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From: forman@netcom.com (frank forman)
Subject: Re: Open Letter to Professor Penrose
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[cross-posted to groups other than
alt.philosophy.objectivism in hopes of getting
better calculations than those I have given
below]

Joel Katz has given us a thought-provoking open
letter to Roger Penrose, arguing against his
attempt to use Go"del's Incompleteness Theorem
to show that our minds cannot be reduced to
computers. He then tells us that Objectivism can
supply such an argument, but does not give us
the details.

Now Penrose's arguments were discussed in
postings that came in at the rate of over two
dozen a day in various UseNet groups for several
weeks after his _Shadows of the Mind_ came out.
As usual everyone interested has his say, his
counter-say, his counter-counter-say, and then
the discussion petered out. I wearied of it
fairly quickly, after reading the first fifty
postings.

Kurt Go"del's paper, "U"ber formal
unentscheidbare Sa"tze der Principia Mathematica
und verwandter Systeme I" (1931) is, of course,
of the greatest achievement of the human
intellect, but it actually says something quite
precise. That is that certain mathematical
systems, like the _Principia Mathematica_ of
Albert North Whitehead and Bertrand Russell
(basically those that have sentences of finite
length and are powerful enough contain algebra),
contain statements that cannot be proven one way
or the other in a finite number of steps from
the axioms, but statements that are true
nevertheless. Go"del achieved his result by
stepping outside the system and reasoning
*about* how things get proven.

I can illustrate how this might be. There is a
very simple theorem in algebra called Goldbach's
Assertion, which says every even number greater
than two is the sum of two prime numbers. Thus 4
= 2+2, 6 = 3+3, 8 = 3+5, and (say) 100 = 17+83.
No one has ever found a counter-example, but no
one have ever proven the assertion, either. It
*might* be that proof either way is possible,
leading to a formally undecidable proposition
("formal unentscheidbare Satz"). Recall that a
proof is a chain of sentences that begin with
the axioms and proceed link by link until the
QED is reached. There are lots and lots of these
chains but no automatic guarantee that one of
them will result in either in Goldbach's
Assertion or its negation. Now, if we can show
(by cogitating on the theorem-proving machinery)
that none of them will, we also come immediately
to know that the assertion is true, for the
reason that if it were false, there would be a
counterexample, which of course could be reached
from the axioms very simply. What Go"del did was
to prove that a highly artificial sentence he
constructed was undecidable, but it was a
sentence nevertheless. (Since then, some
sentences that have more ordinary meanings have
been shown to be undecidable. The first such is
Jeff Paris and Leo Harrington, "A mathematical
incompleteness in Peano Arithmetic," in Jon
Barwise, ed., _Handbook of Mathematical Logic_
(Amsterdam: North-Holland, 1977).)

Less powerful systems do not suffer such
limitations. In the propositional calculus (A
and B, A or B, not A, etc.), all statements are
decidable, and indeed can be decided by means of
truth tables. In the predicate calculus (the
propositional calculus, plus sentences about
properties, such as A is a positive number, X
lies between Y and Z), all statements are still
decidable, but there is no mechanical
(algorithmic) method of generating a proof or
disproof. This was established by Alonzo Church
(who died just a couple of months ago) in 1936
in a paper called "An Unsolvable Problem of
Elementary Number Theory." Alan Turing
established a rather similar result the
following year in "Computable Numbers, with an
Application to the Entscheidungsproblem," when
he tackled the so-called Halting Problem and
showed that any given computer program designed
to crank out proofs in the predicate calculus
would often times start going on a proposition
that is, in fact, true but would keep on going
and never halt.

Unhappily, Mr. Katz' letter mixes up Go"del,
Church, and Turing, but what very few observers
have noticed is that the finite size of the
visible universe is more than enough to
guarantee limitations on our minds. 

What limitations? I'll generously calculate some
maximum numbers.

