Hilbert's program failed to provide a secure foundation for mathematics because the axiomatic method is not an adequate account of mathematical truth. The upshot of Godel's incompleteness theorem is that there will always be true theorems in mathematics that are undecidable in the system because they cannot be proven from its axioms. When a theorem is undecidable within a formalization, it cannot be proven true or false, so mathematicians must decide whether to add the theorem or its negation as an axiom.
Hilbert's formalism fails to provide an adequate philosophy of mathematics, for it ignores how mathematicians actually write and do mathematics. Mathematicians do not follow the axiomatic method when publishing proofs of theorems. Proofs are written with a mixture of natural and formal language. According to Hilbert's formalism
[t]he authenticity of a mathematical proof is established by verifying that a sequence of atomic symbol strings is legitimate. In point of fact, proofs are not written in terms of atomic symbol strings. They are written in a mixture of common discourse and mathematical symbols. Definitions are made to serve as abbreviations for longer combinations of words and symbols [...] Corollaries are introduced for the psychological lift of obtaining deep theorems cheaply. Splicing two theorems is standard practice. In the course of a proof, one cites Euler's Theorem, say, by way of authority [...] If splicing is common to lend authority, then skipping is even more common. By skipping, I mean the failure to supply an important argument. (Davis 171)
As an example of a proof--which will forego all the above shortcuts with the exception of definitions--take Euclid's proof of the Prime Number Theorem.
Definition: A number is prime if it has no divisors, eg. 2, 3, 5, 7, 9, 11, 13, 17, 19, 23.
Definition: A number is composite if it can be divided by prime numbers, eg. 6 (2 x 3), 9 (3 x 3), 30 (2 x 3 x 5).
Theorem: There are infinitely many prime numbers.
Proof: Assume, on the contrary, that there are not an infinite number of primes. With a finite number of primes, there must be a last prime, call it N. Multiply all the primes from 1 up to N together, call that number P:
1 x 3 x 5 x 7 x 9 x 11 x ... x N = P
Now add 1 to P. Is P + 1 composite or prime? If we divide P + 1 by any of the primes that preceded it, we are left with a remainder of 1. So P cannot be divided by any of the primes that preceded it, so P + 1 is prime. This contradicts our assumption that there are not an infinite number of primes. If an assumption leads to a contradiction, it must be false (mathematics retreats from paradox). So we conclude that there are in fact an infinite number of primes.
Euclid's proof is regarded as one of the most beautiful proofs in mathematics. When proofs are short and incorporate ingenious reasoning, they are said to be elegant and beautiful. Just as poets are esthetes of language, mathematicians are esthetes of reason:
The mathematician's patterns, like the painter's or the poet's must be beautiful; the ideas, like the colors or the words must fit together in a harmonious way. Beauty is the first test: there is no permanent place in this world for ugly mathematics. (Hardy)
Aesthetics are the epistemological foundation of mathematics. The truth of a proof relies on the beauty of its argument: "Beauty is the first test." To understand the implication of that statement, we need to visit the mathematician at the moment he takes his pleasure, we must become voyeurs.
Jacques Hadamard's study The Psychology of Invention in the Mathematical Field divides the methods of the creative mathematician into 4 stages: preparation, incubation, illumination, and verification. When trying to prove a theorem, the preparation stage involves studying the problem with intense concentration, and considering the use of various approaches common in the field. Once the problem is thoroughly understood, and known approaches have been exhausted, the mathematician may experience a sense of anxiety or frustration at being unable to attack the problem head on, deductively. The second phase of incubation begins, where the mathematician lives with the problem and allows unconscious processes to operate. If the mathematician is lucky, illumination occurs, which is a momentary flash of insight that provides the solution.
The illumination does not appear while the mathematician is thinking about the problem directly. Hadamard tells the story of Henri Poincare, who worked arduously at a problem for months without any success. The moment of illumination arrived while on vacation: "Poincare was not working when he boarded the omnibus of Coutances: he was chatting with a companion; the idea passed through his mind for less than one second; just the time to put his foot on the step and enter the omnibus." (Hadamard, 31). In that moment of illumination, in that fraction of a second when the solution presents itself unexpectedly, that is where beauty serves as the first test of truth. The solution to the proof does not appear to the mathematician as a chain of deductions, with the moment of illumination occurring at the last axiomatic transformation. In the moment of illumination, the entire solution can be checked in an instant, through a mental process similar to the pattern recognition involved in recognizing someone's face.
The last phase of invention is verification, which is when the mathematician writes out the deductive argument whose structure appeared in the moment of illumination. This can be a long and arduous process, but deductive reasoning cannot begin until after the mathematician's intuition has provided the solution. This written argument has to allow the reader to get the same insight as the author "all at once":
Indeed, every mathematician knows that a proof has not been 'understood' [or verified] if one has done nothing more than verify step by step the correctness of the deductions of which it is composed and has not tried to gain a clear insight into the ideas which have led to the construction of this particular chain of deductions in preference to every other one. (Bourbaki)
To this end, proofs must be short to be grasped. Philip Davis has noted that overly long proofs have a lesser chance of being verified correctly: "Proofs cannot be too long, else their probabilities go down, and they baffle the checking process." (173) We have seen that beauty is the first test of truth, and both the beauty and truth of a proof are proportional to its length. Beauty and truth are so intertwined that all mathematicians' espouse Keats' maxim without the slightest trace of irony:
Beauty is truth, truth beauty -- that is all
Ye know on earth, and all ye need to know
The literary analogue for the work of the creative mathematician while trying to prove a theorem would not be a text written according to Oulipian constraint. It would begin with the poet reading a triple-decker detective novel, at the end of which the poet is convinced he knows the killer. Upon discovering in a moment of illumination the irrefutable pattern that connects the evidence to convict the killer, the poet inscribes the elegant proof as a haiku in the margin of the last page.
