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On teaching mathematics (1997)
- hervature 6y agoI don't know if things changed since then, but the people I know who went through the French math system had some of the strongest technical abilities in math I've seen. They also would have exactly 0 difficulty drawing that parametric equation so I don't know what to take away from this as it seems overly sensational.
- koalafied 6y agoMy understanding is that the article mainly concerns the education of (pure) mathematicians and not engineers, say. Maybe that explains the discrepancy between your experience and the article’s claims.
- Agingcoder 6y agoThe people you met probably went through the 'classe preparatoire aux grandes écoles' https://en.m.wikipedia.org/wiki/Classe_pr%C3%A9paratoire_aux_grandes_%C3%A9coles https://en.m.wikipedia.org/wiki/Classe_pr%C3%A9paratoire_aux... It's brutally efficient at selecting the best math students (including the ones who actually want to study engineering, since the entrance exams are about maths and physics ) , and the sheer amount of work you put in two years is absolutely staggering. It's obviously not perfect, and people regularly talk about getting rid of this system, but it works well for the students who survive.
- User23 6y ago> Mathematics is a part of physics. It's probably treading close to the don't be dismissive rule, but I confess it's really hard to keep reading when the piece opens with something so obviously wrong.
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- bitdizzy 6y agoWhy is this obviously wrong?
- drdeca 6y agoBecause mathematics doesn’t have to be about the world?
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- bitdizzy 6y agoFor Arnold, mathematics is rooted in physical intuition and experimental inquiry. Can you name some math that is completely disconnected from that intuition?
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- dooglius 6y agoMathematics may be rooted in that, in a historical or pedagogical sense, but areas of math can certainly be disconnected from physical intuition. Non-measurable sets (e.g. those in Banach-Tarski) and transfinite numbers cone to mind.
- bitdizzy 6y agoNon-measurable sets are precisely the kinds of things Arnold wanted marginalized in mathematical pedagogy, instead of placed front and center. They are necessary auxiliaries to the main theory, that of measures and integration but auxiliary nonetheless. I disagree that transfinite numbers are detached from physical intuition since most of the ones you or I could write down can be easily visualized with a few ellipses here or there. But i do think Arnold would consider them marginal players. Perhaps he thought set theory was a formalist distraction from the main of mathematics!
- idolaspecus 6y agoToo close, in my opinion. If I said "Communication is a part of physics", it would be clear that I don't mean "all of communication is a subset of physics" and that I do mean "doing physics involves communicating your findings". I think this is the author's intention. Edit: Actually, it does seem like the author claims mathematics is a proper subset of physics.
- erdos4d 6y agoHistorically, this is accurate. The whole concept of "pure math" is actually very recent and before that math was almost exclusively tied to real world problems that often came directly from physics.
- mettamage 6y agoHow about having some understanding of complex numbers? Wasn't it an Italian who started to use sqrt(-1)? I don't see how that'd be practical. Disclaimer: I'm not a mathematician.
- bitdizzy 6y agoThe first use of complex numbers was as intermediate quantities in the calculation of the roots of cubic polynomial functions. This use is analogous to using negative numbers in a ledger even though negative amounts of physical things don't make sense. Edit: I meant physical things like apples fam. This is an important philosophical point we don't appreciate because we are so used to them. De Morgan once wrote: "It is not our intention to follow the earlier algebraists through their different uses of negative numbers. These creations of algebra retained their existence, in the face of the obvious deficiency of rational explanation which characterized every attempt at their theory."
- Koshkin 6y agoThe complex numbers enjoy a widespread use in physics (and electrical engineering).
- Koshkin 6y agoFor geometric algebraists, mathematics is part of physics; for algebraic geometers, it is the other way around.
- cassepipe 6y agoI am fascinated by geometric algebra but what I have seen so far seems like mathematicians trying to convince physicists and programmers to use their system, which to be honest seems much better, but is also harder to build an intuition of. I would love to read a geometric algebra for dummies/from the ground up.
- dang 6y agoIf curious, past threads: “On teaching mathematics by” by V.I. Arnold - https://news.ycombinator.com/item?id=21353855 https://news.ycombinator.com/item?id=21353855 - Oct 2019 (1 comment) On teaching mathematics by V.I. Arnold (1997) - https://news.ycombinator.com/item?id=17209444 https://news.ycombinator.com/item?id=17209444 - June 2018 (21 comments) On teaching mathematics, by V.I. Arnold (1997) - https://news.ycombinator.com/item?id=12994218 https://news.ycombinator.com/item?id=12994218 - Nov 2016 (14 comments) V.I. Arnold, On teaching mathematics (1997) - https://news.ycombinator.com/item?id=8441682 https://news.ycombinator.com/item?id=8441682 - Oct 2014 (9 comments) V.I. Arnold: On teaching mathematics - https://news.ycombinator.com/item?id=619346 https://news.ycombinator.com/item?id=619346 - May 2009 (19 comments) I feel like there has been at least one significant thread on the Arnold-Serre debate but I can't find it.
