In his introduction to projective geometry for Scientific Americans, Morris Kline recounted the theory's artistic origins. Renaissance painters solved the problem of foreshortening by a skillful use of projection and section. Over two centuries later, mathematicians developed a theory of perspective.

This example contrasts two forms of mathematical activity: experimental vs. theoretical, inductive vs. deductive, or analog vs. analytic. The development of perspective painting and projective geometry is more complex (Filippo Brunelleschi was both an architect and engineer, and Desargues was both an architect and a mathematician).

But it's worthwhile considering how mathematics develops in the individual and the profession in multiple ways. And that historically these different cognitive modes often emerge within hierarchies. A physics professor of mine in university once confided that he chose experimental over theoretical physics because he wasn't the top student in his class. 

Last week we learned there's a geometry without angle and distance. This week we'll encounter a mathematics without symbols and equations.