The experimental effectiveness of mathematical proof

The aim of this paper is twofold. First, it is an attempt to give an answer to the famous essay of Eugene Wigner about the unreasonable effectiveness of mathematics in the natural sciences [25]. We will argue that mathematics are not only reasonably effective, but that they are also objectively effective in a sense that can be given a precise meaning. For that—and this is the second aim of this paper—we shall reconsider some aspects of Popper’s epistemology [23] in the light of recent advances of proof theory [8, 20], in order to clarify the interaction between pure mathematical reasoning (in the sense of a formal system) and the use of empirical hypotheses (in the sense of the natural sciences). The technical contribution of this paper is the proof-theoretic analysis of the problem (already evoked in [23]) of the experimental modus tollens, that deals with the combination of a formal proof of the implication U ⇒ V with an experimental falsification of V to get an experimental falsification of U in the case where the formulæ U and V express empirical theories in a sense close to Popper’s. We propose a practical solution to this problem based on Krivine’s theory of classical realizability [20], and describe a simple procedure to extract from a formal proof of U ⇒ V (formalized in classical second-order arithmetic) and a falsifying instance of V a computer program that performs a finite sequence of tests on the empirical theory U until it finds (in finite time) a falsifying instance of U .