[HN Gopher] On teaching mathematics (1997)
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On teaching mathematics (1997)
Author : agnosticmantis
Score : 40 points
Date : 2021-03-31 17:22 UTC (5 hours ago)
(HTM) web link (www.uni-muenster.de)
(TXT) w3m dump (www.uni-muenster.de)
| dang wrote:
| If curious, past threads:
|
| _"On teaching mathematics by" by V.I. Arnold_ -
| 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 - June 2018 (21
| comments)
|
| _On teaching mathematics, by V.I. Arnold (1997)_ -
| 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 - Oct 2014 (9
| comments)
|
| _V.I. Arnold: On teaching mathematics_ -
| 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 wrote:
| Dang, 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 - Dec 2020 (87
| comments)
| dang wrote:
| There have been lots of Bourbaki stories. The one involving
| Arnold is more specific.
| hervature wrote:
| I 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.
| Agingcoder wrote:
| The people you met probably went through the 'classe
| preparatoire aux grandes ecoles'
|
| 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.
| koalafied wrote:
| My 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.
| mr_gibbins wrote:
| No 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 wrote:
| My 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 wrote:
| I 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 wrote:
| I 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 within the boundary. 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.
| zabzonk wrote:
| I agree completely. When I was taking an A-level course (UK)
| in Pure Maths (circa 1970) I had the hardest time working out
| why I should care about calculus at all, except to hopefully
| pass an exam (which I did, with the lowest possible grade).
| If I had taken the Applied Maths course, which applied maths
| to physics, I'm sure I would have done much better than I
| did, but I couldn't because of timetabling constraints with
| my other two A-level subjects.
| oytis wrote:
| Wonder who put it on WWU Munster website.
|
| Arnold's views on mathematics might be curious, but are far from
| being mainstream.
| Glavnokoman wrote:
| I 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.
| ggm wrote:
| Davis and Hersh?
| 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=re...
| User23 wrote:
| > 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.
| bitdizzy wrote:
| Why is this obviously wrong?
| drdeca wrote:
| Because mathematics doesn't have to be about the world?
| [deleted]
| bitdizzy wrote:
| For Arnold, mathematics is rooted in physical intuition and
| experimental inquiry. Can you name some math that is
| completely disconnected from that intuition?
| [deleted]
| dooglius wrote:
| Mathematics 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 wrote:
| Non-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!
| oytis wrote:
| E.g. logic? A pretty important part of mathematics that
| is hard to marginalize, but that can't be observed
| experimentally. Rather scientific observation presupposes
| logic ability.
| [deleted]
| Koshkin wrote:
| Because physics is part of mathematics.
| [deleted]
| erdos4d wrote:
| Historically, 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 wrote:
| How 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.
| erdos4d wrote:
| I am, have a PhD in it at least, and complex numbers are
| 100% required to do quantum mechanics, so are physically
| motivated. In fact, many physicists consider the complex
| numbers to be the preferred number system of the universe
| for this reason.
| Koshkin wrote:
| The complex numbers enjoy a widespread use in physics (and
| electrical engineering).
| bitdizzy wrote:
| The 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 wrote:
| Depends on the "thing." We do use negative values for,
| say, degrees of temperature.
| jlg23 wrote:
| > even though negative amounts of physical things don't
| make sense.
|
| I beg to differ, just today, on the road, accelerating by
| some negative amount made a lot of sense.
| 0--__-_-__--0 wrote:
| Complex numbers are practical, they're everywhere in
| classical mechanics and circuit design, off the top of my
| head
| Koshkin wrote:
| Curiously, many mathematical curricula used to include
| analytical mechanics, and some of the mathematics departments
| of universities around the world had the word "mechanics" in
| their names.
| oytis wrote:
| Historically already ancient Greeks knew pure mathematics
| (e.g. Euclid's Elements), and differentiated it from its
| applications as far as I'm concerned. That made them pretty
| distinct from earlier Egyptian mathematicians who indeed were
| only concerned with solving practical problems without much
| attention to logical rigor.
| aaplok wrote:
| Historically mathematics was part of accounting, both in
| Ancient China where it was used to calculate taxes, in Egypt
| and in Babylon. People used numbers to keep count and then
| developed advanced techniques to do sophisticated things like
| split non-rectangular land in equal parts.
|
| It wasn't until much later, and only in a tiny part of the
| world called Europe, that physics started using mathematical
| models. Even at that time and in that place, there were other
| disciplines making advanced use of mathematics and leading to
| exciting discoveries, starting from Economics. Jacobi who
| developed utility theory around the time of Newton.
|
| Mathematics was never a subdiscipline of physics. Important
| parts of mathematics were developed to support models of
| physics, but claiming that the parts should not exist is
| intellectually dishonest.
| Koshkin wrote:
| For geometric algebraists, mathematics is part of physics; for
| algebraic geometers, it is the other way around.
| idolaspecus wrote:
| Too 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.
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