[FOM] 1 = 1

[Vaughan Pratt]

If Ron Pisaturo's axiomatization

>I state this axiom in the form, "1 = 1", which means not that some
>abstract object named "1" is equal to itself, but rather that each unit
>being considered is interchangeable with each other unit.

is to be understood as coming under the jurisdiction of FOM, then his
conclusion

>I claim that, from this one mathematical axiom, and using the principles of
>inference known from philosophy (which is more basic than mathematics), all
>of mathematics is developed and can be validated.

follows straightforwardly.  Just consider the multiplicative units of Z_2
and Z_3 as rings.  With 1 "interchangeable with" (congruent to) 3 in the
former and to 4 in the former, you are now within easy reach of any true
sentence of arithmetic.  (And as an overlooked bonus, its negation.)

If however the question of the utility and/or meaning of "1 = 1" is to be
understood as lying more in the realm of Philosophy of Mathematics, then
one could reasonably suppose Ron to be promoting unity as a natural kind in
mathematics in the sense of the book "Naming, Necessity, and Natural Kinds"
(ed. Stephen P. Schwartz).  This is a fair and (to me) very interesting
question that one could say several paragraphs about before becoming
repetitive.

However my sense (besides that of tiptoeing through a minefield) is that
FOM emphasizes foundations over philosophy.  In that case it would seem
that one could not say say much more than that any decent
axiomatization of mathematics supplemented with "1 = 1" as interpreted by
Ron is inconsistent.  The only FOM-type question I can think of as
being raised by this is, what consistent weakenings exist for this version
of 1 = 1?

If I've misconstrued the scope of FOM I'm happy to be set straight.

Vaughan Pratt