Proof-Theoretic Reduction As A Philosopher's Tool

Abstract

1. PROOF-THEORETIC REDUCTION AND HILBERT ’ S PROGRAM Hilbert’s program in the philosophy of mathematics comes in two parts. One part is a technical part. To carry out this part of the program one has to prove a certain technical result. The other part of the program is a philosophical part. It is concerned with philosophical questions that are the real aim of the program. To carry out this part one, basically, has to show why the technical part answers the philosophical questions one wanted to have answered. Hilbert probably thought that he had completed the philosophical part of his program, maybe up to a few details. What was left to do was the technical part. To carry it out one, roughly, had to give a precise axiomatization of mathematics and show that it is consistent on purely finitistic grounds. This would come down to giving a relative consistency proof of mathematics in finitist mathematics, or to give a proof-theoretic reduction of mathematics on to finitist mathematics (we will look at these notions in more detail soon). It is widely believed that Godel’s theorems showed that the technical part of Hilbert’s program could not be carried out. Godel’s theorems show that the consistency of arithmetic can not even be proven in arithmetic, not to speak of by finitistic means alone. So, the technical part of Hilbert’s program is hopeless, and since Hilbert’s program essentially relied on both the technical and the philosophical part, Hilbert’s program as a whole is hopeless. Justified as this attitude is, it is a bit unfortunate. It is unfortunate because it takes away too much attention from the philosophical part of Hilbert’s program. And this is unfortunate for two reasons. First, because it is not at all clear what the philosophical part of Hilbert’s program comes down to. What of philosophical importance about mathematics would Hilbert have shown if the technical part of his program would have worked? Secondly, because even though the technical part of Hilbert’s original program is hopeless, there are modified versions of Hilbert’s program that do not have the technical difficulties that Hilbert’s original program has. In fact, for so-called relativized versions of Hilbert’s program the technical parts are known to work. 1 However, the people working on the technical results related to relativized versions of Hilbert’s program have