[JohnVideogames]

I love persistent homology; it's such a weirdo bit of maths that topologists keep trying to make useful. Similarly its bigger cousin, Topological Data Analysis, which as far as I can tell is used by people who love weirdo maths but also want to be paid.

I don't see how much good the persistent topology adds here, though. Why bother to do the abstract birth-death diagram and measure the variance in lifetime of these simplices, when the variance and median of the areas in the Voronoi diagram will give you a much easier and more interpretable result?

[vouaobrasil]

> I don't see how much good the persistent topology adds here, though. Why bother to do the abstract birth-death diagram and measure the variance in lifetime of these simplices, when the variance and median of the areas in the Voronoi diagram will give you a much easier and more interpretable result?

Well, my personal feeling as someone who has published math papers is that there is a bit of a pressure in mathematics to continually do things in new ways, even if they are not the best ways of doing things. Topological data analysis seems to be of that sort.

Of course, there could be debate here...but I do think that a lot of mathematics has gone past the point of diminishing returns when it comes to its usefulness or even artistic elegance as an abstract art of the intellect. Yet, the show must go on I suppose.

[xanderlewis]

> a lot of mathematics has gone past the point of diminishing returns when it comes to its usefulness or even artistic elegance as an abstract art of the intellect.

This reads like a quote from 200 years ago.

[vouaobrasil]

Just because something might have been incorrect 200 years ago does not mean it is not correct now. At some point, the false claims of diminishing returns will turn out to be true. It is foolishness to believe in an infinite supply of wealth from anything, including scientific research.

[xanderlewis]

I didn’t claim it was incorrect. I was just observing that it sounded like an echo of the past, which it does.

…however, surely if we’re reaching some sort of plateau in mathematical understanding, it’s the end of all research — and that is slightly hard to believe.

Do you think we’re nearing a proof of the Collatz conjecture? Or is the theory we have just totally inadequate?

[vouaobrasil]

> Do you think we’re nearing a proof of the Collatz conjecture? Or is the theory we have just totally inadequate?

Well, Terrence Tao made a recent advance towards it. It's hard no doubt, but I don't think recent theory is inadequate, except maybe that no one has found any structural link to a body of more advanced math like it was done with Fermat's last theorem, which is why some people think we need new theory to solve it.

Also, I definitely didn't mean to imply we are reaching a plateau of mathematical understanding: there's still a long way to go in proof theory, K-theory, Langlands (a better understanding of orbital integerals, how to compute them symbolically...), nonabelian cohomology even (lots of stuff there seems ad-hoc like definitions of higher nonabelian cohomoogy groups).

What I do mean to imply is that it is getting to the point where none of the discovers at that level really lead BACK to anything that the vast majority of scientists or even math graduate students have much interest in. That's different in cases like evolutionary biology: even the advanced discoveries can still be tied back to things that are interesting for the "every day scientist".

Math has lost that link. It's not running out of problems, but it's running out of interesting problems IMO.

[lencastre]

Meet my friend “it’s what you know for sure that it isn’t so (…)”

We are clueless of our ignorance, and of how much there is still to discover in all domains of science.

IMHO I think we are accelerating discovery and we do not know yet what to do with all the things we invent and discover.

[vouaobrasil]

> We are clueless of our ignorance, and of how much there is still to discover in all domains of science.

Perhaps, but I do not believe we SHOULD discover that knowledge. So far, we have not had a great track record of developing wisdom along side that.

> IMHO I think we are accelerating discovery and we do not know yet what to do with all the things we invent and discover.

IMHO you are right and the knowledge is simply too powerful for us.