Does all of math, even the most abstract, have a physical analog...

does all of math, even the most abstract, have a physical analog... meaning all of math DESCRIBES in some way the physical universe, no exceptions?

t. Banach-Tarski paradox

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No.

prove it.

of course not. There's nothing stopping you from writing out some equations for a rocket ship traveling at 3000 times the speed of light but that doesn't describe any physical reality.

first of all, mathematics is based on axioms, we must choose these axioms carefully to keep things sensible. unfortunately the tendency in modern mathematics is to introduce completely unjustified axioms that allow you to do things that are anything but obvious. most egregious is the pernicious axiom of choice which is one of the biggest cop outs in modern maths, and leads to shit like banach-tarski

Math is a description, not the underlying framework itself. Even something as basic as Complex numbers don't have a particularly meaningful or even relevant physical meaning (no, Fourier analysis and it's derivatives don't count).

are you trolling or serious retard?

Obviously not.
You can pick different versions of Euclid's 5th postulate and get different versions of geometry, all equally consistent. Only one (at most) can describe the reality we live it.

BT fails because matter can't be infinitely subdivided into mathematical points. Same with Gabriel's Horn. mathworld.wolfram.com/GabrielsHorn.html

>does all of literature, even the most abstract, have a physical analog... meaning all of math DESCRIBES in some way the physical universe, no exceptions?
>t. retard

You have a poor understanding of axioms and set theory. I recommend Naive Set theory by Halmos.

That's not accurate. You can use a spherical interpretation for air travel, the standard for construction, and a Poincare perspective for parts of marine biology.

i know them, and i have worked through that very book
this is a meta-mathematical point in any case

>in some way

Yes, everything you can possibly imagine is in "some way" related to the physical universe. Math related or not.

So, why do you have a problem with the Axiom of Choice? It's necessity comes naturally and it is consistent. How is it a cop out?

it's an abomination. a tasteful set of axioms should be self-evident. axiom of choice isn't. not that zf was tasteful to begin with.

Yes, yes, you can find examples where all have applications. Maybe I should have picked a better example.
How about en.wikipedia.org/wiki/Skewes's_number
It would be nice to know it but I can't imagine there's a Skewes' number of anything in the material universe.
Math is undeniably useful but not everything has a physical analog. Which doesn't mean that math which doesn't have a physical interpretation is necessarily any less important.

What system has a nice set of axioms in your opinion?

Wildberger's Math Foundations

there is no fucking way the Banach-Tarski paradox could happen is because the cardinality of units of matter, like particles, cant go above countably infinite.

>Assumes a Euclidean proof for a nonphysical reality based on pure logic will apply to a physical reality based on different logic

no, basic arithmetic does, nothing else. if it does then it's just a coincidence of intuition (which we derive from reality), and due to mathematics being consistent and logical (reality is too).

Essentially yes. Check out Godel's Completeness theorem.

There is nothing wrong with it, or any axiom. It's just wrong to assume that anything you derive using this particular axiom will have some implications for the physical universe.

Your logic is only as good a predictor of any real results as your axioms are an approximation of the thing you're studying. There's no contradiction between being logically consistent and completely wrong.

The Banach-Tarski paradox works just fine with things that don't have matter.

>prove it
zero
infinity
pi

even the most rudimentary concepts have no practical application in reality (at least, for now).

I can tell you're a pure bro. I hate all these applied guys. They're like the soft science of math.

no but it's entirely based on physical things

professors often bring up how bases haven't been found in nature which i assumed was true but was always skeptical. i can't provide an example where they were found

imaginary numbers pop up everywhere in electrical engineering and they are real and detectable in transmission lines for instance

actually infinity doesn't exist in nature technically as everything is quantifiable just massive unless i'm ignorant to some quantum shiz