FOM: A comment on an old question asked by Professor Pratt

[Matt Insall]

I am somewhat new to the FOM list, and I stayed away from the electronic
(web) revolution until fairly recently, but I began today reading some of
the older postings.  The following is a 1997 question of Professor Pratt on
which I would like to comment:


To: fom at math.psu.edu
Subject: Re: FOM: basic concepts; structuralism
From: "Vaughan R. Pratt" <pratt at cs.Stanford.EDU>
Date: Tue, 21 Oct 1997 02:13:01 -0700
Cc: pratt at cs.Stanford.EDU
In-reply-to: Your message of "Mon, 20 Oct 1997 23:47:11 EDT."
<199710210347.XAA11400 at boole.math.psu.edu>
Sender: owner-fom at math.psu.edu

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<snip>

But why is real analysis your litmus test of the worth of a
foundation?

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My response (in case no one has already replied with this type of response)
is that the whole bit began essentially with ``(real) analysis''.  Going
back to Cantor, the question of what constitutes a set was actually ``What
constitutes a set *of real numbers*?''  He realized the need for such an
increase in the level of formalism for the notion of a set of real numbers
as being basic to the understanding of the behaviour of trigonometric
series.  Thus, the reason the ``litmus test'' should be (real) analysis is
that the original problems of foundations relating to set theory, out of
which the formalizations of category theory, etc., grew, were related to
analysis.  Those questions are not all answered, so to those who wish to
answer those particular analysis questions, the ``litmus test'' of any
theory they will take an interest in will be real analysis.

It is the case that there was mathematics before Cantor, and the ancient
Greeks, in particular, formalized geometry.  Of course, as we know, later
non-Greeks came to question the foundations of geometry, leading to the
non-euclidean geometries.  However, these geometries were classically
founded on the presupposition that there exists a system of mensuration,
which is essentially the real number system.  (Some of the ancient greeks
thought all these numbers were rational, but this came to and end with the
discovery that there is no rational square root of two.)  Even before the
realization that one needed to deal with properties of real numbers to do
geometry, it was realized that properties of integers rational numbers
needed to be addressed, and these properties rightly form the subject matter
of analysis.




 Name: Matt Insall
 Position: Associate Professor of Mathematics
 Institution: University of Missouri - Rolla
 Research interest: Foundations of Mathematics
 More information: http://www.umr.edu/~insall