Given that it was a research frontier where arguments assume an educated audience, it's probably very difficult to formalize.
The purpose of a proof is to show yourself and someone else why something is true. I don’t know what it would mean to be writing them for computers to verify. Unless the only thing you are interested in is y/n
Human verification can never be ruled out entirely with these sorts of systems: you always have to check that the definitions used in the final statement mean what you think they mean, and that all of the base axioms are acceptable.
And of course, there's always the possibility of bugs in the kernel. I even recently found a bug [0] in a verifier for Metamath, which is designed to be so simple that its only built-in logic is typed string substitution. But such bugs should hopefully be unlikely in non-adversarial settings.
In other words, the chance that we find gaps and mistakes in the written proof? 100% - the chance we find out it's false due to sloppy logic? 0%.
There are multiple reasons for formalizing the proof in Lean ... see https://github.com/ImperialCollegeLondon/FLT/blob/main/GENER...
P.S. "So that patching is exactly what I’m referring to."
No, it isn't.
"The mathematicians can see the idea that’s true"
Mathematicians could not "see" that FLT was true, and they could not "see" that Wiles' original proof demonstrated it because it didn't. His original flawed proof showed how certain tools could be used, but it didn't establish the truth of FLT. There long had been speculation that it was undecidable, and it might still have been until Wiles and Taylor provided a correct proof.
From a previous comment by the same user: "The purpose of a proof is to show yourself and someone else why something is true."
The purpose of a proof of an assertion is to demonstrate that the assertion is true. Once that is done, the assertion can be treated as a theorem and other results can be built upon it.
The purpose for digitally formalizing a proof of a theorem that has already been accepted as proven by the mathematical community is multifold, as laid out at the link above.
Lean helps with none of that. It doesn’t help you find proof ideas and it doesn’t help you communicate them,
No. The purpose of math is to increase our understanding, not check off boxes.
In your model you might as well have a computer brute force generate logical statements and study those. Why would that be less valuable then an understanding of differential equations?
Some proof assistants contribute more directly to understanding by making proofs easier to study.
In fact, Buzzard has an "existence theorem" of this exact thing. Annals of Mathematics (one of the top mathematics journals) has published one paper proving a theorem, and another paper proving the opposite result of a theorem: https://www.andrew.cmu.edu/user/avigad/meetings/fomm2020/sli...
"It will not be the original Wiles/Taylor-Wiles proof, but rather a "newer" proof which takes into account more recent developments due to Khare-Wintenberger, Kisin and many other people."