V.Sazonov at csc.liv.ac.uk
Fri Dec 24 14:06:52 EST 2004
Quoting Matt Insall <montez at fidnet.com>:
> Dear Vladimir,
> Your original posts, if I recall correctly, indicated that you consider
> ``infinity'' to be less than or equal to 2^1000. Now, it is known that no
> natural number is infinite, so this indicates to me that you do not believe
> that numbers larger than 2^1000 exist. You seemed to give ``feasibility''
> as a justification for this conclusion. I do not agree that feasibility of
> notation should be used as the basis for determining what exists and what
> does not exist, but if we momentarily adopt your position, that feasibilty
> of _notation_ should be used in this metaphysical way, to determine what we
> consider to exist, then the main question becomes not just feasibility of
> the notation you prefer, but also why should one prefer such notation.
I actually do not know what do you mean by existence.
When I use the term "feasible", I actually mean the relation
of natural numbers to the real world via counting some real
physical objects like the written symbol "|". This leads to
considering unary notation system. Of course, physical "size"
of these bits (which you suggest to discuss) does not actually
matter because of the fact that even the number of electrons
is less than 2^1000, nothing to say about the number of such
physical macro objects as "|" whichever way we will represent
them (on a sheet of paper or in computer memory).
Is it now clear what do I mean? (Non-)efficiency of unary
notation system playes no role here. (It would be an issue
when considering calculations. Then decimal or binary notation
system will come to the scene, of course.) A natural and
simple relation to the real world - this is the point at
this moment. If numbers are intended to count some objects,
let them count real (macro) objects of our world. Those numbers
which "count" are considered as feasible. It looks plausible
that 2^1000 is not feasible in this sense. At least we, human
beings, will hardly be able to count till this number, to make
so many derivation steps in mathematical proofs, etc.
No question, the idea of feasible numbers is extremely
vague - it is unclear where is the "border line" between
feasible and non-feasible, although it is sufficiently clear
and certain that 2^1000 is non-feasible. But let us try to
put this idea in a formal framework: write down axioms
governing feasible numbers and try to elaborate what the
appropriate underlying logic could be so that these axioms
would be meaningful and consistent (in a suitable sense)?
Say, postulate that (i) the "set" F of feasible numbers is
closed under +1 (and even under +), once there is no clear
boarderline between feasible and non-feasible, but that
(ii) 2^1000 is not in F.
What I assert, there is really some formal framework (see
http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps) where these
axioms are consistent (actually, feasibly consistent.)
Once it is a formal framefork, everything deduced about
feasible numbers here is quite rigorous. Thus, we started
with a highly vague idea (but the idea which is clear enogh
to initiate first steps of its formalization) and came to
mathematically rigorous style of reasoning on feasibility.
I believe that the resulting formal approach (actually initiated
by Rohit Parikh for some other version) gives a new insight.
To give more details how can it be done means actually to
present the wole paper. Thus, let me stop here. I do not
think it makes sense to continue the discussion without
invoking more formal aspects. (One such aspect appeared
in posting of Arnon Avron.) I will also be busy with some
other activity and, unfortunately, will hardly be able
to reply actively.
> PS: Have a happy holiday, if this is to you a holiday season, as it is for
> us in the USA. (Have a happy season, in any case.)
Thanks! Merry Christmas and happy New Year to you and
everybody here in FOM.
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