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Will Our Understanding of Math Deteriorate Over Time?

blog.computationalcomplexity.org

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Re: Will Our Understanding of Math Deteriorate Over Time?

#51

Post by tel »

GFK_of_xmaspast wrote: post #42
tel wrote: post #12

Earlier quoted context omitted.

I think this is an interesting perspective to think about. I certainly remember a long time where I felt this way—that calculus was the entire reason for learning mathematics and the rest are a little silly or outdated. It took a long time for me to catch on to why the simple, silly stuff is where all of the fun is. Today, calculus feels boring and dead to me. Obviously useful, but a mere tool instead of something gr…

Are you excluding things like 'continuity' from calculus because sure, you get topologies on discrete spaces and such, but on the other hand: manifolds.

I'm being a little poetic and talking a little limitedly about calculus. Obviously it's got big connections elsewhere, but my perspective on it has shifted a lot.

Re: Will Our Understanding of Math Deteriorate Over Time?

#52

Post by jacobolus »

arh68 wrote: post #27
eli_gottlieb wrote: post #15

Earlier quoted context omitted.

>"The best way to teach real mathematics, I believe, is to start deeper down, with the elementary ideas of number and space. Everyone concedes that these are fundamental, but they have been scandalously neglected, perhaps in the naive belief that anyone learning calculus has outgrown them. In fact, arithmetic, algebra, and geometry can never be outgrown, and the most rewarding path to higher mathematics sustains thei…

> I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus Do you have recommendations for other books? I stopped at multivariable calculus as well. For what it's worth those yellow Graduate Texts in Maths books feel like reading TaoCP or CLRS; I'm looking for more approachable textbooks. I feel like I'm not even up to the 1800s, math-wise,…

There are hundreds of good math textbooks to recommend, it really depends on your interests.

For a broad overview at an undergraduate level, with a great job explaining the context of various mathematics topics, these Russian books from the 50s, Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov, and Lavrentiev, are pretty fun. Amazon link to the one-volume Dover reprint (but I’d recommend finding a used three-volume hardback copy): http://amzn.com/0486409163

Or check out John Stillwell’s Mathematics and its History: http://amzn.com/144196052X

Re: Will Our Understanding of Math Deteriorate Over Time?

#53

Post by j2kun »

reagency wrote: post #50
anatoly wrote: post #36

Earlier quoted context omitted.

The theorem the SA article is talking about - CFSG, the Classification of Finite Simple Groups - is somewhat special in that respect. Lots of things in math fall out of fashion and get forgotten, often whole subfields. CFSG is different because the theorem itself is so basic and important that it's not likely to be forgotten in any foreseeable future. But its proof is so long and complicated that it's not even clear…

How are categories equivalent to group representations?

I agree. To some extent new generations invent notations that subsume existing ideas, but I don'tthink they are claiming to reinvent symmetric functions.

Re: Will Our Understanding of Math Deteriorate Over Time?

#54

Post by sdenton4 »

coliveira wrote: post #32
Houshalter wrote: post #29

Earlier quoted context omitted.

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc... The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC. Stuff like prime…

> Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC. That's the worst example you could find, because Einstein didn't develop the mathematics for general relativity. He relied on the math invented in the XIX century for non-Euclidian geometry. If nobody had though about such a "sillY' geometry with "no practical value" it would probably take much longer because the…

Riemann's contribution is overlooked far far too often. The early non-Euclidean geometries were spaces of constant curvature - spherical and hyperbolic - and Riemann brought the idea of a manifold, and the notion of having a geometry that changes as you move around the space. And he did it in a fantastic lecture with only one equation in 1854, a good 50 years before special relativity.

Einstein was also definitely familiar with the work of Helmholtz, who did some fascinating work on non-Euclidean geometry in the context of ophthalmology: Lenses change the amount of curvature we perceive in space (think of fish-eye lenses), and provide a great jumping off point for the notion that the universe might not be as flat as it appears.

The Dover book 'Beyond Geometry' collects a bunch of the major papers in non-Euclidean geometry leading up to relativity, and is a fantastic read.

Re: Will Our Understanding of Math Deteriorate Over Time?

#55

Post by blintzing »

Houshalter wrote: post #29
blintzing wrote: post #23

Earlier quoted context omitted.

