## Abstract

This paper pursues two goals. Its first goal is to clear up the “identity problem” faced by the structuralist interpretation of mathematics. Its second goal, through the consideration of examples coming in particular from the theory of permutations, is to examine cases of *mis*understandings in mathematics fit to cast some light on mathematical understanding in general. The common thread shared by these two goals is the notion of setting. The study of a mathematical object almost always goes together with the choice of a particular setting, and the understanding of the workings of mathematical settings is an essential component of mathematical knowledge. It is claimed that the recognition of mathematical settings, as features distinct from both mathematical structures and the systems which instantiate those structures, allows one to classify most of understandable misunderstandings in mathematics, and also to solve the identity problem.

## Keywords

*Ante rem*structuralism Identity problem Theory of permutations Mathematical settings Labels Parameters Mathematical cognition Misunderstandings in mathematics

## Notes

### Acknowledgements

I wish to thank Denis-Charles Cisinski, Étienne Fieux, Gerhard Heinzmann and Marco Panza, as well as anonymous referees, for very valuable suggestions and comments.

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