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The Philosopher's Axe [Mar. 21st, 2006|05:31 pm]
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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 Trek-style 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 n-dimensional 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 2-category, 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 ice-axe 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 half-an-hour over time...]

[User Picture]From: pozorvlak
2006-03-22 01:29 pm (UTC)
Hmmm, interesting approach. But don't you then have the problem that if you hack the ship's mast off it loses the information contained in the mast and thus becomes a different ship?
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[User Picture]From: elvum
2006-03-22 01:47 pm (UTC)
I don't think the mast contains any useful information. I may be drifting from the information-theoretic definition of the word now though. Perhaps we need a new term, analogous to energy:

Q. What is energy?
A. Something that is conserved during physical processes - it doesn't really exist but we claim it does because it's useful.

Q. What is quintessence?
A. That which makes an item what it is - it doesn't really exist but we claim it does because it's useful.

Quintessence is like information, but weighted by the importance that is placed on the information content of a component. eg I don't think that the information content of the mast of a ship is important at all in the general case, so its loss doesn't alter the quintessence of the ship. If a particular mast has survived deadly storms, then perhaps the sailors would start ascribing importance to it, saying things like "the old girl wouldn't be the same ship without it". In which case I rest mine. ;-)
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[User Picture]From: pozorvlak
2006-03-24 04:21 pm (UTC)
That's just like saying that an ice-axe is its handle - what if you replace the smaller "important" bit of ship piece-by-piece?

The quintessence idea is more interesting, however. We could think of an item's quintessence as being removed by damage, and slowly recharging with experience - if the new mast survived many storms, say, it could grow to be an important part of the ship in its own right. However, if too much is removed at once the ship's quintessence drops too low, and it can no longer be considered the same ship.

Energy - are you aware of Noether's Theorem? Essentially, symmetries of a physical theory are in one-to-one correspondence with conserved quantities. Conservation of energy is equivalent to time-invariance of Newtonian physical laws, conservation of momentum is equivalent to position-invariance, etc. Of course, it's more general than that, and holds in non-Newtonian cases - I believe conservation of the stress-energy tensor is given by Noether's theorem and relativity's invariance under diffeomorphisms of spacetime. But this is way beyond my actual understanding of either Noether's theorem or general relativity.
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[User Picture]From: elvum
2006-03-24 08:00 pm (UTC)
Yes, Noether's theorem was one of the cooler (and easier to understand, conceptually) things I learned in the first term of my PhD. Cool but scary ;-)
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