FOM: January 1 - January 31, 1998

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FOM: Mathematics as governing intuition by formal methods

This is continuation of a discussion in FOM on what is 
Mathematics (after questions by Solomon Feferman and by 
Stephen Cook).

Moshe' Machover:

> What characterizes mathematics in general is not only, or even mainly, its
> subject matter (which can be taken or borrowed from the most diverse
> sources) but its unique standard of argument: deductive and (ideally)
> conclusive.

Solomon Feferman says about _objectively subjective_ character of 
Mathematics and defines:

> 1. Mathematics consists in reasoning about more or less clearly and
> coherently groups of objects which exist only in our imagination.
> 2. The reasoning of mathematics is logical, but mathematics is not the
> same as logic since logic concerns the nature of correct reasoning applied
> to any subject matter, whether or not that is clearly conceived.

I like these definitions.
Let me give also my own version with somewhat different nuances:

   Mathematics (in a wide sense) deals with governing our intuitions
   (abstractions, idealizations, imaginations, illusions, fantasies,
   abilities to foresee and anticipate, etc.) on the base of appropriate
   formal/deductive/axiomatic methods/systems/theories/rules.

So, the subject matter of Math. are ARBITRARY INTUITIONS and even 
ILLUSIONS related to ANY kind of human activity provided they can be 
governed by FORMAL METHODS.

Also this definition of Math. suggests to consciously *manipulate* 
our illusions instead of canonizing them. (Illusion is a very useful 
thing if not absolutized and if WE govern it, but not conversely!) 
As a result, our initially "ungoverned", "raw" intuitions become much 
STRONGER (this is the goal of Mathematics).  May be the greatest 
example to illustrate this is numerous applications in physics, 
engineering, astronomy of differential and integral calculus (which 
is just an advanced formalism).

The traditional Mathematics deals with rather *specific* and stable 
versions of our "basic" intuitions on numbers and geometric figures 
and with their numerous combinations. There is some illusion that 
these intuitions are strongly fixed and that we get them and 
corresponding deductive rules of reasoning "with the milk of our 
mothers".  However, this is rather a social-cultural process as it 
was argued by Reuben Hersh:

> mathematical reality is social-cultural

and therefore intuitions may be changed or "elaborated" by any one of 
us (either successful or not). What is specific in the above 
definition(s?), it is that no intuition is declared as a basic or 
universal one for the whole Mathematics. Only the deductive method, 
i.e. using formal (or semi-formal, almost formal) systems is 
considered as basic.  Choose any intuition which you like and which 
is formalizable in (more or less) reasonable way. 

It may happen that some intuition and its formalization (like set 
theory) proves to be universal in *some* respect. Then we may get 
some (may be extremely strong) illusion that we have an approach to 
the unique "TRUE" Mathematics.  (In the Soviet Union it was the 
unique TRUE doctrine of Communism.  Does anybody want such one in 
Mathematics?)  However, according to the above definition we should 
rather discuss on *adequateness* of formalisms to corresponding 
intuitions or something like, and on *formal provability* instead of 
"truth" (an extremely overloaded term with oversimplified two-valued 
meaning having also a numerous technical counterparts in Mathematical 
Logic such as forcing, etc.).  Also, not all intuitions (say, of 
Constructive Math.) are based on *such* a concept of truth. 

We could admit, at least in principle, that some formal mathematical 
system based on some intuition may have the form essentially 
different from the ordinary predicate calculus + some special axioms. 
What would the "truth" of a provable statement mean in this case? Thus, 
it is more safe to use this word mainly in a *technical* sense so that 
no confusion arise. The "intimate" correspondence of a formal system 
to some intuition is something different and should be discussed and 
investigated each time in specific appropriate (formal or informal) 

Moreover, it seems (and actually, we should know this after Goedel!) 
that there is no reason to believe that each such intuition will be 
formalized in a complete way (who ever knows, what does it mean 
"complete" in this context?). We just *govern* the intuition by a 
formal method to reach some *specific* (not necessarily all 
potentially possible and recognized some later) goals. 

