21–30 of 73 posts
Most 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.
Earlier quoted context omitted.
Most of the Wikipedia articles on technical subjects, and especially on mathematical topics, are terrible as introductory exposition. They are jargony, highly technical, and self referential. They usually contain much that is irrelevant, and they almost never properly explain the context for an idea. The main problem is that Wikipedia articles are tiny and atomic, so it’s difficult to synthesize and organize ideas in…
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.
Reference material or not.
“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…
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, before the advent of computing, was possibly the most useless kinds of mathematical 'art' that could have existed. I imagine it was the mathematical equivalent of frolicking in the fields.
Mathematicians explored Fermat's little theorem starting in 1640, but they didn't do it because they knew it'd be useful several hundred years later in RSA. They did it simply because math is worth exploring in itself.
Even if you don't subscribe to the idea that we should pursue math for math's sake, history shows us that it's very difficult to know what parts of math will be useful to humanity, especially hundreds of years later. Since people work best on what they find interesting, mathematicians should continue exploring the topics that most interest them, because we really can't say with any certainty what will prove useful (or even essential) to future generations.
PDF (of print from 1979): http://www.evolocus.com/Textbooks/Fleck1979.pdf
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.
I still think the unpublished problem ie "publication bias" is a bigger issue which I suppose is somewhat in similar vain. Supposedly google was working on that.
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…
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, not even up to Gauss.
“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…
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 much too complicated to allow anything but approximations—that mathematical ideas originate in empirics, although the genealogy is sometimes long and obscure. But, once they are so conceived, the subject begins to live a peculiar life of its own and is better compared to a creative one, governed by almost entirely aesthetical motivations, than to anything else and, in particular, to an empirical science. There is, however, a further point which, I believe, needs stressing. As a mathematical discipline travels far from its empirical source, or still more, if it is a second and third generation only indirectly inspired by ideas coming from "reality" it is beset with very grave dangers. It becomes more and more purely aestheticizing, more and more purely I'art pour I'art. This need not be bad, if the field is surrounded by correlated subjects, which still have closer empirical connections, or if the discipline is under the influence of men with an exceptionally well-developed taste. But there is a grave danger that the subject will develop along the line of least resistance, that the stream, so far from its source, will separate into a multitude of insignificant branches, and that the discipline will become a disorganized mass of details and complexities. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. At the inception the style is usually classical; when it shows signs of becoming baroque, then the danger signal is up. It would be easy to give examples, to trace specific evolutions into the baroque and the very high baroque, but this, again, would be too technical.
In any event, whenever this stage is reached, the only remedy seems to me to be the rejuvenating return to the source: the re-injection of more or less directly empirical ideas. I am convinced that this was a necessary condition to conserve the freshness and the vitality of the subject and that this will remain equally true in the future.
“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…
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…
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 numbers have eaten up millions of brain hours of highly intelligent people. I remember thinking it was weird that so many project Euler problems were about prime numbers. And I looked up what the applications of them were and couldn't find anything significant beyond cryptography.
And they seem to have been chosen for cryptography simply because it was a well studied problem with certain properties. Not because cryptography inherently needs prime numbers and would be impossible without centuries of previous work studying them.
To make it readable, paste this in the console: document.querySelector('.date-outer').style.backgroundColor = 'white';