[OPE-L:7035] [OPE-L:529] How mathematicians think about equality [OPE 523]

Alan Freeman (a.freeman@greenwich.ac.uk)
Fri, 26 Feb 1999 14:12:25 +0000

Here's what Birkhoff and MacLane (1963:155), in one of the standard
undergraduate works on algebra, say about equality and equivalence.

"We have asserted that any reflexive, symmetric and transitive relation
might be regarded as a kind of equality. We shall now formulate the
significance of this assertion. For convenience, a relation R which has the
reflexive, symmetric and transitive properties,

aRb, aRb->bRa, aRb and bRc -> aRc

for all members of a set S, will be called an *equivalence* relation.
If...we are willing to treat suitable *subsets* of S as elements, such an
equivalence relation becomes an ordinary equality"

The way this works is as follows: we define an equivalence class R(a) as
the set of all elements R-equivalent to a. (eg everything that exchanges
for a coat). We then find that

aRb -> R(a)=R(b)

that is, equivalence between the original elements a,b translates into
equality between the subsets.

If, now, the original elements a, b etc have a structure, the following
applies (Birkhoff and MacLane p157)

"In discussing the requisites for an admissible equality relation we also
demanded a certain 'substitution property' relative to binary operations.
In terms of the equivalence relation R and the binary operation a*b = c on
the set S, this property takes the form

aRa' and bRb' imply (a*b)R(a'*b')"

Let me repeat my axiom of basket decomposition [OPE 516] to compare it
with the above:

"(4 )if bRx and cRy then (b U c) R (x U y), where U is set union."

The operation * in the standard algebraic formulation is thus the set union
operation in my formulation. This thus conforms to the textbook definition
of equality.

"Equality" in Marx, as among the mathematicians, means the following: we
can partition the space of baskets of use-values into subsets of baskets
that exchange for each other. Two baskets are 'equal' under this relation
if they belong to the same subset. We can then identify the price of a
basket with the subset to which it belongs. In common language, two baskets
are equal if they have the same price.

That's what 'equal' means. Nothing more, and nothing less. It has no
implication that their values are equal, it is completely independent of
the supposition of equilibrium, and it is a coherent, scholarly definition.
As far as I can see Marx's analysis conforms to it with almost textbook

Of course if the 'vast majority of economists' wish to think differently
they are entitled to do so. They are not, however, entitled to tell us that
this is the only way to think, and they are not entitled to tell us that
Marx must have thought like they do.


Birkhoff and MacLane (1963) 'A Survey of Modern Algebra', New