/sci/ - Science & Math

>>12510620
I think for finite-dimensional linear representations, there's uniqueness theorems.

There's certainly a lot of be done in terms of non-linear as well as infinite-dimensional representations. In fact I think this subject is so vast, we'll never get it done.

That said, in some way the axioms of trying to squeeze things into a Lagrangian framework is arbitrary anyway, and somewhat ties to perturbative field theory.

>>10938587
I don't think you can do much better than embedding the nats as a semiring into the reals, so I guess you mean just in bijection? I think even in the most restrictive schools, the reals and the naturals are just in subcountable relation to each other.
Don't fight boys.

Coming back on topic - what's a nice book on order or lattice theory. Let's start with something not all too dry.

Well for one, your two examples aren't "true".

Fix a language (may it be first order language of English), and consider any property [math] \Phi [/math]. Classically, we take the law of excluded middle (LEM) as axiom so that for any such property,

[math] \Phi \lor \neg \Phi [/math]

holds.

Now take any predicate [math] \phi [/math], and for any [math] y [/math], the claim [math] \phi(y) \lor \neg \phi(y) [/math] says that the property either holds for it or not. And indeed, by LEM, we have

[math] \forall x.\ \phi(x) \lor \neg \phi(x) [/math]

If you adopt a mathematics with this sort of rule, then you force a non-constructive interpretation on the subject, because we can (as we know since Turing) come up with predicates that can't be evaluated by humans. E.g. the mortal matrix problem
>Given two arbitrary 15-by-15 matrices A,B with integer entries, can they be multiplied in some way, possibly with repetition, such that they give the zero matrix.
I.e. naively you'd start trying out
>A, B, AB, BA, ABA, ... ABBBABAABAAA,...
and this makes for a word problem that Emil Post and those early CS guys have shown is such that there can't possibly be a clever algorithm that is able to, for any given two such matrices, correctly compute whether they can be multiplied to a zero matrix.
https://en.wikipedia.org/wiki/List_of_undecidable_problems

You can be an ultrafinitists or at least a hard formalists like the Russian school, but even they considered e.g.
https://en.wikipedia.org/wiki/Markov%27s_principle

As far as axioms are concerned to which you can easily find models (e.g. the axiom of groups, where 0, 1 and addition mod 2 is a first model), then those axioms are mere specifications with what you want to deal with.
This example is basic one from universal algebra, which are theories you can cast positively in terms of positive constants (symbols) that don't even need existential quantifiers.
https://en.wikipedia.org/wiki/Universal_algebra

The main mathematical concept - understanding of which implies you got all the tool to understand theoretical physics - are Lie groups.