Can there be a different kind of mathematics with different sets of rules? That is without inference and axioms. One attempt might be Stephen Wolfram's "New Kind of Science", but I am not sure that the rules of automata can be called mathematics.
PS. I'm sure I can ask the question more clearly, however, I can't think of a better way for now!
If applied science were a house, physics would be the roof halting the rain, mathematics would be the bricks holding up the roof, and logic would be the earth it is built on. It isn't just a foundation, it is the basis for which anything can be proven or established.
Without which, you cannot prove anything. If we're working with the assumption that nothing is proven, prove 0 = 0. You can't. Technically one is on the left and one is on the right, so they're technically in different positions. You'd have to establish the axiom that 0 = 0 and 0 + 1 = 1 before you can establish that 1 = 1, etc. Heck, prove that you exist without using any axiom or assumption. Decartes did so, but with the assumption that God exists and would not lead us astray (which is no small axiom).
Your Title postulates no rules at all, but the first sentence posits (merely) different rules.
There are different formalisms for reducing math to a few primitives; e.g. ZFC.
When it comes to using the principles to build up something, there are different kinds of logic with differing levels of power.
Is it possible for logic itself to be based on totally different rules? Well, you'll have a hard time getting anywhere without Modus Ponens. But in the most abstract terms any number of different formulations exist with the same expressive power.
I suggest you read GEB, post haste! You seem to have the appetite for it right now, even without knowing it.
Say hi to the Tortoise for me.
Your question may have a point.
Our mathematical thinking is build on the foundation our brain provides. So objects, sets and logical connections may be the preferred way mathematical problems are tackled, but it is completely open if there is an alternative way by different, alien informational systems.
It could be that alien organisms use something we can call "mathematics", but they come to their conclusion in a completely inscrutable, non-human approach. They could vice versa not decipher our strange "mathematics". It is also possible that we both share some mathematical objects, but do not understand some concepts of the other worldview. It could also be that the differences are not so huge and we can in fact understand each other.
Given the lack of experience, I consider that an open question.
Mathematics without a basis in inference and rules? At first, you may think that pure human intuition might fit the bill - but it falls apart because our brains are finite and so can be modeled with complicated-enough Turing machines, which are based on the same rules of inference. Gödel's Incompleteness Theorem puts logic on third base, and Turing bats the run in.
However, one of the hidden clauses buried in the fine print of the Turing machine concept (at least for the purpose of finding computable numbers) is that the tape starts out blank - if there are already marks on the tape, there's no way to tell whether or not the existing marks already constitute a noncomputable sequence. Think about it, if the input sequence is computable, it was generated by an upstream Turing machine, and if so, you can combine the two machines into a bigger one that starts out blank. Even if the sequence might be comptable, you can't prove it; this is equivalent to the infamous, mis-named Halting Problem.
So, without logic, we have to be starting with marks on the tape our brains work on whose nature we know nothing about. Maybe even a noncomputable sequence. There are two candidates for placing such a sequence on the tape:
So, we're essentially left with supernatural, or even divine, inspiration for our non-logical mathematical discoveries.
After all, the most frighteningly gifted mathematician of our time, Srinivasa Ramanujan, believed that he got his mathematical intuitions from the goddess of Namagiri, and who's to say he wasn't right?
