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Care to expand? Do you know of a physical theory that would not be affected by new results in number theory? That is, a refutable, physical theory that is not

Care to expand? Do you know of a physical theory that would not be affected by new results in number theory? That is, a refutable, physical theory that is not based at all on integers?

saulrh 11y ago
The way this works is: * We take a set of axioms, just like any other mathematics. Say ZFC. * We construct an isomorphism - a bijective mapping - between states of universes and the reals. This step is honestly pretty trivial. Our laws of physics, as far as we can tell, are just big continuous-ish differential equations, so it's just going to be a serialization. * We construct the reals from our axioms. * By the isomorphism from before, we have constructed all possible universes which run under our laws of physics. * We have constructed our universe. Epistemologically speaking, mathematics doesn't work like anything else. Just about everything else you ever do is fundamentally bayesian in nature - we form hypotheses and attempt to disprove them by comparing them to our observations, and all we can do is disprove hypotheses, and even then not particularly strongly. Consult Karl Popper's works on empirical falsification for deeper discussion. Mathematics, in contrast, is inductive. We can prove that a statement is True by combining other True statements, and once we've done this it's done and cannot be "disproven" by the addition of more information. Well, unless you count Godel's result, but that's a bit of a special case and we usually ignore it because it's annoying.
- jsprogrammer 11y ago
  Sure. I'm actually reading through (a translation of) The Logic of Scientific Discovery right now, but I am already familiar with the ideas, conclusions, etc of Popper, Godel, and others whom have done related work. However, mathematical induction is different from "inductive reasoing" (as opposed to deductive, or "the deductive theory of testing", as Popper's translators call it). In fact, I don't believe it is a contrast to Popper's work; mathematical induction is actually a form of deductive reasoning (of course, this makes it rather confusing to talk about). Popper's method requires first formulating a system (axioms) in pure mathematics (logic). Then deriving the logical conclusions from the system and determining tests that could be performed that could contradict those conclusions. Popper also provides a schema for comparing systems, deriving contradictory conclusions, and designing experiments to distinguish between the various theories under consideration. >The way this works is: I don't mean to be obtuse, but it's not immediately clear to me at the moment what "this" refers to (although, from the context of the prior post, I would assume it refers to Inter-universal Teichmuller Theory). Is there another way you could clarify what we are talking about? Edit: Ok, I read my last post again and I think maybe you are giving a physical theory that is not based on integers or number theory? My concern with that interpretation is that I'm not sure how meaningful of a distinction can be made between Integers, ZFC, or Logic within the context of physical theories. Doesn't Shannon's work allow us to collapse all writable (a necessary precondition to execute Popper's method) theories to integers?

Care to expand? Do you know of a physical theory that would not be affected by new results in number theory? That is, a refutable, physical theory that is not based at all on integers?

saulrh 11y ago

The way this works is: * We take a set of axioms, just like any other mathematics. Say ZFC. * We construct an isomorphism - a bijective mapping - between states of universes and the reals. This step is honestly pretty trivial. Our laws of physics, as far as we can tell, are just big continuous-ish differential equations, so it's just going to be a serialization. * We construct the reals from our axioms. * By the isomorphism from before, we have constructed all possible universes which run under our laws of physics. * We have constructed our universe. Epistemologically speaking, mathematics doesn't work like anything else. Just about everything else you ever do is fundamentally bayesian in nature - we form hypotheses and attempt to disprove them by comparing them to our observations, and all we can do is disprove hypotheses, and even then not particularly strongly. Consult Karl Popper's works on empirical falsification for deeper discussion. Mathematics, in contrast, is inductive. We can prove that a statement is True by combining other True statements, and once we've done this it's done and cannot be "disproven" by the addition of more information. Well, unless you count Godel's result, but that's a bit of a special case and we usually ignore it because it's annoying.

jsprogrammer 11y ago
Sure. I'm actually reading through (a translation of) The Logic of Scientific Discovery right now, but I am already familiar with the ideas, conclusions, etc of Popper, Godel, and others whom have done related work. However, mathematical induction is different from "inductive reasoing" (as opposed to deductive, or "the deductive theory of testing", as Popper's translators call it). In fact, I don't believe it is a contrast to Popper's work; mathematical induction is actually a form of deductive reasoning (of course, this makes it rather confusing to talk about). Popper's method requires first formulating a system (axioms) in pure mathematics (logic). Then deriving the logical conclusions from the system and determining tests that could be performed that could contradict those conclusions. Popper also provides a schema for comparing systems, deriving contradictory conclusions, and designing experiments to distinguish between the various theories under consideration. >The way this works is: I don't mean to be obtuse, but it's not immediately clear to me at the moment what "this" refers to (although, from the context of the prior post, I would assume it refers to Inter-universal Teichmuller Theory). Is there another way you could clarify what we are talking about? Edit: Ok, I read my last post again and I think maybe you are giving a physical theory that is not based on integers or number theory? My concern with that interpretation is that I'm not sure how meaningful of a distinction can be made between Integers, ZFC, or Logic within the context of physical theories. Doesn't Shannon's work allow us to collapse all writable (a necessary precondition to execute Popper's method) theories to integers?