[FOM] The deductive paradigm for mathematics
vladimir.sazonov at yahoo.com
Tue Aug 10 19:11:52 EDT 2010
This is a reaction to the posting of Walt Read
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
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?
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.
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