But the proofs survive because they are proofs; if they don't communicate the proof of the result then they have failed and should not be accepted by journals. At the extreme end, machine-checkable proofs are in standard, documented formats; an alien reading them in ten thousand years should still be able to understand what's going on, at least if they understand the notation and the axioms.
Will Our Understanding of Math Deteriorate Over Time?
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Re: Will Our Understanding of Math Deteriorate Over Time?
#12This 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where I felt this way—that calculus was the entire reason for learning mathematics and the rest are a little silly or outdated.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 greater. I spend my time thinking about things like topology where I work really hard to think about what it means for things to be close to one another and nothing more.
A younger me would not have understood. Which is a little scary.
Re: Will Our Understanding of Math Deteriorate Over Time?
#13But the proofs survive because they are proofs; if they don't communicate the proof of the result then they have failed and should not be accepted by journals. At the extreme end, machine-checkable proofs are in standard, documented formats; an alien reading them in ten thousand years should still be able to understand what's going on, at least if they understand the notation and the axioms.
And codes does what code does. Given some binary executable, an alien reading it far in the future should be able to understand what's going on, at least if they understand the architecture. :P
I've written a few machine checked proofs, and there's really two ways that I've seen, either writing it for the next human to read, or just enough that the checker accepts it. The latter makes free use of tactics like `crush`, which brute force solutions out of current assumptions, exploring the search space automatically. That's really convenient, but can make reading the proof very un-enlightening.
Re: Will Our Understanding of Math Deteriorate Over Time?
#14This 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…
> 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 their development alongside the 'advanced' branches such as calculus.
I doubt one's "outgrowth of certain branches of math" is the reason math as taught to non-math majors is a castrated mess it is. It's probably the market forces that reject real analysis, abstract algebra or anything at that level or higher. It's the same reason "some programming language du jour > fundamentals of CS, IRL".
Re: Will Our Understanding of Math Deteriorate Over Time?
#15This 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 their development alongside the 'advanced' branches such as calculus. Also, by maintaining ties between these disciplines, it is possible to present a more unified view of mathematics, yet at the same time to include more spice and variety."
While I do agree, we have to remember why most math classes actually exist: to teach calculus to physicists and engineers, and, as my stepfather's undergraduate advisor once said, "to keep the children from running in the halls".
(For the mathematician's extremely self-centered view of "children" as "anyone who has yet to ace two semesters of real analysis".)
I've been starting into real analysis myself via Pugh's textbook[1] after not taking a serious math class since multivariable calculus, and found that, once I get past the applied stuff, I really like the approach of building up calculus from its foundations in real numbers (taken as Dedekind cuts), limits (Cauchy-convergent sequences), the set-theoretic construction of functions, and the construction of topological and metric spaces "from scratch". But I can tell that I like it because, deep down, I have the mind of a theoretical computer scientist (which is what I like to be when I'm not writing firmware), which is a kind of mathematician. I appreciate that someone has to teach the applied classes to the people who aren't going to kvetch about "how can I trust that works!?" and who demand to just get their math over with as quickly as possible.
[1] -- http://www.amazon.com/Mathematical-Analysis-Undergraduate-Te...
Re: Will Our Understanding of Math Deteriorate Over Time?
#16But the proofs survive because they are proofs; if they don't communicate the proof of the result then they have failed and should not be accepted by journals. At the extreme end, machine-checkable proofs are in standard, documented formats; an alien reading them in ten thousand years should still be able to understand what's going on, at least if they understand the notation and the axioms.
And codes does what code does. Given some binary executable, an alien reading it far in the future should be able to understand what's going on, at least if they understand the architecture. :P I've written a few machine checked proofs, and there's really two ways that I've seen, either writing it for the next human to read, or just enough that the checker accepts it. The latter makes free use of tactics like `crush`…
Chlipala, if you're on here, I want to be the second person to say: `crush` is the least communicative tactic I have ever seen. Where the hell is `unsafePerformIO` when I want some reporting on exactly how you crushed my proof goals!?
Re: Will Our Understanding of Math Deteriorate Over Time?
#17Of course. Most modern mathematicians aren't fluent with half the material in (the ~100 year-old text) Whittaker and Watson "A Course of Modern Analysis". This was standard material even 60 years ago. You can get a PhD in mathematics today without once seeing an elliptic function, because computers are good enough at numerically solving the problems they were once used to solve symbolically.
On the other hand, most modern mathematicians know stuff that would blow Whittaker and Watson's socks off.
Much mathematics is obsoleted. For example, there was a lot of incredibly difficult mathematics for finding areas under curves, which was all completely obsoleted with the discovery of the fundamental theorem of calculus. Nothing of value was lost.
Individual pieces of mathematics come and go from general awareness, but the overall trend is definitely one of increasing, not decreasing, understanding.
Re: Will Our Understanding of Math Deteriorate Over Time?
#18To make it readable, paste this in the console:
document.querySelector('.date-outer').style.backgroundColor = 'white';Re: Will Our Understanding of Math Deteriorate Over Time?
#19Exactly, this unfortunate "saturate and move on" can be observed in pretty much any area.
Re: Will Our Understanding of Math Deteriorate Over Time?
#20Most people use math less and less (even your average cashier will have issues if the machine isn't working).
Will Myron Aub give us the feeling of power back?
http://downlode.org/Etext/power.html by Isaac Asimov on just this topic.