What is a casual thought that you cannot explain in math?

That question makes no sense. You can explain anything in math, because math is a language and lets you define whatever terms and axioms you need at a given moment.

(Whether or not such explanation is useful for anything is another issue entirely.)

Can you explain how intuition led you to try a certain approach?

Is it enough if I hand-wave it with probability distributions, or do you want me to write out adjacency search in a high-dimensional space?

It's totally different: those formalisms are in a workbench, following a set of rules that either work or not.

So, yes, that (math) is representative of the actual process: pattern recognition gives you spontaneous ideas, that you assess for truthfulness in conscious acts of verification.

Math comes from brains.