FOM: Jurassic pebbles (more on Davis/Hersh)
Vladimir Sazonov
sazonov at logic.botik.ru
Tue Mar 17 11:05:02 EST 1998
Martin Davis wrote:
>
> At 09:25 PM 3/16/98 +0300, sazonov at logic.botik.ru wrote:
>
> >But I am very unsatisfied with this irrational explanation because this
> >explains almost nothing. This seems to be the best way to find
> >ourselves in the prison of our own illusions (about the absolute truth,
> >about the standard model for PA, and the like).
> >
> >
> Dear Dr. Sazonov,
>
> I understand from some of your previous postings that you take a skeptical
> view of the infinitude of the natural numbers (except in the context of a
> formal system with exact finitary syntax). So, I'm not surprised by what
> you've written.
>
> For me, it has been clear since I was a boy (a very long time ago) that an
> acceptable account of G"odel's incompleteness theorem would necessarily take
> the natural numbers as given in their totality with objective properties
> beyond what could be derived in any particular formal system. As my teacher
> Emil Post put it (even longer ago): "this ... must result in at least a
> partial reversal of the entire axiomatic trend of the late nineteenth and
> early twentieth centuries, with a return to meaning and truth as being of
> the essance of mathematics."
No doubts, we deal in mathematics with the meaning and truth
(essentially in the context of application of math. either outside
math.
or inside it in a different domain). I also have some imagination on the
infinitude of the natural numbers. We are working with various
formalizations of some ideas (which allow some formalization). We
discuss whether our formalisms or definitions are adequate. (Just for
example, this way Church-Turing thesis arises.) I have no problem with
this. I only have a problem with the mysterious *absolute* mathematical
truth or *standard, fully determinate* model for arithmetic, and the
like.
Note, that this does not prevent me also to find a meaning to (what is
proved
in) G"odel's incompleteness theorem. I do not know where and why I will
need all these phantoms. But I feel that they are a hindrance to me and,
it seems, to many others, whether they realize this or not, as I see
sometimes in discussions on f.o.m. It seems that some discussions would
disappear or change the direction to more fruitful way if we replace
phantoms by something more real.
Martin Schlottmann wrote:
>
> Vladimir Sazonov wrote:
> >
> > [...]
> >
> > I would rather say that it was always true that Lagrange's theorem is
> > PROVABLE in a formal system. [...]
> Now, that (the provability of LT in, e.g., PA) is, by itself, a
> nice piece of mathematics which is, according to you, an eternal
> truth.
Rather, *provability* of it is an "eternal truth" because the statement
on provability is rather concrete, simple and, I hope, everybody here
understand its meaning.
> So, take this as an example for discussion; any example
> does the job, there is no need to quarrel over a particular one.
Well, LT is provable, its "pebble" version is true, LT, as it is, is
also true in some sense and possibly false in some other sense. I did
not think about this. It is easier to illustrate such possibility on the
experimental arithmetical (also Pi^0_1-) low: forall x (log log x < 10).
Its "pebble" version is true. This statement itself is true in an
abstract idealized "set" (or "semiset") of feasible numbers which I have
in my imagination and for which I have corresponding formal intuitively
sound theory. It is just an axiom of that theory and therefore is
provable. Simultaneously, it is also disprovable in PA and therefore
false in corresponding idealized model which I, as all of us, have in
mind. (Probably each one have in its mind its own model. I do not know
and I am not very interested in this. It is only strange to me when
somebody says that he has "standard model" which is the privileged one.)
Moshe' Machover wrote:
>
> Quiz about mathematical truth
> and social construction/destruction/deconstruction.
> ========================================================================
> Let P be the proposition that the well-ordering theorem is first-order
> provable in (formalized) ZFC.
>
> Q1. Is P absolutely true now/always?
Yes, provable, even feasibly provable (i.e. with sufficiently short
proof
so that there is no problem whether it is a proof or not; I think all of
us understand this statement analogously).
> Q2. Assume that on 1 January 2023 a big astroid hits Planet Earth and
> obliterates all human life and all human-made material artefacts. Will P be
> true on 2 January 2023?
Yes. But what will we do with this "yes" in comparison with the above?
I see no essential problem with this purely existential finitary
statement. It was a problem at that time when it was not proved jet.
(But I would rather say that it was *provable* at that time, too,
because now we definitely know this.) Here I see almost no heavy
idealization and complication, except feasibility, which is worth to be
discussed.
On the contrary, the well-ordering theorem itself is, of course much
more problematic, independent on the date 2 January 2023 or any other
date. Why do we need (to postulate) it or the equivalent Choice Axiom?
By which reasons we sometimes reject it? There are numerous
considerations
on this subject together with proofs of related deep results in f.o.m.
such as G"odel's on constructible sets. Probably somebody could say
something essentially more new than it was known on this subject before
so that we will learn more on the relation of AC to other mathematical
subjects, notions and will get some new insights, etc.
Note, that I said nothing on whether the well-ordering theorem is
absolutely true. I feel myself very comfortable without thinking about
this at all, or rather I think that it is not true nor false, or that it
is true in one sense and false in another; it is interesting, useful,
etc.
When I am reading some proof based on classical logic where AC
participates,
I of course think in terms of truth because classical logic "provokes"
to
do this. Intuitioniustic logic "provokes" to think somewhat differently.
But all of this this has nothing general with the *absolute* truth.
Do anybody think essentially differently? It seems many of us could
have agreement if the dangerous words like mentioned above are not
mentioned at all. Of course, there is somewhat different flavour in
what I said, just little bit more freedom in choosing our axioms and
formalisms. Is anything "destructive" in this point of view?
> Q3. Assume that on 1 January 2023 a big astroid hits Planet Earth and
> obliterates all human life; but one library escapes desruction. In this
> library there is a book about ZFC (but there is no-one left to read it).
> Will P be true on 2 January 2023?
I see essentially no difference with Q2.
> Q4. Assume that on 1 January 2023 a big astroid approaches Planet Earth; on
> that day, absolutely no-one thinks about mathematics because everyone is
> terribly worried about the impending catastrophe. But on 2 January 2023 it
> transpires that the astroid has missed Planet Earth by 1/10 of a
> light-second. Was P true on 1 January 2023, when no-one thought about it?
Same as above.
> NOTE: You could play the same game with the proposition that 2 + 3 = 5; but
> then someone might ask: `in what axiomatic theory?'
Yes, your explicit fixing the formal system ZFC was very appropriate.
Vladimir Sazonov
--
Program Systems Institute, | Tel. +7-08535-98945 (Inst.),
Russian Acad. of Sci. | Fax. +7-08535-20566
Pereslavl-Zalessky, | e-mail: sazonov at logic.botik.ru
152140, RUSSIA | http://www.botik.ru/~logic/SAZONOV/
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