FOM: determinate truth values, coherent pragmatism

V. Sazonov V.Sazonov at
Thu Sep 7 14:14:04 EDT 2000

John Steel wrote:
>   I would like to add a few little comments about pragmatism vs. realism.
>   First, I consider myself both a pragmatist and a realist. When I say
> what it is I believe (e.g. there are sets, or there are measurable
> cardinals) I speak as a realist. When I describe why I believe it (the
> simplicity, scope, power, coherence with previously accepted principles,
> and independently verified predictions of the theory these assumptions are
> part of), I speak as a pragmatist. I see no inconsistency here.

If "there are sets", etc. means simply the ordinary intuition 
(a very strong and very useful illusion of existance, but without 
pretensions on anything more) which one has when working with 
a consistent formalism then, as I understand, this is not called 
"realism" and this is, of course, consistent with the coherent 
pragmatism as it was described above or by Prof. Friedman (or 
with rationalism of Prof. Mycielski, or with formalist view on 
mathematics; I do not see any essential difference).  

Realism (which seems to me a very unnatural term - the reason of 
a lot of misunderstanding) means considering imaginary mathematical 
objects as real, as if they existed before and independently of 
any formalism or a family of interrelated formalisms (due to which 
they actually arose and without which they cannot exist in our 

I ask the FOMers to correct me if this is a wrong understanding 
of realism. The example of discussion on SOL shows that we 
sometimes do not understand one another because we may use non 
coherently even technical terms. 

This position contradicts to pragmatist (and/or rationalist,  
formalist) point of view. Moreover the realist position seems 
to me extremely doubtful as a philosophy of mathematics because 
it is essentially based on beliefs, instead of mathematical 
knowledge (i.e. formal deductions, constructions). Otherwise 
it would be called formalism or coherent pragmatism, or the 
like, and would avoid using the term `belief' or at least would 
be especially careful with its using. Say, one can believe that 
every natural set-theoretical statement (such as CH) will be 
eventually resolved in some reasonable extension of ZFC and 
all these extensions will be (practically) consistent and 
adopted by mathematical community. This is sufficiently 
understandable (however, doubtful for me), but needs in further 
comments, clarifications, explanations "why", etc. Beliefs in 
an absolute existence of imaginary and illusory objects do not 
seem to me as a solid or scientific ground for any philosophy 
of mathematics. 

Vladimir Sazonov

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