[FOM] real numbers

Bill Taylor W.Taylor at math.canterbury.ac.nz
Mon May 12 02:31:17 EDT 2003

"Miguel A. Lerma" <mlerma at math.northwestern.edu> wites:

->In the realm of Mathematics there are no cabbages, nor apples, nor bricks, 
->nor planets, nor anything "material" at all. 

Correct; no physical material.  Nonetheless, it is my view and that of 
the mainstream (I think) that it still has "objects" - numbers & sets and
what-have-you; it is just that these objects are *abstract* not physical.

But as Godel said, we have as much, maybe more, reason to believe they exist
as we do for physical objects.  They are at least intersubjective, AND they
"kick back".

->But in Mathematics there are not "things" to start with.

So I disagree.  There ARE things, just not physical things.  Of course,
these abstract things can MODEL physical things, as we all know, including
Hartley Slater's greengrocer.  But (apart from initial motivations, which
are quickly superceded) they have no essential connection with them.

->The real problem of Foundations is how to build the entire 
->building of Mathematics without any material bricks at all.

The "Platonist" (I prefer "realist") says the bricks ARE there, we're just
ascertaining their properties in detail.  The FOM problem is not thereby
removed, but it HAS had its philosophical uncanniness reduced somewhat.

-> the problem 
->remains how to provide a self-sufficient foundation for Mathematics 
->without any references to the real world to start with.          

I would like to gently chastise Miguel for his common but unhelpful conflation
of "real" with "physical".  The physical world is real enough, but so is
(IOHO) the abstract world of mathematical objects.

             Bill Taylor            W.Taylor at math.canterbury.ac.nz
       MATH:     the discovery, clarification and rigorous study of
            precise relationships in number, pattern, and structure.

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