Rants of the Undead God: Instrumentalism. Thought of the Day 68.1

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Hilbert’s program has often been interpreted as an instrumentalist account of mathematics. This reading relies on the distinction Hilbert makes between the finitary part of mathematics and the non-finitary rest which is in need of grounding (via finitary meta-mathematics). The finitary part Hilbert calls “contentual,” i.e., its propositions and proofs have content. The infinitary part, on the other hand, is “not meaningful from a finitary point of view.” This distinction corresponds to a distinction between formulas of the axiomatic systems of mathematics for which consistency proofs are being sought. Some of the formulas correspond to contentual, finitary propositions: they are the “real” formulas. The rest are called “ideal.” They are added to the real part of our mathematical theories in order to preserve classical inferences such as the principle of the excluded middle for infinite totalities, i.e., the principle that either all numbers have a given property or there is a number which does not have it.

It is the extension of the real part of the theory by the ideal, infinitary part that is in need of justification by a consistency proof – for there is a condition, a single but absolutely necessary one, to which the use of the method of ideal elements is subject, and that is the proof of consistency; for, extension by the addition of ideals is legitimate only if no contradiction is thereby brought about in the old, narrower domain, that is, if the relations that result for the old objects whenever the ideal objects are eliminated are valid in the old domain. Weyl described Hilbert’s project as replacing meaningful mathematics by a meaningless game of formulas. He noted that Hilbert wanted to “secure not truth, but the consistency of analysis” and suggested a criticism that echoes an earlier one by Frege – why should we take consistency of a formal system of mathematics as a reason to believe in the truth of the pre-formal mathematics it codifies? Is Hilbert’s meaningless inventory of formulas not just “the bloodless ghost of analysis? Weyl suggested that if mathematics is to remain a serious cultural concern, then some sense must be attached to Hilbert’s game of formulae. In theoretical physics we have before us the great example of a [kind of] knowledge of completely different character than the common or phenomenal knowledge that expresses purely what is given in intuition. While in this case every judgment has its own sense that is completely realizable within intuition, this is by no means the case for the statements of theoretical physics. Hilbert suggested that consistency is not the only virtue ideal mathematics has –  transfinite inference simplifies and abbreviates proofs, brevity and economy of thought are the raison d’être of existence proofs.

Hilbert’s treatment of philosophical questions is not meant as a kind of instrumentalist agnosticism about existence and truth and so forth. On the contrary, it is meant to provide a non-skeptical and positive solution to such problems, a solution couched in cognitively accessible terms. And, it appears, the same solution holds for both mathematical and physical theories. Once new concepts or “ideal elements” or new theoretical terms have been accepted, then they exist in the sense in which any theoretical entities exist. When Weyl eventually turned away from intuitionism, he emphasized the purpose of Hilbert’s proof theory, not to turn mathematics into a meaningless game of symbols, but to turn it into a theoretical science which codifies scientific (mathematical) practice. The reading of Hilbert as an instrumentalist goes hand in hand with a reading of the proof-theoretic program as a reductionist project. The instrumentalist reading interprets ideal mathematics as a meaningless formalism, which simplifies and “rounds out” mathematical reasoning. But a consistency proof of ideal mathematics by itself does not explain what ideal mathematics is an instrument for.

On this picture, classical mathematics is to be formalized in a system which includes formalizations of all the directly verifiable (by calculation) propositions of contentual finite number theory. The consistency proof should show that all real propositions which can be proved by ideal methods are true, i.e., can be directly verified by finite calculation. Actual proofs such as the ε-substitution procedure are of such a kind: they provide finitary procedures which eliminate transfinite elements from proofs of real statements. In particular, they turn putative ideal derivations of 0 = 1 into derivations in the real part of the theory; the impossibility of such a derivation establishes consistency of the theory. Indeed, Hilbert saw that something stronger is true: not only does a consistency proof establish truth of real formulas provable by ideal methods, but it yields finitary proofs of finitary general propositions if the corresponding free-variable formula is derivable by ideal methods.

Epistemological Constraints to Finitism. Thought of the Day 68.0

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Hilbert’s substantial philosophical claims about the finitary standpoint are difficult to flesh out. For instance, Hilbert appeals to the role of Kantian intuition for our apprehension of finitary objects (they are given in the faculty of representation). Supposing one accepts this line of epistemic justification in principle, it is plausible that the simplest examples of finitary objects and propositions, and perhaps even simple cases of finitary operations such as concatenations of numerals can be given a satisfactory account.

Of crucial importance to both an understanding of finitism and of Hilbert’s proof theory is the question of what operations and what principles of proof should be allowed from the finitist standpoint. That a general answer is necessary is clear from the demands of Hilbert’s proof theory, i.e., it is not to be expected that given a formal system of mathematics (or even a single sequence of formulas) one can “see” that it is consistent (or that it cannot be a genuine derivation of an inconsistency) the way we can see, e.g., that || + ||| = ||| + ||. What is required for a consistency proof is an operation which, given a formal derivation, transforms such a derivation into one of a special form, plus proofs that the operation in fact succeeds in every case and that proofs of the special kind cannot be proofs of an inconsistency.

Hilbert said that intuitive thought “includes recursion and intuitive induction for finite existing totalities.” All of this in its application in the domain of numbers, can be formalized in a system known as primitive recursive arithmetic (PRA), which allows definitions of functions by primitive recursion and induction on quantifier-free formulas. However, Hilbert never claimed that only primitive recursive operations count as finitary. Although Hilbert and his collaborators used methods which go beyond the primitive recursive and accepted them as finitary, it is still unclear whether they (a) realized that these methods were not primitive recursive and (b) whether they would still have accepted them as finitary if they had. The conceptual issue is which operations should be considered as finitary. Since Hilbert was less than completely clear on what the finitary standpoint consists in, there is some leeway in setting up the constraints, epistemological and otherwise, an analysis of finitist operation and proof must fulfill. Hilbert characterized the objects of finitary number theory as “intuitively given,” as “surveyable in all their parts,” and said that their having basic properties must “exist intuitively” for us. This characterization of finitism as primarily to do with intuition and intuitive knowledge has been emphasized in that what can count as finitary on this understanding is not more than those arithmetical operations that can be defined from addition and multiplication using bounded recursion.

Rejecting the aspect of representability in intuition as the hallmark of the finitary; one could take finitary reasoning to be “a minimal kind of reasoning supposed by all non-trivial mathematical reasoning about numbers” and analyze finitary operations and methods of proof as those that are implicit in the very notion of number as the form of a finite sequence. This analysis of finitism is supported by Hilbert’s contention that finitary reasoning is a precondition for logical and mathematical, indeed, any scientific thinking.