FOM: "Relativistic" mathematics?
Vladimir Sazonov
sazonov at logic.botik.ru
Mon Oct 12 06:59:05 EDT 1998
First, a reply to Charles Silver.
Charles Silver wrote:
>
> Vladimir Sazonov:
> > > > 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.
>
> Take these second-order axioms:
>
> 1) (Ax)Sx <> 0
> 2) (Ax)(Ay)(Sx = Sy --> x = y)
> 3) (AX)(X0 & (Ay)(Xy -> XSy) --> (Ay)Xy)
>
> What is wrong with the proof that any structure satisfying these
> axioms must be isomorphic to <N,0,S>, the standard structure? Or is it
> that you accept this proof, but want to raise the question: Just what *is*
> this standard structure? Or are you asking just what is it to *be* a
> structure in the first place? Or,...?
Any (mathematical) proof assumes (at least implicitly) a formal
system where it is written. Let us assume that this formal system
is ZFC. Then I have NO problem with such a proof of the statement
"all structures satisfying the axioms 1) - 3) are isomorphic (to
some fixed, so called, standard structure)". Thus, in the
framework of ZFC the notion of standard model is completely legal
one. But if anybody asserts anything on "standard model" in a
*general context*, without assuming that he deals with, say, a ZFC
statement, then I completely do not understand what is happening.
Moreover, I seriously suspect that he also does not understand
what he is asserting because my question "what does it mean"
or, alternatively, "which is the assumed framework?" is usually
hanging in the air as if it is a silly question, or it is said
something about the God who created natural numbers or some other
mysticism or deviltry (or holiness, which is even worse)
[of which, I am sorry, ushi vjanut (in Russian); Can anybody
translate in English? For CENSOR: it is decent, but expressive;
Please omit if you have any doubts.].
It seems that there exists indeed a good analogy with relativistic
physics.
In Physics:
Q. What does it mean *absolute* space-time (simultaneity notion)?
A. This notion is meaningless. Relativise to a rigid system of
physical bodies [a *frame*? Is this correct English term from
relativistic physics?]
In Math.:
Q. What does it mean *absolute* mathematical (arithmetical) truth?
A. The genearl notion of absoulte truth is meaningless. Relativise
it to a rigid *frame*(work), i.e. to a formal (or axiomatic) system.
There exists no unique mathematical (or arithmetical) world or a
universe (or standard model for arithmetic). Each (meaningful)
formal system has his own meaning which is a delicate and very
informal matter and should be considered for each formal system
separately.
It proves that some formal systems, like ZFC, are sufficiently
universal. They were especially created to reach some degree of
universality. But, being a happy and even the GREAT occasion,
this does not have any *absolute* character.
Thus, again, (rigorous!) mathematics is a science on
(meaningful!!!) formal systems, not about a specific unique
universal or absolute mathematical world.
Now we should reconsider "from the beginning" the notion of
formal system. We cannot suppose that we have any *advanced*
notion of finite object to discuss formalisms (finite
formulas, terms, definitions, proofs). We should explain this
by concrete examples by writing some symbols on real sheets of
paper. I.e. we should deal with *feasibly* finite strings of
symbols, and, in particular, we will inevitably come "exactly"
to the notion of *feasible consistency* of a formal system.
Then let us consider various formal systems (believably feasibly
consistent and having any reasonable informal, intuitive meaning):
ZF, PA, some versions of feasible arithmetic, etc. Prove some
theorems, correct axioms and proof rules to get a better
correspondence with our intuition, prove again, compare
various systems, try to find some applications outside or inside
mathematics, etc. This is a normal everyday work in mathematics
without too big "absolutistic" ambitions.
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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