There are some reasonable points in here (problems with abstraction for abstraction's sake) but on the whole this reads as someone who simply cares about geometry and analysis, but not algebra or discrete mathematics.
Which is fine, he's entitled to his opinion. But the stronger argument that "mathematics is only good when it's tied to physics" argument is wrong and outdated -- belongs to the pre-computer, pre-information age.
Even on a purely mathematical level, some of his arguments don't hold. For instance, for all his complaining about how abstraction doesn't contribute anything -- Cayley's theorem on groups, Whitney's theorem on manifolds -- there ARE categories where the abstract definition is needed or is more natural, like algebraic varieties (which can't always be embedded in complex N-space for any N) and rings (I have no idea what better definition he'd be proposing).
To me, it sounds like he doesn't like or know much about the algebraic side of math, and resents how much it dominates in math today.
> To me, it sounds like he doesn't like or know much about the algebraic side of math, and resents how much it dominates in math today.
Given that Arnold was a student of Kolmogorov, was awarded the Wolf prize, and was denied the Fields medal due to some Soviet-era politics (https://en.wikipedia.org/wiki/Vladimir_Arnold#Honours_and_aw...), I think it's safe to say he understood algebra. You can also read his books, and find out that he used abstract algebra quite well. That's not what he's saying here, I think you misread.
> Which is fine, he's entitled to his opinion. But the stronger argument that "mathematics is only good when it's tied to physics" argument is wrong and outdated -- belongs to the pre-computer, pre-information age.
I think you're missing the point he's making and cherry-picking examples from the article. Indeed, much of the article is directed at how very conventional mathematics is taught at the school and undergraduate level, and one key point is that the way it is taught is too far divorced from how it was discovered a long time ago. He claims that teaching mathematics in the algebraic way makes it unnecessarily more difficult to solve mathematical problems that used to be essentially trivial, making people worse at mathematics for no good reason.
So you picking out the example of categories of algebraic varieties is misleading because that's not at all the focus of the article, it seems like an incidental example. I think the determinant and group examples are a much clearer illustration of the difference he's talking about.
I also have no idea what you mean by your comment about pre-computer, pre-information age. If anything, mathematics became much more of an experimental science (thus closer in spirit to physics in Arnold's view) now that you can run computational experiments so easily with this much computing power. Note his example of the difference between observation - model - investigation of the model - conclusions - testing by observations and definition - theorem - proof.
Whilst some of the replies rebuke large part of your points well, I agree that the argument "mathematics is only good when it's tied to physics" is wrong.
I'd reformulate his point to: Mathematics, especially when being taught, benefits from intuitive examples that motivate the abstract definitions.
A point that is quite obvious. Whilst certainly much intuition concerns physics, that need not be the underlying source of the intuition.
It's interesting comparing Arnold's very anti-abstract approach to Gromov's (another great Russian mathematician from the same era and now a professor at the IHES). Gromov in his later writings take a very categorical approach in his work. For example in the following paper he attempts to define entropy from a very abstract view point
http://www.ihes.fr/~gromov/PDF/structre-serch-entropy-july5-...
Gromov's approach is not abstract, it's extremely concrete. It uses abstractions for a very precise treatment, and it's probably not the best text for an introduction to entropy, but it's essentially a very concrete treatment.
It analyses very concrete observations using abstract methods in order to construct an abstract definition. It doesn't simply spew out abstract definition. Everything is motivated by concrete facts.
I would not say it's against Arnold's method at all, if anything, it uses the same concrete to abstract treatment. It is, however, not the best way to learn about entropy, but it's a good way to formalise it once you have become familiar with it. It's poor for undergraduate level material, but it's good as graduate level material.
> A smooth k-dimensional submanifold of the Euclidean space RN is its subset which in a neighbourhood of its every point is a graph of a smooth mapping of Rk into R(N - k) (where Rk and R(N - k) are coordinate subspaces). This is a straightforward generalization of most common smooth curves on the plane (say, of the circle x2 + y2 = 1) or curves and surfaces in the three-dimensional space.
I just wanted to provide a somewhat humorous convergece between super-abstract mathematics and the concrete mathematics VI Arnold advocated: The category of smooth manifolds is the idempotent completion of the category of open subsets of Euclidean spaces:
When you unravel the definition you get a very concrete idea of what a manifold is without the bureaucracy of charts and atlases and whatnot. In general, Lawvere's mathematical career is an ambitious program to cast classical mechanics in categorical terms. I see this as the synthesis of the two traditions.
Another way in which abstract mathematics is converging with concrete mathematics is the Curry-Howard-Lambek-... correspondence between logics, programming languages, and category theory. The most abstract branch of mathematics happens to be very amenable to the very physical, concrete process of computation!
I think abstract definitions that are far divorced from the concrete objects they are trying to capture is a deficiency in the abstraction. The correct abstraction somehow miraculously becomes far more concrete than any intermediate definition. For example, homotopy theory has a very ugly definition in set based mathematics, but by abstracting the essentials, homotopy type theory looks very promising at reconciling the bureaucracy of rigor required to do homotopy theory in sets and the very geometric intuition one has for homotopical objects.
