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

[Vladimir Sazonov]

Hi, Vladik!


Of course I agree with your notes 

http://cs.nyu.edu/pipermail/fom/2010-June/014808.html

that other sciences can contain 
mathematical formal fragments. This confirms that formalisation 
makes our thought powerful and that is why mathematics is so widely 
applicable. Also other scientists can contribute to mathematics 
by creating some formalisms even not necessary satisfying strong 
mathematical criteria. The greatest example is Newton who for 
the needs of physics (if I correctly describe the history of 
sciences) created the beginnings of Analysis/Calculus. 
It took hundreds years to make Newton-Leibniz Analysis to be 
formalised according to mathematical standards. But even 
in not so perfect form, the Calculus (yet without proper 
mathematical foundation) was from the early times at large 
degree quite a formal system as I tried to argue in a 
recent posting: 

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


I only want to emphasise that sciences other than mathematics 
either use existing mathematical formalisms or create their own 
only sporadically and ***without*** such maniacality as mathematics 
does. In general, they have gaps in their quasi-formal considerations 
which cannot be eliminated (as Hilbert eliminated some rare gaps 
in the geometry of Euclid). In general, other sciences do not 
pursue mathematical rigour with such a passion as mathematics 
does it. This would be actually impossible because in general 
other sciences have different goals than mathematics. 
Studying the Nature cannot be done in fully mathematical way 
but can benefit from using mathematics in some particular cases. 
Even when they use mathematics, they usually do this in a different 
style because their goal is truths (on the nature) rather than proofs. 
So they can often neglect the full rigour and it is quite usual that 
the full rigour is impossible because they rely on experiments 
and on other style of argumentation concerning the real world. 


In mathematics it is just vice versa - the goal is proofs rather 
than truths. 


I am even very suspicious concerning using the word truth in 
mathematics and in general non-technical discussions try 
to replace it by something scientifically more adequate like 
formalising our intuition, reflecting some kind of reality, etc. 
Mathematics cannot pretend that proved theorems are true 
in the ordinary sense of the real world because it rather 
formalises our fantasies. It is better to avoid the unnecessary 
temptations on relations with the real world. This relation 
is in general not compulsory and only occasional (and in this 
case highly valuable). This is another serious difference with 
other sciences. They can have fantasies as well, but their goal 
is still approaching to the real world. Mathematics is free 
in this respect. Its general goal is creating formal tools 
for our free and pure thought. Quite happily, these tools 
are sometimes applicable in sciences. But forcing mathematics 
by our society to create ***only*** applicable formalisms 
would kill it. 


Best wishes,  
  
Vladimir Sazonov