Doing and Showing

TL;DR

This paper critiques the traditional axiomatic approach in formal mathematics, arguing it inadequately captures the informal understanding of mathematical theories and proposing a need for a new conceptual framework.

Contribution

It identifies limitations of the current axiomatic method and suggests developing an alternative notion of mathematical theory to bridge the gap between formal and informal mathematics.

Findings

Current axiomatic methods are insufficient for capturing informal mathematical reasoning

A new conceptual framework for mathematical theories is proposed

The gap between formal and informal mathematics is attributed to the inadequacy of existing notions

Abstract

The persisting gap between the formal and the informal mathematics is due to an inadequate notion of mathematical theory behind the current formalization techniques. I mean the (informal) notion of axiomatic theory according to which a mathematical theory consists of a set of axioms and further theorems deduced from these axioms according to certain rules of logical inference. Thus the usual notion of axiomatic method is inadequate and needs a replacement.