1. I only need normal congruence
2. I only need perfect information games
Under problems that are solvable using these two assumptions, there is little benefit in tying proofs back to an axiomatic basis. Once you drop one of these two assumptions, proofs get much harder and a solid foundation gets more important.
But this is very different to suggesting that most regular mathematics will switch to using formal proofs. There's a big ergonomics gap at the moment.
An analogy could be to look how pure mathematicians look down on applied mathematicians' work, the applied mathematicians don't care, they just get on with their own standards. You need regular mathematicians to choose to switch over en masse, what will compel them to do that?
Which is happening right now — and the younger mathematicians who are supporting those efforts (and more broadly, things like Lean libraries) are gaining the experience while making the ergonomic changes.
That is, the person you’re replying to isn’t unaware of the historic problems: they’re pointing out that migration is starting now with early adopters like Tao and Scholze.
But let's say they make huge amounts of progress - they might improve the ergonomics enough that the formal foundations of mathematics will be brought into mathematics departments as a standard, legitimate and popular subject. But I still can't see how this would affect most mathematics.
It won't, in the same way that introducing physics didn't affect philosophers. They were still free to do philosophy. But most new innovation and innovators came up doing physics instead - and that's been where all the growth had been for centuries.
In the end ungrounded theory is better than nothing. But grounded theory beats out ungrounded theory hands down.
In the old days there were philosophers. They sought truth through thought and wisdom. Some started to measure reality and compare thought against that measurement, but real philosophers paid little heed for it didn't matter.
Testing their theory became known as physics. It turned out the testing against reality can be used without the theory, which we call engineering. We still have philosophers and they still pay little heed to measurement.
There is a similar disaster inbound for mathematicians. Formal mathematics is tedious and mostly only shows things that you were pretty sure were true anyway, so why bother.
It turns out the physicists and engineers have unsportingly built machines which are really, really good at the tedious bookwork required by formal mathematics. That is going to go exactly the same way that introducing measurement to philosophy went for the philosophers.
> It turns out the physicists and engineers have unsportingly built machines which are really, really good at the tedious bookwork required by formal mathematics. That is going to go exactly the same way that introducing measurement to philosophy went for the philosophers.
Formalized mathematics is just a very small subset of what mathematicians work on: in my observation people who work on (computer) formalized proofs often rather sit in CS departments.
Social Processes and Proofs of Theorems and Programs
The definition of a valid mathematical proof that I've heard mathematicians use is if it convinces other mathematicians. I think there's integrity in depth in mathematical proofs for lots of reasons, tying back to an axiomatic basis is a lot of extra work for likely no benefit.
On formal mathematics, almost no mathematicians pay attention to this afaik. It's something that lives in philosophy departments, not mathematics.