FOM: Re: Arithmetic/Geometry
Vladimir Sazonov
sazonov at logic.botik.ru
Sat Oct 17 12:10:41 EDT 1998
Harvey Friedman wrote:
>
> Sazonov 7:15PM 10/16/98 writes:
>
> >I agree only that once a formal system is fixed mathematician
> >(if he is indeed a mathematician) should deduce theorems in this
> >system using physical and any other consideration *only* as
> >motivations or a help for the intuition to find a proof.
> >However, when a formal system arise it may be based on physical
> >experiments or anything else. The best example is geometry!
>
> But I made a distinction between these four:
I do not see the reason for this "but". What I said is related
just with (various forms of) 1. and 2. (mathematics). The cases
1'. and 2'. are quite different, analogous to (experimental and
may be theoretical) physics.
> 1. Arithmetic as motivated by casual considerations of physical reality.
> 1'. Arithmetic taken as statements about physical reality.
> 2. Geometry as motivated by casual considerations of physical reality.
> 2'. Geometry taken as statements about physical reality.
>
> So I see two Geometries. But I go on to question that 2' is appropriate
> given the current status and modus operandi of physics. E.g., maybe now
> time is linearly ordered, with a first element. Maybe next week, it has no
> first element, and going backwards approahes a point 0. Maybe the week
> after next, time isn't even linearly ordered. Then a week later, it is
> linearly ordered again. Finally, another week later and time has finitely
> many points, which may or may not be linearly ordered. And how good are the
> experiments to determine the nature of time?
>
> >Does not this mean that mathematics suggests to physics
> >fundamental concepts of finite/infinite, discrete/continuous
> >which are not sufficiently appropriate to its contemporary
> >needs? It seems that if we will continue to stay on our
> >*traditional* fundamental concepts without any attempt of
> >variations to find more *realistic* versions then mathematics
> >may loose some of its positions in science.
>
> As I indicated above, I don't think that the situation in physics is stable
> enough these days for mathematicians to feel confident in a pursuit of
> radically new conceptions that are not definable using the ordinary
> concepts of mathematics, and which are to be justified by their
> applications to physics.
When the traditional mathematical approach to natural numbers
and to geometry was developed it was used, as is implicit in 1.
and 2., only the most basic features of physical reality.
Probably we need not to take from physics too much. May be some
very primitive new for mathematics knowledge will be enough *to
start*. Anyway, as mathematicians, we will deduce in a purely
formal manner from this some new knowledge which probably will
shed a light on some physical questions. However, at this step,
it is only a fantasy.
> >Just one example. The astro-physical fact that the number of
> >electrons in our Universe is < 2^1000 is usually interpreted
> >according to traditional mathematics, as that the Universe is
> >finite so that seemingly it is meaningful considering the
> >*exact* number of all electrons. This seems to be very
> >non-plausible conclusion. I understand the situation quite
> >differently. The Universe is infinite, but bounded (by 2^1000).
>
> So basically what you are saying is that 2^1000 is not feasible. Of course
> I like this sort of comment, since it is much more exciting than saying
> that 2^1000 is finite. However, I think you must recognize that the burden
> of proof is on you to back up what appears to be an awkward position.
I hope you do not ask about a formal mathematical proof! Axioms
and fundamental notions are not proved, but just introduced and
motivated.
> To
> illustrate this, let me start by asking you which of the following
> "numbers" are feasible?
>
> 2^500
> 2^250
> 2^125
> 2^63
> 2^32
> 2^16
> 2^8
> 2^4
> 2^2
> 2
>
> And also, what is the intellectual process by which you can, will, or have
> determined whether these "numbers" are feasible or not?
First, as I mentioned there are some astrophysical facts. I am
not a specialist and cannot comment which kind of physical
theorizing and experimenting are involved here. Anyway, we know
very well that exponential is "very" rapidly growing and,
respectively, logarithm is growing very slowly (practically, the
double logarithm is bounded by 10 - it is an experimentally true
universal statement "like" commutativity of addition). Also,
physicists use only numbers with about 30-40 decimal digits
(after the point, in the case of micro-physics). This roughly
corresponds to the number 2^100. Thus, even this number is
infeasible if to consider capabilities peoples (or even
computers). Depending on various generations of computers we
can take as infeasible probably even 2^50 or even less. Also
note that the precision of calculations needed in engineering is
not necessary to be very high. In a sense we "need not" very big
numbers at all and these numbers are actually only imaginary
theoretical fictions having no real (physical) meaning. I think
there is No precise answer to the question on what is feasible
and what isn't. It is an essentially *vague* notion which also
depends on some our concerns. But this does not mean that it is
absolutely "irrational".
Second, we know very well (but do not take it into account
seriously) that induction axiom is sometimes false (the heap
paradox and the like). Let me recall also one example which was
presented by Poincare, but rather, for a different conclusion.
Equality predicate is non-transitive: Take three pebbles A,B and
C and weight them by hands. It is quite possible that we will
find that A = B, B = C, but A \not= C. Analogously the logical
implication may be non-transitive. Which is the proper solution
of these "paradoxes"? Traditional approach to mathematics gives
one (advocated by Poincare). It seems we need also some other,
more "realistic" solution. All of this evidently should be
related in some way with the above concept of feasibility. Let
me interrupt discussion on this topic now until other more
appropriate occasion.
Third, we may consider formally, as Rohit Parikh did, that there
is a feasibility predicate F(x) such that F(0) and F(x) =>
F(x+1) is "true", but F(t) is "false" for some primitive
recursive term t having no free variables. For t = exponential
tower of the height which is also exponential (just about
2^1000) Parikh showed that this is "feasibly" consistent with PA
(where Induction Schema, of course, should not involve F!).
Thus, it was demonstrated that the concept of feasibility may be
considered in the framework of a rigorous mathematical
formalism. Another somewhat different attempt which I did
(essentially) gives rise to a different, more realistic upper
bound t for feasible numbers, just to t = 2^1000 (or to 2^100,
if we believe that this number is infeasible). (Feasible)
consistency of such formal systems is guaranteed by the physical
considerations I mentioned above.
One reaction on this formalizations of feasibility may be that
they are some anomalies which do not deserve much attention.
Alternatively, we may try to continue going this or some
analogous way. (This is the work for proof-theorists!) I prefer
the latter because I do not see a very big fundamental
difference here with the general mathematical approach:
formalize what can be formalized, investigate the formalism, try
to apply it to physics, etc, if possible. Just somewhat new, but
still rigorous mathematics. No mystery at all!
Finally, there may be the following objection: consistency of
mentioned above formalizations of feasibility concept is based
again on the intuitive concept of feasibility; is not this a
vicious circle? My answer is "NO", because the initial intuitive
concept is very primitive in comparison with what we get as
formalized. There are formally deduced some inexpected, (but
technically rather simple) theorems on feasible numbers which it
was difficult even to imagine to hold a priori. Now our vague,
amoeba-like intuition on feasibility becomes organized,
regulated and strengthened by the formalism. This is the general
valuable outcome of *any* reasonable formalization.
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/
More information about the FOM
mailing list