[FOM] The deductive paradigm for mathematics

[Vladimir Sazonov]

This is a reaction to the posting of Walt Read

http://cs.nyu.edu/pipermail/fom/2010-August/014976.html

Dear Walt, 

It seems you are trying to decide what is mathematics about. 
Is it, like PA or ZFC, studying numbers or sets  
"or are these things in fact defined by, and therefore 
essentially the same as, their theories?" 

A good question essentially containing the answer!

You also compare mathematical theories with theories of 
natural sciences. 

I suggest the following view. Say, PA is NOT about natural numbers 
as something existing in some reality. Thus it is not like 
and should not compare directly with natural sciences. Of course, 
we have some concrete examples of the natural numbers like unary 
strings of symbols empty (zero), |, ||, |||, ... ||||||||||||||||||, 
or the like. But "all" natural numbers do not really exist, except 
in our fantasies which are INEVITABLY something vague. On the other 
hand, PA itself is quite a concrete finite object with a "rigid" 
formal "behaviour". It is THIS finite object (PA) which we can 
consider as representing "all" natural numbers. (This seems coherent 
with my citation of you above.) Predictability of PA theorems can be 
validated (as you write) only on some particular examples of numbers. 
It is impossible and even meaningless to validate this on "all" "genuine" 
numbers (what does it ever mean?). Also some indirect witnesses of 
predictability of PA system are possible. That is, some coherence of 
various theorems with our vague intuition (and other formal theories 
like PRA or ZFC and with sometimes in the real world) can be demonstrated. 

What I suggest, is to forget about any Platonist (quasi-religious 
and so having no SCIENTIFIC meaning) ideas on existence of numbers 
sets, etc. This is an awfully misleading misconception. We can use 
the term "all natural numbers, or N" only in some technical contexts 
like in the quantifier \forall x \in N, etc. We know how to work 
with these quantifiers formally. We have our (inevitably vague) 
intuition which is coherent with these formal rules, and that is 
all what we really need. There is actually NO NEED to pretend on 
anything more. There is also no need of any religion (Platonist 
or any other style) in mathematics as in any science. We are not 
in the middle ages to resolve questions like how many angels can 
fit on the needle end! Considering formal tools as strengthening 
our thought is quite a different, actually an experimental question 
of applying formal theories in the real (or in any imaginary) world.  
We just apply the rules and see how they are miraculously efficient, 
how they economise our thought, how much we can get with using these 
formal rules in comparison with the situation when we had not invented 
these rules yet. 


But then, what is mathematics about? Just a system of meaningless 
formal derivation rules? Of course NOT! Mathematics is a kind of 
ENGINEERING devoted to devising formal systems as tools strengthening 
our THOUGHT (quite a meaningful, useful and so respectful enterprise!). 
Particular formal foundational systems like PA, ZFC, etc. are supporting 
other formal systems like the Newton-Leibniz Calculus which already have 
demonstrated their role as extraordinary strengthening our thought 
(say, concerning movement of mechanical bodies in the space). 

Take this view on mathematics as a specific kind of engineering and 
reconsider your questions again. Mathematics has NO AIM to study 
IMMEDIATELY the real world and is not a science in the ordinary sense 
of this word, but it helps so much by strengthening our thought 
(about the real world or about any our imaginations about imaginations 
... on the real world) with the help of formalisms (formal tools or 
formal "levers", or formal "engines" for our thought) it creates! 
Let us judge mathematics from this point of view. 

A mathematical formal theory is good if it serves the above purposes 
well. Say, ZFC is good because it serves as a unique foundation 
of the Calculus and of a lot (I even do not say "all") of other 
formal tools of thought and also because it is coherent with some 
our (vague) intuitions on sets and with a lot of other intuitions, 
e.g. on continuity (by defining topological spaces in ZFC). 
Any other formal version of set theory should be judged similarly. 


If it is sufficiently good in this sense and simultaneously resolves 
CH with some new intuition on sets, that is great. 


If another version does the same for "not CH" (with a different 
interesting intuition on sets), that is also great. 


Thus, potentially, both CH and not CH can find its (alternative) ground. 
What is the problem? Just to find any formal system that do this well 
in some way. If we cannot devise it, CH becomes "unresolved" in this sense. 
That is the life... Of course there is no unique or objective criterion 
of what is "well" or "great". But when it is "really well" we see that 
quite clearly and can give an appropriate argumentation. 


Do we need anything more from mathematics (than devising new powerful 
formal tools for our thought about ANYTHING having any intellectual 
or practical interest)? Do we need any essentially different criteria? 


Regards, 

Vladimir Sazonov 


P.S. 
Finally, as to the announced solution of the P=?NP problem, if it 
is really so, the main outcome of this will be not the mere fact 
that P \neq NP but some new ideas and related formal tools which 
will be extracted from this solution. The very fact that P=?NP was 
resolved positively or negatively is not so interesting (non-informative) 
in itself. Mathematics is not about truth. It is about proofs and formal 
systems where we derive these proofs because "formal" makes our 
thought so powerful.