FOM: Silly questions (Reply to Tennant and Silver on arithmetic v. geometry)

[Vladimir Sazonov]

Charles Silver wrote:

...

> 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, then, you *do* have a
> "very informal intuitively understood structure behind it."  I'm not sure
> what you're objecting to that I said.  Is it the word 'unique'?  Are you
> saying that when you think of PA, you juggle *multiple* possible
> structures?

I work with one of them, but I cannot definitely say with which
exactly because I do not know which is the priviledged (standard)
one and what does it mean at all. I also do not know what is the
"set" ("class"?, "semiset"?) of all these models. All of this is
somewhat fuzzy, 'sophisticated' and I do not see how to make it
precise and clear and do not believe that in this general context
it could be done. If your word 'unique' is just replacing the
linguistic definite article 'the' and we actually have analogous
understanding then I could ask you what is so puzzling to your
here?  On the other hands, your words 'unique' and 'puzzling'
demonstrated me that you indeed believe in the 'standard mopdel'.
(Now you seems do not confirm this.) But such a belief is puzzling
to me. I think that such an uncritical belief is allowed only for
beginners. Or I do not understand something very essential.

> > 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" something.

I would like to see such a mathematician who faithfully behaves
in this way! Also this would be like using, say, a clock for
hammering nails.

> > Note also that formal rules of classical
> > logic are (I would say, especially) accommodated to this way of
> > thinking.
>
>         I'm sorry, but I just don't see this.  I don't see that the
> "formal rules of classical logic" require you to think of the standard
> model.

I did not say here anything on the 'standard model'. Formal 
rules of classical logic (especially quantifier rules) rather 
'suggest' to think or are in a good agreement with thinking of 
an (informal, I do not know which exactly) model. By the way, 
if a formal theory is (seemingly) consistent then this suggests 
to think that an informal model exists. This is a *naive* form 
of Goedel completeness theorem.

> > 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).

OK! [But the word 'unique' which you use seems should assume here
'up to isomorphism'. Thus, I should ignore your 'unique'.]

Tell me
> whether you do or you do not think that PA is "about" a certain standard
> structure.

Certain in only the sense that I fixed my attention on *a* 
structure satisfying the axioms of PA. Omit "standard" because 
there could be undesirable association with the set-theoretic 
ordinal omega and it would be necessary to ask in turn whether 
the intended model of set theory is standard and unique.

I deliberately enclosed 'about' in scare quotes because this
> is a difficult notion to unpack.

I would rather enclose in the quotes the word 'certain'.

My claim was that mathematicians *think*
> the the formalism of PA (to take an example) is "about" a certain unique

Again this dangerous word! I just ignore it and put 'certain' in quotes.

> structure. I think you agreed with my claim when you said that you "have a
> very informal intuitively understood structure behind it, say, "natural
> numbers" in the case of PA."  I am not sure what it is I've said that you
> think is wrong.

It seems that the agreement may be achieved. I would be gald!

> > 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?

This was essentially a question to everybody (in the light of
Tennant's note on education) on what does it mean the "standard
model of PA (which is unique up to isomorphism)".

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?

Let these people think what they want and like. We probably 
should not distract them from their business by our discussions 
on f.o.m.  Also they probably know nothing about (the technical 
set-theoretic notion of) nonstandard models.  As to a 
metatheory, it seems to me the only reasonable way to make a 
sense of the notion of standard model. Only in such a metatheory 
we could formulate and porove (if metatheory will allow this) 
the statement

\exists N |= PA s.t. [\forall M |= PA (M is end extension of N)
		and, moreover, such N is unique, up to isomorphism].

and then call this N a *standard model*. But now we are not in that 
context where the question arose. 

>
> > May be illusion of understanding of the
> > unique standard model is anyway really helpful?
>
>         Yeah.  I think it helps to suppose there is one.  Don't you think
> so?

It helps that there is (even only in our not very sober 
imagination) *some*, I do not know which exactly, informal model.

> > What real is staying behind this illusion?
>
>         What kind of answer are you looking for here?

May be there is some (which?) kind of stability of what is true 
or false in this "intended" model M, what in case of a metatheory 
would be called Th(M), so to speak, "absolute truth". This would 
be interpreted as an *indirect* witness of existence of some
priviledged  model. Say, it is usually said that Consis(PA),
however inprovable in PA, is "of course true" in the intended
model. This "of course" is insufficient to me. I would like to
have more explanations, "why?".  I also would present my doubts
that Consis(PA) "really" expresses the consistency statement. 
(It is much stronger than consistency.) I need some elaborated 
picture. May be in terms of potential feasibility. (What it is? 
Just ability of add 1, or somethinfg more? How more?) No pure faith!  
I do not like when it is said that natural numbers are from the 
God or the like. They are (were) created by peoples and I would 
like to understand how. I am trying myself to do something 
alternative with feasible numbers. 

> > 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 this your question: What makes us (apart from our education,
> consisting of anecdotes, etc.) think that the standard model is *really*
> unique?

Yes, especially, what does it mean completely unexplained but 
permanently used term "standard model"? 


Vladimir Sazonov