I feel there needs to necessarily be a separation of the doing of mathematics and the teaching of mathematics in these kinds of matters. In the teaching of mathematics, especially in the more 'abstract' areas, there is not nearly enough driving of intuition, and the 'problems' students are given often are unrelated to the 'problems' the theory they are learning about was created to solve. Pushing things in the concrete direction is probably the right direction for pedagogy.
But in the doing of mathematics is where I think Arnold is looking back at the past too romantically. Prior to programs like Bourbaki, the lack of unity in mathematical notation, definitions, ideas in methods was very present. Often if you were not a student of one of the masters of the time, you would have an incredibly difficult time understanding their work. The tradition that has lead to an 'over-abstraction' of things, as Arnold would have seen it, also lead to a much more widespread accessibility of contemporary mathematics. when you can count Kolmogorov among your teachers, this is less relevant to you. But for the 'unwashed masses' of mathematics this has made a world of difference.
> I feel there needs to necessarily be a separation of the doing of mathematics and the teaching of mathematics in these kinds of matters. In the teaching of mathematics, especially in the more 'abstract' areas, there is not nearly enough driving of intuition, and the 'problems' students are given often are unrelated to the 'problems' the theory they are learning about was created to solve. Pushing things in the concrete direction is probably the right direction for pedagogy.
Indeed. By far the best teachers I've had for math courses spent a good deal of time discussing the history of the topic and motivating its creation.
I've read this article countless times when looking for his textbook recommendations, cited at the end. All excellent.
While I also appreciate abstraction, very often mathematics is taught at the wrong level of abstraction. A book I like a lot, in the spirit of his recommendations, is Hubbard & Hubbard.
Growing up I found Mathematics difficult to grasp for various reasons. One of them required mental gymnastics of "hidden" rules and exceptions to the rules. There would always be rules that were often forgotten. And one of the bigger problems I had was forcing myself to practice these exercises. In order to store this information one must actually practice these problems many times to have it lodged into your brain for later use. Being a kid made it even more of an obstacle since I always wanted to play outside or hangout with friends.
It seemed that mathematics required more of my time than any of the subjects. So wouldn't spend the necessary time needed to actually store it in my head. Later in life, I realized how fascinating math was, and fell in love with it. Even though I am still not good at it, it's fun to solve puzzles.
> One of them required mental gymnastics of "hidden" rules and exceptions to the rules. There would always be rules that were often forgotten.
Could you elaborate on this? Mathematics, perhaps uniquely among human intellectual endeavours, has no hidden rules. Its rules may be complicated and unintuitive at times, but they can be listed; and whatever follows from these listed rules is true (to be precise: whatever follows from the axioms is true in any model of the axioms), and whatever contradicts them is false, regardless of how natural or un- it may seem.
In particular, I am curious what you mean by "There would always be rules that were often forgotten." Do you mean that you forgot them, or that your teachers forgot them (both of which were eminently believeable!), or that practicing, professional mathematicians forget them? I am sceptical of the last (although plenty of, indeed probably all, professional mathematicians don't need to keep all the rules explicitly in their heads at all times), and, although I am eminently willing to believe the first two, I think that that shows a fault in the teacher, not the subject.
This article raises an important and interesting problem, but would benefit from a lot more nuance. It seems to claim there is only one ubiquitous way people teach, learn and do mathematics, but that is not the case.
In my view, using abstract notions for describing is in no way incompatible to motivating them with concrete, visual notions, and that is what good teachers do and good students seek. This is done imperfectly by many teachers, but there is nothing new in stating that not everyone teaches perfectly.
It also really depends on the particular skills and desires of the student. I'm a more abstraction-oriented person, and as such when studying physics I often wished the teachers would rely more on abstract mathematics because in my view it made the phenomena clearer; I have no doubt that many students feel otherwise.
14 comments
[ 3.2 ms ] story [ 17.7 ms ] threadWhich is fine, he's entitled to his opinion. But the stronger argument that "mathematics is only good when it's tied to physics" argument is wrong and outdated -- belongs to the pre-computer, pre-information age.
Even on a purely mathematical level, some of his arguments don't hold. For instance, for all his complaining about how abstraction doesn't contribute anything -- Cayley's theorem on groups, Whitney's theorem on manifolds -- there ARE categories where the abstract definition is needed or is more natural, like algebraic varieties (which can't always be embedded in complex N-space for any N) and rings (I have no idea what better definition he'd be proposing).
To me, it sounds like he doesn't like or know much about the algebraic side of math, and resents how much it dominates in math today.
That sounds unlikely given who he was and the fact that he's now dead.
Given that Arnold was a student of Kolmogorov, was awarded the Wolf prize, and was denied the Fields medal due to some Soviet-era politics (https://en.wikipedia.org/wiki/Vladimir_Arnold#Honours_and_aw...), I think it's safe to say he understood algebra. You can also read his books, and find out that he used abstract algebra quite well. That's not what he's saying here, I think you misread.
> Which is fine, he's entitled to his opinion. But the stronger argument that "mathematics is only good when it's tied to physics" argument is wrong and outdated -- belongs to the pre-computer, pre-information age.
