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

**Date:** Tue Sep 21 2004 - 07:04:55 EDT

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________________________________ From: OPE-L [mailto:OPE-L@SUS.CSUCHICO.EDU] On Behalf Of Gerald A. Levy Sent: 20 September 2004 14:13 To: OPE-L@SUS.CSUCHICO.EDU Subject: (OPE-L seminar] Paul Cockshott's "Hibert Space Models Commodity Exchanges" Hi Paul C. I expect to be busy later today so I'm going to just jump in and ask a couple of questions about your paper while I have a few minutes: 1) You suggest that it is possible to posit an "underlying linear vector space of which commodity space is a representation" (p. 3) and "*unlike commodity space itself*, this space, is a true vector whose evolution can be modeled by the application of linear operators" (Ibid, emphasis added, JL). This is repeated in the last sentence of the paper where you write that commodity amplitude space "*unlike commodity space*, is a linear vector space within which angles of rotation have a clear meaning" (p. 6, emphasis again added, JL). I am confused by this. a) If commodity "space" is *not* a linear space, *why* is it reasonable to suggest that the "underlying" representation can be linear? -------------------------- The analogy here is between classical probability space which is non linear, and the Hilbert space of quantum amplitudes which is linear. One observes phenomena that can not be accounted for classically (Bells Inequality) but which can be accounted for by positing an underlying linear space. There is obviously a lively debate about the ontological status of this linear space, but what I was consciously doing was using this model applied to commodity exchange. It struck me that there has been a wealth of models applying linear algebra techniques to examining the magnitude of values, but so far no attempt to use linear algebra to analyse the value form. Since the value form is presented in the context of the exchange of equivalents, how can we model this in linear terms - obviously it has to be in terms of unitary, and thus value conserving, operators. The problem is that one can not apply these directly to exchange values but one can posit a dual space in which linear unitary operations are possible. ------------------------------------------ b) If "commodity space" is non-linear shouldn't methods of measurement which take into account this non-linearity be utilized? ------------------------------- I think this is a very valid point. It indicates that if one is for example looking at the fit between a price predictor and actual prices it may be better to look at the sum of absolute deviations rather than the sum of square deviations. ----------------------------- c) are there any tests that you can use which use non-linear methods which can be applied to the data to confirm or not confirm the empirical propositions about the closeness of market prices to vertically integrated labour values? ---------------------------------- See the above --------------------------------- 2) For those who wish to do further research in this area, what do you identify as the major directions requiring further research? ---------------------------------- I am interested in the possibility of analysing the creation of money in terms of creation and annihilation operators so that the analysis extends to monetary circulation. I think one may be able to make analogies between such operators and the cancellation of debts in a system of credit money. In solidarity, Jerry http://ricardo.ecn.wfu.edu/ope/seminar/id18.htm

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