The Philosopher's Axe

[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 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...]

Comments

[the-barlow.livejournal.com]

This reminds me of the lectures I had on Philosophy of Identity.

If you replace the ship a piece at a time, until none of the original pieces remain is it the same ship?

If you then put the old pieces back together, is it the same ship?

If they are both the same ship then do you have the same ship in both places at once?

Intruiging...

And then there's the question of what happens if you swap half of your brain with half of someone else's brain... Then who's who?

E
x

[johnckirk.livejournal.com]

The ship always seemed like an easier version of this puzzle - I would say that yes, they are both the same object, in the same way that they would be if time travel was involved (e.g. if today's ship is sitting in the harbour next to last week's ship).

[the-barlow.livejournal.com]

I'm not so sure it is analogous to time travel. In a time travel example a continuous link could be traced between the two bodies, thus showing them to be the same thing (the earlier body will become the later one which has been the earlier one). However there is no such direct continuation between ship reconstructed from the original pieces, and the ship made of new pieces.

E
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[elvum.livejournal.com]

I would suggest the question "is the information content the same, or a superset of the information content the item in question possessed the last time it was examined?"

[pozorvlak.livejournal.com]

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?

[elvum.livejournal.com]

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. ;-)

[pozorvlak.livejournal.com]

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 (http://en.wikipedia.org/wiki/Noether%27s_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.

[elvum.livejournal.com]

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 ;-)

[michiexile]

There is an absolutely lovely monologue held by The High King in The Fifth Elephant by Terry Pratchett, where, upon being asked by Commander Vimes about the Scone of Stone being cast in Ankh-Morporkian clay, he explains the consistency of the Scone regardless of the actual matter in terms of his family axe. The axe, as it were, has needed replacement of the blade, and replacement of the handle, and added decorations, and engravings, but is this not my father's axe regardless of these minor fixes?

You should read it. It's a lovely exposition. And it's Pratchett!

[pozorvlak.livejournal.com]

I'd forgotten that bit - it's been far too long since I read The Fifth Elephant. Dammit, I must re-read some Pratchett, but almost all of my Pratchett books are in Oxford!

Transitions

[(Anonymous)]

In terms of human identity, I think the first differential (if you like) of the information in a person is at least as important as the static information.

Hm - mathsey words to make me feel clever (and, hopefully to encourage pozorvlak to write some more meaningful ones).

More intuatively, I think my identity is more closely bound up with where I'm going than where I am. At the moment, I'm a student of CS, and I go to the gym on a regular basis. 2 years ago, I was (officially) an engineer - not a student, and I didn't use a gym - I did karate. A couple of years before that I was a different sort of student, and I did dancing. I've picked a couple of vague areas of my life, and traced their evolution for you.

If the thing that makes me who/what I am is the common denominators in those things, then I'm a bloke who works with computers and does vaguely active things. That's not a very unique bloke.

I think that the more important part of who I am is that I'm not staying this way for long. I'm always looking to learn more - hence the career changes, and a similar itchey feet logic could be applied to my hobbies (though, in part, they're driven by other things too - but I digress).

I rarely meet static people. I suspect that all my friends may well be less static than most, in fact... But anyway - I reckon that the identifying feature of your axe is the path of its change (and the reasons for it), not any subset of its arbitrary components.

--gds (who doesn't have an LJ)

Re: Transitions

[pozorvlak.livejournal.com]

Now that is interesting. I'm reminded of the Zetetic Elench in Iain M. Banks' book Excession who try to embrace every new experience and be changed by it, to the extent that if you meet an Elencher again after some years apart, you're expected to treat them as an entirely new person.

[andustar.livejournal.com]

Haha. I was just googling 'Vimes' and 'the same axe' to see if anyone had the quote. And I got a familiar journal! Funny.

[(Anonymous)]

This comes up constantly in the world of collector's items, as you might well imagine. If an object has been repaired, or had parts replaced, it is obviously not 'original', but had it not been repaired, it would have been in poor condition, and possibly junked or cannibalized. But then nothing, not even so-called 'NOS' items are as they once were, regardless of appearance. Complex items may have dried out or degraded components and will not live up to original standards. I am a watchmaker, and it constantly amazes me how wrapped around the axles people get when they find out their flea market 'find' has had it's movement or case replaced, albeit with a correct version of either.

[pozorvlak.livejournal.com]

Interesting, thanks!