On teaching mathematics (1997)(uni-muenster.de)
uni-muenster.de
On teaching mathematics (1997)
https://www.uni-muenster.de/Physik.TP/~munsteg/arnold.html
44 comments
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)
The Secret Math Society Known as Nicolas Bourbaki - https://news.ycombinator.com/item?id=25042327 - Dec 2020 (87 comments)
There have been lots of Bourbaki stories. The one involving Arnold is more specific.
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.
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.
What do you mean by Stoll, Codd and others?
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.
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.
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.
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. 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.
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.
>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!
Things like this sound awesome! Is there a book you can recommend? Or are you writing this book? Because I want to pre-order it.
Things like this sound awesome! Is there a book you can recommend? Or are you writing this book? Because I want to pre-order it.
The book was written by Arnol'd. I recommend reading his opus magnum, written mostly as he was commuting on the Moscow Subway. It's called "The Mathematical Methods of Classical Mechanics". https://www.amazon.com/Mathematical-Classical-Mechanics-Grad...
but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out.
The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that
Integral over the boundary = Integral over the interior of the divergence, or more poetically
Int_(dA)A = Int_A dA
Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total.
So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a).
The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b).
I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone.
I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
but be careful, it is a bit terse. You have to spend a lot of time with it and some paper and pencil, working things out.
The reason why Gauss's principle is just a generalization of the fundamental theorem of calculus is that this general result is that
Integral over the boundary = Integral over the interior of the divergence, or more poetically
Int_(dA)A = Int_A dA
Assume you have some fluid flowing down the number line, where f(t) is the amount of fluid flowing through t. And this number line has some fluid sources and sinks in (things that add or subtract fluid). For an incompressible fluid, you will only get more fluid at f(t+h) then you have at f(t) if there some fluid producing source between t and t+h that adds a bit of fluid, df, to the total.
So the total amount of fluid flowing past b will be the fluid that enters the interval at a, f(a), together with the sum over all the divergences (sources) between a and b. Thus f(b) = Int(df) + f(a).
The reason the one dimensional analogue of divergence is just the derivative should be clear enough, the divergence is the rate of change in all directions (gradient) but in one dimension, the gradient is just the derivative. In fact you can prove the multi-dimensional version from the one dimensional version via slicing and applying the one dimensional argument, taking into account the linear properties of the gradient (e.g. rate of change along some vector given by the sum of directions a + b is the sum of the partial derivatives along a and b).
I unfortunately am not writing any books, I am cranking out code for work and hot takes on hackernews for fun. I wish I had time to write a book, but I have often fantasized about writing math books for kids, especially parents homeschooling kids, but it could be anyone.
I would also recommend the following (Russian) books by Kolmogorov and Aleksandrov: https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
Thanks for the recommendation. I'd probably need the undergraduate version unfortunately. But this internet stranger wants to encourage you write the book for kids someday.
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.
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.
The people you met probably went through the 'classe preparatoire aux grandes écoles'
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.
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.
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.
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...
Or just Rueben Hersh maybe?
https://books.google.com.au/books?id=cocpm4oBKqwC&sitesec=re...
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.
Wonder who put it on WWU Münster website.
Arnold's views on mathematics might be curious, but are far from being mainstream.
Arnold's views on mathematics might be curious, but are far from being mainstream.
> 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.
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.
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.
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.
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.
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.
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.
Disclaimer: I'm not a mathematician.
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."
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."
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.
Complex numbers are practical, they're everywhere in classical mechanics and circuit design, off the top of my head
The complex numbers enjoy a widespread use in physics (and electrical engineering).
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.
For geometric algebraists, mathematics is part of physics; for algebraic geometers, it is the other way around.
I 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.
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.
Edit: Actually, it does seem like the author claims mathematics is a proper subset of physics.
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Why is this obviously wrong?
Because mathematics doesn’t have to be about the world?
For Arnold, mathematics is rooted in physical intuition and experimental inquiry. Can you name some math that is completely disconnected from that intuition?
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.
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!
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!
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.
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Because physics is part of mathematics.
“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)
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I feel like there has been at least one significant thread on the Arnold-Serre debate but I can't find it.