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gb7nash

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Hurkyl

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I'm sure there are things you can do in the sense of category theory to create new objects and morphisms between them so that it would make sense to talk about R

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Deveno

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in the absence of any other space to interact with, R

R

i'm not sure how you could make this mathematically precise, or even what uses it might have, but it seems to me you could formulate it as some quotient of the tensor algebra (after all, we have annihilator spaces).

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chiro

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What can you decompose points into? If you have a way of defining that, you might have a definition for R^(-n) but visually I can't think of such a decomposition.

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disregardthat

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If you think of it: R^1 x R^-1 = R^0 would violate cardinality of sets.

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Deveno

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i don't think it's a matter of decomposing points. a zero-dimensional vector space is just the origin. there should be a symmetry in the construction of R

i mean think of how we construct the real projective plane from the sphere: first we construct an isometry that exchanges antipodal points. the antipodal isometry actually turns the ball of the sphere "inside out". well you can extend that to all of R

so two spinning objects that collide, one from the original space, and one from the antipodal one, their spins cancel.

now, normally, mathematicans will say something like: "by convention, we take e1 to be..." to indicate a choice of orientation is something of an arbitrary choice. but maybe it's not, maybe orientation is just as important as whether or not an integer is a natural number or not. not because of vector properties, but because of some "super-vector" properties that come about when one considers additional structure.

and yes, if you start reducing "anti-vector spaces" to "anti-bases", one has to consider the implication of "anti-sets" (sets with negative cardinality). obviously if one says A "U" -A = Ø, it's not ordinary union we're talking about. but hey, we have a natural structure on P(A), the power set of A, so we ought to be able to create a structure on P(A) x P(A) with a suitable equivalence.

as i pointed out before, it's not immediately clear how useful this is. but i don't think it's entirely nonsensical.

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Deveno

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http://www.csz.com/cyber/html/negsets.pdf

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