What, it may be asked, has all this to do with value? Well, value is a metric
on commodities. To apply the previous concepts we define commodity bundle
space as follows.
A commodity bundle space of order 2 is the
set of pairs (ax, by) whose elements are a
units of commodity x and b units of commodity y.
A commodity bundle space of order 3 is the
set of triples (ax, by, cz) whose elements are
bundles of a units of x, b units of y, c units
of 
and so on.
Consider for example the commodity bundle space of order 2 composed of bundles of iron and corn. The set of all points equidistant with (e iron, f corn) from (a iron, b corn) under the Manhattan metric is shown in Figure 3.
Figure 3: Points equidistant with
(e iron, f corn) from (a iron, b corn) in Manhattan space
We have a distinct equality operator, 
, for each
point 
iron, 
corn) in our corn-iron space.
Let us consider one particular equality operator,
that which defines the equality set of points equidistant from the origin, 
.
Whichever metric we take, so long as we use it consistently each
point in the space belongs to only one such equality set under the given metric.
These equality sets form an ordered set of sets of the space.
It follows that any of the metrics could serve as a system of valuation,
conceived as a partial ordering imposed upon all bundles.
This is shown in Figure 4. Both the diamonds
and the conventional circles are, in the relevant space,
circles: the diamonds are circles in Minkowski or Manhattan space.
Figure 4: The ordering of equality sets under possible metrics
We now advance the hypothesis that if the elements of a set of commodity bundles are mutually exchangeable--that is, if they exchange as equivalents--then they form an equality set under some metric. If this is valid, then by examining the observed equality sets of commodity bundles we can deduce the properties of the underlying metric space.