I think you're missing the point he's making and cherry-picking examples from the article. Indeed, much of the article is directed at how very conventional mathematics is taught at the school and undergraduate level, and one key point is that the way it is taught is too far divorced from how it was discovered a long time ago. He claims that teaching mathematics in the algebraic way makes it unnecessarily more difficult to solve mathematical problems that used to be essentially trivial, making people worse at mathematics for no good reason.
So you picking out the example of categories of algebraic varieties is misleading because that's not at all the focus of the article, it seems like an incidental example. I think the determinant and group examples are a much clearer illustration of the difference he's talking about.
I also have no idea what you mean by your comment about pre-computer, pre-information age. If anything, mathematics became much more of an experimental science (thus closer in spirit to physics in Arnold's view) now that you can run computational experiments so easily with this much computing power. Note his example of the difference between observation - model - investigation of the model - conclusions - testing by observations and definition - theorem - proof.
I'd reformulate his point to: Mathematics, especially when being taught, benefits from intuitive examples that motivate the abstract definitions.
A point that is quite obvious. Whilst certainly much intuition concerns physics, that need not be the underlying source of the intuition.
You do realize who you are talking about here right?
It analyses very concrete observations using abstract methods in order to construct an abstract definition. It doesn't simply spew out abstract definition. Everything is motivated by concrete facts.
I would not say it's against Arnold's method at all, if anything, it uses the same concrete to abstract treatment. It is, however, not the best way to learn about entropy, but it's a good way to formalise it once you have become familiar with it. It's poor for undergraduate level material, but it's good as graduate level material.
I just wanted to provide a somewhat humorous convergece between super-abstract mathematics and the concrete mathematics VI Arnold advocated: The category of smooth manifolds is the idempotent completion of the category of open subsets of Euclidean spaces:
http://mathoverflow.net/questions/224569/smooth-manifolds-as...
https://ncatlab.org/nlab/show/smooth+manifold#patching_as_id...
When you unravel the definition you get a very concrete idea of what a manifold is without the bureaucracy of charts and atlases and whatnot. In general, Lawvere's mathematical career is an ambitious program to cast classical mechanics in categorical terms. I see this as the synthesis of the two traditions.
Another way in which abstract mathematics is converging with concrete mathematics is the Curry-Howard-Lambek-... correspondence between logics, programming languages, and category theory. The most abstract branch of mathematics happens to be very amenable to the very physical, concrete process of computation!
I think abstract definitions that are far divorced from the concrete objects they are trying to capture is a deficiency in the abstraction. The correct abstraction somehow miraculously becomes far more concrete than any intermediate definition. For example, homotopy theory has a very ugly definition in set based mathematics, but by abstracting the essentials, homotopy type theory looks very promising at reconciling the bureaucracy of rigor required to do homotopy theory in sets and the very geometric intuition one has for homotopical objects.
But in the doing of mathematics is where I think Arnold is looking back at the past too romantically. Prior to programs like Bourbaki, the lack of unity in mathematical notation, definitions, ideas in methods was very present. Often if you were not a student of one of the masters of the time, you would have an incredibly difficult time understanding their work. The tradition that has lead to an 'over-abstraction' of things, as Arnold would have seen it, also lead to a much more widespread accessibility of contemporary mathematics. when you can count Kolmogorov among your teachers, this is less relevant to you. But for the 'unwashed masses' of mathematics this has made a world of difference.
Indeed. By far the best teachers I've had for math courses spent a good deal of time discussing the history of the topic and motivating its creation.
While I also appreciate abstraction, very often mathematics is taught at the wrong level of abstraction. A book I like a lot, in the spirit of his recommendations, is Hubbard & Hubbard.
It seemed that mathematics required more of my time than any of the subjects. So wouldn't spend the necessary time needed to actually store it in my head. Later in life, I realized how fascinating math was, and fell in love with it. Even though I am still not good at it, it's fun to solve puzzles.
Could you elaborate on this? Mathematics, perhaps uniquely among human intellectual endeavours, has no hidden rules. Its rules may be complicated and unintuitive at times, but they can be listed; and whatever follows from these listed rules is true (to be precise: whatever follows from the axioms is true in any model of the axioms), and whatever contradicts them is false, regardless of how natural or un- it may seem.
In particular, I am curious what you mean by "There would always be rules that were often forgotten." Do you mean that you forgot them, or that your teachers forgot them (both of which were eminently believeable!), or that practicing, professional mathematicians forget them? I am sceptical of the last (although plenty of, indeed probably all, professional mathematicians don't need to keep all the rules explicitly in their heads at all times), and, although I am eminently willing to believe the first two, I think that that shows a fault in the teacher, not the subject.
In my view, using abstract notions for describing is in no way incompatible to motivating them with concrete, visual notions, and that is what good teachers do and good students seek. This is done imperfectly by many teachers, but there is nothing new in stating that not everyone teaches perfectly.
It also really depends on the particular skills and desires of the student. I'm a more abstraction-oriented person, and as such when studying physics I often wished the teachers would rely more on abstract mathematics because in my view it made the phenomena clearer; I have no doubt that many students feel otherwise.