Comment by ngriffiths | Hacker News Reader

Comment by ngriffiths | Hacker News Reader

> We might say that generating ideas is the real intellectual work of mathematics.

The article talks about how many new ideas are relatively worthless and the real goal is to find the "concepts that 'carve nature at its joints.'" I think this is the crux of the whole thing and I haven't seen a satisfying discussion of it anywhere.

I mean, FLT is mentioned. Is that an accessible proof to humans? Is it full of these high value, refined concepts or is it more like a bunch of little hacks that at least dozens if not hundreds of people randomly stumbled upon, in an all out attempt to solve one of the most famous math problems?

I'm not totally convinced what value math concepts have beyond "you can use them to solve even more math problems." I really want to believe there is. But if not, it's just a pure benefit to have faster ways to solve them, no?

There are definitely some results which have this "a bunch of little hacks" quality you're describing, and while opinions differ I share your intuition that there's something a little disappointing about solving a big problem that way.

But I think FLT is about as far as one can get from that situation! Wiles's work was the culmination of centuries of theory-building work, and the concepts that were developed over that time are far more important than FLT; the thing Wiles actually proved (a special case of something called the "Modularity Theorem", the full version of which was proved a bit later) is itself much more valuable to human understanding of mathematics than FLT. It's certainly very cool that it can be used to answer such a simple question that was open for so long, and it makes for a great headline, but I think if you asked number theorists working in the area they would almost all tell you that they're much more grateful for the theory that came out of this quest than for the mere fact that the quest was completed.

Agreed! There has not been enough appreciation for re-organization efforts of the dependency-tree of knowledge. Now that expansion of the frontier has been somewhat trivialized, it should be an indication that our priorities have not been entirely wise.

The value of these math concepts is in their explanatory power. We want to understand! Insofar as we want to maximize the diffusion of powerful thought-tools, ironing out the wrinkles at the frontier is needed to be able to fold everything up neatly. The benefit we should seek is wide diffusion of powerful theories, and the folding and the neatness has been long undervalued.

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