Dear all:
Originally, the present text was inspired by a private message (see
below), but it quickly turned into a more general discussion regarding
the Church-Turing thesis...
I hope the text below might be of more general interest to newsgroups
like comp.theory, comp.theory.cell-automata or sci.physics.discrete --
that's why I decided to post it here.
Thank you in advance for your time and attention.
Yours truly,
---
Plamen Petrov
https://digitalphysics.org
-----Original Message-----
From: Joel Isaacson
To: stephenk1@c...
Cc: ppetrov@m...; kitada@k... ; integrity@c...
Date: Thursday, March 11, 2004 5:34 AM
Subject: RE: The T-shaped Universe
> if the Universe is thought to be one huge, global T-shape it
> doesn't follow that every part of it is also T-shaped. The way
> out was to assume that there may be some elementary T-shaped
> fragments, that -- if assembled into larger chunks thru fractality
> and self-similarity (a BIG "if") -- then it may build up into
> successive T-shaped hierarchies.
Dear Joel Isaacson:
First of all, let me say that I agree with you that, "If the Universe
is thought to be one huge, global T-shape it doesn't follow that every
part of it is also T-shaped."
Of course, nothing of this sort follows.
And if you think the simple argument presented in my work:
https://digitalphysics.org/Publications/Petrov/Pet02a1/Pet02a1.htm
has anything in common with this style of thinking, then let me say
that we have a situation usually referred to as, "it is not this way,
but the OTHER way around."
In other words, the argument is not that:
(1) "If the Universe is thought to be one huge, global T-shape then it
follows that every part of it is also T-shaped." (which is wrong)
but rather:
(2) "If some *specific* part of the Universe is thought to be unable
to produce anything but T-shaped fragments, then the Universe itself
*must* be T-shaped". (Which is, surprisingly, correct -- if we choose
a *very* specific part of the Universe, indeed!)
In case someone does not understand what I am talking about:
The *specific* part of the Universe I am talking about is the human
*mind*...
* * *
Since you prefer to express yourself in parables, let me also use this
style of communication and reply to you in a similar way...
Once upon a time, somewhere in 1935, a C-shaped professor made a bold
proposal that all mathematical systems are C-shaped.
(The C-shaped professor actually had a vague idea why this must be so,
i.e. his proposition was not backed up by any philosophy or argument;
it was merely based upon his *intuition* and his long time
mathematical studies of his favorite C-shaped systems...)
A G-shaped professor, who made an important contribution to a similar
area of study in 1931 that shocked the mathematical world and made him
famous, objected.
The G-shaped professor stated that he is not convinced that our way of
thinking (i.e. all mathematical systems) must be exactly C-shaped. He
said this may be correct, but he does not see *why* this is
necessarily so.
A year later, in 1936, a T-shaped professor submitted a wonderful
paper to a respectful scientific journal. The highly original paper of
the T-shaped professor offered a similar argument, but this time for
T-shaped systems.
Sadly enough, the editors of the journal rejected that paper, and it
was returned back to its author (the T-shaped professor).
The editors pointed out that someone (the C-shaped professor) had
already succeeded to publish a similar result.
Fortunately enough, after reading the work of his colleague, the
T-shaped professor was able to show quickly how C-shaped systems can
be simulated by his T-shaped systems.
So, at the very end of 1936, the paper finally appeared.
After reading this work, the G-shaped professor finally admitted that
he "is now convinced" that the T-shaped systems do capture all kinds
of mathematical systems.
The C-shaped professor was also impressed by the ideas of his young
colleague; he was nice enough to admit that the T-shaped professor
arrived at his idea independently and that the argument used in that
paper was much better than his own...
* * *
Well, enough parables...
In case someone was unable to recognize the characters of this story:
The C-shaped professor was Alonzo Church. The G-shaped professor was
Kurt Godel.
And, finally, the T-shaped professor was Alan Turing.
