I decided to read a chapter of 'Software Engineering Mathematics' every morning before I get up, to ease me gently into the day. In fact, the title is misleading as it's actually about 'Formal Methods Mathematics'.
It introduces Formal Systems, Propositional Calculus, Predicate Calculus, Set Theory, Relations, Functions, Sequences and Algebras to build up to case studies and formal methods.
In 1982/3 I was asked a question by a Cambridge graduate about the metaphysics of maths. It came from one of his professors: "Is maths a belief system?" He said it can be seen as a set of rules that we made up and follow strictly or as an embodiment of great truths about the universe but that requires faith. At the time, I think it made me glad I didn't go to Cambridge but I think I'm far more curious now than I was at 22. I think I needed to know more first.
I need a bit more time to think about it but I think Chapter 1 may have finally provided an answer: maths is a language in which we can express some of our belief systems. We project meaning onto all language but maths is particularly abstract. In maths, we can reason without meaning.
As 'half a physicist' this is particularly profound because it implies that if the universe is made of maths, as some have suggested, then it has no meaning beyond that which we project onto it. My computer scientist half prefers the theory that the universe is made of information, even if it is wrong.