X-Message-Number: 4135
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Date: Sun, 2 Apr 1995 19:11:20 -0400
Subject: SCI. CRYONICS Goedel
Following is part of a draft of a chapter in one of my books in progress
(which I hope to
complete some year this side of the freezer). There are some changes in print
style because of the restrictions on the Net.
GOEDEL, SHMOEDEL & SHMETE
This chapter on the "logical paradoxes" and Goedel's "Incompleteness Theorem"
is included for several reasons. For starters, it's fun. Also, it's good for
ordinary people to realize that they too can walk on clouds. Finally--clouds
or no--there may be genuine, practical importance in these questions; they
may be related to serious matters of the construction of reality.
The discussion below could have been much shorter--or much longer. But making
it too concise would have put too heavy a burden on the reader to absorb it;
making it much longer would wear out the welcome with most readers and the
publisher. As usual, we makes our choices and takes our chances.
EASY STUFF FIRST:
Some "paradoxes" are so tansparent, so frivolous, one wonders how they ever
made it into print, let alone into professional journals of mathematics and
philosophy. Goes to show you. One such is that of Jules Richard (who can't
sue me because he's dead). It goes something like this:
Consider the expression, "the least natural number not nameable in fewer than
twenty two syllables." This nevertherless DOES name in twenty one syllables
something that by "definition" cannot be named in less than twenty two
syllables.
Is this really hyper-frivolous and easy? In the terms given, yes: it is just
a trick involving the different ways of "naming" things, or the different
meanings of "name." I could devise a partly similar one: "There is a certain
quality which is unnameable; nevertheless, I shall now name it--Melvin."
(Maybe you will later decide--if you aren't afraid of being hanged for
heresy--that Goedel's Incompleteness Theorems are in essence just as stupid
as this.)
On another level, however, the picture may not be quite so clear. If we work
with the symbols and rules of formal logic and mathematics, instead of
ordinary words, it often becomes easier to make things clear, but sometimes
also MUCH harder to prove or disprove them. I suspect also that sometimes,
instead of clarifying things, it misleads the worker into thinking he is on
solid ground instead of quicksand: his symbols look so simple and comforting
he may fail to notice that they don't stand for anything well defined--or
even well undefined! ("Undefined" quantities may be admitted in formal
science, in the sense for example that a "point" in Euclidean geometry cannot
be defined explicitly in terms of anything more fundamental in the
system--even though, of course, it has a more or less definite meaning in our
minds.)
This point is worth belaboring. When we work with ordinary language, and use
words like "true" and "included" and "less than" and "or" and so on, we may
(if we are very careful) pause now and then to remind ourselves of what these
really mean, in the context of the current discussion, or whether indeed they
mean anything. But if we are working with symbols it is EXTREMELY unlikely
that we will ever look back or remember to reassess our assumptions. Hence we
will repeatedly be hit by the old "garbage in, garbage out" sandbagger.
Let's look first at one of the easiest.
THE BARBER
A variety of "self-reference" paradoxes are similar to that of the barber,
which says: "In a
certain town, the barber shaves everyone who does not shave himself." Now we
are supposed to ask, "Who shaves the barber?"--and run around in mental
circles. "If he shaves himself, then he cannot shave himself; if he does not
shave himself, then he must shave himself," etc.
The stupendous number of hours devoted to this "paradox" by towering
intellects of history is not TOTALLY unjustified; the form of the statement
is similar to those of certain important symbolic sentences in logic or
mathematics. Bypassing for the moment the question of these latter
statements, the Barber puzzle in the form given is easily resolved.
Clearly, the "everyone" mentioned in the statement is just a distraction; the
only question
concerns the barber himself. In other words, the equivalent statement is
just: "The barber shaves himself if and only if he does not shave himself."
Since this is a clear logical contradiction, the statement is simply false,
and nothing more need be said. (A really "paradoxical" statement, by
contrast, cannot be shown--or its implications cannot be shown--to be either
true or false.)
Now we look at the "Liar" paradox in its simplest and purest form, after
first laying out the
assumptions or premises.
