**From:** Ian Wright (*iwright@GMAIL.COM*)

**Date:** Wed Sep 22 2004 - 21:03:24 EDT

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Hi Paul Thanks for the answer, which I think I understand. If correlation assumes a special kind of vector space then I'd expect that to be discussed in theoretical statistics. It should be a known issue. The argument revolves around the following inference: "... this space is not a vector space and it is questionable whether measures of similarity based on vector space metrics are appropriate for it". If it is questionable, then I'd guess some mathematicians will have made it precise. It would seem important to get to the root of this one. There's some confusion of terminology regarding vector spaces. Vector spaces are not defined by their metric. A vector space with a metric is a normed vector space. Euclidean space is one kind of normed vector space. Manhattan distance will also yield a vector space. So I am guessing that your references to "vector space" need to be replaced with "Euclidean space" in order to convey your intention more precisely. This confused me to begin with. You label the metric actually observed in the space of bundles of commodities d_{b} and go on to state: "Because of its metric, this space is not a vector space ..." I agree that this space is not Euclidean, but it may yet still be a vector space. First, we need to check whether the observed metric d_{b}(p,0) satisfies the three conditions for a norm. It is zero when p is zero. If we multiply p by a scalar and get ap then the norm of ap is equal to a times the norm of p. Finally, the metric must satisfy the triangle inequality, which it does by equality. Therefore, I think it safe to conclude that the observed metric is consistent with a vector space, contrary to what is stated. But yes it is not Euclidean, so if the similarity measures are based on an underlying Euclidean space then your argument proceeds unaltered. The observed metric is unusual because it has weights. What is the ontological status of commodity amplitude space? Given that the observed metric is consistent with a vector space then maybe this suggests that the commodity amplitude space is a convenient mapping to a Euclidean vector space, i.e. a useful re-representation of the problem, rather than pointing to some hitherto unnoticed ontology. (You state that commodity amplitude space is a "true vector space", so I guess the norm is Euclidean, although I don't think you state it). The analogy with quantum mechanics suggested to me that maybe this is pointing to some hidden thing, but now I think maybe not. The important thing is still the holdings, and the commodity amplitudes are a convenient transformation of the situation. I need to look it up, but my memory has been jogged that in some geometries straight lines in fact represent circles, which is suggestive, but this may be red-herring. I think all this is interesting and stimulating. I always return to the "Value's Law, Value's Metric" paper, basically because it intrigues me and because I honestly am not sure what to think of it. I feel there is something important there, but don't know what it is. -Ian.

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