Formal Proof Systems Reveal Overlooked Ambiguities in Advanced Mathematics

Table Of Links

Abstract

1. Acknowledgements & Introduction

2. Universal properties

3. Products in practice

4. Universal properties in algebraic geometry

5. The problem with Grothendieck’s use of equality.

6. More on “canonical” maps

7. Canonical isomorphisms in more advanced mathematics

8. Summary And References

Summary And References

Whilst I am not making any claims about errors in the literature or holes in arguments which cannot be filled in after some work, I am arguing that these holes do exist, and that new mathematics might need to be done by formalisers in order to fill in these holes in an efficient way. The holes are of two kinds. Firstly there is the issue of people making constructions or proving theorems which make essential use of a model of a mathematical object which is defined up to unique isomorphism; the hole here is that it needs to be checked that the argument does not depend on the explicit details of the model.

\ Mathematicians are well aware of this when it comes to, say, picking a basis for a vector space and then checking that nothing important depended on the choice, or picking a representative for an equivalence class and then checking that nothing important depends on the representative. However they seem to be less careful when doing more advanced mathematics, confusing “a” localisation with “the” localisation or “a” pullback with “the” pullback, and leaving to the reader the details of checking that many diagrams commute.

\ One useful trick is to abuse the equality symbol, making it mean something which it does not mean; this can trick the reader into thinking that nothing needs to be checked. Sometimes such checks can be surprisingly painful, and it may be easier to restructure a mathematical argument than to actually make these checks. The second kind of hole is the issue of various maps (like boundary maps in exact sequences) being regarded as “canonical” where now they are in fact not unique, and there are implicit choices of sign being made.

\ Unfortunately it is not at all difficult to point to explicit examples in the literature where an author does not state precisely which convention they are using when it comes to things like the theory of Shimura varieties, or homological algebra. This puts an unnecessary burden on the careful mathematician (for example Conrad, or a computer theorem prover) who is attempting to use or verify the work. Both of these issues have shown up in my formalisation work, and I expect them to show up more often as we go deeper into the formalisation of modern mathematics.

References

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:::info Author: KEVIN BUZZARD

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:::info This paper is available on arxiv under CC BY 4.0 DEED license.

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