What the fuck is mathematics? semantic architecture...

what the fuck is mathematics? semantic architecture? is it an application of just raw instinct to quantify notions or ideas and give them structure so that they can be consistent? where does consistency come from then, and why is it desired? where does mathematics stop being consistent? is there a limiting case or boundary where we stop having stable models for things like finding solutions to differential equations? is it always connected to the fundamental axioms or can they be surgically removed to be with something more supporting to ideas that aren't reliant on completion, consistency, structure, or quantity?

completion, consistency, structure, or quantity; where do all these properties intersect? can it be given a fundamental description?

mathematics is purity. its about what the world should be. a sphere is what we see when we look at a boulder or ball because we desire purity. the concept of smoothness, continuity, and diffusion all break down when we talk about reality because there is a discrete structure called atoms that make up our visible world. mathematics is archaic when encountering these models and is now self destructing due to the smoothness needed by relativity and the discreteness needed by quantum mechanics due to eigenvalues replacing degrees of freedom for elementary particles.

something newer will come that will treat mathematics like how mathematics treats philosophy; an idea that helps guide it but is too weak to bolster it against the relentless reality.

>what the fuck is mathematics?

Mathematics is logic of the highest level. It is when you turn off your human brain and wake up your inner eternal all powerful god and close your eyes to see a world much bigger than anything you will ever be, take some bits from it, and bring it to life.

Do you know what philosophers do? Well, imagine a philosopher but instead suppose he is actually intelligent. That is a mathematician.

>semantic architecture?
This is part of mathematics, though this is usually just contained in algebra.

>just raw instinct to quantify notions or ideas and give them structure so that they can be consistent?
In mathematics in general you cannot make things consistent because you are given consistent theories to begin with. But in meta-mathematics, where you look to create a theory in which mathematics can exist, the entire goal is to make it consistent so again, this is true about only a subfield of mathematics.

>where does consistency come from

Any mathematical logic book will define consistency. Consistency comes from that definition and that definition came from our desire to not have to worry too much because...

>why is it desired?
Because there is this thing called logical explosion that basically says that if you have an inconsistent theory, then everything is true and everything is false at the same time. Making any knowledge you extract from that theory literally meaningless. Therefore, only consistency can yield proper results.

>where does mathematics stop being consistent?
If you mean this literally then whenever we invent an inconsistent theory. If you mean this figuratively, when we poke at the outermost edge of mathematics, the fundamental axioms. Nobody really agrees on what the fundamental axioms should be, some mathematicians use some axioms, other mathematicians reject those same axioms. That's pretty inconsistent.

>is there a limiting case or boundary where we stop having stable models for things like finding solutions to differential equations?

Yeah, pretty trivial. As you know, there is no general formula to solve most differential equations. You can only reach general theorems when there is enough structure and with differential equations it is pretty easy to find something that has no structure at all and has to be tackled on its own.

>is it always connected to the fundamental axioms or can they be surgically removed to be with something more supporting to ideas that aren't reliant on completion, consistency, structure, or quantity?

Define completion.
And for the rest, yes it is.

completion such that you won't rely on a theorem to claim that another theorem is proven such that the proof relies on another theorem ad infinium until it just turns into a circular argument.

I see nothing wrong with a countable chain of proofs.

But if it is a circular argument in the sense that theorem A is proven by applying theorem B and then theorem B is proven by applying theorem A, that is just a fallacious argument.

So yeah, if you define completion like this then we need it and it is an important part of mathematics.

At the very bottom are the axioms of logic, so unless you want to prove something is true without relying on the validity of things like modus ponens, some circularity is going to be unavoidable.

>countable chain of proofs
You mean "finite".

A simple counterexample demonstrating that a countably infinite chain of proofs is invalid:

""Theorem"": The set of all natural numbers N is finite.
Proof 1: If N \ {0} is finite, then N is finite. (Here \ denotes set difference.)
Proof 2: If N \ {0, 1} is finite, then N \ {0} is finite.
Proof 3: If N \ {0, 1, 2} is finite, then N \ {0, 1} is finite.
...
Proof [math]\omega[/math]: If N \ N is finite, then N \ (N \ {0}) is finite.
Since N \ N is the empty set which is finite, N is finite.

Not writing it in TeX.

>muh syntax
Representing the set of natural numbers as N rather than [math]\mathbb{N}[/math] is not going to make the argument any less valid.

Mathematics is [math]not[/math] about syntax, although it is often [math]represented[/math] as a syntactic game of manipulation of formal symbols.
Just about every "paradox" of mathematical logic (Russell's, Cantor's, Girard's, Curry's, Skolem's, Burali-Forti's, Godel's, Tarski's, ...) arises from a conflation of the two.