The Philosopher's Axe 
[Mar. 21st, 200605:31 pm]
pozorvlak

Another idea/problem of which everyone should be aware.
The problem is simple: a philosopher has an axe (bear with me here). He uses it for a while, until the blade wears out and has to be replaced. He uses it for a bit longer, until the handle breaks. The head's still OK, so he puts it on a new handle.
Is the axe he has now the same axe as the one he started with?
Wikipedia calls this the Ship of Theseus paradox, and gives examples from fields as diverse as rock music and automobile registration fraud, some with serious consequences. It's a problem that often comes up in discussions of the feasibility of Star Trekstyle matter transference. Like so many important philosophical problems, the answer depends on what your definition of "is" is, and these are seriously deep waters. As a mathematician, it depends on what context you're operating in: sometimes two things can be the same for one purpose but different for another. Two groups can have the same elements, and thus be the same for the purposes of set theory, but have quite different algebraic structure. Or vice versa  two groups might happen to have different names for their elements, but exactly the same structure, in which case we'd call them the same group. Topologists will frequently say that two objects are "the same" when they're homotopic, which is a very weak condition  for instance, Euclidean ndimensional space is homotopic to a single point for all n. This can all be made precise using category theory: each category will have a different notion of isomorphism, and you say that two things are "the same" when they're isomorphic in the appropriate category. But then you get into higher categories, and it all goes wrong: in a 2category, the moral notion of "the same" isn't isomorphism but rather equivalence...
By the way, I once mentioned the Philosopher's Axe to an ice climber. He said that iceaxe heads wear out all the time, and that when you replace the handle then you've got a new axe, regardless of whether you then put an old head on it :)
[By the way, it seems my Lie Algebras lecture wasn't as bad as I'd thought. Catharina said I'd done well to get the whole theorem into one lecture, albeit one that ran nearly halfanhour over time...] 

