction.” Gödel also wrote to Abraham Robinson, a young mathematical logician of whom Gödel thought very highly, that Wittgenstein’s comments on his proof constitute a “completely trivial and uninteresting misinterpretation” of his results.
II
Hilbert and the Formalists
A Mathematician’s Intuition
We return to the subject of the tantalizing uniqueness of mathematics, pursuing its truth through a priori reason, establishing its conclusions so firmly that no empirical discoveries as to the nature of the world can overturn them.
Since the earliest days of the ancient Greeks, mathematical knowledge has seemed to be on the one hand the least problematic area of human knowledge, in fact the very model toward which all knowledge ought to aspire: certain and unassailable, in short, proved. No wonder that epistemological utopians, from Plato onward, have urged that the standards and methods of mathematics ought to be applied, insofar as is possible, to all of our attempts to know.
Yet, on the other hand, mathematical knowledge has seemed, to darker-souled epistemologists, highly problematic, its very certainty, which emboldens the utopians, making it suspect in warier eyes. How can any knowledge be certain and unassailable, in short: proved? Perhaps, some epistemologists of the darker cast argue, it is because mathematical knowledge is not really knowledge at all; perhaps it is simply a game, played by stipulated rules, telling us nothing about anything. “There is no there, there,” Gertrude Stein famously said of her birthplace, Oakland, California. So it is with mathematics, at least according to some.
So the question is: Whence certainty? What is our source for mathematical certainty? The bedrock of empirical knowledge consists of sense perceptions: what I am directly given to know—or at least to think—of the external world through my senses of sight and hearing and touch and taste and smell. Sense perception allows us to make contact with what’s out there in physical reality. What is the bedrock of mathematical knowledge? Is there something like sense perception in mathematics? Do mathematical intuitions constitute this bedrock? Is our faculty for intuition the means for making contact with what’s out there in mathematical reality? Or is there just no “there�
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Mathematical proofs must start from somewhere. Often proofs start with conclusions from other proofs and then deduce further conclusions from these. But not everything can be proved, otherwise how can we get off the ground? There must be, in mathematics just as in empirical knowledge, the “given.” Given to us through what means? Mathematical intuition is often thought of as the a priori analogue to sensory perception.
Intuitions. They are a tricky business, and not only in mathematics. An intuition is supposed to be something that we just know, in and of itself, not on the basis of knowing something else. (Sometimes, of course, the word is used in a weaker way, conveying the sense of having a vague feeling, lacking in any certainty. But it is in the stronger sense that it functions in epistemological debates.) Obviously intuitions—or, more precisely, claims to intuitions—vary greatly among people; there are places on the planet right now where people are slaughtering one another because they make claims to fundamentally differing intuitions, patently nonmathematical. All genuine intuitions are (tautologously) true (tautologously, because we would not call them “genuine” unless they were true). But not all putative intuitions are genuine intuitions; and how is one to tell when one is in possession of the genuine article? Murky motivations—to believe, for example, propositions that would, if true, conduce to one’s own self-importance, notoriously propositions asserting the innate superiority of one’s own kind—not only abound but also tend to hide themselves. The resulting beliefs can feel intuitively obvious precisely because we are not prepared to face their real and suspect source in our own personal situations and egos.
You might think that in mathematics—perched on its topmost turret of Reine Vernunft, far from the madding human scene below—murky motives for beliefs are at a minimum. Still, even in mathematics we can get suckered. Accidental features can insinuate themselves into our most pristine mathematical reasoning, presenting us with propositions that seem intuitively obvious when they are not obvious at all—maybe not even true at all.
Incompleteness Page 10
