Sometimes, the key to an invention is a selection of two (or more) systems and linking them together. Linking primarily means links via logical connectives, i.e. selecting premises, forming theorems.Sometimes, we already know the fields (systems!) but we need to find winning connection between the two (or more) of them. Where intuition fits in? It fits in selecting appropriate fields and selecting correct and useful links between them. Don't forget, an axiom is not provable within the system it defines, i.e. within the system it is developed from them. Choosing right premises and choosing to search for axioms is usually inspired by the linkage to the world outside the one that we look to find the axioms for.
Here are some examples. Each field, or system, is assumed to have its own set of axioms, postulates, and theorems, whatever that means in that system. As you will see, the system does not have to be mathematics. Note the selection and links.
Music -> Emotions.
Instrument -> Music -> Emotions
Physics -> Mathematics -> Human Language
Engineering Design Requirements -> Physics -> Mathematics -> Human Language.
Emotions, morals -> Paints, canvas -> Painting
Electric Power Systems -> Economic dispatch
For readers' exercise, try to define axioms in each of the systems and illustrate how the postulates, theorems in one system dictates premises in the other.
The power of a good question is that it can point to the areas of knowledge you need to familiarize yourself with. It can also initiate effective knowledge filtering and selection of the right facts that will be connected in a new, original way, to answer your question.
You probably got your engineering degree for knowing how to solve differential equations, not how to select useful and innovative initial and boundary conditions.
A mathematician and an artist. An accomplished NASA and IBM statistician and scientist talks about his sculptures. http://www.bbc.co.uk/news/magazine-16976145