V.Sazonov at csc.liv.ac.uk
Fri Nov 3 19:41:37 EST 2006
Quoting Rupert McCallum <rupertmccallum at yahoo.com> Fri, 03 Nov 2006:
> I'm interested in understanding the formalist position better. One
> version of it is that mathematics is "the science of formal systems",
I consider myself to be a formalist. My understanding of formalism
differs from the traditional caricature view assuming just a play with
symbols and nothing else. Moreover, I have no idea about any real
mathematician having a formalist view who would have such a caricature
view. I believe that this is rather "invention" of platonists or so
called "realists" (an awfully misleading term) to show that formalism
is something stupid.
Let me just repeat the following definition of mathematics from my last
posting (I have presented in FOM various versions of this) which
describes my formalist view:
Mathematics, "by definition", deals with our (inevitably vague)
imagination and fantasies governed/restricted/strengthened by FORMAL
axioms and rules.
This definition is very general, but it does not change the nature of
mathematics. It only changes some traditional angle of view on
mathematics - a wrong angle assuming beliefs in fictions having no
scientific grounds. Of course this definition is not about the truth of
mathematical results (which ever truth if we are dealing with fantasies
and imaginations? recall, e.g. Imaginary Geometry of Lobatshevsky).
> but then the question arises in what metatheory do we study these
> formal systems.
The above definition does not require any metatheory. Formal systems
are assumed to be considered in a naive manner (to avoid the evident
vicious circle) as I described in another recent posting answering to
Timothy Y. Chow. Of course, any metatheoretical considerations are not
excluded by this formalist view as it does not restrict mathematics in
any way, just vice versa!
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