[FOM] formalism

V.Sazonov@csc.liv.ac.uk 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!

Vladimir Sazonov

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