[danidiaz]

I'm not a mathematician, but I feel the "syntethic"/"analythic" distinction in mathematics is an interesting and useful concept.

"In modern mathematics, an analytic theory is one whose basic objects are defined in some other theory, whereas a synthetic theory is one whose basic objects are undefined terms given meaning by rules and axioms"—Michael Shulman

In programming terms, I guess "synthetic" mathematics feels a bit like programming to an abstract interface.

https://ncatlab.org/nlab/show/synthetic+mathematics

In the first part of this video Cédric Villani gives (in French) a nice explanation of the distinction: https://youtu.be/xzVk56EKBUI?t=258

Edit: an English explanation, also by Villani: https://www.youtube.com/watch?v=AIrLXbwyYXQ