theorem (being a solution to Hilbert’s 17th problem13). It can be written as a It seems that Hilbert would be satisfied by such results!
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1 Detlefsen (1990, p. 374) writes that “Hilbert did want to preserve classical mathematics, but this was not for him an end in itself. What he valued in classical mathematics was its efficiency (including its psychological naturalness) as a means of locating the truth of real or finitary mathematics. Hence, any alternative to classical mathematics having the same benefits would presumably have been equally welcome to Hilbert”.
2 ,,Schon Kant hat gelehrt – und zwar bildet dies einen integrierenden Bestandteil seiner Lehre –, dass die Mathematik über einen unabhängig von aller Logik gesicherten Inhalt verfügt und daher nie und nimmer allein durch Logik begründet werden kann, weshalb auch die Bestrebungen von Frege und Dedekind scheitern mußten. Vielmehr ist als Vorbedingung für die Anwendung logischer Schlüsse und für die Betätigung logischer Operationen schon etwas in der Vorstellung gegeben: gewisse, außer-logische konkrete Objekte, die anschaulich als unmittelbares Erlebnis vor allem Denken da sind. Soll das logische Schließen sicher sein, so müssen sich diese Objekte vollkommen in allen Teilen überblicken lassen und ihre Aufweisung, ihre Unterscheidung, ihr Aufeinanderfolgen oder Nebeneinandergereihtsein ist mit den Objekten zugleich unmittelbar anschaulich gegeben als etwas, das sich nicht noch auf etwas anderes reduzieren läßt oder einer Reduktion bedarf. Dies ist die philosophische Grundeinstellung, die ich für die Mathematik wie überhaupt zu allem wissenschaftlichen Denken, Verstehen und Mitteilen für erforderlich halte. Und insbesondere in der Mathematik sind Gegenstand unserer Betrachtung die konkreten Zeichen selbst, deren Gestalt unserer Einstellung zufolge unmittelbar deutlich und wiedererkennbar ist”.
3 In some of Hilbert’s publications (cf., e.g. Hilbert 1926, 1927) both aspects are stressed but usually (cf. Hilbert and Bernays 1934–1939) the one-sided emphasis is put on the consistency problem.
4 Both those aspects are interconnected – as was indicated by G. Kreisel. He showed that if φ is a
5 Hilbert rejected the opinion that Gödel’s results showed the non-executability of his programme. He claimed that they have shown “only that for more advanced consistency proofs one must use the finitistic standpoint in a deeper way than is necessary for the consideration of elementary formalisms” (cf. Hilbert and Bernays 1934–1939, vol. I). Gödel wrote in (1931) that “Theorem XI [i.e. Gödel’s second theorem for arithmetic P where P denotes the arithmetic of Peano extended by simple type theory – my remark, R.M.] (and the corresponding results for M and A) [where M is the set theory and A is the analysis – my remark, R.M.] do not contradict Hilbert’s formalistic viewpoint. For this viewpoint presupposes only the existence of a consistency proof in which nothing but finitary means of proof is used, and it is conceivable that there exist finitary proofs that cannot be expressed in the formalism of P (or Mor A)”.
6 It seems that Bernays was among the first who recognized this need. He wrote: “It thus became apparent that the ‘finite Standpunkt’ is not the only alternative to classical ways of reasoning and is not necessary implied by the idea of proof theory. An enlarging of the methods of proof theory was therefore suggested: instead of a restriction to finitist methods of reasoning, it was required only that the arguments be of a constructive character, allowing us to deal with more general forms of inference” (cf. Bernays 1967, p. 502).
7 For the description of PRA see, e.g. Smoryński (1977).
8 This was first observed by Hilbert and Bernays. Weyl (1918) had shown that a substantial part of ordinary mathematics can be developed within a certain “predicative” subsystem of Z2 (allowing ω-iterated arithmetical definability).
9 Recall that H. Poincaré saw the source of antinomies in mathematics just in impredicativity and therefore demanded a restriction to predicative methods only.
10 Drake claims even that the implications of the results of reverse mathematics “make much of what was written in the past on the philosophy of mathematics, obsolete” (cf. Drake 1989).
11 For the definition of and basic information on the arithmetical and analytical hierarchies of formulas and relations see, e.g. Shoenfield (1967).
12 Denote by WO(α) the sentence stating that the ordinal α is a well-ordered set. One can prove in RCA0 that WO(ωω) is equivalent to Hilbert’s basis theorem. But the sentence WO (ωω) is incomparable with WKL0.On the other hand, for any given natural number n one can prove in RCA0 that WO(ωn) is equivalent to Hilbert’s basis theorem for n. What more, WO(ωn) is provable in RCA0. So we have here certain analogy with the ω–incompleteness of Peano arithmetic (cf. Gödel 1931, see also Mendelson 1970).
13 Hilbert asked in his 17th problem “whether every definite form [of any number of variables with real coefficients – my remark, R.M.] may not be expressed as a quotient of sums of squares of forms” (cf. Hilbert 1901, see also Browder 1976). Recall that a form is called “definite” if it becomes negative for no real values of the variables. In 1926, Artin answered this question positively.
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On the Distinction Proof–Truth in Mathematics
Concepts of proof and truth are (even in mathematics) ambiguous. It is commonly accepted that proof is the ultimate warrant for a mathematical proposition, that proof is a source of truth in mathematics. One can say that a proposition A is true if it holds in a considered structure or if we can prove it. But what is a proof? And what is truth?
The axiomatic method was considered (since Plato, Aristotle and Euclid) to be the best method to justify and to organize mathematical knowledge. The first mature and most representative example of its usage in mathematics were Elements of Euclid. They established a pattern of a scientific theory and a paradigm in mathematics. Since Euclid till the end of the 19th century, mathematics was developed as an axiomatic (in fact rather a quasi-axiomatic) theory based on axioms and postulates. Proofs of theorems contained several gaps – in fact the lists of axioms and postulates were not complete, one freely
