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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 geomet

by jl274 10y ago

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 But the stronger argument that "mathematics is only good when it's tied to physics" argument is wrong and outdated -- belongs to the pre-computer, pre-information age.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.

pvg 10y ago
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.
conistonwater 10y ago
> 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_awards 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 But the stronger argument that "mathematics is only good when it's tied to physics" argument is wrong and outdated -- belongs to the pre-computer, pre-information age.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.
rocqua 10y ago
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.
francasso 10y ago
> 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. You do realize who you are talking about here right?

by jl274 10y ago

pvg 10y ago

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.

conistonwater 10y ago

> 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_awards 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.

rocqua 10y ago

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.

francasso 10y ago

> 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. You do realize who you are talking about here right?