# Formalism and Pluralism

## Abstract

This chapter introduces the reader to pluralism from the starting point of formalism. Formalism is in some ways the closest position to pluralism. The characterisation of formalism is taken from Detlefsen. Adding support to the pluralist’s argument in Chap. 3 against Maddy, about the philosophical conceptions of mathematicians not always being realist, we give support to the claim that many mathematicians see themselves as formalist. We also find support for this claim from the practice of mathematicians. We look at three test cases: the classification of finite simple groups, renormalisation and Lobachevsky’s model for indefinite integrals. With this *de dicto* and *de re* evidence, we then argue that pluralism reaches beyond formalism, and better fits the *de dicto* and *de re* evidence. In particular, we argue for a pluralism in methodology which is not permitted under the structures of formalism, as we characterise it.

## Keywords

Mathematical Practice Hyperbolic Geometry Proof Theory Finite Simple Group Adaptive Logic## References

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