Hilbert’s Program: An Essay on Mathematical Instrumentalism (Synthese Library, 182) 1986th Edition by M. Detlefsen (PDF)

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Hilbert’s Program was founded on a concern for the phenomenon of paradox in mathematics. To Hilbert, the paradoxes, which are at once both absurd and irresistible, revealed a deep philosophical truth: namely, that there is a discrepancy between the laws accord­ ing to which the mind of homo mathematicus works, and the laws governing objective mathematical fact. Mathematical epistemology is, therefore, to be seen as a struggle between a mind that naturally works in one way and a reality that works in another. Knowledge occurs when the two cooperate. Conceived in this way, there are two basic alternatives for mathematical epistemology: a skeptical position which maintains either that mind and reality seldom or never come to agreement, or that we have no very reliable way of telling when they do; and a non-skeptical position which holds that there is significant agree­ ment between mind and reality, and that their potential discrepan­ cies can be detected, avoided, and thus kept in check. Of these two, Hilbert clearly embraced the latter, and proposed a program designed to vindicate the epistemological riches represented by our natural, if non-literal, ways of thinking. Brouwer, on the other hand, opted for a position closer (in Hilbert’s opinion) to that of the skeptic. Having decided that epistemological purity could come only through sacrifice, he turned his back on his classical heritage to accept a higher calling.

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Editorial Reviews: Review `The book is well written and recommended reading for everyone interested in the philosophical dimensions of technical results from foundations of mathematics and pure logic.’ S. Gottwald, Zentralblatt für Mathematik und ihre Grenzgebiete, Vol. 641.

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⭐Three arguments are raised against the standard view that Gödel’s theorem proves the unfeasibility of Hilbert’s programme:(1) Gödel proved the nonprovability of a particular sentence (Con(T)) expressing the consistency of the system (T), but this does nothing to exclude the possibility that “there might still be some formula other than Con(T), expressing the same proposition that Con(T) expresses, that is provable in T.” “Were this the case, the unprovability-in-T of Con(T) and its expression of T’s consistency would best be taken as sheer coincidence.” (p. 81).(2) Gödel’s result shows only that the consistency proof cannot be carried out *within the system itself,* but Hilbert is not committed to this. As Gödel himself said: his results “do not contradict Hilbert’s formalistic viewpoint. For this viewpoint presupposes only the existence of a consistency proof in which nothing but finitary proof is used, and it is conceivable that there exist finitary proofs that cannot be expressed in the formalism [in question]” (p. 91). (Detlefsen tries to put his own spin on this argument by motivating it in terms of his “instrumentalist” interpretation of Hilbert, but it comes to the same thing.)(3) Gödel’s result shows that any system containing number theory will be unable to prove its own consistency. But to Hilbert formulas are largely “ideal” (as in “ideal numbers,” “point at infinity,” etc.), i.e., they are instruments invented to aid human thought. Therefore the consistency of this “ideal” formalism is of no interest in itself. The part of the formalism that is of any value excludes, for example, “all ideal proofs of real formulae that are to long or complex to be of any human epistemic utility” (p. 89). And since “‘elementary number theory’ … designates an infinite system of proofs,” “not every appreciable system of ideal proofs contains elementary number theory” (p. 87). “Thus, despite the Gödelian challenge, it may still prove possible to give both a finitary and a feasible demonstration of the soundness of the useful ideal methods.” (p. 90). This is a weak argument, since it ignores the fact that Hilbert’s programme is a mathematical research programme, and although Detlefsen’s move may be philosophically safe it is obviously disastrous from this point of view.

⭐この本では，まず有名なポアンカレの数学的帰納法に関する循環を指摘した批判や，フレーゲ的な論理的推論に対する考えから来る形式主義に対する批判が扱われる．しかしながら，この本で1番大事なのはここではないと思う．1番大事なのは，ゲーデルの不完全性定理からヒルベルト・プログラムの実行不可能性を論証するにはどんな論理的仮定が必要で，それは正当化可能なものなのかについて議論している後半であろう．大雑把に言ってしまえば，すぐに思いつく論理的仮定はderivability conditions(以下DCs)の正当化であろう．ディトレフセンはこのDCsの正当化を試みている（と思われる）モストフスキ，クライゼル-竹内，プラヴィッツらの議論を批判し，ヒルベルト・プログラムの実行不可能性はいまだに論証されていない，と結論づける．さらには，ヒルベルト・プログラムに対する独特の理解（厳密道具主義，といわれる）によってその実行可能性を論じていく．個人的には，こうした形でヒルベルト・プログラムを考えるのは不可能であるとは思うが，綿密な議論が詰まっているのでとても興味深い．（ただし，その綿密さ故に読むのは結構しんどい．）数学の定理から哲学的帰結を出すには，いかに哲学的な前提が隠されているかが明瞭にわかるよい例であろう．

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