Against Hilbert's emphasis on mathematical discourse, the mathematician Philip Brouwer focuses on mathematical intuition: "The first act of intuitionism completely separates mathematics from mathematical language, in particular from the phenomena of language which are described by theoretical logic, and recognizes that intuitionist mathematics is an essentially languageless activity of the mind" (Brouwer 141-2).
Jacques Hadamard's survey of mathematicians in Psychology of Invention largely confirms that mathematical intuition is a languageless activity, and subsequent writers have been content to cite him in this regard. Language only enters the scene after the moment of illumination. The movement from intuitive revelation to deductive reason is seen as the movement from spirit to body:
Mathematical objects are perceived by the soul. (Plato)
[H]ow does geometrical ideality (just like that of all the sciences) proceed from its primary intrapersonal origin, where it is a structure within the conscious space of the first inventor's soul, to its ideal objectivity? In advance we see that it occurs by means of language, through which it receives, so to speak, its linguistic living body. (Husserl 161)
Our souls perceive metaphysical objects, and then language makes them physical. This passage is necessary for the mathematician to verify his results, but it comes with some anxiety: "The process of verification can be very painful: one's terribly afraid of being wrong" (Connes). Language enters the scene as the Law, disciplining insight according to the dictates of deductive reasoning.
At the conclusion of his study, Hadamard declares "In conformity with a rule which seems applicable to every science of observation, it is the exceptional phenomenon that is likely to explain the usual one" (136). In taking his advice with regards to his own text, we find ourselves confronted with exceptional mathematicians who do use language in the process of creative mathematics.
Jessie Douglas provides Hadamard's first exception: "Jessie Douglas generally thinks without words or algebraic signs. Eventually, his research thought is in connection with words, but only with their rhythm, a kind of Morse language where only the numbers of syllables of some words appear." Language in Douglas' research thought loses all semantic effects in preference to its rhythmic properties. By noting only the syllables of a string of words, Douglas performs a mathematical analysis of language, abstracting a pattern from the sensual phenomena. This pattern is nothing but poetic meter.
Hadamard devotes more space to the second exception:
G. Polya's case [...] is different. He does make an eventual use of words. "I believe, [...] that the decisive idea which brings the solution of a problem is rather often connected with a well-turned word or sentence. The word or sentence enlightens the situation, gives things, as you say, a physiognomy. It can precede by little the decisive idea or follow on it immediately; perhaps it arises at the same time as the decisive idea... The right word, the subtly appropriate word, helps us to recall the mathematical idea, perhaps less completely and less objectively than a diagram or a mathematical notation, but in an analogous way... It may contribute to fix it in the mind." Moreover, he finds that a proper notation - that is, a properly chosen letter to denote a mathematical quantity- can give him similar help; and some kind of puns, whether of good or poor quality, may be useful for that purpose. For instance, Polya, teaching in German at a Swiss university, usually made his junior students observe that z and w are the initials of the German words "Zahl" and "Wert", which precisely denote the respective roles which z and w play in the theory which he was explaining.
The language that proves effective in aiding the creative mathematician in the incubation stage is 'well-turned', and 'subtly appropriate', its most concrete exemplar being a mathematical pun. To understand how well-turned phrases and puns aid Polya in his mathematical research, we must turn our attention to the role analogy plays in mathematics.
Polya provides a definition in his book Induction and Analogy in Mathematics: "two systems are analogous, if they agree in clearly definable relations of their respective parts". In talking of specialization, generalization, and analogy, Polya posits analogy as the most essential operation in mathematics: "There is perhaps no discovery either in elementary or in advanced mathematics or, for that matter, in any other subject that could do without these operations, especially without analogy." Many of the most astonishing proofs in mathematics connect disparate systems through analogy so that once the necessary features of the systems are shown to be analagous, a problem in one discipline of mathematics can be solved using methods of another discipline. A pun creates an analogy between the system of expression and the system of semantics. The pun posits a clearly defined relation between the sonic or visual form of words and their meanings: a creative mathematics of language.
The language present for Douglas and Polya while the creative mathematician is at work in the business of mathematical intution is not the formal language of mathematical discourse, but the fanciful language of poetic play. Not the axioms and inferences of verification, but the puns and rhythms of versification. The reason? Polya's puns are more mathematical than Hilbert's axioms. Hence, by inspection
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Connes, Alain and Jean-Pierre Changeux. Conversations on Mind, Matter, and Mathematics. Princeton: Princeton University Press, 1995.
Davis, Philip J. "Fidelity in Mathematical Discourse: Is One and One Really Two?" New Directions in the Philosophy of Mathematics. Princeton: Princeton University Press, 1998. 163-176.
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Husserl, Edmund. "The Origin of Geometry". Edmund Husserl's Origin of Geometry: An Introduction by Jacques Derrida. tr. David Carr. Lincoln: Univeristy of Nebraska Press, 1989.
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