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:
- An equally-mystical object called an oracle which resembles a Turing machine but can't possibly be one.
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?