You can make positions relative in ways other than using pairwise distances as your coordinates. For instance, just take R^4n (or R^11n or whatever) and quotient by the appropriate group of isometries of R^4 or R^11 or whatever. That way you get a dimension linear in the number of particles. The space might be more complicated topologically, but if you take general relativity seriously then I think you have to be prepared to cope with that anyway.
So, in Eliezer’s example of triangles in 2-space, we start off with R^6; letting E be the group of isometries of R^2 (three-dimensional: two dimensions for translation, one for rotation, and we also have two components because we can either reflect or not), it acts on R^6 by applying each isometry uniformly to three pairs of dimensions; quotienting R^6 by this action of E, you’re left with a 2-dimensional quotient space.
Of course you end up with the same result (up to isomorphism) this way as you would by considering pairwise distances and then noticing that you’re working in a small subset of the O(N^2)-dimensional space defined by distances. But you don’t have to go via the far-too-many-dimensional space to get there.
But … suppose the laws of physics are defined over a quotient space like this. From the anti-epiphenomenal viewpoint, I wonder whether we should consider the quantities in the original un-quotiented space to be “real” or not. Consider quantum-mechanical phase or magnetic vector potential, which aren’t observable (though other things best thought of as quotients of them are). Preferring to see the quotiented things as fundamental seems to me like the same sort of error as Eliezer (I think rightly) accuses single-world-ists of.
But … the space of distance-tuples (appropriately subsetted) and the space of position-tuples (appropriately quotiented) are the same space, as I mentioned earlier. So, how to choose? Simplicity, of course. And, so far as we can currently tell, the laws of physics are simpler when expressed in terms of positions than when expressed in terms of distances. So, for me and pending the discovery of some newer better way of expressing the state space that supports our churning quantum mist, sticking with absolute positions seems better for now.
I agree that taking quotients of the configuration space is a more natural way of doing things. But, when you say
quotienting R^6 by this action of E, you’re left with a 2-dimensional quotient space
don’t you mean you’re left with a 3-dimensional quotient space? Counting degrees of freedom: wherever we put A, that eats the translation. Wherever we put B, that eats the rotation and we’re left with the distance |AB| (one dimension). Wherever we put C, that eats reflection and we’re left with the position of C up to reflection. So, the space of triangles ends up as R x R x (R / ~), where a~b iff |a|=|b|.
But then, this space should be homeomorphic to the one Eliezer gives, with the relative distances. We’ll take a point (x,y,z) in R x R x (R / ~). Then |AB|=x, |AC|=hypot(y, z), |BC|=hypot(y-x, z), clearly this is continuous and nice, and also clearly the image doesn’t change if we replace z by -z (so the function is well-defined despite the domain being a quotient space, which generally needs to be checked). Showing that the mapping is invertible, with continuous inverse, is left as exercise for the reader.
Consider now the apparent boundary when we embed this in R³; it’s z=0, which corresponds to “A, B and C form a straight line”, which (triangle inequality) corresponds to the boundary of the subset of distance-space. But if you imagine the particles moving, it’s a lot more obvious that you should bounce off the ”/ ~” surface than that you should bounce off the “if you cross this surface you get a distance-tuple that’s un-geometric” surface. Similarly, straight lines in R x R x (R / ~) correspond to fixing any two particles and moving the third in a straight line.
I would conclude from this that the equations of physics in the quotient space are likely to be much nicer than the equivalent equations in distance-tuple space.
So why bother formulating the relational configuration space in distance-tuples? After all, with the distance-tuples, you still end up having to quotient afterwards on particle-swapping to get the quantum-mechanical picture. Isn’t it easier to just use quotients, rather than an odd mix of quotients, new bases, and subsets?
(Note 1: “g” = me; I had to change my username when Less Wrong started, but existing Overcoming Bias comments kept their existing commenter names. Note 2: I only just saw this.) Yes, I meant 3-dimensional. Sorry. (And I think we are agreed that absolute space quotiented by symmetries is likely to be a nicer thing to work with than a space parameterized by relative distances.)
