(Original poster here.) If you are interested in this sort of thinking, I recommend checking out the page I found it on: https://www.dpmms.cam.ac.uk/~wtg10/vsipage.html , including the last link on the page that delves a bit into the philosophy behind mathematics and its implications.

That page in turn was found at the end of the author's book 'Mathematics: a Very Short Introduction', which is pretty good exploration of what goes into mathematical definitions and how modern mathematical thinking works, addressed towards the layman but without excessive watering down of the subject.