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Why does a^2+b^2=c^2 for a right triangle? It's just axiomatic, right? Not exactly. It's only true in flat euclidean geometry. If General Relativity holds then

Why does a^2+b^2=c^2 for a right triangle? It's just axiomatic, right? Not exactly. It's only true in flat euclidean geometry. If General Relativity holds then we're only in a flat spacetime if the total energy content of the universe is in some range of values, otherwise the universe could be the 3d equivalent of a sphere or it could be a hyperbolic shape. In neither of these configurations do right triangles behave the way you learnt in primary school. So a^2+b^2=c^2 if and only if the big bang happened to have some particular initial conditions. It's not axiomatic, and studying _if it's even true_ tells you a lot more than you'd expect.

deltaonefour 5y ago
>Why does a^2+b^2=c^2 for a right triangle? It's just axiomatic, right? Pythagorean theorem is not an axiom. It's a theorem. >Not exactly. It's only true in flat euclidean geometry It's a theorem for Euclidean geometry. >So a^2+b^2=c^2 if and only if the big bang happened to have some particular initial conditions. It's not axiomatic, and studying _if it's even true_ tells you a lot more than you'd expect. For theorems and the entire concept of math to even work all forms of mathematics from Euclidean geometry to hyperbolic geometry rest on universal axioms of logic. At best we have built higher forms of logic from the universal axioms and mathematics from arbitrary assumptions but we have never delved deeper than the 3 "laws" of logic. The other weird thing that we don't understand, and this is likely connected with logic is probability. We don't know why it works. It seems foundational just like logic but the weird thing is, formally probability is just like another math theory with random axioms at it's foundation and the rest of the theorems built up from logic. >If General Relativity holds then we're only in a flat spacetime... This is part of science. Observed phenomena then statistically matched to a arbitrary mathematical model. The OP's article is talking about something more fundamental... he asks "why?" When you ask that kind of question you have to really deep. Deeper than the science, deeper then the math all the way to the bottom. Everything stops at logic and probability. There is nowhere left to go to even move the needle closer to answering the question "why?"
- grog454 5y ago
  > The other weird thing that we don't understand, and this is likely connected with logic is probability. We don't know why it works. Can you elaborate on this or provide further reading?
  - deltaonefour 5y ago
    Probability theory is just a bunch of theorems and axioms, it's a logical game we play just like graph theory or any other mathematical theory. If we roll a dice many times the dice follows a probability distribution. We don't know why this occurs in reality. Or in other words we don't know why this random mathematical theory some how is applicable to repeatedly observing a "random" process many times. We're not even sure what "random" is as we can't even formally define a random function in a computer. Probability seems like it's connected to logic. For example "if A then B" is a statement that is part of logic. But if you say "if A then 40% chance of B" then the statement is still logical but you can now see the connection with probability. It could be probability is the foundation of all causal connections and the logic itself is just a a special case of: "if A then 100% chance of B." But the weird part it, probability theory itself is built upon logic. You can't have probability as a formal mathematical theory without logic to derive theorems from axioms.
    - grog454 5y ago
      I've always thought of probability, teleologically, as an approximation of deterministic processes that are too complicated to calculate explicitly. Under very carefully controlled circumstances, even a die becomes deterministic and feasible to predict. Just drop it sufficiently low to the ground with as close to 0 angular and linear velocity as you can. Does something intrinsic or fundamental change as you increase the height at which you drop the die? I'd say no.
      - deltaonefour 5y ago
        >Under very carefully controlled circumstances, even a die becomes deterministic and feasible to predict. You aren't thinking deep enough. WHY is it an approximation? How come this random mathematical set of theorems and axioms happens to approximate these things? When we DON'T look at the experiment in a controlled circumstance what is the fundamental process driving it to follow the laws of probability? We in fact do not know. The phenomenon of probability is assumed... aka axiomatic. We have no further explanation for it. In fact it's encoded into a fundamental physical law. The second law of thermodynamics: Entropy must increase. It's actually a misnomer. Increasing entropy is not the law, it is the consequence. Entropy increases because disordered configurations of particles are more probable then ordered configurations. Thus over time the more probable configuration dominates the system. Hence the system becomes more and more disordered. Note the word "Probable." Thus Probability is the fundamental axiomatic law here. I understand why this is hard to process. It's because seemingly it seems like that if we understand the exact physics of a single particle it means we will understand the aggregate macro behavior of a billion particles by deriving the macro behavior from the physics of each individual particle. Turns out the aggregate behavior obeys axiomatic laws of entropy that we have no deeper explanation for. Even if you have a physics engine simulate the rolls of 10000000 physical dice approximately 1/6 of those dice come out to be 6. Why? We don't even know why the simulation behaves that way let alone the real world.