WAS IT THE GOAL to prove (or disprove) that all sets on the real line 
are measurable when set theory (and Choice Axiom) was created and 
formalized?  WAS IT THE GOAL to use Induction Axiom *especially* for 
proving that logarithm function is unbounded and exponential function 
is total (instead of *partial recursive*) **despite** the real everyday 
computational experience when this Axiom was first consciously or 
non-consciously used (postulated)?  (Cf. my previous postings to FOM 
starting from 5 NOV 1997.) Of course, there were some different goals 
of these formalizations and we often (if not always!) have some 
undesirable *side effects* of any formalization. Why to pay so mach 
attention to achieving the whole mathematical "TRUTH", as if we are 
able to understand what does it ever mean?

In particular, I cannot agree with Martin Davis (or understand him

> Let me say it. There is mathematical *truth*. It is genuine and objective.
> It is when consensus arises *because* mathematical truth has been attained
> that the subject advances. Some may believe the dogmas of a religion as
> fervently as Lagrange's theorem. But when pressed it will be admited that it
> is a matter of "faith". My belief in the truth of Lagrange's theorem is not
> a matter of faith.

Of course no "faith", no "dogma"!. But what about using the term 
"provability" which is quite "genuine and objective" notion (wrt any 
fixed formal system) instead of "truth"? Even if the majority of our 
"questions" to a formal system on provability of a sentence or its 
negation usually have a positive answer (as for PA vs.  ZFC), why 
should we make the conclusion that we are much more near in this 
system to corresponding mathematical TRUTH and to give to this "fact" 
too high value?  Who of working mathematicians *really* needs this 
phantom of FULL MATHEMATICAL TRUTH? There are two things to say. 

First, I think that we *actually need* 
only the *illusion* of full mathematical truth as a 
very comfortable for our mind psychological support or a kind of 
doping.  This illusion is based mainly on *formal* rules of the 
classical logic (such as the lows of excluded middle, double 
negation, etc.) which all mathematicians use everyday even if they know 
nothing on its existence. But they, of course, know!  We start to 
learn these lows at school, say, when proving first time the simplest 
geometrical theorems, and even earlier.  (However, these lows are 
rather specific, almost not used in our everyday, non-mathematical 
life or too artificial as A => B iff ~A v B.) Say, I myself recall 
to my students their own mathematical experience from school when 
teaching them to (natural deduction rules of) logic and argue that 
they actually know these rules and have used them many times. Is not 
using classical (or, in principle, any other) logic the "consensus" to 
which Hersh appealed in his description of Math.? 

Second, mathematicians are rather dealing with some *narrow class of 
specific problems* grouped according to some, also specific goals.  
What for then we need the FULL MATHEMATICAL TRUTH, if not as an 
illusion for comfortable work? 

It was extremely important to understand that (and especially how and 
why) CH is independent of ZFC. Of course, there are many interesting 
and really great, deep and fascinating problems related with more 
proper and deep understanding the place of CH in set theory and in 
Mathematics, whether its alternatives are true/false, 
inevitable/refutable in that or other specific technical (natural or 
artificial) sense, to find some reasons pro and contra, etc.  All 
this experience may prove to be very useful even for quite different 
researches in f.o.m.  (Note, that I myself do not work in this very 
direction and therefore cannot properly evaluate what is happening

However, there is also somewhat alternative way to f.o.m: use all the 
gained experience and highly elaborated techniques and ideas to 
develop various NEW (i.e. not related directly with traditional systems 
PA or ZFC, etc) "basic" intuitions and formalisms, say, in connection 
with Theoretical-Mathematical Computer Science (as it were NEW and 
non-traditional at that time the Cantorian (or ZFC) Set Theory or 
Robinson's Nonstandard Analysis playing the role of NEW foundations for 
basic mathematical, intuitions, notions and approaches).  

For example, I believe, it is important to work on reasonable 
*foundational* approaches related to computational complexity theory 
which deals with bounded resources. On the foundational level this would 
mean *reconsidering* our intuitions and formalisms concerning to the 
very nature of finite objects (say, via systems of Bounded Arithmetic 
or Arithmetic of "Feasible" natural numbers; cf. my previous postings to 
FOM starting from 5 NOV 1997). 

Vladimir Sazonov
Program Systems Institute,  	| Tel. +7-08535-98945 (Inst.), 
Russian Acad. of Sci.		| Fax. +7-08535-20566
Pereslavl-Zalessky,		| e-mail: 
152140, RUSSIA			|

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