Jaynes' interpretation of probability theory as an extension of logic that calculates probability of propositions conditioned on other propositions is much more straightforward that the usual set theory formulation, and the notation he uses is enormously helpful.

Alfred Korzybski, of The Map is the Territory fame, had some chapters on the concepts of differential calculus in Science in Sanity which way back when I found provided key insights (as all the General Semantics literature does, and should get more attention here.)

Judea Pearl is the main game in town for a clear notation for the mathematical analysis of causality.

David Wolpert's theoretical framework for analyzing generalization algorithms (in which he produced Stacked Generalization) is a little more obscure, but very useful for learning theory.