FOM: Silly questions (Reply to Tennant and Silver on arithmetic v. geometry)
Vladimir Sazonov
sazonov at logic.botik.ru
Sat Oct 10 11:15:25 EDT 1998
Neil Tennant wrote:
>
> On Fri Oct 9 09:51 EDT 1998, Joe Shipman wrote:
...
> > It is perhaps underappreciated that the empirico-physical mode of
> > investigation is relevant to arithmetic as well. It all started with
> > counting pebbles and the like; ...
>
> What started? Presumably, given the context, our hominid ancestors' slow
> ascent to full conceptual grasp of the role that natural numbers can
> and must play in the mature thought of all rational creatures.
>
> > and although the principles we have
> > abstracted from this mode and rendered in pure mathematics are quite
> > compelling *to those of us whose intuition has been developed by a
> > modern mathematical education*,...
>
> I'm afraid, intellectual elitist that I am, that I am not much
> interested in the intuitions of those who happen to *lack* a proper (I
> hesitate to say: "modern") mathematical education. Indeed, what one
> really yearns for is someone who has not only got the mathematics
> under their belt, but the necessary *philosophical* education as
> well. (No ad hominem implicatures intended!)
>
> > we should remember that the physical
> > universe may be finite and Friedman's theorem "there exists n such
> > that all sequences from {0,1,2} of length n have i < j <= n/2 with
> > s(i)...s(2i) a subsequence of s(j)...s(2j)" may be FALSE in some
> > reasonably defined physical interpretation.
>
> I confess you have lost me here. What would a "reasonably defined
> physical interpretation" BE? How do *theorems* about the natural
> numbers get to be *falsified* in "reasonably physical defined
> interpretations" within an ontologically challenged physical universe?
This is a typical example of how mathematical and especially
philosophical education may sometimes prevent an "intellectual
elitist" to see even the trivial point on some (let imaginary)
physical experiment.
Consider another well-known, I think instructive example when it
was necessary to *overcome* the education.
The meaningfulness of the ABSOLUTE notion of simultaneous events
in spatially different places (the God sees what is simultaneous
and what isn't!) was considered dogmatically as non-questionable
before Einstein. But he dared to ask a "silly" question "WHAT
DOES IT MEAN?" and gave a definition in terms of an experiment
(no philosophy! no religion!) relativised to a rigid system of
material bodies. Even only realizing that this notion is
definable *in such a way* leads to *possibility* of a theory of
relativistic space-time.
In the case of arithmetic we may analogously ask: WHAT DOES IT
MEAN the unique up to isomorphism standard model and absolute
truth in it? I think this should be a legal question of f.o.m.
But it seems that our education does not allow us to ask it.
Instead of direct answering, desirably in simple and clear terms
(as Einstein did in his case), we are going around the question
by drawing away our attention to Clinton or to some anecdote or
by appealing just to the great authority of Goedel. (Cf. posting
of Tennant from Fri, 9 Oct 1998 08:55 EDT.) It seems that this
question is really considered by Tennat and probably by many
others as indecent (or religious?) one.
On Fri Oct 9 06:28 EDT 1998, Charlie Silver wrote answering to Tennant:
> I take it that Hersh and others would think you are radically
> misdescribing mathematical activity, though I believe in their
> descriptions of mathematical activity they'd have to include somewhere the
> fact that mathematicians may *think* there's a unique structure out there.
> I'm a little puzzled as to how a sophisticated version of this would go.
I am not sure about Hersh, but I, as any mathematician, when
considering a (meaningful!) formal theory based on classical
logic think that I have a very informal intuitively understood
structure behind it, say, "natural numbers" in the case of PA.
How else it would be possible to do mathematics (the science on
*meaningful* formalisms)? Note also that formal rules of classical
logic are (I would say, especially) accommodated to this way of
thinking. (In contrast to this, intuitionistic logic gives a bit
more *complicated* model of thinking which is the reason that this
logic is actually not used by the majority of working
mathematicians. Who want to think in terms of a Kripke model or
Kleene realizability? There should be a sufficiently serious
reason. If possible, we prefer simple and convenient in use
instruments.) But there is nothing here (except possibly of my
education) what forces me to think that the structure I am working
with is mysteriously unique one, even up to isomorphism, because
I even do not know WHAT DOES IT MEAN *any* "isomorphism" and *any*
structure in this general context.
Is anybody here sufficiently educated who knows this (without
appealing to anecdotes or to authorities and of course to a
metatheory allowing to consider formally both formal systems
and their models)? May be illusion of understanding of the
unique standard model is anyway really helpful? What real is
staying behind this illusion? Will you, fomers, excuse me,
please, for these silly questions. May be I do not know
anything important and evident. Could anybody give a serious
answer?
Vladimir Sazonov
-- | Tel. +7-08535-98945 (Inst.),
Computer Logic Lab., | Tel. +7-08535-98953 (Inst.),
Program Systems Institute, | Tel. +7-08535-98365 (home),
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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