6 ms·

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 the

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
      You define, but the choice is in fact not quite arbitrary; rather, it is guided by something objective, i.e. by something that exists outside of your own mind.
      - danbruc 3y ago
        What are the constraints? I can make up any set of axioms I like and see what they imply, there is no need that they are in any way related to the physical world. Sure, there is a lot of mathematics that was specifically invented in order to deal with the real world, but that is not a general requirement.
        Koshkin 3y ago
        > I can make up any set of axioms Yeah, but that wouldn't be mathematics.
        danbruc 3y ago
        Why not? If I decide I want to study the properties of the space of total function from vectors of octonions with prime dimension to the surreal numbers, who is to say that this is not mathematics?
        Koshkin 3y ago
        What you are describing is not "any set of axioms." (Rather, it is a pre-existing concept.)
        danbruc 3y ago
        Yes, but that is a technical detail, it was just easier for me to come up with that than making up a set of axioms. How is an obscure combination of existing axioms still mathematics but not any set of axioms I make up? And how would we ever extend mathematics if coming up with new axioms is not mathematics? But let us just take the integers with the common definitions for addition, subtraction and multiplication, but then redefine them so that every operation first performs the usual operation and then increments the result by one. 1 + 1 = 3 1 * 1 = 2 (1 + 2) * 3 = 13 (a + b) * c = a * c + b * c + c - 4 Still not quite what I had in mind, nothing completely new, but maybe at least different enough from the normal integers to have some weird properties. Not the numbers themselves, they are still just the integers, but the algebraic expressions involving the redefined operations.

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
    You define, but the choice is in fact not quite arbitrary; rather, it is guided by something objective, i.e. by something that exists outside of your own mind.
    - danbruc 3y ago
      What are the constraints? I can make up any set of axioms I like and see what they imply, there is no need that they are in any way related to the physical world. Sure, there is a lot of mathematics that was specifically invented in order to deal with the real world, but that is not a general requirement.
      - Koshkin 3y ago
        > I can make up any set of axioms Yeah, but that wouldn't be mathematics.
        danbruc 3y ago
        Why not? If I decide I want to study the properties of the space of total function from vectors of octonions with prime dimension to the surreal numbers, who is to say that this is not mathematics?
        Koshkin 3y ago
        What you are describing is not "any set of axioms." (Rather, it is a pre-existing concept.)
        danbruc 3y ago
        Yes, but that is a technical detail, it was just easier for me to come up with that than making up a set of axioms. How is an obscure combination of existing axioms still mathematics but not any set of axioms I make up? And how would we ever extend mathematics if coming up with new axioms is not mathematics? But let us just take the integers with the common definitions for addition, subtraction and multiplication, but then redefine them so that every operation first performs the usual operation and then increments the result by one. 1 + 1 = 3 1 * 1 = 2 (1 + 2) * 3 = 13 (a + b) * c = a * c + b * c + c - 4 Still not quite what I had in mind, nothing completely new, but maybe at least different enough from the normal integers to have some weird properties. Not the numbers themselves, they are still just the integers, but the algebraic expressions involving the redefined operations.