Boston resident here, so I thought I’d add some more points and further emphasize some things.

The bike infrastructure is

**really**good, and rapidly improving. In fact, there’s so much bike infrastructure that I want to make the converse warning: if you are a nervous driver, driving around here can be terrifying because of the bikers.The winters can be quite brutal (though they seem to be getting milder). And since Boston is way too far east for its timezone, this means that the winter sun sets very early (think ~4:30pm).

New England in general, and Boston in particular, is very lovely. If you like the European town aesthetic, this is probably the closest you can get in the U.S.

The food scene is pretty bad—food which is both good and cheap basically doesn’t exist.

People here are very young, especially when all the students are in town. Whenever I leave Boston, I’m shocked at how old the people are.

Marijuana is legal here. However, the dispensaries can be inconvenient to get to: none have opened yet in Boston or Cambridge.

I really love living here, and almost everyone I know also likes living here. The exceptions tend to be Californians, though. Did I mention how brutal the winter is?

Whatever those Intangible Qualities of a Happy Place are, Boston has them. I’m not sure what gives Boston this feel; I think it’s some mixture of excellent green space, good walkability, a sense of history, small-town aesthetic blended with big-city resources, and generally well-educated and competent populace. Think of the anxious feeling you get when you feel like the world is falling apart and there are a million little things coming apart at the seams—the felt sense of Boston (at least for me) is the polar opposite of that.

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.