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"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 catego

by 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
    I don't know the answer to your question but want to state the a differing point of view in the same vein which is how I understood it as the contrast might be something I learn from. There can never be practically useless mathematics. Someone will always find a practical use in the physical world for any new mathematics that is developed. The story I heard illustrating this was that in the shadow of WW1 various mathematicians who lived through that horror became (understandably) pacifists (Hardy?) and decided they wanted to make sure their work would never be useful in weaponry and so devoted themselves to the utterly useless number theory. Sadly for them in a world of crypto number theory has a major use and that is military.
    - perl4ever 8y ago
      This is a layman's view, but I thought Gödel's incompleteness theorem means there must be in principle mathematics that can never be connected to other parts of mathematics. And if something is part of the universe, it must be connected to something we can perceive, so I conclude that not all math can be connected to that which concerns something real. In other words, even though math that was thought to be useless has repeatedly been applied to physics, I think the claim that it is "always" applicable is too strong and in fact proven false in principle.
      - entrypoint0 8y ago
        This may or may not be helpful: Geometry is decidable per tarski's axioms of elementary geometry. As far as I know physical theories are dominantly geometric. Unfortunately I know little in the topic, so I cannot add anything beyond noting his work.
      - joker3 8y ago
        That's not what GIT says. See https://plato.stanford.edu/entries/goedel-incompleteness/ https://plato.stanford.edu/entries/goedel-incompleteness/ for a description of what it does say.
effie 8y ago
> Try telling that to a statistician, or a computer scientist, or an economist, or.... It is easy to miss Arnold's point here due to imperfect expression. Which is that without reality checks and intuition about the math concepts all the elaborate abstract thought frameworks and rigorous results may be very misleading and counter-productive. > central field of study in math and is also becoming a major part of theoretical physics It is in vogue, perhaps, but it is hardly a major part of theoretical physics.

by joker3 8y ago

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
  I don't know the answer to your question but want to state the a differing point of view in the same vein which is how I understood it as the contrast might be something I learn from. There can never be practically useless mathematics. Someone will always find a practical use in the physical world for any new mathematics that is developed. The story I heard illustrating this was that in the shadow of WW1 various mathematicians who lived through that horror became (understandably) pacifists (Hardy?) and decided they wanted to make sure their work would never be useful in weaponry and so devoted themselves to the utterly useless number theory. Sadly for them in a world of crypto number theory has a major use and that is military.
  - perl4ever 8y ago
    This is a layman's view, but I thought Gödel's incompleteness theorem means there must be in principle mathematics that can never be connected to other parts of mathematics. And if something is part of the universe, it must be connected to something we can perceive, so I conclude that not all math can be connected to that which concerns something real. In other words, even though math that was thought to be useless has repeatedly been applied to physics, I think the claim that it is "always" applicable is too strong and in fact proven false in principle.
    - entrypoint0 8y ago
      This may or may not be helpful: Geometry is decidable per tarski's axioms of elementary geometry. As far as I know physical theories are dominantly geometric. Unfortunately I know little in the topic, so I cannot add anything beyond noting his work.
    - joker3 8y ago
      That's not what GIT says. See https://plato.stanford.edu/entries/goedel-incompleteness/ https://plato.stanford.edu/entries/goedel-incompleteness/ for a description of what it does say.

effie 8y ago

> Try telling that to a statistician, or a computer scientist, or an economist, or.... It is easy to miss Arnold's point here due to imperfect expression. Which is that without reality checks and intuition about the math concepts all the elaborate abstract thought frameworks and rigorous results may be very misleading and counter-productive. > central field of study in math and is also becoming a major part of theoretical physics It is in vogue, perhaps, but it is hardly a major part of theoretical physics.