Number theorist fears many proofs widely considered to be true are wrong

the more important thing is that the number of humans aware of the existence of this tech and the more actually use it the better society will be off in the long run

the following may seem totally impossible today but seems a lot less impossible to people who have been thinking about this for a long time: once sufficient mathematics has been translated, people can start adding physics postulates, one most physics has been inserted we can start formally verifying (up to validity of postulates on top of axioms) not just proper operation of digital circuits etc but also power plants and so on, and once interacting with proof systems becomes as widespread as alphabetic literacy, people might one day be able to formalize and vote on their requirements of society and policy, see predicted perverse incentives, or decide that policy can only be enacted when facets of their consequences can be proven. We might one day not vote on promises of strategy, also not vote on strategy, but rather on outcomes!

I know it sounds absurd.

I do agree that it's pretty hard to be rigorous without setting up all of this edifice, but I've also found that it's very easy (at least, as an applied mathematician) to take an idea and continue generalizing it until the original application becomes a very small corollary of the new statement—one that appears usually unmotivated and "magical."

In some sense, this is a wonderful thing: you can usually prove a much stronger statement using the same tools you used to prove the weaker one, but in some sense it makes many things unbelievably difficult to understand as a first or even second-time reader.

To give an example, I was reading a book on dissipative PDE-solving as an optimization problem. The book's main chapter starts with a theorem that is so general and so far removed from any one specific application that, even as someone who works on several related fields, I struggled to understand its significance. The book then spends the next several chapters showing how existence/uniqueness of solutions to a bunch of PDEs are all corollaries (with some work) of this theorem.

The problem I have is that the presentation would've been much clearer going the other way. Start with the easy-to-prove existence theorems for the PDEs you care about (which mostly relied on simple tools and could've been proven in a few lines) and build up the generalization to the "main theorem." There, the reader doesn't have to guess the meaning (and/or marvel at the "ingenuity") of the original statement since the motivation is perfectly natural and clear, and they certainly don't have to struggle through a proof without a clear ending. Plus, you usually get the nice side-benefit that, by the time you present the most general statement, the proof is almost always immediate ± some small tricks.

While I do think that math requires a large amount of definitions to work and to put everything on rigorous footing, I also think there's this kind of weird way of writing where definitions and proofs are presented as "magical" and then shown to be right or useful later, rather than the other way around. This is fine for a few things, but I've found that most presentations that work don't follow this template (yet most mathematical papers I've read sadly do).

For every complaint about a textbook, there is an equal and opposite complaint if it had done the opposite thing. At some point we have to accept that the underlying concepts inherently take time and effort to assimilate, that the easiest style to learn from is different for different people, and that the authors really did think about all these issues when they were writing. They clearly care about exposition, otherwise they would not have written a book in the first place, a thankless and financially useless task.