Concrete Mathematics. Finitistic Approach to Foundations

摘要

We discuss the idea of concrete mathematics inspired by Hilbert's idea of finitistic mathematics as the part of mathematics not engaged into actual infinity. We explicate it as the part of mathematics based on △_2~0 arithmetical concepts. The explication is justified by equivalence of △_2~0 definability with algorithmic learnability (an epistemic argument) and with FM-representability (representability in finite models, an ontological argument). We show that the essential part of classical mathematics can be interpreted in the concrete framework. We claim that current mathematics is a social game of proving theorems on some axiomatic set theoretic • background. On the other hand, concrete mathematics is the reality on which our mathematical experience is based. This is what makes the game intersubjective. Nevertheless, this game is one of the most efficient methods of building our mathematical knowledge.