FOM: Re: Arithmetic/Geometry

Sat Oct 17 03:52:10 EDT 1998

I mistakely sent (on Fri, 16 Oct 1998 19:15) the following 
posting only to Harvey Friedman personally. Now I am sanding 
it also to fom. 

Harvey Friedman wrote:

> I think
> making some distinctions would help get to the bottom of the disagreement,
> if there is any.
>
> Consider the following 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.
>
> Mathematicians are generally concerned only with items 1 and 2. They focus
> successfully on 1 and 2 because of the startling fact that everything they
> want to know is based on only such casual considerations, and the great
> power of deductive reasoning. The main point is that for 1 and 2, physical
> experimentation seems completely useless and irrelevant.

Completely useless and irrelevant? 

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! 

Is this correct interpretation of your point of view? At least, 
after making these notes I see (almost) no contradiction with you 
and will follow below to this interpretation of 1. and 2. 

Both 1. and 2. are based on "casual" considerations of 
physical reality and, I would add, different such considerations 
may lead, in principle, to different versions of 1. and 2.  

For the clarity I should say that I personally am interested 
mostly in *various* forms of 1. (and 2., which is actually 
sufficiently elaborated if not take into account influence of 
other possible forms of 1.) because these are *mathematical* 
subjects. 2'. (and 1'.?) are rather physical subjects (or 
subjects of experimental "computational" mathematics?) and 
should be investigated by corresponding experimental methods 
in close cooperation with appropriate mathematical formalisms.

Although, even
> this point is subtly misleading or incorrect. The running of computer
> programs can be viewed as a kind of physical experimentation, which is now
> a common way of establishing new results in 1 and 2. However, it is also
> "clear" that such results are always deductive consequences of the basic 1
> and 2, but the deductions are themselves too large to be obtained by humans
> without resorting to the "physical experimentation of actual computing." I
> could follow this line of thought further, to good effect, but I will stop
> here. E.g., how do I know that this physical experimentation can be
> "replaced" by deductions? That involves a whole host of issues such as "why
> should I believe a computer?" In any case, it seems inconceivable that
> physical experimentation - in the broadest possible sense - is ever going
> to falsify results in 1 and 2.

Yes, of course!! If a *formal* proof is written on a paper 
nothing can make it incorrect with respect to fixed axioms and 
proof rules. But experiments or general physical considerations 
may suggest us to change "the rules of game", i.e. axioms and 
proof rules to get some more appropriate formalism (as an 
*instrument* of knowledge).

> One can follow this train of thought, back
> and forth, with a determined sceptic - like Wittgenstein(?)(!) - and
> generate state of the art f.o.m.
>
> As far as 2' is concerned, it has become part of the folklore of physics
> that this kind of geometry is affected by physical experimentation and
> observation, and their is an underlying objective reality to this. I think
> that the situation in contemporary physics is so conceptually murky - with
> such things as quantum mechanics, string theory, etc. - that the content
> and status of 2' is rather unclear. For example, in light of modern
> thinking in physics, what exact meaning, operationally and otherwise, can
> really be assigned to such questions as "is the parallel postulate true?"
> "is space indivisible?" I have my doubts about the meaningfullness and/or
> objectivity of this sort of thing.

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.

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). 
Compare with the case of standard natural numbers imbedded into 
a nonstandard model M and bounded by a nonstandart m\in M.  What 
does it mean the nonstandard cardinality (in the sense of M) of 
the set of all standard numbers? Is it meaningful and < m?  
Analogously, 2^1000 serves rather as "non-standard number" for 
feasible numbers (playing the role of standard ones) by which 
(i.e. by feasible numbers) we probably could count electrons or 
some other "objects" of our Universe. There is no biggest 
feasible number, and the Universe is, "therefore" infinite.

> And for 1', I don't see people even considering it. However, one could
> interpret 1' to involve something like this: Is the axiom of successor
> true? Another way one may want to put this is: are there infinitely many
> physical objects? Of course, I don't know what a physical object is, and
> that seems also to be very murky right now. Spinoff for f.o.m.: study
> arithmetic without the axiom of successor.

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?  
By the way, such a studying arithmetic with changed axiom(s) of 
successor as well as any reasonable arithmetics of feasible 
numbers is of course a subject of 1., *rather than* of 1'. We take 
only *some* aspects of reality, formalize them by some appropriate 
axioms and proof rules and then work as in the ordinary mathematics: 
infere theorems by using considerations from the reality *only* 
as motivations or a help for the intuition to find a proof. 

Do you agree? 

> I don't want to say more about
> this right now, except that this also leads to some specific state of the
> art projects in f.o.m.
>
> Now there are profoundly interesting similarities and difference and
> relationships between 1 and 2. A lot of this quickly leads to unexplored
> state of the art f.o.m. if systematically pursued. But before going further
> into this on the fom, I'll stop here and see whether people in this thread
> find it useful to cast the issues in this general framework. In particular,
> in the light of this framework, what precisely is the disagreement about?

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.

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
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