What is the point of computers? A question for pure mathematicians

Abstract

We discuss the idea that computers might soon help mathematicians to prove theorems in areas where they have not previously been useful. Furthermore we argue that these same computer tools will also help us in the communication and teaching of mathematics. Contents 1.5. The Kepler conjecture 10 1.6. Perfectoid spaces 11 1.7. Condensed mathematics 12 1.8. Other results 13 2. mathlib 14 3. A brief guide to type theory 15 3.1. What is a type? 16 3.2. Inductive types 17 3.3. Dependent types 18 3.4. Examples 19 3.5. Foundations 20 4. The Future 21 4.1. A new kind of mathematical document 21 4.2. Semantic search in a mathematical database 21 4.3. Checking proofs 23 4.4. Teaching 24 4.5. Other ideas 24 References 24

Formalising Szemerédi’s Regularity Lemma and Roth’s Theorem on Arithmetic Progressions in Isabelle/HOL

“…The final stage of our proof matches Step (3), showing that this cleaned graph must be triangle-free. Again, this required some fiddly counting reasoning using the bounds established in Step (2). To help structure this reasoning, we show that the new graph obtained is regular, dense, and decent (as per our earlier Isabelle definitions), with Bell and Grodzicki's notes proving particularly useful here.…”

“…Szemerédi's Regularity Lemma and Roth's Theorem on Arithmetic Progressions are regarded as major results and our announcement of their formalisation was greeted enthusiastically [2]. And yet, the formalisation was almost straightforward, the main difficulties stemming from ambiguities in our sources compounded by our unwise refusal to consult available experts.…”

Finite field models of Raleigh-Akiyama polynomials for Hecke groups

“…Let tφ ´1, φ 0 , ...u " t1, 24, ...u be the McKay-Thompson series of class 4A [21]. For some degree 2n, irreducible, monic polynomial γ n pxq in Qrxs we have 6 :…”

“…5 [3], notebook "conjecture 1.nb". 6 [3], notebook "conjecture 1 clause 2.ipynb' 7 [3], notebook "conjecture 1 cause 3.nb".…”

“…Coercion is a concept from type theory. For a recent discussion, see [6]. For SageMath's own explanations of its coercion routines, see [22] and [11].…”