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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,

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 6y ago
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 6y ago
  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 6y ago
    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.
    - floxy 6y ago
      >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.
      - rsj_hn 6y ago
        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-Graduate-Mathematics/dp/0387968903 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-Volumes/dp/0486409163 https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
- zabzonk 6y ago
  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.
Rerarom 6y ago
What do you mean by Stoll, Codd and others?

agnosticmantis 6y ago

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 6y ago
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 6y ago
  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.
  - floxy 6y ago
    >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.
    - rsj_hn 6y ago
      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-Graduate-Mathematics/dp/0387968903 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-Volumes/dp/0486409163 https://www.amazon.com/Mathematics-Content-Methods-Meaning-V...
zabzonk 6y ago
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.

Rerarom 6y ago

What do you mean by Stoll, Codd and others?