FOM: Re: Arithmetic/Geometry

[Harvey Friedman]

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:

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.

>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. 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?

>Could you be a bit more explicit? Do you mean to postulate
>existence of a biggest number (note that in some my posting I
>discussed on such a \Box-arithmetic or []-arithmetic), or what?

Not yet. Work in progress.

>Say, my disagreement with Neil Tennant concerns
>
>"1. Arithmetic as motivated by casual considerations of physical
>reality."
>
>This is also my point of view on arithmetic (and on its possible
>reasonable variations), but Tennant, on the contrary, mysteriously
>believes in "absolutely true" arithmetic which arose completely
>independently of our reality.

Is this how Tennant would characterize the disagreement?