[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