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Lógos and Máthēma 2

Studies in the Philosophy of Logic and Mathematics

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Roman Murawski

The volume consists of thirteen papers devoted to various problems of the philosophy of logic and mathematics. They can be divided into two groups. The first group contains papers devoted to some general problems of the philosophy of mathematics whereas the second group – papers devoted to the history of logic in Poland and to the work of Polish logicians and math-ematicians in the philosophy of mathematics and logic. Among considered problems are: meaning of reverse mathematics, proof in mathematics, the status of Church’s Thesis, phenomenology in the philosophy of mathematics, mathematics vs. theology, the problem of truth, philosophy of logic and mathematics in the interwar Poland.
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On the Philosophical Meaning of Reverse Mathematics

On the Philosophical Meaning of Reverse Mathematics

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The aim of this chapter is to discuss the meaning of some recent results in the foundations of mathematics – more exactly of the so-called reverse mathematics – for the philosophy of mathematics. In particular, we shall be interested in implications of those results for Hilbert’s program.

One of the reactions on the crisis in the foundations of mathematics on the turn of the 19th century was Hilbert’s program. Hilbert’s aim was to save the integrity of classical mathematics (dealingwith actual infinity) by showing that it is secure.1 He saw also the supra-mathematical significance of this issue. In 1926 he wrote: “The definite clarification of the nature of the infinite has become necessary, not merely for the special interests of the individual sciences, but for the honor of human understanding itself ”. Being first of all a mathematician, he “had little patience with philosophy, his own philosophy of mathematics being perhaps best described as naïve optimism – a faith in the mathematician’s ability to solve any problem he might set for himself ” (cf. Smoryński 1988).

Hilbert’s program of clarification and justification of mathematics was Kantian in character (cf. Detlefsen 1993). Following Kant, he claimed that the mathematician’s infinity does not correspond to anything in the physical world, that it is “an idea of pure reason” – as Kant used to say. On the other hand, Hilbert wrote in (1926):

Kant taught – and it is an integral part of his doctrine – that mathematics treats a subject matter which...

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