THE "LIAR"
Classical logic deals with "propositions" which MUST be either WHOLLY "true"
or WHOLLY "false;" no shadings or other possibilities are allowed. (If a
proposition says more than one thing, or contains sub-propositions, then it
is true only if all the sub-propositions are true; otherwise it is considered
false.) Logicians have never been able to agree on a clear-cut definition of
a "proposition," or of any reliable way to distinguish a proposition from a
non-proposition, but common sense tells us that, at a minimum, the
"proposition" must be clear-cut and unambiguous.
Consider this sentence (The Liar):
"This statement is false."
According to the usual analysis, the "proposition" cannot be true, because
then what it SAYS must be true--and it SAYS it is false. But it can't be
false either, because it SAYS it is false, and if that were correct, that
would make the statement true. Thus--it is said--the rules of logic appear to
be in trouble; we may have to throw out the requirement that every
proposition is either true or false.
This conclusion--that the rules of logic come into question--should
immediately be seen as ridiculous. Obviously a much simpler and more
satisfactory conclusion is that the sentence above is not a proposition.
Looking for independent verification of this alternative, we find it
instantly. The sentence isn't even MEANINGFUL. It is a string of words with
apparent syntax, but no content. There is no root referent, nothing on which
to hang your hat. At bottom, it just doesn't SAY anything, although it may at
first give that illusion.
To see this even more clearly, we can put the statement in slightly different
words, which can easily be seen as equivalent:
"I am lying."
Any ordinary, sensible person will respond by saying, "Oh? You are lying, eh?
About WHAT?"
The smug puzzle-poser may answer, "Er, well, I'm just lying. Right now. I'm
lying about lying. That is, the statement I'm making right now is false--I'm
saying--so it therefore can't be, and..." Etc.
He can't get off the hook. He can't actually be lying unless we know the
content of the lie, and there isn't any.
Another, related avenue is to ask about verifiability of the "proposition."
It is still an unsettled question in philosophy whether a statement, to be
meaningful, must be verifiable or falsifiable in principle; but at least we
must look with great suspicion on any statement that obviously, by its very
nature, can never be verified or refuted.
SUMMING UP: The "paradox" of The Liar, in the form above, is easily resolved
by refusing to acknowledge that the statement is a proposition, and this
refusal has strong underpinnings. (Aristotle said the same thing, and so have
others from time to time, although not a majority.)
Next we look at a different form of the puzzle.
ARE SHMETANS LIARS?
The best known form of the "Liar" paradox is that of Epimenides:
"All Cretans are liars."
We understand it is a Cretan talking, rather unpatriotically; also that a
"liar" ALWAYS lies and a non-liar NEVER lies. Thus the supposed paradox: Is
the speaker lying? Working through the implications, we are supposed to
conclude that the statement cannot be either true or false, hence the rules
of logic are in trouble. (If all Cretans always lie, then the statement
above--spoken by a Cretan--must be false; but since it SAYS that all Cretans
always lie.....and so on.)
Again the simple resolution is that the statement is not really a proposition
at all. To see this, it helps to strip the statement down to its essentials.
Obviously Epimenides was not talking about the real island of Crete or its
real inhabitants, none of whom was or is a pure liar or pure truth-teller. So
we are dealing with an imaginary or hypothetical island; let's call it
Shmete.
Clearly any hypothetical Shmetans other than the speaker are irrelevant; any
and all others could indeed, for all we know, be either pure liars or pure
truth-tellers. So we are really interested only in the present speaker. In
fact, we are only interested in that single statement--the one above--since
any other statements, past or future, are unknown and irrelevant.
Thus we are left with just one Shmetan uttering just one statement:
"I am lying."
As noted, this is equivalent to our first formulation, "This statement is
false," so we reach the same conclusion: it just isn't a proposition.
Now we proceed to a more difficult stage of the investigation and try to
discover whether our simple resolution (the non-proposition) can apply also
to a mathematical or symbolic form of the Liar paradox, or something similar.
GOEDEL'S THEOREMS
In 1931 and thereafter Kurt Goedel and others produced the famous
Incompleteness Theorems which say, in effect, that there are true statements
in mathematics which cannot be proven. (Our ill-used Everyman will
immediately interrupt and growl, with his usual bad temper, "If it can't be
proven, how do we know it's true?" Down, boy; the big brains will tell us.)
