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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?

#41

Post by Houshalter »

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…

He figured out that spacetime may be non-Euclidean before he was aware that the math had been extensively studied. Then he learned about the preexisting work.

It's true it probably would have been much harder for him to work out the math on his own. But he and/or others eventually would have done it.

Re: Will Our Understanding of Math Deteriorate Over Time?

#42

Post by GFK_of_xmaspast »

tel wrote: post #12
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…

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where I felt this way—that calculus was the entire reason for learning mathematics and the rest are a little silly or outdated.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.

Re: Will Our Understanding of Math Deteriorate Over Time?

#43

Post by zardo »

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…

>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.

Re: Will Our Understanding of Math Deteriorate Over Time?

#44

Post by eli_gottlieb »

stuxnet79 wrote: post #39
asanagi wrote: post #10

Earlier quoted context omitted.

I think language and symbology is at the core of why they are so impenetrable. One major sin is taking new concepts and ideas and putting the primary discoverer's name on them. Such names yield no clue as to the interpretation or application of the idea itself. Another problem is the symbols used in certain mathematical texts. Everyone who uses them treats them like they're universally understood, but in reality the…

I was extremely frustrated with my machine learning class because of this. The notation used was hazy. Also if you ask me, probability theory should just be squashed and the ideas reformulated with new more consistent syntax / symbols, because the entire thing is so damn inconsistent and disorganized at present.

>Also if you ask me, probability theory should just be squashed and the ideas reformulated with new more consistent syntax / symbols, because the entire thing is so damn inconsistent and disorganized at present.

How do you mean? Probability doesn't even have that much complicated symbolism... although I do wish we would teach in probability courses how to translate between random-variable "distributed according to" notation and actual density functions. As in, I wish I knew how to do that.

Re: Will Our Understanding of Math Deteriorate Over Time?

#45

Post by [deleted] »

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…

[deleted]

Re: Will Our Understanding of Math Deteriorate Over Time?

#46

Post by LimpWristed »

coliveira wrote: post #35
pc2g4d wrote: post #4

Integrate concise and effect explanations into the relevant Wikipedia articles and you at least give future generations a good head start on understanding these things.

Wikipedia is a horrible way to learn math. At most it works as a way to get initial pointers for literature. In most specialized fields, don't expect wikipedia to provide any understanding of mathematics beyond a summary of formulas without much explanation of what they are.

Wikipedia has the same definitions that are in your textbook, but often provides much more readable proofs as compared to the very short and "elegant" ones or unwieldy monstrosities in your usual textbooks. At least that's true for textbooks at the level of, say, Dummit&Foot's Abstract Algebra or Baby Rudin.

Re: Will Our Understanding of Math Deteriorate Over Time?

#47

Post by reagency »

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…

The real shame is that academia funds a lot of research, but not a lot of scientific "scholarship". We should have academics dedicated to high quality exposition of advanced material. You don't see much of that beyond the never ending rewrites of basic calc and algebra textbooks.

Re: Will Our Understanding of Math Deteriorate Over Time?

#48

Post by reagency »

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 :)

You wrote the same quote twice.

Re: Will Our Understanding of Math Deteriorate Over Time?

#49

Post by reagency »

marcosdumay wrote: post #22
vezzy-fnord wrote: post #9

Earlier quoted context omitted.

The Wikipedia pages are then useful later, as a reference, for people who already understand their content. Which, if I'm not wrong, is exactly the intent of an encyclopedia. It's a reference work.

If the only people capable of understanding what you are saying are the people that already know it, saying it is quite useless. Reference material or not.

A reference is a brief reminder and links to primary sources. We don't hate cars because they aren't boats.

Re: Will Our Understanding of Math Deteriorate Over Time?

#50

Post by reagency »

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…

How are categories equivalent to group representations?
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