On teaching mathematics (1997)
uni-muenster.de
uni-muenster.de
“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.
The Secret Math Society Known as Nicolas Bourbaki - https://news.ycombinator.com/item?id=25042327 - Dec 2020 (87 comments)
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.
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.
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.
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.
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...
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.
Or just Rueben Hersh maybe?
https://books.google.com.au/books?id=cocpm4oBKqwC&sitesec=re...
Arnold's views on mathematics might be curious, but are far from being mainstream.
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 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.
Disclaimer: I'm not a mathematician.
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 beg to differ, just today, on the road, accelerating by some negative amount made a lot of sense.
Edit: Actually, it does seem like the author claims mathematics is a proper subset of physics.
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!