Some scientists believe this implies, or at least suggests, that there are
PHYSICAL facts in the real world that cannot--even in principle--ever be
proven. In parallel with the math, such facts can be KNOWABLE, but not
provable! (I hope you have a cast iron stomach.)
Some of these theorems are long and difficult. I am not bright enough nor
sufficiently well
trained to follow them quickly, and I have never had allocable time to work
through them slowly; so perhaps I could be missing something. However, a
number of mathematicians who ought to know tell us that certain simplified
versions, including English-language versions, represent the main ideas
faithfully enough. See e.g. INTRODUCTION TO METAMATHEMATICS, S.C. Kleene, Van
Nostrand, N.Y., 1952; or GOEDEL'S INCOMPLETENESS THEOREMS, R.M. Smullyan,
Oxford U. Press, 1992. I am relying mainly on these.
Generally an "incompleteness" theorem says that certain statements in a
mathematical language or system cannot be decided within the system--i.e.
cannot be proven true or false.
In the simplest cases, the theorems are closely related to the first Liar
paradox above, "This statement is false." But there is an important
difference: instead of the statement asserting that it is FALSE it asserts
that it is UNPROVABLE. Supposedly the effect of this is to produce, not a
paradox or threat to the foundations of logic, but new information about the
nature of the world and its accessibility to investigation.
As far as I can see, all of these self-reference statements or theorems
suffer at least from the same basic defect as the Liar paradoxes, viz., that
they are basically meaningless--not propositions at all in any acceptable or
useful sense. But there are some new considerations also; let's take a look.
UNPROVABILITY THEOREM, ENGLISH VERSION
It is claimed that in every mathematical system or language one can make a
true statement corresponding (more or less) to the following:
(1) This statement is unprovable.
The reasoning--"proof" of the statement--seems to go like this:
There are only two possible initial assumptions--that the statement is true
or that it is false.
If we assume it is true, then it is unprovable, since that is what it says;
thus it is true but
unprovable.
If we initially assume it is false, on the other hand, then it is still
unprovable, since no false
statement is provable (in any fully satisfactory system of mathematics or
logic). But if it is
unprovable, and it says it is unprovable, then what it says is true; i.e.,
the statement is true. Thus, once more, we are led to conclude that it is
true but unprovable.
Now our Everyman is really howling. "You are saying that the statement is
unprovable, and in the same breath you say you have proved it--proved that it
is true! How can you claim it is unprovable, and also claim you have proved
it?"
The mathematician's answer seems to be that such a statement, represented
symbolically in a particular mathematical or logical language, is not
provable "within the system"--but by going outside the system, working in a
larger universe of discourse, we can prove it. But then--they claim--one can
still make similar statements in that larger language, which again cannot be
proven except by recourse to a still larger language, etc. But even if that
were true, it does not address the problem of a valid proposition, nor would
it appear to ungibberish the English version. My best guess is that Everyman
is right, and the geniuses have simply forgotten the foundations and become
lost in the superstructure.
However, let's look at a simplified symbolic version. [To accomodate to the
Net, I use the symbol == instead of the triple line to mean "identically
equal to" and the symbol n. to mean "not."]
(2) A == n. PA
The symbol "==" signifies "is identically equal to" or "is equal to by
definition" or "means."
The symbol "n." signifies "not."
"P" signifies "provable." "PA" means "provable A" or "A is
provable."
Thus we read (2) as, "A means not-provable A," or in other words, "A means
A-is-not-provable." (Do you see the parallel to "I am lying"? And do you see
that the fatal flaw is right here in the very beginning, in this inadmissible
"definition?")
Again, a mathematical innocent may protest: "How can A mean something else
than just A--and doesn't it matter what A IS?" Down, boy, again; quit
whining. Shut up and listen to the mathematicians.
We set out to prove that the statement A [not the statement (2), which is a
definition, but the statement A] is true but not provable, and the usual
exposition again goes like this (with some redundancy added):
1. If A is false, it is not provable, since no false statement is provable.
2. If A is true it is not provable either, since it SAYS it is not provable.
3. Thus it is not provable, hence true (since that is what it says).
4. We thus have a statement or proposition which is true but not provable.
Naturally, at this point our beaten-down Everyman will again have the feeling
he is being flim-flammed, and indeed he is being both flimmed and flammed.
