FOM: Silly questions (Reply to Tennant and Silver on arithmetic v. geometry)
Charles Silver
csilver at sophia.smith.edu
Sat Oct 10 15:17:44 EDT 1998
On Sat, 10 Oct 1998, Vladimir Sazonov wrote:
[I'm including only the most recent remarks and snipping lots of
earlier stuff]
> Neil Tennant wrote:
> > 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.
> On Fri Oct 9 06:28 EDT 1998, Charlie Silver wrote answering to Tennant:
> > ...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.
Vladimir Sazonov:
> 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.
You say that when you are thinking of PA, you *do* have a "very
informal intuitively understood structure behind it." So, what it is that
I said that you're objecting to? Is it the word 'unique'? Are you saying
that when you think of PA, you juggle *multiple* possible structures?
> How else it would be possible to do mathematics (the science on
> *meaningful* formalisms)?
I think there are lots of ways. One way would be to just look at
the formalism and not think that it's "about" anything.
> Note also that formal rules of classical
> logic are (I would say, especially) accommodated to this way of
> thinking.
I don't see that the "formal rules of classical logic" require you to
think of the standard model at all. What do you mean by "accommodated"?
Do you mean that the formal rules make it "natural" to think of standard
models for theories? I don't see that this is so.
> 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.
Forget 'isomorphism' (which I never mentioned, anyway). Tell me
whether you do think that PA is "about" a certain standard
structure. I deliberately enclosed 'about' in scare quotes because this
is a difficult notion to unpack. My claim was that mathematicians *think*
that the formalism of PA (to take an example) is "about" a certain unique
structure. I think you agreed with my claim when you said that you "have a
very informal intuitively understood structure behind it" (say of the
natural numbers in the case of PA). I am not sure what it is I've said
that you think is wrong.
> 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)?
Are you here objecting to Neil Tennant's "intellectual elitism" or
to something I said? I don't get the role of the metatheory here, since I
think that people who work in number theory and never ever look at formal
logic still have in their minds that there is a unique structure to the
natural numbers. Are you disagreeing with this?
> May be illusion of understanding of the
> unique standard model is anyway really helpful?
I think it helps to suppose there is one. Don't you think
so?
> What real is staying behind this illusion?
What kind of answer are you looking for here? What would make
you believe that "the standard model" is real?
> 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?
Is your question: "What makes us (apart from our education,
consisting of anecdotes, etc.) think that the standard model is *really*
unique?"
Charlie Silver
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