But sometimes the connection between "beautiful art project" and "practical tools" is totally unexpected. We often invest time in projects that seem simply like "beautiful art", and then much later stumble upon something practical. I think a great example of this is cryptography. The foundations of it come from number theory (prime numbers, modular arithmetic, elliptic curves), but the subject of number theory, befor…

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc... The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC. Stuff like prime…

> Stuff like prime numbers have eaten up millions of brain hours of highly intelligent people

I think the idea that brilliant minds have been 'wasted' on prime numbers is nonsense. Don't 'highly intelligent people' have the right to pursue what interests them, and even disregarding that, won't they do their best work on problems that interest them?

Even further, is learning anything that is not practical or useful a 'waste'? Certainly not. Calculus might not be of the utmost importance career-wise for an aspiring musician, but learning it helps us think in new ways.

> The vast majority of "useless" mathematics really do turn out to be useless.

That's fine! So long as we strike gold every once in a while (cryptography, which is pretty essential to the internet functioning as anything more than a bulletin board), math is doing it's job.

Re: Will Our Understanding of Math Deteriorate Over Time?

#56

Post by LGBT_2000 »

reagency wrote: post #48
chestervonwinch wrote: post #37

Earlier quoted context omitted.

"I'art pour I'art" should be "l'art pour l'art". I'm only nitpicking because I recently started compiling a list of interesting quotes and intend to save this one :)

You wrote the same quote twice.

I had to read it over and ended up pastiing it into IPython before I realized that.

Re: Will Our Understanding of Math Deteriorate Over Time?

#57

Post by tsotha »

Houshalter wrote: post #29
blintzing wrote: post #23

Earlier quoted context omitted.

But sometimes the connection between "beautiful art project" and "practical tools" is totally unexpected. We often invest time in projects that seem simply like "beautiful art", and then much later stumble upon something practical. I think a great example of this is cryptography. The foundations of it come from number theory (prime numbers, modular arithmetic, elliptic curves), but the subject of number theory, befor…

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc... The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC. Stuff like prime…

>And I looked up what the applications of them were and couldn't find anything significant beyond cryptography.

Cryptography is a pretty big deal, though. You can't run a modern economy without it.

Re: Will Our Understanding of Math Deteriorate Over Time?

#58

Post by moonchrome »

zardo wrote: post #43
moonchrome wrote: post #38

Earlier quoted context omitted.

>But sometimes the connection between "beautiful art project" and "practical tools" is totally unexpected. We often invest time in projects that seem simply like "beautiful art", and then much later stumble upon something practical. That's like saying that we should randomly start drilling holes in the ground because sometimes we will strike oil. people arguing for it usually ignore the silent evidence of research th…

>That's like saying that we should randomly start drilling holes in the ground because sometimes we will strike oil. Unless you have better tools for locating the oil, that's not a bad strategy.

I would argue that if you didn't have better tools for locating oil or determining viable research areas you shouldn't be digging/funding it in the first place because unless you have a high probability of finding something (and we've left the times of early science when there was a bunch of low hanging fruit) the cost of failed attempts will outweigh the benefits of the success stories.

Re: Will Our Understanding of Math Deteriorate Over Time?

#59

Post by goolulusaurs »

Houshalter wrote: post #29
blintzing wrote: post #23

Earlier quoted context omitted.

But sometimes the connection between "beautiful art project" and "practical tools" is totally unexpected. We often invest time in projects that seem simply like "beautiful art", and then much later stumble upon something practical. I think a great example of this is cryptography. The foundations of it come from number theory (prime numbers, modular arithmetic, elliptic curves), but the subject of number theory, befor…

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc... The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC. Stuff like prime…

That is nonsense. We have have no idea what math will turn out to be useful in the future. To say that it has already turned out to be useless presupposes that we already know all uses we might put it to in the future, which we clearly don't.

Re: Will Our Understanding of Math Deteriorate Over Time?

#60

Post by Houshalter »

goolulusaurs wrote: post #59
Houshalter wrote: post #29

Earlier quoted context omitted.

That's a popular meme but its mostly false. See http://lesswrong.com/lw/4kt/the_value_of_theoretical_researc... The vast majority of "useless" mathematics really do turn out to be useless. In the rare exceptions, there's not much evidence that doing the work beforehand is actually an advantage. E.g. Einstein wasn't aware of most of the work on non-Euclidian geometry before developing relativity IIRC. Stuff like prime…

That is nonsense. We have have no idea what math will turn out to be useful in the future. To say that it has already turned out to be useless presupposes that we already know all uses we might put it to in the future, which we clearly don't.

Well empirically most math developed in the past turned out to be useless up until now. Are you suggesting that it will suddenly become useful in the (near) future? Are the past few centuries not enough time for you to generalize from?
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