deltaonefour 5y ago

>Why does a^2+b^2=c^2 for a right triangle? It's just axiomatic, right? Pythagorean theorem is not an axiom. It's a theorem. >Not exactly. It's only true in flat euclidean geometry It's a theorem for Euclidean geometry. >So a^2+b^2=c^2 if and only if the big bang happened to have some particular initial conditions. It's not axiomatic, and studying _if it's even true_ tells you a lot more than you'd expect. For theorems and the entire concept of math to even work all forms of mathematics from Euclidean geometry to hyperbolic geometry rest on universal axioms of logic. At best we have built higher forms of logic from the universal axioms and mathematics from arbitrary assumptions but we have never delved deeper than the 3 "laws" of logic. The other weird thing that we don't understand, and this is likely connected with logic is probability. We don't know why it works. It seems foundational just like logic but the weird thing is, formally probability is just like another math theory with random axioms at it's foundation and the rest of the theorems built up from logic. >If General Relativity holds then we're only in a flat spacetime... This is part of science. Observed phenomena then statistically matched to a arbitrary mathematical model. The OP's article is talking about something more fundamental... he asks "why?" When you ask that kind of question you have to really deep. Deeper than the science, deeper then the math all the way to the bottom. Everything stops at logic and probability. There is nowhere left to go to even move the needle closer to answering the question "why?"

grog454 5y ago
> The other weird thing that we don't understand, and this is likely connected with logic is probability. We don't know why it works. Can you elaborate on this or provide further reading?
- deltaonefour 5y ago
  Probability theory is just a bunch of theorems and axioms, it's a logical game we play just like graph theory or any other mathematical theory. If we roll a dice many times the dice follows a probability distribution. We don't know why this occurs in reality. Or in other words we don't know why this random mathematical theory some how is applicable to repeatedly observing a "random" process many times. We're not even sure what "random" is as we can't even formally define a random function in a computer. Probability seems like it's connected to logic. For example "if A then B" is a statement that is part of logic. But if you say "if A then 40% chance of B" then the statement is still logical but you can now see the connection with probability. It could be probability is the foundation of all causal connections and the logic itself is just a a special case of: "if A then 100% chance of B." But the weird part it, probability theory itself is built upon logic. You can't have probability as a formal mathematical theory without logic to derive theorems from axioms.
  - grog454 5y ago
    I've always thought of probability, teleologically, as an approximation of deterministic processes that are too complicated to calculate explicitly. Under very carefully controlled circumstances, even a die becomes deterministic and feasible to predict. Just drop it sufficiently low to the ground with as close to 0 angular and linear velocity as you can. Does something intrinsic or fundamental change as you increase the height at which you drop the die? I'd say no.
    - deltaonefour 5y ago
      >Under very carefully controlled circumstances, even a die becomes deterministic and feasible to predict. You aren't thinking deep enough. WHY is it an approximation? How come this random mathematical set of theorems and axioms happens to approximate these things? When we DON'T look at the experiment in a controlled circumstance what is the fundamental process driving it to follow the laws of probability? We in fact do not know. The phenomenon of probability is assumed... aka axiomatic. We have no further explanation for it. In fact it's encoded into a fundamental physical law. The second law of thermodynamics: Entropy must increase. It's actually a misnomer. Increasing entropy is not the law, it is the consequence. Entropy increases because disordered configurations of particles are more probable then ordered configurations. Thus over time the more probable configuration dominates the system. Hence the system becomes more and more disordered. Note the word "Probable." Thus Probability is the fundamental axiomatic law here. I understand why this is hard to process. It's because seemingly it seems like that if we understand the exact physics of a single particle it means we will understand the aggregate macro behavior of a billion particles by deriving the macro behavior from the physics of each individual particle. Turns out the aggregate behavior obeys axiomatic laws of entropy that we have no deeper explanation for. Even if you have a physics engine simulate the rolls of 10000000 physical dice approximately 1/6 of those dice come out to be 6. Why? We don't even know why the simulation behaves that way let alone the real world.