So Andrew Wiles’s genius was in showing there were no unexpected obstructions for the “likely” outcome to be true. That’s why the proof is so hard: he was trying to prove something very “likely”, and show an absence of structure, rather than a presence, without knowing what that structure could be.

This is a poor description of Wiles’s proof; in fact, I would call it diametrically wrong. Wiles proved the presence of a very rigid structure—not the absence—and the presence of this structure implied FLT via the work of other mathematicians.

I don’t have a great understanding of the broader point you are making, so I don’t know how big an issue this mistake presents. However, be aware that the paradigm you’ve extracted from the ideas in this post has lead to at least one incorrect prediction.

I’ll try to explain how Wiles’s proof diverges from your model of it by way of analogy. Suppose that we instead wanted to prove Fermat’s first theorem:

Fermat’s first theorem: For every even integer n≥4 there are no nontrivial integer solutions to the equation xn+yn=−zn.

Further suppose that in our world, mathematicians know about the notion of positivity and absolute values, but the proof of the following fact has long evaded them.

Positivity conjecture: For every integer n, we have n2≥0.

The positivity conjecture is a very important structural fact about the integers. And it immediately implies Fermat’s first theorem (since the left-hand side must be positive, but the right-hand side must be negative unless x,y,z are all 0). So Fermat’s first theorem follows from an important structural fact.

However, in our supposed world, mathematicians don’t have access to the positivity conjecture. They might perform the exact same analysis in your post (it goes through verbatim!), and conclude that if you check Fermat’s first theorem for enough n, then it is probable to be true. However, it is not true that the proof of FFT via the positivity conjecture is “proving an absence of structure”—quite the opposite!

The analogue of the positivity conjecture in the real world is the Modularity theorem. This is what Wiles actually proved, and it was already known that the Modularity theorem implies FLT. And as with the positivity conjecture, the Modularity theorem is a very powerful structural result. To give a slogan, it says that every elliptic curve over Q is “modular,” meaning that it relates in an appropriate way to an object called a modular form.

Wiles proved the presence of a very rigid structure—not the absence—and the presence of this structure implied FLT via the work of other mathematicians.

If you say that “Wiles proved the Taniyama–Shimura conjecture” (for semistable elliptic curves), then I agree: he’s proved a very important structural result in mathematics.

If you say he proved Fermat’s last theorem, then I’d say he’s proved an important-but-probable lack of structure in mathematics.

So yeah, he proved the existence of structure in one area, and (hence) the absence of structure in another area.

And “to prove Fermat’s last theorem, you have to go via proving the Taniyama–Shimura conjecture”, is, to my mind, strong evidence for “proving lack of structure is hard”.

This is a poor description of Wiles’s proof; in fact, I would call it diametrically wrong. Wiles proved the

presenceof a very rigid structure—not the absence—and the presence of this structure implied FLT via the work of other mathematicians.I don’t have a great understanding of the broader point you are making, so I don’t know how big an issue this mistake presents. However, be aware that the paradigm you’ve extracted from the ideas in this post has lead to at least one incorrect prediction.

I’ll try to explain how Wiles’s proof diverges from your model of it by way of analogy. Suppose that we instead wanted to prove Fermat’s first theorem:

Fermat’s first theorem:For everyeveninteger n≥4 there are no nontrivial integer solutions to the equation xn+yn=−zn.Further suppose that in our world, mathematicians know about the notion of positivity and absolute values, but the proof of the following fact has long evaded them.

Positivity conjecture:For every integer n, we have n2≥0.The positivity conjecture is a very important structural fact about the integers. And it immediately implies Fermat’s first theorem (since the left-hand side must be positive, but the right-hand side must be negative unless x,y,z are all 0). So Fermat’s first theorem follows from an important structural fact.

However, in our supposed world, mathematicians don’t have access to the positivity conjecture. They might perform the exact same analysis in your post (it goes through verbatim!), and conclude that if you check Fermat’s first theorem for enough n, then it is probable to be true. However, it is

nottrue that the proof of FFT via the positivity conjecture is “proving an absence of structure”—quite the opposite!The analogue of the positivity conjecture in the real world is the Modularity theorem. This is what Wiles actually proved, and it was already known that the Modularity theorem implies FLT. And as with the positivity conjecture, the Modularity theorem is a very powerful

structuralresult. To give a slogan, it says that every elliptic curve over Q is “modular,” meaning that it relates in an appropriate way to an object called a modular form.If you say that “Wiles proved the Taniyama–Shimura conjecture” (for semistable elliptic curves), then I agree: he’s proved a very important structural result in mathematics.

If you say he proved Fermat’s last theorem, then I’d say he’s proved an important-but-probable

lack of structurein mathematics.So yeah, he proved the existence of structure in one area, and (hence) the absence of structure in another area.

And “to prove Fermat’s last theorem, you have to go via proving the Taniyama–Shimura conjecture”, is, to my mind, strong evidence for “proving lack of structure is hard”.