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On teaching mathematics by V.I. Arnold (1997)

43 points by adamnemecek 8y ago

projectileboy 8y ago
He lost me at the beginning. Paraphrasing Wolfgang Pauli, “That is not only not right; it is not even wrong.” Certainly much of math was born from science. But the notion of pure mathematics goes back to antiquity.
- ahartmetz 8y ago
  I think his point is that mathematics has always been an abstraction of physical things before the effort to basically invent first principles of mathematics, axioms.
- romwell 8y ago
  I would advise you to go and read his statement in full. His point is quite valid, even if some of the phrasing is not well-perceived.
  - projectileboy 8y ago
    Thanks for the response. I took your advice and was glad I did. After reading the entire piece, I agree with your statement.
    - romwell 8y ago
      Thank you for following through!
ahartmetz 8y ago
I can ecpecially relate to the part where intuitively obvious things were probably invented for intuitively obvious reasons, then the derivation was thrown away and the thing was described axiomatically like the blind man describes an elephant. The non-explanation of the determinant in my linear algebra textbook was especially infuriating: "There is this thing, right? We will later see its uses. It happens to have the following properties: ..., If you swap two columns its sign changes, ..."
- gizmo686 8y ago
  The important thing is that a reasonable derivation is presented. It need not be historically accurate. For instance, if I was teaching category theory to CS students, I would try to motivate it with topology. Similarly, if I were teaching group theory, I would not try motivating it with solving polynomial equations.
  - jacobolus 8y ago
    > Similarly, if I were teaching group theory, I would not try motivating it with solving polynomial equations. Why not? That’s a fine (and historically important) reason to study group theory, and everyone who goes through high school is fairly deeply familiar with polynomial equations. While we are discussing Arnold, you might like this book created out of Arnold's lectures to some bright high school students in the 60s, https://www.mathcamp.org/2015/abel/abel.pdf https://www.mathcamp.org/2015/abel/abel.pdf
    - gizmo686 8y ago
      >Everyone who goes through high school is fairly deeply familiar with polynomial equations. A high schooler is not nearly familiar enough with polynomials to understand the group theory component of Galois Theory. The central observation of Galois theory is that, in certain circumstances, it is possible to permute values without affecting the correctness of any polynomial (with coefficients in a certain field). For example, suppose you have two unknown values, A and B, and knew that the following polynomial equations held: A + B = 0 AB = 1 A^2 = -1 B^2 = -1 There are two possibilities: A=i and B=-i or A=-i and B=i. Further, no matter what polynomials (with rational coefficients) I give you, it is impossible to distinguish these two possibilities. The group in this case is all the possible ways of permuting A and B which do not effect any polynomials (with rational coefficients). There is a lot of work involved in actually developing the theory I summarized above. EDIT: Perhaps more damning for using this approach to teaching group theory is that it is, in general, difficult to actually compute what the Galois group is for a given polynomial.
      - 8y ago
joker3 8y ago
"Mathematics is a part of physics." Try telling that to a statistician, or a computer scientist, or an economist, or.... I think it's worth noting that category theory, which was initially derided as abstract nonsense even by other mathematicians, is now a central field of study in math and is also becoming a major part of theoretical physics. If we only studied things that are immediately applicable to some particular science, we would not be able to advance the sciences as quickly as we can now. I'm not that wild about the Bourbaki style of exposition, which does emphasize rigor over intuition, but I think that there is a lot of value in having students go through that phase. Terry Tao has a good explanation of why at https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-..., and I'd encourage anyone interested in the topic to read it.
- gizmo686 8y ago
  I assure you, "abstract nonsense" is a term of affection.
- gowld 8y ago
  What Arnold is saying is that mathematics is part of the natural world, with immutable truths that can be discovered. He is saying that the science of mathematics is part of the natural sciences. There's no reason for a statistician, or a computer scientist, or an economist to agree with that -- these fields depend on the natural truths discovered by mathematical science just as surely a biologists depends on the natural truths discovered by chemists.
  - perl4ever 8y ago
    Hasn't it been proven that there must be areas of math that can never be connected to other areas of math? Saying it's part of science, or the natural world, seems like saying all math is part of one great whole. That's something that's been passé since 1931 or so, I thought.
    - harry8 8y ago
aphextron 8y ago
This has hit the front page at least 3 times in the last year. Why?

