‘Logic is to Philosophy what Mathematics is to Nature’. So goes the line. For me the distinction has always been fuzzy. But I will use both headings.
The problem-zone is the point of intersection between Mathematics and Philosophy. Or more precisely, the Mathematician’s application of philosophical notions, the noble intent to come to terms with issues better dealt with if the researcher was familiar with the evolution of philosophical presumptions over the millennia.
First, the Mathematician’s interpretation of the idea of ‘Unity’ and secondly, his application of the idea of ‘Truth’. Philosophers have racked their heads for a very long time over these two words. So how does a Mathematician resolve them?
‘Truth’ is largely captured in Logic in the concept of ‘Proof. There are various levels of ‘Proof’ and numerous interpretations of what exactly the word means . But what we do know is that they all take life upon a central principle, the Principle of Contradiction. I’ll get to it shortly.
[There are others: the notion of Finiteness, for example, a central theme of the Hilbert Program; the notion of the Observer: the emergence of Metamathematics and its arrangement of hierarchical statements; the blithe takes on the notion of ‘Isness’ or Ontological Presence: ‘There is an X such that…’]