Hilbert's Ideal Elements
Hilbert's philosophy of mathematics arose in part out of his assessment of Russell's, and that assessment was strongly influence by Behmann. Behmann takes Russell's theory to be one that takes concrete objects, and nothing else, to exist. That is not quite correct about Russell clearly in one way---what Russell takes to exist is sense data, not concrete objects. The rejection of classes by Russell is formally complete, but the rejection of concrete objects is programmatic, and Russell often acts as if he takes concrete objects seriously (the king of England) for the sake of examples. That is important because it is reasonable to take Hilbert to think of himself as maintaining that aspect of Russell when he finally rejects other details of Russell's theory. It is also not quite what one expects to hear about Russell's theory in another way: although Russell officially rejects classes (and all mathematical objects), he minimizes or covers up that fact much of the time. Standard descriptions of Russell's theory do not typically emphasize his official nominalism. That kind of ambiguity is taken up by Hilbert in spades. It is, however, an important continuity between Russell and Hilbert that has often been overlooked, since the nominalist aspect of Hilbert's theory is central while the nominalist aspect of Russell's theory is secondary---it is in the service of allowing him to use axioms that can be taken to be logical.
My hobbyhorse: Hilbert thinks that the paradoxes pose a serious problem for Cantor's theory of sets. They
do pose a problem for Frege and for Russell, but Cantor's theory is quite different and was never subject to paradox. Hilbert takes the problem for Russell's theory to be a problem for Cantorian foundations of analysis (arithmetic), but he is wrong, and most mathematicians who used Cantorian ideas in analysis never worried in the slightest about the paradoxes. Hilbert is responsible for what is largely the myth that the paradoxes created a crisis in the foundations of analysis.
Russell's foundations were not accepted by Brouwer or Weyl. They are influential mathematicians who are actually influencing mathematical research to move away from Cantorian foundations. That, as opposed to some deep philosophical analysis, is an important part of Hilbert's motivation. Russell starts with assumptions Brouwer and Weyl reject, and so Hilbert cannot rest content with Russell's theory. Hilbert, instead, must start on common ground and work out from there. He takes his finitist metamathematics to be common ground. We can see that, in part, as follows: Hilbert attempts to justify the use of the law of the excluded middle when reasoning about infinite domains on the basis of the unargued assumption that it applies on finite domains. Brouwer rejects the law of the excluded middle on infinite domains but accepts it on finite ones. He also, though this is less obvious from what we have read, accepts predicative arguments as finitist and rejects impredicative ones. Weyl accepted only predicative arguments.
Hilbert takes the justification of his finitist metamathematics to be that it is "contentual," that is that it involves only reasoning about concrete objects, the sort for which our modes of reasoning were developed. Ironically, the so-called concrete objects are signs, that is, types, not tokens. Charles Parsons says that they are "quasiconcrete." Hilbert associates being concrete with being present to intuition, and if that is what is meant by concrete, then signs are a pretty reasonable choice.
Since I take it that Hilbert's motivation for his finitist metamathematics is to be on common ground, I take it that the central epistemological role finitism plays for him tells us nothing about his ontological predilections.
Hilbert's most important contribution to the philosophy of mathematics predates all of this: it is in his foundations of geometry (1899). The contribution is one that is now often inculcated as early as the fourth grade, and so it is hard to see what a departure it was from the other views we have been discussing. Hilbert takes a system of axioms to "implicitly define" its subject matter. Euclid, Dedekind, Peano, Frege, and Russell all took axioms to be about a particular antecedently understood subject matter. Hilbertian axioms are uninterpreted (or at least officially uninterpreted) principles in search of an interpretation. The axioms of Euclid, Dedekind, Peano, Frege, and Russell are true of their subject matter.
