FOM: Formalists (reply to V. Sazonov)

V. Sazonov V.Sazonov at
Mon Jun 12 15:30:07 EDT 2000

Dear Matthew Frank, 

Thanks for your reply! It essentially coincides with what I think 
on formalism. 

> On Monday (June 5), Vladimir Sazonov asked if formalists discussed
> questions like:  "why such and such formalisms are considered?", "could
> they be related one with another and with the real world?", "how it is
> possible that some of formalisms are so useful, say, in physics,
> engineering?"
> I was never satisfied with what I saw from the historical formalists on
> these issues.  However, as a contemporary formalist, I have some
> responses: 
> Often we consider a formalism because we have a good set of intuitions
> somehow associated to it:  we have a picture of the mathematical objects
> to which it is supposed to apply, we have an idea of the sorts of
> arguments it is supposed to encode, etc.  

As I understand, these appeals to intuitions have nothing to do with 
Platonism or realism in mathematics. It is rather *naive* Platonism 
of working mathematicians. 

> (One can also start with the
> formalism and try to develop the intuition from it, but that generally
> seems to be more difficult and less rewarding.)  

Formalisms usually arise supplied with some intuitive appeals. 
Only in badly written papers the reader is forced to guess 
what was the real intention of the author. 

> Then we can create new
> formalisms by generalizing or reformulating old ones; and we can decide
> not to work on certain formalisms because we find them inelegant or
> isolated from other formalisms of interest, etc.
> Formalisms can certainly be related to each other; indeed, this is a major
> subject of proof theory, and is important in model theory in the context
> of interpretability.  As for whether formalisms can be related to the real
> world, I prefer to consider the next question....
> The usefulness of various mathematical formalisms in physics or
> engineering is largely an evolutionary phenomenon:  people create many
> formalisms, and the ones which do not seem useful are not studied, and
> disappear from view.  It is also unsurprising given the strong historical
> connections between math and physics or engineering:  through the 18th
> century, these subjects were not distinguished in the way that we
> distinguish them now; in the 19th century, most work on analysis or
> differential geometry was connected with these areas.  Even in the 20th
> century, the idea of an abstract Hilbert space was created by von Neumann
> in a 1927 paper on the mathematical foundations of quantum mechanics, and
> affine connections were introduced into differential geometry by Weyl (and
> others) in order to elucidate Einstein's general theory of relativity.  I
> find that this retelling of the history (and corresponding devaluation of
> the "unreasonable effectiveness of mathematics in natural science") lends
> considerable support to formalism.

I completely sympathizing to all written above. 

Also, some addition on meaning of formalisms (which I already 
mentioned in FOM). It is also desirable to give a technical 
explanation what in formal rules could make them meaningful just 
because they are formal: they can be used "mechanically" as a 
technical devices (like any other device such as bicycle, etc.) 
to make us - our thought and intuition - stronger and more 
organized (like rules on the street do), etc. Moreover, they 
can be implemented on a computer (however they work very well 
even without a computer). Such and may be some other relevant 
considerations could be put into the base of formalist philosophy 
of mathematics which actually seems has not been sufficiently 
elaborated. A lot to do for philosophers of mathematics who 
consider themselves as *scientific* realists (cf. my separate 
today's posting to FOM). 

As to the very well known bad reputation of formalism, it is some 
of its opponents who are responsible for that, not formalism itself. 

Vladimir Sazonov

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