[FOM] Re: {FOM] Platonism and Formalism

[Karlis Podnieks]

----- Original Message ----- 
From: "Dmytro Taranovsky" <dmytro at MIT.EDU>
To: "Karlis Podnieks" <Karlis.Podnieks at mii.lu.lv>
Cc: "FOM" <fom at cs.nyu.edu>
Sent: Wednesday, October 08, 2003 3:44 AM
Subject: A reply to Podnieks on Platonism and Formalism

>...

KP> It seems, the two extremes could reach an agreement... Thank you for a
very valuable letter.

I hope, the following will be interesting at least for students.

Now, from my extreme formalist point of view, the contents of the modern
set-theoretical Platonism seems to be as follows.

1) There are axioms of ZF, and there is AC.

2) There is a growing collection of the so-called large cardinal axioms.

3) A few of these axioms contradict with ZF+AC, and the corresponding proofs
are easy.

4) But the rest does not yield easy contradictions. This allows to think
that they are consistent with ZF+AC, so let us use these large cardinal
axioms as normal axioms of set theory.

Does the real Platonist position contain more than this (except
"metaphysics", used for inspiration)?

If not, then formalists could safely participate in this kind of activity.
The world of large cardinals is exciting, indeed.

Easy adopting of new axioms the (relative) consistency of which cannot be
justified by a formal proof  - this new way of developing mathematical
theories was invented after the discovery of the incompleteness phenomenon.
Who made the first step?

Now, two questions inspired by the formalist angle of view:

a) Of course, cardinals (large cardinals included) are linearly ordered.
Thus, if we have two large cardinal axioms, LCA1 and LCA2, then, "of
course", either, lc1<lc2, or lc1=lc2, or lc1>lc2 (lci denotes the least
cardinal, large according to the axiom LCi). Couldn't we design LCA1 and
LCA2 in such a way that we will never know, which of the three possibilities
is "true"?

b) LCAs possess the following properties: b1) Con(ZF) implies
Con(ZFC+notLCA), b2) Con(ZF) does not imply Con(ZFC+LCA). Couldn't we design
a plausible set-theoretical hypothesis HH such that Con(ZF) does not imply
neither Con(ZF+HH), nor Con(ZF+notHH)? For example, we will never know, is
HH true in the minimal inner model of ZF, or not?

Please, excuse me, if the answers are trivial for you. My experience in set
theory is more than restricted. But, perhaps, students could benefit from
the explanation.

Finally, an interesting idea due to Joe Shipman (see
http://www.cs.nyu.edu/pipermail/fom/1998-March/001641.html):

"... Mathematics on other planets may be very different, ... They may well
contradict our theorems about real numbers, for example if they work from
the Axiom of Determinacy instead of AC."

Thus, one could imagine a different scenario of set theory development.
Assume, Georg Cantor would be educated "in a culture obsessed with games or
religious predestination" (Joe Shipman, again, see
http://www.cs.nyu.edu/pipermail/fom/2000-April/003966.html). Then, in his
informal reasoning about sets, he would, perhaps, use an implicit equivalent
of AD instead of an implicit equivalent of AC. And, after this, instead of
ZFC, we would now be teaching our students ZF+AD as the "true world of
sets". And, perhaps, then Platonists would call investigation of ZFC simply
a study of interesting sets?

This is intended not as a critique. The answer is: yes, we could develop an
AD oriented Platonism instead of the current AC oriented one.

(Joe Shipman is not responsible for these my formalist relativistic
experiments.)

Now, some (less important?) questions.

>...
> Investigation of mathematical theories amounts to investigation of the
> (metaphysically) existing objects that satisfy them.

Where could this kind of metaphysics be coming from? Will the future robots
be able to reproduce it? (They will be able to reproduce the above-stated
formalist view of the Platonist position.)

>...
> ZF+V=L is true in the minimal inner model of ZF.  ZF+AD is true in
> L(R).  The two theories are seriously investigated by mathematicians as
> they reveal facts about constructible sets and about sets constructible
> from the reals.

To "verify" that ZF+AD is true in L(R), currently, we need to postulate that
"there is a supercompact cardinal".

> > From my sideview point, these new "worlds of sets" look even more
> > interesting than the older mainstream "world".
>
> These "new worlds of sets" are simply various interesting sets (or
> classes). Platonism in no way constrains the study of interesting sets.
>

Good reply, "full contact".

>...
>  Fortunately, we know a priori that every axiom of PA is true,
> and hence PA is consistent.

For an extreme formalist, this "a priori knowledge" is an illusion. But I
agree that "every axiom of PA is true" could be proved, if we used (a subset
of) ZF as a metatheory of PA. If not from (a subset of) ZF, where could this
a priori knowledge be coming from?

>...
> Best Wishes,
> Dmytro Taranovsky

Best wishes,
Karlis.Podnieks at mii.lu.lv
www.ltn.lv/~podnieks
Institute of Mathematics and Computer Science
University of Latvia