Mathematician Claims Proof of Connection between Prime Numbers
September 14, 2012 at 8:28 pm
(This post was last modified: September 14, 2012 at 8:28 pm by Categories+Sheaves.)
It's not so much that the rules of logic are being redrawn--mathematicians LOVE stuff like proof-by-contradiction (we aren't going to have to give that up--at least not yet, anyway). If you're familiar with the classic paradoxes of naive set theory, weird stuff happens (e.g. Russell's, a relative of the liar) if you don't restrict the domain you're quantifying over. Non-well-founded sets have this eerie infinite regress quality to them, (even if they don't necessarily lead to contradictions) and I remember coming across some links to papers/sites trying to rehabilitate non-well-founded sets, so this stuff isn't completely out there.
And I guess the other point: mathematicians don't seem to use the phrase 'foundational' (at least in terms of set theory) in the same way philosophers do. The axioms don't have any particular epistemic priority or apodictic quality to them. We're adding some extra objects to our mathematical universe, so we have some new truths (or at least, some previously unprovable truths may now be provable). The more you assume (or the more you admit--in the sense of things being admissible or not) the more you can prove. Are you cool/familiar with paradigmatic descriptions of math/science?
And I guess the other point: mathematicians don't seem to use the phrase 'foundational' (at least in terms of set theory) in the same way philosophers do. The axioms don't have any particular epistemic priority or apodictic quality to them. We're adding some extra objects to our mathematical universe, so we have some new truths (or at least, some previously unprovable truths may now be provable). The more you assume (or the more you admit--in the sense of things being admissible or not) the more you can prove. Are you cool/familiar with paradigmatic descriptions of math/science?