Age of universe: 2x10^10 years
Radius of universe (max): 2x10^10 light years
Seconds per year: 3x10^7
Meters per second at light speed: 3x10^7
Radius of universe (max): 2x10^25 meters
(assumes universe has been expanding at light
speed from the beginning)

Volume of universe (max) = 4pi/3 x radius^3 =
3x10^76 cubic meters

Density of universe = 1 atom/cubic meter
(or so I recall)

Atoms in universe = 3x10^76

Now squash all this together into a giant
computer than can crunch numbers at the rate of
light speed across the radius of an electron
(3x10^-15 meter), which will take 3x10^-15
divided by 3x10^7, or 10^-22 seconds. And then
let all these 3x10^76 electrons make computer
calculations at the rate of 10^22 per second.
Let the process rattle on for the age of the
universe (2x10^10 x 3x10^7 seconds).

Multiplying these last four figures gives us a
*maximum* (by a long way!) number of computer
calculations in our universe since the Big Bang
of 2x10^115.

One more calculation: 10^3 = 2^10. So 2x10^115
is between 2^384 and 2^385.

My point is simply that solving a truth table
that has a mere 385 variables in it would take
longer than the entire universe, run as a
computer, could possibly have generated from the
Big Bang to date.

[By the way, the number of possible chess games
is also a huge number. Recall the fifty move
rules that says every fifty moves, either a pawn
must be moved or a piece captured. Well, there
are 6x16 pawns moves and 30 pieces besides the
Kings, which makes for 126, multiplied by 50, or
6300 as the maximum length of a chess game. I'm
not going to try to calculate the number of
possible moves each time, since it will diminish
as the game goes on, but will rarely be less
than eight except under certain forced
conditions to keep the King from being captured.
So there is something on the order of 8^6300 =
2^18900 possible chess games. We DO know that
there is a maximin strategy for chess, which
ought to be "obvious," but some method other
than brute force will have to be made to find
it!]

Now, given my calculations, why resort to
Go"del's theorem, which can involve proofs of
*any* finite length, to show limitations of
computability? Yet, somehow, these limitations
are used to claim that the human mind
*transcends* mere computability. Recall that our
humble brains have something on the order of
10^10 neurons only.

What Mr. Katz does, in his posting here, but not
in his letter to Penrose, is claim that
Objectivism can show the creativity of mind
*beyond* computability, since what minds do is
invest sentences with *meaning*, something no
mere machine can do. Now he could have said this
all along, and this may be what the Objectivist
doctrine about consciousness consists of. But it
looks like we may have to step outside the
physical universe to get a justification.

Is Objectivism really a variety of spiritualism?
Reading essays in Mr. Katz' home page made me
wonder further. He inveighs against the
existence of the actual infinite on the grounds
that it would conflict with "identity." I once
studied a great deal of mathematics and became
quite familiar with mathematical structures that
employ infinities of many sizes. I find the
concept quite specific and not so murky as to
lack "identity." Indeed, infinity is my
*friend*, for in many cases in real analysis and
algebra, it is easy to prove things in the
infinite case, while doing so for the finite
case takes extra work. So I had developed some
pretty good intuitions about the infinite (much
lost since, as my last math course was in 1966).

What I strenuously object to is the very idea
that we can sit here and introspectively
cogitate about whether the universe is infinite
in space or time, or even whether particles,
subparticles, and sub-sub particles can go
infinitely deep. (One of the axioms for set
theory, called regularity or foundations, says
that we cannot have an infinite series of
descending sets: A contains B as a member, B
contains C as a member, and so on. But there are
some versions of set theory where this is
allowed, apparently without contradiction.) Now
it will certainly be difficult to collect and
interpret evidence about whether space and time
is finite or infinite (and you'll note that I
assumed it finite in both respects in the
calculations I presented above), but it is the
worst sort of presumption to say we can solve
the issue by looking at how *we* formulate
concepts. If Objectivism insists on propounding
a solution, I say we should put Objectivism on
the junk heap. Or at least it should be
reformed, as in the Reformed Church of Latter
Day Saints, or whatever. Too much reformulation,
however, and we are left with only a bunch of
platitudes, albeit ones well worth heeding. We
should indeed strive to get at the truths of the
matters and at the (temporary!) bottoms of
things and not just go mushy.

Mr. Katz seems to know a good many things about
mathematical logic and set theory, but let me do
up a reading list for him and for others.

Patrick Suppes, _Axiomatic Set Theory_ (1960).
An exellent first book, though there may be
better ones.

Keith Devlin, _The Joy of Sets_. An allegedly
first book, but gets difficult quickly.