You can make positions relative in ways other than using pairwise distances as your coordinates. For instance, just take R^4n (or R^11n or whatever) and quotient by the appropriate group of isometries of R^4 or R^11 or whatever. That way you get a dimension linear in the number of particles. The space might be more complicated topologically, but if you take general relativity seriously then I think you have to be prepared to cope with that anyway.
So, in Eliezer’s example of triangles in 2-space, we start off with R^6; letting E be the group of isometries of R^2 (three-dimensional: two dimensions for translation, one for rotation, and we also have two components because we can either reflect or not), it acts on R^6 by applying each isometry uniformly to three pairs of dimensions; quotienting R^6 by this action of E, you’re left with a 2-dimensional quotient space.
Of course you end up with the same result (up to isomorphism) this way as you would by considering pairwise distances and then noticing that you’re working in a small subset of the O(N^2)-dimensional space defined by distances. But you don’t have to go via the far-too-many-dimensional space to get there.
But … suppose the laws of physics are defined over a quotient space like this. From the anti-epiphenomenal viewpoint, I wonder whether we should consider the quantities in the original un-quotiented space to be “real” or not. Consider quantum-mechanical phase or magnetic vector potential, which aren’t observable (though other things best thought of as quotients of them are). Preferring to see the quotiented things as fundamental seems to me like the same sort of error as Eliezer (I think rightly) accuses single-world-ists of.
But … the space of distance-tuples (appropriately subsetted) and the space of position-tuples (appropriately quotiented) are the same space, as I mentioned earlier. So, how to choose? Simplicity, of course. And, so far as we can currently tell, the laws of physics are simpler when expressed in terms of positions than when expressed in terms of distances. So, for me and pending the discovery of some newer better way of expressing the state space that supports our churning quantum mist, sticking with absolute positions seems better for now.
I agree that taking quotients of the configuration space is a more natural way of doing things. But, when you say
don’t you mean you’re left with a 3-dimensional quotient space? Counting degrees of freedom: wherever we put A, that eats the translation. Wherever we put B, that eats the rotation and we’re left with the distance |AB| (one dimension). Wherever we put C, that eats reflection and we’re left with the position of C up to reflection. So, the space of triangles ends up as R x R x (R / ~), where a~b iff |a|=|b|.
But then, this space should be homeomorphic to the one Eliezer gives, with the relative distances. We’ll take a point (x,y,z) in R x R x (R / ~). Then |AB|=x, |AC|=hypot(y, z), |BC|=hypot(y-x, z), clearly this is continuous and nice, and also clearly the image doesn’t change if we replace z by -z (so the function is well-defined despite the domain being a quotient space, which generally needs to be checked). Showing that the mapping is invertible, with continuous inverse, is left as exercise for the reader.
Consider now the apparent boundary when we embed this in R³; it’s z=0, which corresponds to “A, B and C form a straight line”, which (triangle inequality) corresponds to the boundary of the subset of distance-space. But if you imagine the particles moving, it’s a lot more obvious that you should bounce off the ”/ ~” surface than that you should bounce off the “if you cross this surface you get a distance-tuple that’s un-geometric” surface. Similarly, straight lines in R x R x (R / ~) correspond to fixing any two particles and moving the third in a straight line.
I would conclude from this that the equations of physics in the quotient space are likely to be much nicer than the equivalent equations in distance-tuple space.
So why bother formulating the relational configuration space in distance-tuples? After all, with the distance-tuples, you still end up having to quotient afterwards on particle-swapping to get the quantum-mechanical picture. Isn’t it easier to just use quotients, rather than an odd mix of quotients, new bases, and subsets?
(Note 1: “g” = me; I had to change my username when Less Wrong started, but existing Overcoming Bias comments kept their existing commenter names. Note 2: I only just saw this.) Yes, I meant 3-dimensional. Sorry. (And I think we are agreed that absolute space quotiented by symmetries is likely to be a nicer thing to work with than a space parameterized by relative distances.)