TITLE: What is the endgoal of formalising mathematics?
QUESTION [29 upvotes]: Recently, I've become interested in proof assistants such as Lean, Coq, Isabelle, and the drive from many mathematicians (Kevin Buzzard, Tom Hales, Metamath, etc) to formalise all of mathematics in such a system. I am very sympathetic to some of the aims of this project, such as finding mistakes in published proofs, and definitively verifying complex and controversial proofs [Kepler conjecture (Hales), four-colour theorem (Appel/Haken), Fermat's last theorem (Wiles), ...].
However, I'd like to understand more about the ultimate goals of this project, and the vision for achieving a computer formalisation of all of mathematics. I understand a good portion of core (undergraduate) mathematics has been formalised. Yet, the bulk of research-level mathematics remains unformalised, and it seems to me that we are currently creating new mathematics at a far greater rate than we are formalising mathematics. Are we always going to be "chasing our tails" here? What needs to change so that we can catch up and achieve a complete formalisation of mathematics?
Another issue is, of course, new mathematics is being created all the time. If we achieve complete formalisation, how do we sustain this? Will there be "professional formalisers" whose job it is to formalise the recent results of mathematicians? Will mathematicians themselves be expected to formalise their results as they are proved? What role will proof assistants play in 20 (or 50, or 100) years?

REPLY [16 votes]: Other answers address several aspects of your question, but to add a point not mentioned yet — you write:

it seems to me that we are currently creating new mathematics at a far greater rate than we are formalising mathematics. Are we always going to be "chasing our tails" here? What needs to change so that we can catch up and achieve a complete formalisation of mathematics?
Another issue is, of course, new mathematics is being created all the time. If we achieve complete formalisation, how do we sustain this? Will there be "professional formalisers" whose job it is to formalise the recent results of mathematicians? Will mathematicians themselves be expected to formalise their results as they are proved?

Certainly, at the moment, formalising maths is very laborious and time-consuming.  But I think most people who work the formalisation hope that it won’t stay so bad!  It’s been getting steadily better as proof assistants and their libraries have progressed — “better” both in terms of how quickly experienced users can cover ground, and in terms of the learning curve for new users.  This improvement hasn’t perhaps been as fast as we’d like, or might have hoped — but it’s been steady, and continuing.
If usability continues to improve, then it’s reasonable to imagine a future proof assistant with a role comparable to LaTeX today.  Typesetting our work in LaTeX takes a bit of work, and has a bit of a learning curve — but most mathematicians accept the time and effort as worthwhile for the payoff of good typesetting.  A very few mathematicians outsource this to professional typesetters; a larger minority leave it to their co-authors when possible.  But it’s part of the expectations of the field, and there’s no problem of “tail-chasing” — no question of “can we typeset mathematics as fast as we can produce it?”, or of requiring major grant support to get work typeset.  This is one possible future that I and some others in formalisation envisage, and that it seems reasonable to hope for in a timescale of a few decades.
One likely difference from LaTeX is that this could well arrive in some subfields of maths sooner than others, since (so far) some topics seem much more amenable to formalisation.