FOM: Mathematics as governing intuition by formal methods
Vladimir Sazonov
sazonov at logic.botik.ru
Thu Jan 8 16:10:11 EST 1998
In the previous posting I formulated the following "definition":
Mathematics (in a wide sense) deals with governing our intuitions
(abstractions, idealizations, imaginations, illusions, fantasies,
abilities to foresee and anticipate, etc.) on the base of appropriate
formal/deductive/axiomatic methods/systems/theories/rules.
(This seems to be in some agreement with the definitions of Moshe'
Machover and Solomon Feferman also cited there.)
I also wrote that this definition suggests to consciously *manipulate*
our intuitions or even illusions instead of canonizing them. Let me
illustrate this by showing how intuition of natural numbers may be
changed by a critique of the Induction Axiom (and some related things).
Solomon Feferman wrote:
> I. The positive integers are conceived within the structure of objects
> obtained from an initial object by unlimited iteration of its adjunction,
> e.g. 1, 11, 111, 1111, .... , under the operation of successor.
I have questions. What does it mean "unlimited" explicitly? I.e., what
are these dots "...."? I think it may be understood in various ways.
Also, why "unlimited"? Is this a necessary component of any intuition
on the positive integers? Further, is Induction Axiom which is usually
attributed to natural numbers inevitable? Is it true that we do not
loose anything essential by choosing once and for all the traditional
approach to the natural numbers via Peano Arithmetic? How to formulate
this "traditional approach" and corresponding intuition in a clear way
which is not reduced simply to writing down, say, the axioms of PA (or
PA + ?)?
Actually, I myself have and present below some answers to some of these
questions. Not all of this is probably very original. But usually some
essential points are not taken into account at all at the beginnings of
arithmetic. So let me fix them. I am also interested in the opinions on
these considerations of other participants of the FOM list.
First, everybody know that there are resource bounds. And these bounds
may prevent us to choose or use non-critically the term "unlimited"
above. Thus, we could in principle admit existence of the biggest
natural number which I call Box (and denote as \Box in LaTex or as [])
to emphasize the idea of something "bounded" like a room bounded by the
walls. We may consider corresponding version PA_[] (or
"Box-arithmetic") of Peano Arithmetic (PA) *relativized* (together with
Induction Axiom) to this indefinitely finite row 0,1,2,...,[]-1,[] of
natural numbers. Moreover, it reasonable to consider that this
"uniformly, or globally bounded" arithmetic PA_[] involves symbols for
*all* recursive functions relativized to []. (This is a quite
meaningful notion whose precise definition may be known and clear to
those acquainted with descriptive complexity theory or with
corresponding approaches in finite model theory.) Then this
Box-arithmetic proves to be just an *arithmetic of polynomial time
computability*. (There exists corresponding theorem about this
arithmetic.)
Note, that interrelation of the first-order version of this arithmetic
with the second-order one (both in terms of definability and
proof-theory) is essentially connected with the problem "P=NP?". (Here
a well-known definability result of Ronald Fagin (1974) is implicitly
used.)
Let me formulate some problems: Is induction axiom over second-order
formulas provable in the (first-order) PA_[]? Is second-order version
of PA_[] conservative over PA_[]? What are corresponding approaches to
predicativity notions analogous to the ordinary arithmetic. What about
constructive (second-order) versions of PA_[]? A bit *informally*, are
all (finite!) functions over the row 0,1,2,...,[]-1,[] of natural
numbers recursive in the above sense? What about *fixing* [] = 1000, or
= 2^1000, or = any "nonstandard" number?
By the way, it seems to me that this approach is sufficiently serious
reason to consider the problem "P=NP?" as foundational one, in contrast
to the opposed opinion of Professor Solomon Feferman in his recent
posting on "P=NP". Note, that even with a *fixed* value of []
corresponding
analogue of this problem remains interesting and non-trivial. Moreover,
it looses its traditional asymptotic character. (May be some reasonable
changing this problem will be fruitful for finding its solution?)
Actually, we could say that it is a problem on the *nature* of finite
objects. What are they? Are they *all* ("all"?) recursive, or
constructive, constructible? (Of course, this reminds Kolmogorov's
complexity.)
This is one possibility of formalizing natural numbers,
alternative to PA.
Another possibility:
Let us postulate that there exists no last natural number (as in the
ordinary PA), but restrict Induction Axiom (and probably some other
formal rules) in some way. Why? Because it is unclear that this axiom
is necessarily "true". Consider, e.g. its equivalent reformulation as
the Minimum Principle
\exists x A(x) => \exists x (A(x) & \forall y < x ~A(y)).
