Mathematics as Dialogue

Abstract

This chapter addresses Lakatos’ version of constructivism. Lakatos shows how mathematical notions, theorems and proofs become constructed through a dialogical process. He illustrates how the logic of mathematical discovery operates as an interactive process within the mathematical community. In opposition to logical positivism, Popper, who inspired Lakatos profoundly, moved from being interested in verification to being interested in falsification. In parallel with this, he also turned his attention from “the context of justification” to “the contexts of discovery.” This way scepticism and fallibilism come into focus. Lakatos reinterprets Popper’s ideas, and this way he formulates fallibilism as a position in the philosophy of mathematics.

Lakatos introduces a neo-empiricist perspective on mathematics, which he refers to as quasi-empiricism in the sense that mathematics to some extent resembles natural sciences. Mathematics is about concept development, and it is driven forward by particular observations concerning conceptual connections. With reference to Euler’s polyhedron theorem, Lakatos provides a detailed specification of how this development takes place, which he condenses in terms of the method of proofs and refutations. This method includes many features, as for instance the emergence of proof-generated concepts. Lakatos finds that mathematical statements have their origin in experience, but his naturalism differs from classic versions, as for instance suggested by Mill. Instead of searching for an empirical basis for mathematical notions and theorems in terms of sensory experiences, Lakatos considers the basis of mathematical reasoning to be quasi-empirical objects.