We don't discover mathematics, we invent it to solve problems. Sometimes we invent it out of curiosity or thought experiment, and it later turns out to solve problems;

Please stop talking about things you just don't know about. Ever since axiomatic systems were defined, we

*define* (or to use your words, invent) a system of axioms, which is the set of claims that is to be held as true without any proof. Then we

*discover* theorems, which are consequences of the axioms, and are true if and only if the axioms are true.

Just because mathematics is consistent within a context, does not mean its consistent within itself overall (its not by necessity), or that its consistent with all of nature or all of truth. There would be a lot less disagreement and work within mathematics if there was a consensus on any particular axiom within it. As Descarte's saw, and we see again and again, there is never consensus and the burden of debt for science and math is too high to act reasonably within it, without faith.

Any given axiomatic system in Mathematics, of which there are many, can fall into one of the following categories:

- It has been proven to be consistent (it is impossible to prove a given theorem and the opposite theorem)

- It has been proven to be inconsistent (a theorem and its opposite were both proven)

- It has been proven that the consistency question cannot be answered

- We don't know yet.

Mathematics does not have to be consistent (whatever that means) with nature or with truth (depending on what you call truth). It only has to be internally consistent. The phrase you concocted:

"There would be a lot less disagreement and work within mathematics if there was a consensus on any particular axiom within it"

is the best example I have ever found of a complete ignorant talking about Mathematics. Mathematicians do not need or want consensus about axioms. Each area of Mathematics uses its own set of axioms, and whoever wants to create a new set of axioms is welcome to try.

The consensus you are talking about exists in Science, not in Mathematics, and it is closely related to Scientific Theories. It is only then that abstract mathematical concepts, which are true by definition, get applied to real world observations and experiments and all the need for consensus start to have some sense.

The fact science can be done without mathematics can in one fel swoop remove its necessity and its necessitative connection to the world for most that would will it.

A very small subset of our scientific knowledge can be worked without Mathematics. That does not give any support at all to your claim that... just a moment... what the hell are you trying to say in that phrase? Anyhow, most Science is intimately related to Mathematics, and loses all predictive power if you take the mathematical tools away. Science without Mathematics would be like what you see in this forum.