Mary Tiles, _Philosophy of Set Theory_. This
goes into the historic roots of Zeno's Paradox,
the actual infinite, and many other things. Most
of it is reasonably accessible, though much will
be fully meaningful only to those with a good
graduate school course or two in the foundations
of mathematics under their belts.

Michael Hallett, _Cantorian Set Theory and
Limitation of Size_. Wonderful book that goes
into the main way set theoreticians have used to
sidestep the paradoxes, namely by decreeing that
sets cannot be formed ad libitum.

Peter Aczel, _Non-Well-Founded-Sets_. An
alternative set theory that admits of infinitely
descending member chains, as mentioned above.

N.B. Cocchiarella, "Canotr's power-set theorem
versus Frege's Double-Ocrrelation Thesis,"
_History and Philosophy of Logic_, 13.2 (1992).
This is maddenly badly written, so I could not
make much sense out of it. Denies the axiom of
comprehension (two sets are equal iff they have
the same members) and Leibnitz' "indentity of
indiscernables" to partly get around the Russell
paradox.

Steven Pollard, _Philosophical Introduction to
Set Theory_

S.G. Shanker, ed., _Go"del's Theorem in Focus_

Susan Haack, _Philosophy of Logics_. Splendid
discussion of alternatives to classical two-
valued logic, classical meaning Whitehead and
Russell, not Aristotle. (There's a sizable
literature reformulating, and sometimes
correcting, Aristotle's logic into modern
terms.) But I have no idea what it would take
for me to give up two-valued logic or whether
beings on other planets all use it.

Susan Haack, _Deviant Logic_. An earlier book.
Has an absorbing discussion about whether
Aristotle believed in truth-value *gaps*, in the
case of future contingent events, i.e., it is
neither true nor false that the ship will sail
tomorrow.

Paraconsistent logic. Check out the "yellow
peril" (North-Holland and Springer-Verlag) books
in logic and foundations till you find some
stuff on the Latin American school of logic.
These systems allow for contradictions but
disallow "ex falsio quodlibet" (from a
falsehood, anything goes).

Abraham A. Fraenkel, Yehoshua Bar-Hillel, and
Azriel Levy, _Foundations of Set Theory_ (2nd
rev. ed., 1973). A classic, BADLY in need of
updating.

Jean van Heijenoort, ed. _From Frege to Go"del:
A Source Book in mathematical Logic, 1879-1931_.

Martin Davis, ed., _The Undecidable: Basic
Papers on Undecidable Propositions, Unsolvable
Problems and Computable Functions_. This source
book *being* with the 1931 Go"del paper.

Paul Benacerraf and Hilary Putman, eds.,
_Philosophy of Mathematics: Selected Readings_.
Look at both the 1964 and 1983 editions.

Steven J. Bartlett and Peter Suber, eds., _Self-
Reference: Reflections on Reflexivity_. Not
directly relevant, but I thought I'd throw it in
anyhow.

Abraham Robinson, _Non-Standard Analysis_. Shows
how infinitesimals *can* be used consistently.

AND

Mario Bunge, _Treatise on Basic Philosophy_
(eight volumes). He is the *other* great
philosophical system builder in this century and
knows far more science than Ayn Rand ever did. I
have adopted most of what he says (except the
ethics) as my own, not wanting to work up a
whole system all on my own, even if I had the
brains and the energy. Bunge has a true
scientific spirit and realizes the foundations
he has constructed will change as the overlying
sciences themselves change.

Above all, we should all keep reminding
ourselves that we are evolved creatures and that
evolution does not have the central purpose of
producing Ayn Rand or even just homo sapiens.
Our evolved cognitive machinery can go wrong and
can be corrected (not nec. always) only with
prodigious effort. We should also realize that
there is no "mind" apart from matter and that
our brains will be better and better explained.
One principle reason for accepting an
evolutionary and neurological approach toward
consciousness is to avoid writing thick,
densely-argued tomes denying its reality. Nor
should we write other tomes saying that
consciousness is inherently unexplainable if not
spiritual. Rather, we should be busy as
scientists and work hard to understand it. And
as regards "meaning" as being something outside
the physical realm and therefore not something
computers can do, we should instead inquire as
to how animals evolved the ability to come up
with meaning (or its precursor and thence how
homo sapiens made the final jump).

Frank
