LeanTeach2020

Summer 2020 Illinois Geometry Lab (University of Illinois at Urbana-Champaign) + Uni-High research project

Interactive theorem proving in Lean : teaching mathematics to a computer

Formalizing Euclid's and Hilbert's axioms in Lean

• euclid.lean : Euclid's axioms
• euclid_props.lean : Euclid's propositions using Euclid's axioms
• hilbert.lean : Hilbert's axioms
• hilbert_props.lean : Euclid's propositions using Hilbert's axioms
• tarski.lean : Tarski's axioms

More mathematical details available at the wiki: IllioisWiki for LeanTeach

Project Goals:

Goal 0 : Teach Lean to Humans.

Humans involved : Alex Dolcos, Edward Kong, Lawrence Zhao, Nicholas Phillips, Vaibhav Karve.

We meet 9 hours per week to discuss math, philosophy and programming as it relates to formalizing mathematics in a theorem prover. We will be using Microsoft Research's Lean Theorem Prover and its Mathematical Library (mathlib) as our proof assistant for this. Each of us has experience programming in Python before this. We will explore how things work differently in a purely functional language. We will also learn about tactics in Lean. We will use Slack for text-based communication, Zoom for video meetings and CoCalc for sharing and collaboratively editing code.

Goal 1 : Teach Math to a Computer.

The idea is that we can take a piece of familiar mathematics and translate it into code that is acceptable to Lean. Lean already understands basic logic and does not accept incorrect mathematics as input.

We wanted to choose a topic in mathematics that,

• is known to us with a high degree of familiarity – because intuition comes in handy when we are stuck in a Math → Lean translation,
• is preferably already in an axiomatized form – we wanted to follow the workflow of Definitions → Axioms/Postulates → Propositions/Theorems,
• is not already in mathlib – this ruled out group theory, number theory, category theory ...

We settled on formalizing Axiomatic Geometry of three types:

• Euclid's axioms
• Hilbert's axioms
• Tarski's axioms (partial progress)