[ahhfgshando6698]

This is going to be a classic "computer programmer wants math to be like computers and doesn't get it" kind of take, but In my view the problem of mathematical rigour (or lack thereof) has only become worse over the 20th century and we certainly did not resolve any of the underlying issues in a foundational sense or a practical sense. In a practical sense, it's become much worse and we have many more layers now. In a foundational sense, we succeeded in giving up because we learned that we can in some sense pick and choose whatever is most convenient for our line of research.

That's probably okay if we view mathematics in the way this book (I have not read it, going based on the description here) advocates, as a sort of toy for playing with arguments. And I'm certainly not saying Math should ever be viewed as an empirical discipline nor constrained by that kind of thinking. But I don't think I'm the only one that takes one look at things in the realm of say higher category theory and thinks it's mostly playing word and symbol manipulation games, and lacks any real mathematical content that could not be discovered at a lower and more understandable (and less likely to produce new research) level of abstraction.

I guess I've sort of betrayed that I am pretty firmly a platonist in that respect so make of that what you will.

Like I said, this is not an unusual opinion for a computer person to have and I'm sure it's fairly annoying to any pure mathematician at this point. But I think it's still fair if we want to understand what turns certain people off of pursuing mathematics further.

[jonathanstrange]

I'm not a mathematician so anyone please correct me if I'm wrong, but isn't the point of category theory to provide abstractions so your proofs are very general and cover a lot at once? Of course, it needs to be very abstract to achieve that.

[bananaflag]

This is a classical example of what happens when people give up on rigour:

https://en.wikipedia.org/wiki/Italian_school_of_algebraic_ge...

TL;DR: They started producing false results.

Note that this was not about capital-F Foundations of Mathematics like (arguably) the foundational crisis of math that had its origins in the 19th century, but rather about lowercase-f foundations of a particular field, in this case algebraic geometry.

Weil's foundations of AG in the eponymous 1946 book were horrible and messy but they solved the issue (even today there are a lot of celebrated results that can only be found as expressed in Weil's language) and later in the 1960s Grothendieck provided the elegant language of schemes in which people generally learn and research AG today, and which helped prove long-standing problems like the Weil conjectures and (to some degree) FLT. Category theory was, in this case, essential to proving theorems about "real mathematical content" like numbers and points.

[francasso]

Is that really the case? I'm not an expert in AG, but I would imagine that any major result published in Weil's framework can either be trivially converted to schemes or it has been ported over by now