The important part of this story is that our Universe is T-shaped
thanks to the T-shaped professor, i.e. thanks to Turing.
Actually, he never thought about these things quite this way... And
his contribution was only a small step toward the ultimate truth.
Anyway, Alan Turing gave birth to an important idea, or method, that
even now -- almost 70 years later -- simply HAS NO EQUAL in the
scientific literature.
The important part of his classical paper:
Turing, Alan, "On Computable Numbers, With an Application to the
Entscheidungsproblem", Proceedings of the London Mathematical Society,
Series 2, Volume 42 (1936-37) ; Corrections, Ibid., vol. 43
(1937) .
Online version: http://www.abelard.org/turpap2/tp2-ie.asp
are not the mathematical proofs that accompany the text, but rather
the VERY SPECIAL NATURE OF TURING'S PHILOSOPHY that makes his
arguments so *compelling*...
Turing was lucky enough to come up with an argument that is
*INDEPENDENT* (!) of any physical experiment; in other words, as
someone put it [the original words possibly belong to Kolmogorov]:
"Turing was able to come up with an argument that simply *is* true!"
Almost 70 years after the publication of Turing's work, it is sad to
see his ideas misinterpreted "almost everywhere". Actually, from what
I personally was able read so far, (if we exclude a recent publication
by Yuri Gurevich), I can conclude that almost everyone has problems
with the so called, "Turing's thesis".
For example:
(1) I see that one of Alan Turing's biographers (Jack Copeland) has
decided to devote much of his lifetime to publishing papers about
so-called "hypercomputation", or "super-Turing computation", i.e.
seeking possible ways of doing computation that goes beyond "what can
be computed on a Universal Turing Machine" (UTM).
My only comment is that this is all very sad and it is nothing but a
clear misunderstanding of Turing's ideas that actually do *not* permit
such computations, and there is a VERY GOOD REASON for that!
(2) In a "seminal" publication from 1985, David Deutsch misinterpreted
the original idea behind Turing's thesis *completely*. Right from the
very beginning, this widely cited paper about so-called "quantum
computation" is nothing but a clear misunderstanding of the thesis
(actually, he even calls it "a hypothesis", which is rather
ridiculous...)
Deutsch thinks Turing's argument has something to do with *physical*
computers or calculators, while the original paper from 1936 deals
merely with *human* computers/calculators.
This may seem unimportant, but is actually a *crucial* point of the
affirmation of the Turing's thesis. For Turing studied not digital
computers (as we know them in modern times), but rather the way
human's *read* scientific papers and *write* computations on *paper*!
In other words:
Alan Turing's approach to the whole problem -- believe it or not --
was actually a *FORMALIZATION* (!) OF THE SCIENTIFIC PROCESS as a WAY
OF EXCHANGING IDEAS AMONG *HUMANS* THROUGH SCIENTIFIC PUBLICATIONS
WRITTEN ON *PAPER* that sometimes includes also the checking of some
specific calculations (presented within that paper) on a SUFFICIENTLY
BIG BLANK SHEET OF *PAPER*!
(I suppose the above may come as a "bolt from the blue" for many, but
clearly, this is the case with the original publication of Turing and
his highly *non-trivial* argument that actually *convinced* his
colleagues-mathematicians, including Kurt Godel... )
Turing machines (TMs) really *do* capture the intuitive concept of
"mathematical function" simply because they provide a way to *TRACE*
OUT (!) the human's view (when reading some scientific paper); a TM
"converts" that to bi-directional moves of an abstract device called
"head" (that moves forwards/backwards) and changes its internal
"mental state" in order to "understand" (or "check out") the
"correctness" of some scientific publication [which is, presumably,
mathematical in nature, since mathematics is the final stage of all
possible scientific formalizations].
Sometimes this process also includes not only "reading" (scanning),
but also "writing" of symbols on a separate blank sheet of paper in
order to check out some formulas/results for their supposedly
"unobvious" validity.
