[FOM] Feasible and Utterable Numbers

[Mirco Mannucci]

--- V.Sazonov at csc.liv.ac.uk wrote:
> Quoting Mirco Mannucci <mmannucc at cs.gmu.edu> Sun, 06 Aug 2006:

> I presume that we are here in the context of mathematics and natural
> sciences - not the sociology. So I would not rely on any subjective
> opinions. As I remember, in a known book of Richard Feinman he asserted
> that the number of electrons in the universe is less than a number
> which is in fact less than 2^100. I believe that this was not a
> subjective opinion, but a conclusion from physical experiments. Thus,
> this number is definitely non-feasible (in our current world) in the
> sense that no physical computer could calculate its value in the form
> SSSS...0 (a term a bigger size than our World!). But you seemingly have
> some different idea of (contextual?) feasibility which I do not
> understand.


First of all, I take back what I said: SOME people would  agree that 2^100 is
a feasible number, while some others would not (see the very last postings on
this thread).

So, thanks to Vladimir Sazonov for rectifying my hasty statement (I am not
good at calculations, that is why I am interested in feasibility. Numbers
for which I get a clear sense are VERY small indeed: apparently 2^100 is
not one of them).

                 *************************

Now, to the main point: Feynman was certainly one of the greatest minds of the
20th century, but he was NOT God. In other words, he did not MAKE Nature, nor did he
ever thought he did.

His estimate was based on:

1) available physical models of reality (quantum electrodynamics, general relativity,
etc.)

2) available data from experiments and other controlled and repeatable observations
of the physical world

Both 1) and 2) change over time (in fact, sometimes they change from one day to the next!),
and thus our corresponding estimate of what is actually computable may also get updated.
Feynman's own estimate was based on what we know NOW, not what we knew a 1000 years ago or
what we will know a 1000 years down the line.

A 1000 years ago an arithmetical term that can be calculated on a palm computer today
would have been considered unfeasible (10,000 years ago the number 5000 would be a de
facto infinite number!!!!).

As for 1000 years into the future, I do not know, and I suspect Feynman, Sazonov, our
friends at the Steklov Institute or anyone else would not know either (unless someone
in the FOM list is the next Nostradamus. Then, please call me, I have a few questions
to ask...).

If you like speculation, I suggest reading Greg Egan's "Schild's Ladder", a SF novel that
is (in my opinion) badly written, but packed with a plethora of interesting ideas.
Incidentally, Greg is a contributor on the physics research list, he knows quite a bit
of math and physics, and has learned some Loop Quantum Gravity from John C. Baez and other
theoretical physicists.

In "Schild's Ladder", he suggests that in the future computations will take place not
at the particle's levels, but below Planck's scale, in the very texture of space-time,
modeled by some generalization of spin networks (in this case one could
represent & store numbers that are immensely bigger than molecular or even
atomic and sub-atomic computing would afford).

True? False? I obviously do not know. Either way, it does not matter to me.

What matters, is that I seek a theory of feasible computations that is
not a SERVANT of current scientific theories, but that can at the same time accomodate
ANY constraints and resources (physical, biological, sociological, psychological,
or whatever) one can possibly imagine.

I want a logic that goes hand in hand WITH science, NOT in tow of science.


In this view, the sentence "A is feasible" means absolutely nothing. What is meaningful, is
"A is feasible, given constraints C1, ...., Cn, AND resources R1, ..., Rm). Indeed, even in
common english (or other languages), feasible is a virtual  synonime for "doable".

Now, doable by WHOM?
Using WHAT?
In WHICH context?

Indeed, Sazonov's excellent paper "On Feasible Numbers" itself implicitely points in that direction.
He establishes 3 types of numbers, and opens up the way to further classification.

Just like the theory of classical recursion degrees, one could envision hierarchies of feasibility.

The predicate F itself could be an n-ary one, F(n, x1, x,...., xk), where the x1, ...., x5
are feasibility parameters....

But, once again, I have written a letter of almost unfeasible size.

Apologies & Best wishes

MM






>
> Best wishes,
>
> Vladimir Sazonov
>
>
>
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