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Formalising published mathematics

Optional research direction: turning published mathematics into machine-checked proofs, using blueprints that link informal statements to formal ones, collaborating on large projects, and contributing to libraries such as Mathlib.

math.comp.formalising-published-mathematics

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Builds on

Formalising mathematics in Lean and MathlibType theory and propositions as typesReading the mathematical literature

Required prerequisites, all the way down: 22

You will learn to

  • Write a blueprint that breaks a published theorem into formalisable lemmas and records their dependencies.
  • Formalise a short published result, recording any gaps or implicit assumptions found along the way.
  • Describe completed large formalisations and what they did and did not verify.

Definition

  • A blueprint is a document linking each informal definition and lemma to its formal counterpart and recording the dependencies between them.
  • The trusted base of a formal proof is what must be trusted for the check to be meaningful: the proof checker's kernel, the axioms used and the formal statement.

Theorem

  • The Liquid Tensor Experiment formalised a central theorem of Clausen and Scholze's condensed mathematics in Lean; it was completed in 2022.
  • The Feit–Thompson odd order theorem was formalised in Coq by Gonthier and collaborators (2012).
  • Formalisation verifies the proof of the formal statement; whether that statement faithfully captures the published theorem must be checked by people.

Common misconceptions

  • Thinking formalisation makes peer review unnecessary.
  • Believing a formal proof follows the informal paper line by line; formal proofs often reorganise the argument substantially.

Optional discoveries

Learning pathways

Further reading