From: Paul Cockshott (wpc@DCS.GLA.AC.UK)
Date: Thu Sep 23 2004 - 10:42:14 EDT
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
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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.
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