Dressed up (or down) in symbols instead of words, this bit of business has
the identical trouble as the English-language version above, and for that
matter as the Liar--as well it might, since it says the same thing, or the
same nothing.
It is flummery to say:
(3) "A == A-is-not-provable."
At best, it is a dog chasing its tail:
By substituting for A the meaning of A on the right of (2) above, we get:
(4) "A == [(A-is-not-provable)-is-not-provable]"
At the next stage, we get:
(4) "A == {[(A-is-not-provable)-is-not-provable]-is-not-provable}"
and so on. Forget it.
GOBBLEDEGOOKECTOMY
Perhaps one could further clarify things a bit by just saying, in the English
version,
(5) "A" is a (system) statement that has the property of undecidability.
I believe this says just as much--and just as little--as the standard
(6) "This statement is unprovable."
Of course (5) is not a statement in a mathematical system language; it is
merely a statement in English expressing our belief that there exists a
statement in the system language with the property of undecidability.
Therefore (5) could not in any case be proven (or falsified) in the system.
Nevertheless, since (6) says absolutely nothing about the instant "statement"
except that it is unprovable, I fail to see any substantive difference (at
our level of discourse) between (5) and (6)--except for the fact that (5) may
leave open the possibility of real content in A, which(6) does not.
If this be agreed, it seems to follow that both (5) and (6), as well as (2),
merely lay down a
premise which, if accepted, naturally leads to the conclusion that A
isundecidable--since the conclusion is the same as the premise. In other
words, GIGO--garbage in, garbage out.
CONSEQUENCES
Our common-sense reader asked himself long ago why, if something is
undecidable, it has any practical importance. If it is not verifiable or
falsifiable, how can it have any embodiment in hardware or in any physical
processes--hence how can it have any practical consequences at all, with the
possible exception of psychological consequences?
One answer offered seems to be that quantum theory, to some extent and in
some sense, asserts certain things which cannot be directly and fully
verified; and that quantum theory contains parallels to (1) or (2). (For some
mysterious reason, this seems to give comfort to dualists and
irreductionists.)
I don't want to pursue quantum questions further at the moment, but one of my
guesses is that, if some version of quantum theory truly does contain
parallels to (1) or (2), then that version of quantum theory is in trouble.
SARTOR RESARTUS
If you and I and all those other poor slobs are right, then how did all those
geniuses go wrong? Again, near as I can tell, they just got so caught up in
the details and superstructure that they forgot to look at the foundation, at
the premises.
Georg Cantor established set theory, which has "paradoxes" parallel to that
of the Liar,
especially relating to the "set of all sets" and other self-referents. Set
theory has been extremely useful in many areas of math and science; but it
won't work unless a "set" satisfies certain criteria--WHICH HAVE NEVER BEEN
SATISFACTORILY SPELLED OUT. If you assume something is a set in the
appropriate sense, and it isn't, you are in trouble.
This situation might be vaguely likened to that of a tailor who has developed
great skill and confidence in working with many kinds of cloth in complex
patterns--but one day finds himself with a sheet of isinglass. As long as he
insists on calling the isinglass "cloth" he is going to be frustrated.
Rushing in again where angels fear to tread, I say the incompleteness
theorems are phony.
CAN TOO MUCH KNOWLEDGE BE A DANGEROUS THING?
The well known aphorism says that a little knowledge is a dangerous thing,
and of course it can be, producing the illusion of understanding, begetting
overconfidence. I have been accused of having this problem.
On the other hand, there is ample evidence that "too much" knowledge can have
exactly the same effect, and more so; and this is a more dangerous situation,
since the deluded one may have much more influence and be less apt to
question his own judgments. Among countless other examples, the scientific
and engineering communities heaped scorn on Edison for trying to invent an
electric light, right up until he succeeded....As usual, one cannot overwork
the model; experts and communities of experts are more often right than
wrong. All we can do is try to keep our act clean, and give it our best shot.
R.C.W. Ettinger
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