Some basic theorems on the foundations of mathematics and their philosophical implications
Research in the foundations of mathematics during the past few decades has produced some results, which seem to me of interest, not only in themselves, but also with regard to their implications for the traditional philosophical problems about the nature of mathematics. The results themselves, I believe, are fairly widely known, but nevertheless, I think, it will be useful to present them in outline once again, especially in view of the fact that, due to the work of various mathematicians, they have taken on a much more satisfactory form, than they had had originally. The greatest improvement was made possible through the precise definition of the concept of finite procedure, which plays a decisive role in these results. There are several different ways of arriving at such a definition, which however all lead to exactly the same concept. The most satisfactory way, in my opinion, is that of reducing the concept of finite procedure to that of a machine with a finite number of parts, as has been done by the British mathematician Turing. As for the philosophical consequences of the results under consideration, I don’t think they have ever been adequately discussed or only taken notice of.
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- 7.This last hypothesis can be replaced by consistency (as shown by Rosser in [“Extensions of some theorems of Gödel and Church”, Jrn. Symb. Logic I, pp. 87–91]) but the undecidable propositions then have a slightly more complicated structure. Moreover the hypothesis must be added that the axioms imply the primitive propositions… [?] addition and multiplication and <.Google Scholar
- 10.If he only says “I believe I shall be able to perceive one after the other to be true” (where their number is supposed to be infinite) he does not contradict himself (see below).Google Scholar
- 15.K. Menger’s “… [?]” (cnf. Blatter f. d. Phil 4 (1930) p. 323) if taken in the strictest sense would lead to such an attitude since according to it the only meaningful mathematical proposition (i.e. in any term the only ones belonging to mathematics proper) would be those that assert that such and such a conclusion can be drawn from such and such axioms and rules of inference in such and such manner. This however is a proposition of exactly the same logical character as 2 + 2 = 4. Some of the undesirable consequences of this standpoint are the following: A negative proposition to the effect that the conclusion B cannot be drawn from the axioms and rule A would not belong to mathematics proper. Hence nothing could be known about it except perhaps that it follows from certain other axioms and rules. However a proof that it does so follow (since these other axioms and rules again are arbitrary) would in no way exclude the possibility that (in spite of the formal proof to the contrary) a derivation of B from A might some day be accomplished. For the same reason also the usual inductive proof for a + b = b + a would not exclude the possibility of discovering two integers not satisfying this equation.Google Scholar
- 23.[[I.e. these conventions must not refer to any extralinguistic objects (as does a demonstrat. [ive?] definition), but must state rules about the meaning or… [?] of symbolic expressions solely on the basis of their outward structure. Moreover of course these rules must be such that they do not imply the truth or falsehood of any factual proposition (since in that case they could certainly not be called void of content nor syntactical). It is to be noted that if the term “syntactical rule” is understood in this generality the view under consideration includes as a special elaboration of it the for-malistic foundation of mathematics. Since according to the latter mathematics is based solely on certain syntactical rules of the form: propositions of such and such structure are true (the axioms) and: if propositions of… structure are true, then such and such other propositions are also true. And moreover the consistency proof, as can easily be seen, gives the consequence that these rules are void of content in so far as they imply no factual propositions. On the other hand also vice versa it will turn out below that the feasibility of the nominalistic program implies the feasibility of the for-malistic program. It may be doubted whether this (nominalistic) view should at all be subsumed under the view considering mathematics to be a free creation of the mind, because it denies altogether the existence of mathematical objects. However the relatedness between the two is extremely close since also under the other view the so called existence of mathematical objects consists solely in their being constructed in thought and nominalists would not deny that we actually imagine (non existent) objects behind the mathematical symbols and that these subjective ideas might even furnish the guiding principle in the choice of the syntactical rules. For very lucid expositions of the philosophical aspect of this nominalistic view cnf. H. Hahn, Act. Sci. et ind. 226 (1935) or R. Carnap, Act. Sci. 291 (1935), Erk. 5 (1935) p. 30.Google Scholar
- 25.[[Cnf. Ramsey F. P. Proc. Lond. Math. Soc. II ser 25 (1926) p. 368 p. 382, Carnap R. Log. Synt. of Lang. 1937 p 39 and 110 and 182. It is worth mentioning that Ramsey even succeeds in reducing them to explicit tautologies a = a by means of explicit definitions but at the expense of admitting propositions of infinite (and even transfinite) length which of course entails the necessity of presupposing transfinite set theory in order to be able deal with these infinite entities. Carnap confines himself to propositions of finite length but instead has to consider infinite sets, sets of sets, etc. of these finite propositions.Google Scholar
- 33.Cnf. G. Darboux, Eloges académ. et discours, 1912, p. 142. The passage quoted continues as follows: “qui ne semblent distincts qu’à cause de la faiblesse de notre esprit qui ne sont pour une pensée plus puissante qu’une seule et même chose et dont la synthèse se révèle partiellement dans cette merveilleuse correspondance entre la mathématique abstraite d’une part, l’Astronomie et toutes les branches de la physique de l’autre.” So here Hermite seems to turn toward Aristotelian realism. However he does so only figuratively since Platonism remains the only conception understandable for the human mind.Google Scholar