[FOM] 1 = 1
pisaturo at alum.mit.edu
Tue Mar 4 10:47:34 EST 2003
I would like to call attention to my articles (one co-written with Glenn
Marcus, a former Assistant Professor of Mathematics at Fordham University)
on the philosophy of mathematics, which were published in a non-technical
journal named _The Intellectual Activist_. I would be happy to make my
articles available online to any FOM subscriber who so requests.
In my articles, I take what I think is a distinctive approach to identifying
the axiomatic base of mathematics. I argue that the sole distinctively
mathematical axiom is the premise that all of the units being considered are
equal--i.e., interchangeable--in the context being considered. Of course,
this premise is consistent with the ancient Greek concept of "arithmos"
(which concept has been discussed at length by Prof. John Mayberry on FOM
and in his published works) and with Cantor's idea of "cardinal number" as
the result of a "double act of abstraction"; but I identify this premise
explicitly as an axiom--indeed, as _the_ axiom--of mathematics. 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. (J.S. Mill also claimed that "1 = 1"
is the basic premise of mathematics, but he doubted the truth of the
premise.) E.g., each meter is interchangeable--with regard to length--with
each other meter.
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.
The articles are decidedly non-technical. (My own formal training in
mathematics does not go beyond a B.S. from MIT.) Indeed, undergraduates and
laymen, as well as professors of mathematics, have written to tell me that
the articles have significantly enhanced their understanding of mathematics.
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