[FOM] Platonism and Formalism (a reply to Avron and Franzen)

[Vladimir Sazonov]

Arnon Avron is temporarily absent. I am also participating only 
partly. But I feel that some reaction is necessary. 


Arnon Avron wrote:
> 
..........

> In his posting from August 30 Podnieks wrote:
...........

> I protest that Podnieks  describes the debate whether there is
> true arithmetical sentence unprovable in PA as a clear-cut debate between
> Formalism and Platonism. 


Undoubtedly, it is a good touchstone!


I myself will vote for Formalism rather than
> Platonism with respect to Set Theory, and I always have had strong formalist
> tendencies (because I was, and still am,  expecting 100%
> certainty in  at least some parts of mathematics). 


Which parts of mathematics, if it is really mathematics, 
has not 100% certainty and in which sense? 


> I became a "Platonist"
> with repect to the natural numbers 


I ask very much to excuse me, but I have a strong temptation 
to compare this `to be formalist concerning sets, but not 
concerning natural numbers' with `to be pregnant on a half'. 
Are you really consistent in your views? 


when I realized two things: that
> even a formalist should accept some mathematical statements as true or
> false in an absolute sense, namely: statement of the form that a
> certain formal sentence is provable in a certain formal system (and so also
> sentences stating that a given formal system is consistent or not).
> Otherwise formalism makes no sense. 

You are too categorical. 

If we are "intellectually honest", as you write below, we will not 
mix mathematics with metamathematics, avoiding a vicious circle, 
and will consider mathematical proofs (either in a formal or in 
semi-formal way) as physical objects of feasible length. This 
is quite solid and REAL ground to begin with. It is, therefore, 
quite solid or certain (I will not use the doubtful term "absolute" 
having too bad history in science and in everyday life) to say that 
a theorem has a proof. But the "set" of "all" proofs (in the above 
sense) is by no means anything certain, and, hence, it is rather 
uncertain even the meaning (nothing to say about the "truth") of 
the consistency statement (say, of PA) in its genuine sense. 
Everything may be subject to some doubts. Even if we have a 
simingly rather coherent intuition about numbers and PA, any our 
intuition is inherently vague thing and might be not so reliable, 
and PA could once to be proved inconsistent (who knows?). I already 
wrote about this. This seems to me quite clear view. There is no 
heresy or  extravagance here. Why not comment, if you do not agree 
with something? 


This does not mean that I strongly believe that PA is contradictory. 
I just DO NOT KNOW and UNSURE. Nothing else. I think anyone here 
(at least sometimes) can have analogous doubts. Then, where from 
the intuitively even much stronger arithmetical sentence Con_PA 
can follow to be mathematically true in an "absolute" sense 
(non-relativised to stronger theory ZFC or anything else)? 


Second (I believe that this was
> one of Hilbert's insights): that statements of this sort and
> elementary arithmetical statements (concerning the natural numbers)
> are on exactly the same level (something that Martin Davis has pointed out
> again in his reply to Kanovei). So I found that
> one cannot be both intelectually honest and at the same time a
> fanatic formalist concerning N.  I changed my views accordingly.


Did you forget that Hilbert introduced the very concept of 
metamathematics? It is metamathematically, how these statements 
"are on exactly the same level". Making clear distinctions 
is not a privilege of "a fanatic formalist concerning N" but 
a duty of any serious scientist. There is nothing fanatic here. 
Also, any branding like this is out of a "honest" scientific 
discussion. There are important philosophical questions on the 
nature of mathematics, and, unfortunately, a lot of misunderstanding 
arises which leads the discussion in a wrong direction and gives 
impression on fanatism of some (or even of all, independently of 
views) participants. Please, take the argumentation seriously. 


> 
>  Set Theory, especially Cantor's style, and even Classical analysis
> as it is curently concieved and taught, are completely different stories
> and I am still a formalist with respect to both (not to the same
> degree with respect to both).  So I would like to ask Podnieks
> not to call "platonist" everyone who 
> explicitly expresses opinions different from his opinions, 


I think, it is your personal and wrong interpretation of Podnieks's 
views. 


Torkel Franzen wrote:

A consistently non-Platonistic point of view with regard
> to mathematics is much like a consistently skeptical view of human
> knowledge: it has a certain appeal to the intellect, but it has no
> apparent relation to how people, including professed non-Platonists or
> skeptics, actually go about their business.


Very simple - by using mathematical intuition (including 
applications) and formalisms which strengthen our intuition 
and thought, without appealing to highly doubtful Platonistic 
views. (You yourself confirmed in a posting that the 
Platonistic truth is a controversial idea.)

Incomplete answer? A good task to make this more explicit, 
to describe corresponding mechanisms of interaction of formalisms 
with (inevitably vague - what can we do with this?) intuition 
and applications. I tried to do this to some degree in my postings. 
I see no other alternative. 


Vladimir Sazonov