FOM: Re: Arithmetic/Geometry

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
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