[FOM] Mathematics ***is*** formalising of our thought and intuition

[Vladimir Sazonov]

----- Original Message ----
> From: Shay Logan <logan110 at umn.edu>
http://cs.nyu.edu/pipermail/fom/2010-June/014803.html

> I'm unclear on the meaning of the word formal as it is being used in
> this discussion.
> 


 
> 
> If by formal we mean what the word etymologically should mean; that
> is, that truth is a function EXCLUSIVELY of the form of our sentences,

 
I suggest just this understanding or even a little bit weaker one 
with "EXCLUSIVELY" replaced by "mostly" because in this general 
context, how to check that "EXCLUSIVELY" does hold? This would rather 
applicable for a full formalisation (like FOL and ZFC are formalised 
with axioms and formal proof rules). But we want to also have a more 
flexible concept of "formal". 


Also I suggest not to use the word "truth" here. A mathematical proof 
cannot be true but rather formally correct -- satisfying formal rules 
(or not). 


> it seems odd to me to claim that mathematics is formal. There is far
> more going on in mathematics than merely a verification that the
> shapes (forms) of the sentences of mathematical statements are of the
> right type. 

 
What more? Evidently we assume that we have a formal system 
(such as Euclid's geometry) which *deserves* to be considered 
*by formalising some kind of our intuition* (on the real world 
or on any our fantasies, does not matter). We usually assume 
that the *theorem proved is interesting enough*. Then ANY formally 
correct proof of this theorem is good independently how we obtained 
it or who is its author. We usually prefer an intuitively clear 
well-structured proofs, desirably presented in not too formal 
way, only to be able to evidently see that it is potentially 
formalisable according to standards of the given formal system. 
Contemporary standards of (potential) formalisability are very 
high. 


Additionally, appropriate (subconscious) "maniacality" in getting 
more and more formal versions of formal systems and proofs is assumed 
in mathematics. For contemporary mathematics this means potential 
possibility of absolute formalisation. 

 
This kind of "maniacality" is not a bad thing. It is just a very 
strong adherence to the idea of mathematical rigour. It is because 
of this adherence not very formal proofs of Euclid were shown 
(essentially by Hilbert) fully formalisable with filling several 
clearly localised gaps. 

 
If such consistent filling the gaps could ever be possible in other 
sciences then they would be parts of mathematics. But they are not. 
Even physics! 

 
> However, if by formal we mean something more colloquial, something
> more along the lines of "rigorously thought through and carefully
> argued", then a great deal of political science does seem formal to
> me. So does mathematics. Mathematics may be (somehow) more formal than
> political science when looked at from these lights.

 

Not just "somehow" more formal, but in SO DIFFERENT WAY! 
Mathematical rigour is so strong and so much different from what 
any other sciences do that they are almost incomparable. 

 
By mathematical rigour I understand the formality in the above 
sense (+ "maniacality" in formalising). 

 
See also my previous posting  

http://cs.nyu.edu/pipermail/fom/2010-June/014804.html



Vladimir Sazonov