The thing is, not even a mathematicien thinks always about math that way. In mathematics, one needd to distinguish between a phase of exploration, where you work along a fuzzy, intuitive process, and a phase of justification, where one try to prove one's conjectures. The second part is math-as-a-formal-system, the cliché of mathematics. So it's not a problem of definition, it's a matter of accepting mathematics as a complex process, practiced in various ways and different for each mathematician.
The thing is, not even a mathematicien thinks always about math that way. In mathematics, one needd to distinguish between a phase of exploration, where you work along a fuzzy, intuitive process, and a phase of justification, where one try to prove one's conjectures. The second part is math-as-a-formal-system, the cliché of mathematics. So it's not a problem of definition, it's a matter of accepting mathematics as a complex process, practiced in various ways and different for each mathematician.