[FOM] The deductive paradigm for mathematics

[Walt Read]

> 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.

   I have no problem with this. The concern was, what would it mean
for math to be about forming theories ala natural science?

> 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?).

   Then in what sense can we justify statements about "all" numbers?
As hypotheses based on a few finite examples? Subject to refutation as
we check more cases? (Approximately what is done in natural science.)

> 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.
>

   Coherence with vague intuition is probably what really happens much
of the time but seems to be a shaky basis for any claimed "certainty"
of mathematical knowledge. At times our intuitions disagree, e.g., on
CH. Trying to resolve the disagreements with, e.g., large cardinals,
seems to make math as much about achieving social consensus as about
knowledge. In science social consensus plays a role but they have some
sort of external reality to check their intuitions. What do we have in
math?

> 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).
>

   This view of math as toolsmithing has probably been the dominant
view among scientists at least since the rise of modern science, and
common enough among mathematicians. But tools are validated by their
end-users, not their makers. I don't know many physicists who are
grateful for PA or ZFC. Incompleteness seems to make them
uncomfortable, though a little intrigued. So in this view I think PA
or ZF would have to be considered toolsmiths' tools and their
validation would be how well they helped mathematicians help the users
of math (by building tools), rather than how well they worked as
theories, a pragmatic rather than a "scientific" view of math.

> Vladimir Sazonov

-Walt