FOM: modeling PRA in the physical world

[Vladimir Sazonov]

Stephen G Simpson wrote:
>
> In my posting of 1 Mar 1999 23:35:34 I suggested the following
> approach to f.o.m.:
>
>  > to argue that PRA is consistent because the physical world provides
>  > a model of it, and then to justify at least a significant fragment
>  > of mathematics by reducing it to PRA, a la Hilbert's program of
>  > finite reductionism.
>
> Raatikainen 3 Mar 1999 19:49:54 asked
>
>  > isn't it the received view in it that the universe is finite ?
>  > PRA, on the other hand, requires an infinite universe of discourse.

It seems better to say that the universe is bounded, but 
possibly infinite. (Something like what we can see in 
non-standard models of arithmetic). Otherwise, we could ask 
whether the number of electrons in the universe is even or odd. 
This also shows that it is sometimes meaningless to use our 
mathematical terminology in non-mathematical contexts.

>
> The universe is finite, but Aristotle posits a distinction between
> actual and potential infinity.  The earth has rotated around the sun a
> finite number of times, but this number is potentially infinite.  A
> physical line segment may be finite yet potentially infinite by
> division, i.e. capable of being subdivided an unlimitedly large number
> of times.

What would say about this physicists?

> One can argue that PRA is implicit in the physical fact

is this physical *fact*??

> that many kinds of discrete acts can be repeated a potentially
> infinite (or at least an unlimitedly large finite) number of times.
> The act of adding 1 can be repeated indefinitely, and this gives
> addition.  The act of addition can be repeated indefinitely, and this
> gives multiplication.  Etc.  PRA has been identified with finitism.
> Hilbert described finitism as an unproblematic part of mathematics,
> which eschews the infinite and is indispensable for all scientific
> reasoning.

Thus, despite the physical world is bounded (finite?) it is 
concluded that it models PRA?? I would say, that potential 
infinity (our ability to distract from resource bounds) is 
peoples *invention*. It does no hold in our universe in any 
direct sense. May be only in the weakest sense:  it is unclear 
where is the exact bound of our abilities. Then we, mathematicians, 
*postulate* that successor operation always gives a new number.  
We do this because such a *decision* allows to develop 
mathematics in a reasonable convenient way. Convenience is 
something different from truth or falsity in the real world.  
Yes, we may postulate more: our ability to *iterate* any 
operation which has been defined.  Thus we really come to the 
notion of *total* arithmetical operations of addition, 
multiplication, exponential, and primitive recursive functions. 
But this happens not in the physical world, but in our thought. 
I believe that we should make clear distinctions.

In principle, this is not the only possible way to abstract from 
reality. We could reason as "realists" and take just "negation" 
of the abstraction of potential infinity (feasibility): we 
should not distract from resource bounds, but rather always 
*relativise* our reasoning to these bounds. This would lead us 
to another version of arithmetic with the maximal natural number 
postulated. It is natural that the exact value of this maximal 
number (a possible resource bound; say, the memory of some 
concrete computer) is not specified in such an arithmetic. It 
can be demonstrated (by a theorem) that this approach leads to a 
theory of polynomial time computability which is in a sense 
equivalent to bounded arithmetic. Let me recall also that 
Mycielski have demonstrated how the first basic concepts and 
results of Analysis can be developed in such kind of "finite" 
arithmetic. Recall also his related result that any consistent 
theory (say ZFC) is "isomorphic" (in a reasonable sense) to 
another theory whose each finite fragment has a (possibly very 
large) finite model.  (It seems he called such theories locally 
finite.)

Who knows which could be other possibilities to start 
constructing some (imaginary, but applicable) mathematical world. 
Essentially only one of such possibility was worked out 
extensively by mathematicians.

>
.....
> I'm aware that much more needs to be said, but this seems like a
> promising line for f.o.m.
>
> -- Steve


Vladimir Sazonov

P.S. By the way, as Mycielski was mentioned above, I think it 
will be interesting to FOMers his short abstract of the BEST 7 
conference talk which is related to some discussions on FOM 
and which I have red with the great pleasure:

"To FOM or not to FOM, that is the question"

http://math.idbsu.edu/~best/best7/talks/mycielski.html

It is very pity that he does not participate more explicitly
in FOM ("not to FOM?").