The Posts on Aristotle’s defense of the Principle of Contradiction and the play of the Self-Negating Expression go back over 2,000 years. I’d like to set forth a more recent example, one less than a century old, just to convince you that things haven’t changed much.
‘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’.
[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…’]
The notion of ‘Unity’ [‘One-ness’] in Mathematics is the struggle to define Sets and Classes and Groupings [Cantor, Von Neumann et al; see the next section of Posts].
‘Truth’, a word thoughtful philosophers across cultures have struggled with for millennia 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 [it was this very need that ultimately lead to Kurt Godel’s work in the next section]. But what we do know is that they all take life upon a central principle, the Principle of Contradiction.
Logician’s and Mathematician’s are not expected to know their philosophical ground at the same level of familiarity as the trained philosopher. But at some point, in inquiry directed to the roots of Logic and Mathematics, the questions should converge or at least overlap. And if they don’t, some side has taken a wrong turn.
Georg Cantor accused Immanuel Kant of being a ‘Mathematical Ignoramus’. This is beyond funny. It was Kant who in his Critique of Pure Reason repeatedly warns the reader that he may not use his ‘First Principle of Knowing’ on analyzing ‘The First Principles of Knowing’ [see the Posts for the story]. The mathematicians did exactly the mathematical analogue of that in laying out their Set Theory.
Or in a similar vein, the Logician’s demand for ‘Consistency’ in everything the layman says, when the very notion of ‘Consistency’ remains less than consistent.