Hilbert does not deny, in fact it sometimes seems to be important, that axioms have an intended interpretation. That is what suggests the axioms, that is what lets us find fruitful systems of axioms, but once we find the axioms, we deinterpret them so that we can consider all interpretations on which they are true on an equal footing. Hilbert gave this idea an important application in his geometry: he proves that his axioms for geometry are consistent (relative to the real numbers) by showing that if we interpret "point" as "triples of real numbers," "line" as "set of triples of real numbers that is the solution set of a linear equation," and so forth, the axioms are all true. This very simple idea is new and a real conceptual innovation. There was an extended correspondence between Frege and Hilbert about the idea of taking axioms to be uninterpreted. Frege tries to convince Hilbert he has made a mistake since it the axioms are uninterpreted they can't be true, and so they can't be applied. Hilbert tries to explain that they can be true in an interpretation and hence applied, as truths, given the interpretation, and may have tried to explain that that gives reason to think of the application of the axioms as independent of the particular interpretation. Frege never catches on, and Hilbert eventually stops answering his letters. (That is analyzed, for example, by Hallett.) Brouwer's rejection of Hilbert's finitist theory can be read the same way.
Hilbert's whole justification of infinitary mathematics on the basis of finitary mathematics rests on the idea of taking a single axiom system in various ways. The following theorems of finitary mathematics all have "the same" proof: 2+3=3+2, 4+9=9+4, 58+23=23+58. One can exhibit the form of that proof by replacing the numbers by letters everywhere in the proof to produce a proof form of the theorem form
One can then obtain a proof of any of the above theorems by plugging in the numbers to the form. One can now take that form, that system of signs, deinterpret it, and reinterpret it as a theorem in a new system, one that has
variables. Doing that yields a
new theorem 
In this way, we move from elementary arithmetic to algebra. What is the subject matter of algebra? Are
and
new entities? Hilbert's answer is, modulo the usual nonsense about whether there are really mathematical entities at all, yes. We have moved from the study of numbers as a subject to the study of polynomials as a subject.
One reason it is useful to study polynomials is that we can use them to discover things about our old subject of numbers: If a polynomial is derivable, then we can reinterpret it as a scheme and obtain results about numbers by plugging them in. This is a simple example of the method of ideal elements. (Note that 58+23=23+58
is a polynomial.) The method is general in mathematics, indeed it is the central innovation in mathematics in Hilbert's day that led to an explosion of new mathematics. (The next innovation was Hilbert's axiomatic method.)
Hilbert proposes a new application of the method: Start with logic over finite domains, and add logic over infinite domains as a system of ideal elements. The finite logic has in it
. The new one has
as well, that is, quantification over an infinite domain. Pretty much, we start with sentential logic and add quantifiers. In the new system, we can go from
to
Why would we want to do this? For the same reason as above: if we deduce in this system a result in the language of the old system, it is true in the old system, and the new deduction is much more efficient.
Of course, the example of addition of ideal elements uppermost in Hilbert's mind is adding Cantorian set theory to finitary arithmetic.
I have omitted to answer the obvious question: why does this work, and how do we know that it does. Suppose it didn't work. Then there would be some statement of elementary arithmetic derivable from the system that was wrong, say, 0=1. _Since the system extends elementary arithmetic, it also includes
That is, the system is inconsistent. In order for a system that includes finitary arithmetic to have the desirable property, it is necessary and sufficient that the system be consistent.
In the geometry, Hilbert already lists what he takes to be the characteristics of a good axiom system and gives some others in the surrounding discussion. Here are the listed ones (in our terminology, not his, and hence more definitely specified than his):
- Consistent
- Complete
- Independent
Consistent has come to mean free of contradictions in formal consequences. Complete has come to mean that every question about the objects described has an answer that follows from the axioms. Independent means that no axiom is a consequence of the others. The unlisted criterion in the geometry is that the axioms have as one interpretation the standard or usual one. In later work we have the criterion that the system of axioms should extend an antecedently accepted system.
To prove a system of axioms consistent, one can interpret it in a system already known to be consistent. (Triples of real numbers.) Of course, to get started, you have to do something else. Hilbert's idea is that you take the formal deinterpreted proof theory, take it to be a (contentual) theory of signs, and prove that the sequence "0=1" doesn't occur among the allowed sequences of signs. That is, one gives a "proof theoretic" "finitary" consistency proof.
Note that Julius Caesar is nowhere to be seen.
This program, it seems to me, answers all of our questions, and is completely justifiable, except for one unfortunate problem: there is no complete axiomatization of arithmetic that can be proved consistent in a finitary way. That is the content of Gödel's first and second incompleteness theorems, which we discuss next time.
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ShaughanLavine - 22 Oct 2006