The volume looks to be full of interesting papers, but there's one in particular I've read before, so I'll write a little about that right now.

The paper is Brian Weatherson's "Asymmetric Magnets Problem". The puzzle he sets out is based on a well-entrenched link between intrinsicality and duplication: a property is intrinsic iff necessarily, it is shared among duplicate objects. Weatherson examines an application of this principle to a case where some of the features of the objects we consider are vectorial.

In particular, consider an asymmetric magnet M: one which has a pointy-bit at one end, and is such that the north pole of the magnet "points out" of the pointy end. Intuitively, the following is a duplicate of another magnet M*: one with the same shape, but simply rotated by 180 degrees so that both the north pole and the pointy end are both orientated in the opposite direction to M. (Weatherson has some nice pictures, if you want to be clear about the situation).

Though M and M* seem to be duplicates, their vectorial features differ: M has its north pole pointing in one direction, M* has its north pole pointing in the opposite direction. Moral: given the link, we can't take vectorial properties "as a whole" (i.e. building in their directions) as intrinsic, for they differ between duplicates.

What if we think that only the magnitude of a vectorial feature is intrinsic? Then we get a different problem: for their are pointy magnets whose north pole is directed out of the non-pointy end. Call one of these M**. But in shape properties, and so on, it matches M and M*. And ex hypothesi, in all intrinsic respects, their vectorial features are the same. So M, M* and M** all count as duplicates. But that's intuitively wrong (it's claimed).

Such is the asymmetric magnets problem. The challenge is to say something precise about how to think about the duplication of things with vectorial features, that'd preserve both intuitions and the duplication-intrinsicality link.

Weatherson's response is to take a certain relationship between parts of the pointy magnet its vectorial feature, as intrinsic to the magnet. In effect, he takes the relative orientation of the north-pole vector, and a line connecting certain points within the magnet, as intrinsic.

Ok, that's Weatherson's line in super-quick summary, as I read him. Here are some thoughts.

First thing to note: the asymmetric magnets problem looks like a special case of a more general issue. Suppose point particles a, b, c each have two fundamental vectoral features F and G, with the same magnitude in each case. Suppose in a's case they point in different directions, whereas in b and c's cases they point in the same direction (in b's case they both point north, in c's case they both point south). The intuitive verdict is that a and b are not duplicates, but b and c are. But, if you just demand that duplicates preserve the magnitudes of the quantities, you'll get a, b, and c as duplicates of one another; and if you demand that duplicates preserve direction of vectoral quantities, you'll get none of them as duplicates. That sounds just like the asymmetric magnets problem all over again. Let me call it the vector-pair problem.

What's the natural Weathersonian thought about the vector-pair problem? The natural line is to take the relative orientation ("angle") between the instances of F and G as a perfectly natural relation. (I think that Weatherson might go for this line now: see his comment here).

It seemed to me that a natural response to the problem just posed might be this: require that the magnitude of any quantities is invariant under duplication; also that the *relative orientation* of vectoral properties be invariant under duplication. Thus we build into the definition of duplication the requirement that any angles between vectors are preserved. There's thus no easy answer to the question of whether vectorial features of objects are intrinsic: we can only say that their magnitudes and relative orientations are, but their absolute orientation is not.

This leads to a couple of natural questions:

(A) Why do we demand absolute sameness of magnitude, and only relative sameness of direction, when defining what it takes for something to be a duplicate of something else?

I'm tempted to think that there's no deep answer to this question. In particular, consider a possible world with an "objective centre", and where various natural laws are formulated in terms of whether objects have properties "pointing towards" the centre or away from it. E.g. suppose two objects both with instantaneous velocity towards the centre will repel each other with a force proportional to the inverse of their separation; while two objects both with instantaneous velocity away from the centre will attract each other with a similar force (or something like that: I'm sure we can cook something up that’ll make the case work). Anyway, since the behaviour of objects depends on the "direction in which they're pointing", I no longer have strong intuitions that particles like b and c should count as duplicates (with that world considered counteractually).

I find it harder to imagine worlds where only relative magnitudes matter to physical laws, but I suspect that with ingenuity one could describe such a case: and maybe (considering such a scenario counteractually again) we'd be happier to demand only relative sameness of magnitudes, in addition to relative sameness of orientation of vectoral properties, among duplicates.

**(B)** The above proposal demands invariance of relative orientation of vectoral properties among duplicate entities. But that doesn't straightaway deal with the original asymmetric magnet case. For there we had the orientation of the shape-properties of the object to consider, not just the orientation of the vectoral quantities that the (parts of) the object has.

I'm tempted by the following way of subsuming the original problem under the more general treatment just given: say that some perfectly natural spatial properties are actually vectoral in character. E.g. the spatial property that holds between my hand and my foot is not simply "being separated by 1m" but rather "being separated by 1m downwards" (with, of course, the converse relation holding in the other direction). After all, if in giving the spatial properties that I currently have, we just list the spatial separations of my parts, we leave something out: my orientation. And that is a spatial property that I have (and is coded into the usual representations of location, e.g. Cartesian or polar coordinates. Of course, such representations are all relative to a choice of axes, just as the representation of spatial separation is relative to a choice of unit.)

Now, there might be ways of getting this result without saying that spatial-temporal relations among particulars are fundamentally vectorial. But I'm not seeing exactly how this would work.

(Incidentally, if we do allow fundamentally vectorial spatio-temporal relations, then it's not clear that we need to appeal to spatio-temporal relations among parts of an object to solve the asymmetric magnets problem: appealing to the angle between the "north pole" and the (vectorial) spatio-temporal properties of the pointy magnet may be enough to get the intuitive duplication verdicts. If so, then the Weathersonian solution can be extended to the case where the magnets are extended simples, which is (a) a case he claims not to be able to handle (b) a case he claims to be impossible. But I disagree with (b), so from my perspective (a) looks like a serious worry!)

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2015/09/13

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