This principle is true for *short* initial segments of natural numbers
like 0,1,2,...,10 and for *simple* properties A(x). But, if we are
dealing with numbers more *abstractly*, we have also another,
"negative" experience. E.g., what about existence of "the least natural
number which cannot be denoted by a written phrase containing no more
than one thousand of symbols"? I also doubt that Minimum Principle
should inevitably "hold" for any *pure* arithmetical formula with
quantifiers over "all" natural numbers. ("All" is something extremely
indefinite!)
Yes, it seems reasonable to extrapolate the minimum principle to "all"
natural numbers, whatever they are, and to all arithmetical formulas
A(x), especially because this extrapolation in the form of PA does not
seem to lead to a contradiction. But why we *must* do that? Tradition?
Intuition? Some kind of Mathematical Religion? What about other
possibilities and other intuitions (and even "religions")? Who can say
without sufficient trials that these possibilities do not exist or are
fruitless? It seems that such (and may be some other) alternatives
should be at least formulated explicitly when the nature of natural
numbers is discussed.
Note, that intuitionists and constructivists have explicitly
articulated doubts on meaningfulness of quantification over "all"
natural numbers. But they made somewhat different conclusion, just on
the low of excluded middle, instead of Induction Axiom.
There exists somewhat different way of approaching Induction Axiom
(Rule). If A(n) => A(n+1) is provable then we can prove by using the
cut rule (or transitivity of implication) that A(0) => A(n*) for any
fixed numeral n*. Then we *extrapolate* this metatheorem to the
conclusion A(0) => A(n) where n is a (universally quantified) variable
ranging over "all" natural numbers. This extrapolation is based on
identifying abstract, semantical objects (numbers denoted by the
variable n) with concrete, syntactical ones (numerals and therefore -
via Goedel numbering - with arbitrary terms, formulas, proofs, etc.).
Such identification of entities of very different nature (semantical
and syntactical, abstract and concrete) does not seem in general
sufficiently convincing. It makes the (infinite!) row of natural
numbers much "longer" than it is "intended to be". (By the way, it
follows that the sentence Consis_PA does not have precisely the
intended meaning.)
With the induction axiom (and also with the cut rule and with other
tools which also could be criticized; cf. my previous postings to FOM
starting from 5 NOV 1997) we can prove existence of very big numbers
(like 2^1000) whose meaning and existence may be considered as doubtful
or questionable.
I conclude that in principle there are alternatives to PA and
corresponding traditional arithmetical intuition. One alternative
discussed above is PA_[]. Another example is Feasibly Constructive
Arithmetic of Stephen Cook, and closely related Bounded Arithmetic a
version of which was considered by Rohit Parikh in 1971. (My
own interest to BA (and to Bounded Set Theory) started exactly with
PA_[] whose second-order version is essentially equivalent to BA.) Much
more unusual, even strange (and, I would like to hope, promising too)
is also Feasible Arithmetic (feasible - in an essentially stronger
sense of this word than that of Cook) discussed in my previous postings
in November 1997. The first mathematically rigorous approach to
(almost consistent) Feasible Arithmetic was started also by R.Parikh in
1971. Let me also mention corresponding ideas of Alternative Set Theory
of Petr Vopenka.
All of this is related with paying main attention to resources needed
in computations on the level of mathematical foundations rather than on
the level of the traditional complexity theory.
Besides numerous literature on Bounded Arithmetic, cf. also my papers
related to BA, PA_[], Bounded Set Theory (BST) and Feasible Numbers
[1] "Polynomial computability and recursivity in finite domains".
Elektronische Informationsverarbeitung und Kybernetik, Vol.16,
N7, 1980, p.319--323.
[2] "A logical approach to the problem P=?NP", LNCS 88, 1980.
(See corrections in [3,4])
[3] "On existence of complete predicate calculus in
metamathematics without exponentiation", MFCS'81, LNCS, N118,
Springer, New York, 1981, P.483--490.
[4] Also more late paper essentially subsuming [2]:
"On equivalence between polynomial constructivity of Markov's
principle and the equality P=NP", 1988 (In Russian). Cf. English
translation in Siberian Advances in Math., Allerton Press Inc.,
1991, Vol. 1, N4, 92-121.
[5] "On bounded set theory" in Logic and Scientific Methods, 1997,
Kluwer Acad. Publ., 85-103. (Accessible also by my WWW-page)
[6] "On feasible numbers", in: Leivant D., ed., Logic and
Computational Complexity, LNCS Vol. 960, Springer, 1995,
pp.30--51. (Accessible also by my WWW-page)
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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