**From:** Paul Cockshott (*wpc@DCS.GLA.AC.UK*)

**Date:** Thu Sep 23 2004 - 10:42:14 EDT

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Ian Wright 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. ---------------------- Paul I have done a little digging about but so far have not turned up the appropriate other sources ------------------- Ian 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. ------------------------------ Paul C Ok I think you may be right in making the distinction between vector spaces and normed vector spaces. Since I am not a mathematician myself I usually bounce questions like that off friends in the Maths or Stats department. Yesterdays feedback was that it was redundant of me to talk of linear vector spaces, since linear was implied in vector spaces. Under that interpretation, then a space with a Manhattan metric for example is not a vector space. I am unclear as to what the correct terminology is here. ------------------------------ Ian 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. ----------------------------- Paul I agree for now that one can just view it as an alternative representation within which you can specify exchanges as unitary operations. However, I suspect that there is a virtue in thinking of familiar things in a different form. One may thereby be led to notice things that one would not otherwise recognise. An example is the discontinuity in the unit circle in commodity holding space - the fact that this takes the form of a pair of parallel hyper planes on either side of the origin. I realised that these hyper planes must exist from the definition of the metric on commodity holding space, at first I did not know what the interpretation should be. Then I realised that the plane represents those agents who are in net debt. The topological discontinuity between the two planes reflects the fact that no sequence of equivalent exchanges can ever get an agent out of debt. But I must admit that I do not fully understand the implications of representing debts in commodity amplitude space yet.

**Next message:**Rakesh Bhandari: "Re: Economist slips up over Venezuela"**Previous message:**Gerald A. Levy: "(OPE-L) Re: Seminar: Value redundancy and price value deviations"**Maybe in reply to:**Alejandro Valle Baeza: "Seminar: HILBERT SPACE MODELS COMMODITY EXCHANGES by Paul Cockshott"**Next in thread:**Ian Wright: "Re: Seminar: HILBERT SPACE MODELS COMMODITY EXCHANGES by Paul Cockshott"**Reply:**Ian Wright: "Re: Seminar: HILBERT SPACE MODELS COMMODITY EXCHANGES by Paul Cockshott"**Messages sorted by:**[ date ] [ thread ] [ subject ] [ author ] [ attachment ]

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