It seems there is a wave of interest in formalized mathematics among professional mathematicians, driven by a very successful Lean mathlib project. There is a video of Kevin Buzzard’s talk about the project and an arXiv paper describing it. Kevin Buzzard is an algebraic number theorist and a professor of pure mathematics at Imperial College of London. In the summer 2018 he led a summer project with twenty undergraduates that were were learning Lean and formalizing mathematics in it. He also sometime uses Lean live coding while teaching classes and runs a Lean club at Imperial called the Xena Project. His article in the London Mathematical Society newsletter provides more info on this.

Lean is a programming language and software verification tool from Microsoft Research. I think Thomas Hales’ review is the best source on its strengths and weaknesses from the formalized mathematics point of view. Lean’s logic is calculus of inductive constructions (CIC, similar to what Coq is based on). So Lean mathlib formalizes mathematics in (dependent) type theory. Of course this is a bit unfortunate from my point of view as I am more of a set theory person. There is no doubt though that type theory is nowadays more popular (not to say successful) foundation in formalization efforts. Jeremy Avigad wrote a rare compelling explanation of why this is the case. I personally prefer the opinion of Lawrence Paulson, the original author of Isabelle and its ZF logic (he also implemented a formalization of ZFC set theory in Isabelle/HOL recently). Here is what he says:

I would guess that the amount of effort and funding going into type theory exceeds the amount that went into set theory by an order of magnitude if not two. It’s not unusual to encounter open hostility to set theory and classical logic combined with an air of moral superiority: “Oh, you aren’t constructive? And you don’t store proof objects? Really?” And I have seen “proof assistant” actually DEFINED as “a software system for doing mathematics in a constructive type theory. The academic interest simply isn’t there. Consider the huge achievements of the Mizar group and the minimal attention they have received.(…) I am certain that we would now have highly usable and flexible proof assistants based on some form of axiomatic set theory if this objective had enjoyed half the effort that has gone into type theory-based systems in the past 25 years.

In January 2020 there will be a conference on Formal Methods in Mathematics and Lean in Pittsburgh, Pennsylvania.

Another project related to formalized mathematics that I think is very interesting is Metamath Zero by Mario Carneiro. This is a formal proof verifier, heavily inspired by Metamath, but with a different architecture, some improvements in the metalogic and the ambition to be a “bootstraping theorem prover”, i.e. be able to verify itself. The details are in the arXiv paper, but below is the summary of what I think I understood from it.

Metamath Zero (MM0) separates the specification from the proofs, so there are two inputs to the verifier. The specification is a a human readable text written in mm0 language which is as generic as the Metamath language, i.e. it can specify logics from ZFC on FOL, through Quantum Logic(s), type theories like HOL or CIC, to the Intel x86 instruction set architecture. Anything, really. The proofs are provided separately in a binary whose format (mmb) is optimized for verification speed. This is the format that allowed MM0 to verify the material contained in the Metamath’s set.mm with its 23000 proofs in 195 ms on Intel i7 3.9GHz (the previous record was 0.7s). This part of the architecture can serve as independent backend of any theorem prover that is able to provide the mm0 specification and mmb proof binary.

An example of such theorem prover is included in the MM0 system, together with a proof language mm1 (similar to mm0) and an formal text authoring tool as a Visual Studio Code plugin implementing Language Server Protocol.

I don’t want to write about the MM0 metalogic as there is too much risk to say something wrong here. I only want to mention that the paper contains a very clear explanation of Tarski-Megill metalogic (used by Metamath) that replaces the notion of free and bound variables by the notion of distinct variable groups.

The bootstraping goal which I understand is work in progress consists of specifying Peano Arithmetic, then x86 instruction set architecture and the formal system of MM0 to get a complete specification of the working of the ELF binary file that executes the verifier.

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