For instance, think of a theorem that includes something like "3764 *
2095 = 7885580". Is this correct??
(Of course, it is, but in order to check out this multiplication one
needs to apply a well-known algorithm on a sheet of paper...)
(3) Unfortunately, yet another brilliant intellectual to whom I am
gratefully indebted -- Ed Fredkin -- apparently also has problems with
the Turing thesis...
(I consider Fredkin to be one of the most original thinkers of all
time, and I think his idea about the "Universe as a cellular
automaton" is simply *outstanding*...)
Fredkin advanced the so called "Finite Nature" (FN) assumption, which
is a good thing (to say the least), but unfortunately, he goes
unnecessarily far by proposing the idea that we live in a finite (?)
Universe.
Ed simply *wants* to remove any infinities from the model proposed,
and he mixes his own *desires* with the clever design Mother Nature
has decided to come up for us all.
Indeed, (applying Occam's razor considerations), it is easy to show
that the Universe we live in must be something *much* simpler than any
"standard" physicist currently suspects it is... But still, it is not
*that* simple!
Ed's idea was to conserve some certain parameters of the informational
model (an idea suggested by the so-called "conservation quantities" in
physics), but the final result is wrong.
The problem is that we cannot conserve the total number of all
particles in the system simply because we shall then leave no
possibility for the set of all natural numbers to exists...
In Fredkin's view, there must be some (albeit very large) number M
which is the total amount of "memory" of the digital computer the
Universe is supposed to be isomorphic to.
I know Ed likes to cite the following wise words of the German
mathematician Kronecker:
"God made the integers, all else is the work of man"
(And I like these words of wisdom, too...)
But I think Kronecker (if, by some miracle, he suddenly arose from the
grave) would be vastly disappointed to learn that Fredkin actually
thinks there is no such thing (?) as the set of all integer numbers
(usually represented with the symbol Z).
Obviously, Ed has problems accepting the so-called "potential
infinity" which (unlike "actual infinity") is a quite "normal" thing
that is mathematically and philosophically acceptable and does not
lead to any paradoxes...
Finally, Turing's thesis actually pre-supposes that "potential
infinity" does exist: after all, that is why we refer to that
"possibly infinite on the both ends paper tape" of the UTM!
I suspect that is why Ed feels "uncomfortable" about the
(Church-)Turing thesis and he never uses it as a "sufficiently strong
argument" in his publications...
Yes, the good old Turing's thesis really *is* a *sufficient* argument
[in rigid mathematical terms] in favor of the "Universe as a computer"
idea!
(The last is a *weaker* form of the ingenious Zuse-Fredkin thesis that
says our Universe is not merely "some kind of computer", but rather a
*specific* abstract model for parallel computation known as a
"cellular automaton"...)
* * *
Let me conclude this message in the following way:
We live in a Universe that is T-shaped thanks to the T-shaped man
(Alan Turing).
Some 70 years ago, Turing was able to come up with an argument that is
actually so *stable* that it really does not depend from any
*physical* (i.e. "external") experiments, as we know them.
Personally, I am more than aware that Turing's thesis is a valid
proposition:
For the sake of the joke only, I can tell that in order to "check out"
its correctness, I do not actually need to read any more publications
concerning that matter...
IN ORDER TO ENSURE MYSELF THAT SOME PAPER DOES NOT DISPROVE THE
THESIS, ALL I HAVE TO DO IS SIMPLY "BROWSE" THAT PAPER WITHOUT
*ACTUALLY* READING IT OR UNDERSTANDING ITS CONTENT (!)
For if that paper consists of finite amount of "symbols" (for God's
sake -- it *should*!) :) and if it does use any (finite) alphabet, I
can readily conclude that "Turing's thesis still holds..."! :)
Turing's thesis is here to *stay* -- like it or not.
And despite all possible "rumors" about its "sudden death", it is in
pretty good shape -- especially, concerning its honorable age...
Sincerely,
---
Plamen Petrov
https://digitalphysics.org