FOM: modeling PRA in the physical world
Vladimir Sazonov
sazonov at logic.botik.ru
Sat Mar 6 03:23:01 EST 1999
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?").
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