- happy-go-lucky 6y agoDang, I think this has the thread you are looking for: The Secret Math Society Known as Nicolas Bourbaki - https://news.ycombinator.com/item?id=25042327 https://news.ycombinator.com/item?id=25042327 - Dec 2020 (87 comments)
- dang 6y agoThere have been lots of Bourbaki stories. The one involving Arnold is more specific.
- ggm 6y agoDavis and Hersh? https://books.google.com.au/books?id=I1fj_B60fyoC&printsec=frontcover&dq=inauthor:%22Reuben+Hersh%22&hl=en&sa=X&ved=2ahUKEwi4t6X1rtvvAhUE93MBHT-FAs8Q6AEwAXoECAMQAw#v=onepage&q&f=false https://books.google.com.au/books?id=I1fj_B60fyoC&printsec=f... Or just Rueben Hersh maybe? https://books.google.com.au/books?id=cocpm4oBKqwC&sitesec=reviews https://books.google.com.au/books?id=cocpm4oBKqwC&sitesec=re...
- Glavnokoman 6y agoI think his 100 of 5-minute mathematical problems would be of a more interest. Although maybe to a not so wide audience. I remember solving like 5 of those in 5 minutes, 10 others took much longer, and around 50 I did not understand at all... But I really like the guy, he was great.
- mr_gibbins 6y agoNo matter how clever or acclaimed Arnold is (or was), this speech is nothing but an extended rant on how deficient in skill and intelligence he finds his peers, his students and his contemporaries, and by implication how incredibly clever and acclaimed he is by contrast. As for the argument that mathematics and physics should be conjoined - yes, physics and mathematics are incredibly close. My view, however, is that mathematics is applicable to so much more than physics - in my specialist area of set theory, for example, the underpinnings are Cantor's transfinite numbers and the theory laid down by Stoll, Codd and others in the '60s. There's no quantum about it. What an egocentric piece this was. Hopefully long-forgotten by those who had the misfortune to attend it.
- agnosticmantis 6y agoMy moderated take on Arnold’s piece is that if a piece of math (e.g. calculus) was inspired by physics or created to solve a problem in physics, then at the very least include the motivating physics problem in the curriculum. I think I’d have enjoyed my multivariate calculus much more if they were taught in the context of electromagnetism.
- swiley 6y agoI definitely enjoyed multivariate calculus for its own sake (I too took physics afterwards.) I think part of that though came from reading a linear algebra book on my own time which I did understand the applications for. Part of studying math at a University is learning to learn it without application though. I feel like including applications for everything would make that harder.
- rsj_hn 6y agoI think that's missing the point. The idea isn't to find physical applications of math for each math topic learned, but to rather to teach math ideas as applications of physics. Take, for example Gauss' law about the integral of flux over the surface area being equal to the volume integral of divergence. That wasn't some formula that dropped on Gauss' head. He was working with a physical problem, viewing the flux and divergence as measure of real things, say fluids, passing through a point or emerging out of points, and when viewed in this way, Gauss's theorem is as obvious as the conservation of mass. The total amount of stuff passing across a boundary is the total amount of stuff being generated within the region. But from that, you can ask what is the one dimensional analogue of this conservation of mass principle and you get .. the fundamental theorem of calculus! And then lots of results about topology start becoming clear. All because these results are viewed as applications of simple physical ideas and then mathematically (e.g. formally, logically) the implications of these ideas are deeply examined. That is a much better approach than the Bourbaki style pedagogy where there is an emphasis on formal deduction that strips away, or hides, the underlying physical intuition behind these results. Why would anyone want to hide clear and pedagogically useful explanations of mathematical techniques? Why would we want to treat the Pontryagin principle as some kind of magic formula rather than a fairly straightforward approach in minimizing the action? And why would we view simple variational approaches as something exotic rather than as a generalization of Snell's law? I was privileged enough to take a class with Arnol'd when he was visiting the US, and listening to his lectures was like being transported back into the 19th Century. Deep, modern results were explained in simple terms of balls rolling down incline planes or tangent functions evolving along plane curves. It was an amazing course, and I have to say that one reason why Russian mathematics has been so influential relative to their population size or GDP per capita is because there is still a rich tradition of motivating ideas based on physical or geometric intuition rather than the more western focus that is much more abstract.
- oytis 6y agoWonder who put it on WWU Münster website. Arnold's views on mathematics might be curious, but are far from being mainstream.