Will Our Understanding of Math Deteriorate Over Time? • Hacker News Board

Re: Will Our Understanding of Math Deteriorate Over Time?

#61

Post by antimagic » Fri, Jul 10, 2015, 9:41 AM UTC

chestervonwinch wrote: post #37

rndn wrote: post #28

Earlier quoted context omitted.

> This is why figuring out an elegant, concise, and powerful set of mathematical models which apply to multiple domains, and then devoting effort to simplifying, organizing, and explaining those ideas in an accessible way is so important. This reminds me of this Von Neumann quote about the importance of mathematics having an 'empirical source': — I think that it is a relatively good approximation to truth—which is mu…

"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 :)

Is this some kind of affirmational statement, kind of a counter to "ceci n'est pas une pipe"????

Re: Will Our Understanding of Math Deteriorate Over Time?

#62

Post by clarus » Fri, Jul 10, 2015, 9:46 AM UTC

anatoly wrote: post #36

agentgt wrote: post #26

I read the SA article the blog refers to and I couldn't decide if that particular colossal theory on symmetry was just an isolated incident or that "deterioration" is really happening to many disciplines/theories of math. It is certainly an obvious fact that things become popular and then eventually forgotten and then sometimes brought back. There is also different levels of understanding: breadth vs depth. I recall…

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…

Talking about the finite group classification, there was a project which achieved to formally prove it correct in Coq for the odd case: http://www.msr-inria.fr/news/feit-thomson-proved-in-coq/

However, having a Coq proof does not mean someone human understand the proof, and the software can evolve in incompatible versions unable to recheck the proof.

Re: Will Our Understanding of Math Deteriorate Over Time?

#63

Post by thaumasiotes » Fri, Jul 10, 2015, 10:01 AM UTC

eli_gottlieb wrote: post #15

tokenadult wrote: post #8

This is a very good and thought-provoking essay for a short blog post, and I have already shared it in a Facebook community heavily populated by professional mathematicians (where the moderator, with a Ph. D. in math from Berkeley, has given it a thumbs up). Thanks for sharing. I really like the overall point of the post that mathematics once known can be forgotten or neglected, and mathematics written up for mathema…

>"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…

If you're going to work with Cauchy sequences, why bother defining the reals as Dedekind cuts? You can define them as Cauchy sequences of rationals.

Re: Will Our Understanding of Math Deteriorate Over Time?

#64

Post by erje » Fri, Jul 10, 2015, 10:16 AM UTC

jacobolus wrote: post #2

“In mathematics and theoretical computer science, we read research papers primarily to find research questions to work on, or find techniques we can use to prove new theorems.” This is why figuring out an elegant, concise, and powerful set of mathematical models which apply to multiple domains, and then devoting effort to simplifying, organizing, and explaining those ideas in an accessible way is so important. Incent…

If you haven't already, take a look at Bill Thurston's paper "On proof and progress in mathematics"

http://arxiv.org/abs/math/9404236

Your comment could very well be its abstract.

Re: Will Our Understanding of Math Deteriorate Over Time?

#65

Post by rndn » Fri, Jul 10, 2015, 10:58 AM UTC

antimagic wrote: post #61

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 :)

Is this some kind of affirmational statement, kind of a counter to "ceci n'est pas une pipe"????

It basically suggests an intrinsic motivation in something. The pipe statement suggests that there is a difference between an actual thing and a representation of it.

Re: Will Our Understanding of Math Deteriorate Over Time?

#66

Post by agentgt » Fri, Jul 10, 2015, 1:32 PM UTC

anatoly wrote: post #36

agentgt wrote: post #26

I read the SA article the blog refers to and I couldn't decide if that particular colossal theory on symmetry was just an isolated incident or that "deterioration" is really happening to many disciplines/theories of math. It is certainly an obvious fact that things become popular and then eventually forgotten and then sometimes brought back. There is also different levels of understanding: breadth vs depth. I recall…

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…

I really wish people would explain why they downvoted me on hackernews. Its not that I really care about the points its that I want to know what I did wrong. Like if I accidentally offended someone or it just wasn't an interesting comment or observation or that it is completely wrong statement.

Your response is excellent anatoly and I appreciate it.

Re: Will Our Understanding of Math Deteriorate Over Time?

#67

Post by goolulusaurs » Fri, Jul 10, 2015, 1:58 PM UTC

Houshalter wrote: post #60

goolulusaurs wrote: post #59

Earlier quoted context omitted.

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?

I think it is very plausible that a large portion of mathematics that is not very useful presently could be very useful for problems we have yet to tackle. As science progresses it will be less and less able to make grand unified theories and increasingly focus on the manifold particular. I can imagine much of math being useful only for problems we haven't even identified yet, like algebraic topology being used to study social dynamics, or engineering problems at strange scales. Even beyond that I think it may be the case that much of mathematics will become useful for reasons unforeseen. Unknown unknowns always seem to be where new science pops up.

Re: Will Our Understanding of Math Deteriorate Over Time?

#68

Post by eli_gottlieb » Fri, Jul 10, 2015, 1:59 PM UTC

thaumasiotes wrote: post #63

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…

If you're going to work with Cauchy sequences, why bother defining the reals as Dedekind cuts? You can define them as Cauchy sequences of rationals.

Pugh mostly just wanted to be able to state the theorem that a sequence has a limit iff it is Cauchy convergent.

Re: Will Our Understanding of Math Deteriorate Over Time?

#69

Post by nemoniac » Fri, Jul 10, 2015, 2:13 PM UTC

Whatever about Mathematics, this is certainly true of Computing. Well understood ideas are continually being reinvented, frequently badly. New programming languages and frameworks spring up like mushrooms and everyone wants to jump on board the next big thing.

Nowhere is it more true that those who don't know the past are condemned to repeat it.

Re: Will Our Understanding of Math Deteriorate Over Time?

#70

Post by dllthomas » Fri, Jul 10, 2015, 3:34 PM UTC

moonchrome wrote: post #38

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…

>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…

Your broader point - that we're a lot more likely to find useful things using some guidance as to where useful things are likely to be than in proceeding randomly - is important.

A crucial difference between digging for oil and doing math, however, is the nature of the externalities. In either case, you're burning some work that could be spent somewhere better, but with oil you're left with a hole that you probably want not to be there and there's no good way to put it back. In both drilling and math, "drilling" helps us refine our methods. In the case of math exploring more of the ramifications of our axioms also helps raise our confidence that they're not subtly inconsistent.

And of course, math is generally less expensive than an oil well.

I don't know where the cost-benefit analysis puts work on math when we don't yet see practical application. And I think that's often over-romanticized. However, I do think there are a lot of reasons we should expect the analysis to come out more favorably than for drilling random holes.