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The title is clickbait - they found a connection between two mathematical objects by using a third mathematical object to translate between the two. The third m

The title is clickbait - they found a connection between two mathematical objects by using a third mathematical object to translate between the two. The third mathematical object also happens to be useful in physics.

Koshkin 3y ago
Yeah, I have always wondered, though, how much physics is in the modern theoretical physics.
- danbruc 3y ago
  That probably depends heavily on the field, there is certainly a huge difference whether you are trying to better understand established physics or whether you are trying to come up with a new theory in an area where we are lacking experimental data.
  - Koshkin 3y ago
    Well, what I mean is that (to give you perhaps an extreme example) classical mechanics, while not "lacking experimental data," has for the last 200 years been effectively considered part of (applied) mathematics (known as "analytical mechanics"). Thermodynamics, electromagnetism, quantum field theory, general relativity - they all seem to be purely mathematical now.
    - danbruc 3y ago
      Then I probably misunderstood you, I thought you were talking about how closely the math is tied to the actual world, say all the reformulations of classical mechanics as opposed to some supersymmetric theory. In the end physics is applied mathematics, once you have a theory that describes your observations, you can just study the mathematical structure and look for interesting things, only occasionally going back to the lab to verify your predictions, improving confidence that you picked the correct model.
lukeplato 3y ago
Not sure if it's clickbait, but there is good reason to believe in a deep connection between 4d gauge field theories (i.e. of the standard model) and number theory through representation theory. This involves other work than mentioned in this article though it's still inscrutable for most at this point (motivic cohomology/Hilbert–Pólya in RH, supersymmetry in YM mass gap, cosmic galois in renormalization). From Peter Woit's blog: > It’s worth noting that while there are many connections to the ideas originating with Langlands, this new work shows that the “Langlands program” has expanded into a striking vision relating different areas of mathematics, with a strong connection to deep ideas about quantization and quantum field theory. The way in which these ideas bring together number theory and quantum field theory provide new evidence for the deep unity of fundamental ideas about mathematics and physics. https://www.math.columbia.edu/~woit/wordpress/?p=13578 https://www.math.columbia.edu/~woit/wordpress/?p=13578
- danbruc 3y ago
  [...] provide new evidence for the deep unity of fundamental ideas about mathematics and physics. This makes no sense to me at all. Mathematics is a modeling language and when we use it to model fundamental aspects of the real world, then we call that physics. So there is obviously some relation between mathematics and physics but I do not understand why you would expect some kind of unity, mathematics is much richer than physics.
  - Koshkin 3y ago
    This makes no sense to you precisely because you think that mathematics is (only) a language. It's not. A language is invented; mathematics is discovered. In the US it's not called a science, but for all intents and purposes it indeed is.
    - danbruc 3y ago
      A language is invented; mathematics is discovered. I peg to differ. Mathematics is invented when you define the axioms of some structure, after that you discover the consequences of the axioms you picked. First you define the natural numbers and the operations on them, then you discover the primes. So actually invention and discovery are not mutually exclusive here, but invention is more fundamental as it defines what is there to be discovered.

Koshkin 3y ago

Yeah, I have always wondered, though, how much physics is in the modern theoretical physics.

danbruc 3y ago
That probably depends heavily on the field, there is certainly a huge difference whether you are trying to better understand established physics or whether you are trying to come up with a new theory in an area where we are lacking experimental data.
- Koshkin 3y ago
  Well, what I mean is that (to give you perhaps an extreme example) classical mechanics, while not "lacking experimental data," has for the last 200 years been effectively considered part of (applied) mathematics (known as "analytical mechanics"). Thermodynamics, electromagnetism, quantum field theory, general relativity - they all seem to be purely mathematical now.
  - danbruc 3y ago
    Then I probably misunderstood you, I thought you were talking about how closely the math is tied to the actual world, say all the reformulations of classical mechanics as opposed to some supersymmetric theory. In the end physics is applied mathematics, once you have a theory that describes your observations, you can just study the mathematical structure and look for interesting things, only occasionally going back to the lab to verify your predictions, improving confidence that you picked the correct model.

lukeplato 3y ago

Not sure if it's clickbait, but there is good reason to believe in a deep connection between 4d gauge field theories (i.e. of the standard model) and number theory through representation theory. This involves other work than mentioned in this article though it's still inscrutable for most at this point (motivic cohomology/Hilbert–Pólya in RH, supersymmetry in YM mass gap, cosmic galois in renormalization). From Peter Woit's blog: > It’s worth noting that while there are many connections to the ideas originating with Langlands, this new work shows that the “Langlands program” has expanded into a striking vision relating different areas of mathematics, with a strong connection to deep ideas about quantization and quantum field theory. The way in which these ideas bring together number theory and quantum field theory provide new evidence for the deep unity of fundamental ideas about mathematics and physics. https://www.math.columbia.edu/~woit/wordpress/?p=13578 https://www.math.columbia.edu/~woit/wordpress/?p=13578

danbruc 3y ago
[...] provide new evidence for the deep unity of fundamental ideas about mathematics and physics. This makes no sense to me at all. Mathematics is a modeling language and when we use it to model fundamental aspects of the real world, then we call that physics. So there is obviously some relation between mathematics and physics but I do not understand why you would expect some kind of unity, mathematics is much richer than physics.
- Koshkin 3y ago
  This makes no sense to you precisely because you think that mathematics is (only) a language. It's not. A language is invented; mathematics is discovered. In the US it's not called a science, but for all intents and purposes it indeed is.
  - danbruc 3y ago
    A language is invented; mathematics is discovered. I peg to differ. Mathematics is invented when you define the axioms of some structure, after that you discover the consequences of the axioms you picked. First you define the natural numbers and the operations on them, then you discover the primes. So actually invention and discovery are not mutually exclusive here, but invention is more fundamental as it defines what is there to be discovered.