43 points by adamnemecek 8y ago

He lost me at the beginning. Paraphrasing Wolfgang Pauli, “That is not only not right; it is not even wrong.” Certainly much of math was born from science. But the notion of pure mathematics goes back to antiquity.

ahartmetz 8y ago
I think his point is that mathematics has always been an abstraction of physical things before the effort to basically invent first principles of mathematics, axioms.
romwell 8y ago
I would advise you to go and read his statement in full. His point is quite valid, even if some of the phrasing is not well-perceived.
- projectileboy 8y ago
  Thanks for the response. I took your advice and was glad I did. After reading the entire piece, I agree with your statement.
  - romwell 8y ago
    Thank you for following through!

ahartmetz 8y ago

I can ecpecially relate to the part where intuitively obvious things were probably invented for intuitively obvious reasons, then the derivation was thrown away and the thing was described axiomatically like the blind man describes an elephant. The non-explanation of the determinant in my linear algebra textbook was especially infuriating: "There is this thing, right? We will later see its uses. It happens to have the following properties: ..., If you swap two columns its sign changes, ..."

gizmo686 8y ago
The important thing is that a reasonable derivation is presented. It need not be historically accurate. For instance, if I was teaching category theory to CS students, I would try to motivate it with topology. Similarly, if I were teaching group theory, I would not try motivating it with solving polynomial equations.
- jacobolus 8y ago
  > Similarly, if I were teaching group theory, I would not try motivating it with solving polynomial equations. Why not? That’s a fine (and historically important) reason to study group theory, and everyone who goes through high school is fairly deeply familiar with polynomial equations. While we are discussing Arnold, you might like this book created out of Arnold's lectures to some bright high school students in the 60s, https://www.mathcamp.org/2015/abel/abel.pdf https://www.mathcamp.org/2015/abel/abel.pdf
  - gizmo686 8y ago
    >Everyone who goes through high school is fairly deeply familiar with polynomial equations. A high schooler is not nearly familiar enough with polynomials to understand the group theory component of Galois Theory. The central observation of Galois theory is that, in certain circumstances, it is possible to permute values without affecting the correctness of any polynomial (with coefficients in a certain field). For example, suppose you have two unknown values, A and B, and knew that the following polynomial equations held: A + B = 0 AB = 1 A^2 = -1 B^2 = -1 There are two possibilities: A=i and B=-i or A=-i and B=i. Further, no matter what polynomials (with rational coefficients) I give you, it is impossible to distinguish these two possibilities. The group in this case is all the possible ways of permuting A and B which do not effect any polynomials (with rational coefficients). There is a lot of work involved in actually developing the theory I summarized above. EDIT: Perhaps more damning for using this approach to teaching group theory is that it is, in general, difficult to actually compute what the Galois group is for a given polynomial.
    - 8y ago

joker3 8y ago

"Mathematics is a part of physics." Try telling that to a statistician, or a computer scientist, or an economist, or.... I think it's worth noting that category theory, which was initially derided as abstract nonsense even by other mathematicians, is now a central field of study in math and is also becoming a major part of theoretical physics. If we only studied things that are immediately applicable to some particular science, we would not be able to advance the sciences as quickly as we can now. I'm not that wild about the Bourbaki style of exposition, which does emphasize rigor over intuition, but I think that there is a lot of value in having students go through that phase. Terry Tao has a good explanation of why at https://terrytao.wordpress.com/career-advice/theres-more-to-mathematics-than-rigour-and-proofs/ https://terrytao.wordpress.com/career-advice/theres-more-to-..., and I'd encourage anyone interested in the topic to read it.

gizmo686 8y ago
I assure you, "abstract nonsense" is a term of affection.
gowld 8y ago
What Arnold is saying is that mathematics is part of the natural world, with immutable truths that can be discovered. He is saying that the science of mathematics is part of the natural sciences. There's no reason for a statistician, or a computer scientist, or an economist to agree with that -- these fields depend on the natural truths discovered by mathematical science just as surely a biologists depends on the natural truths discovered by chemists.
- perl4ever 8y ago
  Hasn't it been proven that there must be areas of math that can never be connected to other areas of math? Saying it's part of science, or the natural world, seems like saying all math is part of one great whole. That's something that's been passé since 1931 or so, I thought.
  - harry8 8y ago

aphextron 8y ago

This has hit the front page at least 3 times in the last year. Why?