Dear Alex, dear friends,
On 04 May 2016, at 02:49, Alex Hankey wrote:
Dear Friends,
I was so struck by the group's focus on Gödel's theorems that I went
back to John R. Lucas who originated the idea that Gödel's insights
imply that the human miind is not a machine - and therefore capable
of genuine phenomenal experience. You may find the ideas in the
following informative and useful
https://en.wikipedia.org/wiki/John_Lucas_(philosopher)
https://en.wikipedia.org/wiki/Minds,_Machines_and_Gödel
I noted particularly that I have used complexity (which Lucas
mentions towards the end of MM&G) to establish that organisms are
not machines, and out of that I identify the form of information
which may explain various aspects of experience.
I work on this since a long time. It is my domain of investigation
actually. Indeed, it is Gödel's proof which made me decide along time
ago to become a mathematician instead of a biologist, when I saw that
Gödel's technic gave a conceptually clear explanation of how something
can self-duplicate, self-refers, self-transforms, etc.
The first to get the idea that incompleteness can be used to "prove"
that we are not machine was Emil Post in 1922, wen he anticipated
incompleteness. Then Emil Post was also the first to see the main
error in that argument, and he saw what can still be derived from it
(mainly that iF we are a machine THEN we cannot know which machine we
are: a key that I have exploited in the derivation of physics from
arithmetic and mechanism).
Note that Gödel's first incompleteness theorem can be rigorously
derived in very few line (indeed just one double diagonalization) from
the Church-Turing thesis. This has been seen by Kleene, and is
exploited by Judson Webb in his 1980 book to illustrate that not only
Gödel's theorem does not refute mechanism, but Gödel's theorem is an
incredible chance for mechanism. I have written myself a lot on this,
and my work extends this at the extreme, as it shows that mechanism
makes incompleteness the roots of both the appearance of qualia and
quanta, and this in a precise and unique way, making mechanism
empirically refutable.
I intended to give here the proof in a few line of Gödel's
incompleteness from Church-thesis, but that might wait.
Someone (Lou?) said "Proving” that we are not machines is somewhat
quixotic from my point of view, in that it should be obvious that we
are not machines!"
The statement "we are machine" is ambiguous. Does "we" refers to our
souls or to our bodies? Does it refers to our third person describable
relative bodies or to our private unnameable and non describable first
person view.
Here, what the Gödel-Löbian machine can already prove about themselves
is that IF they are self-referentially correct machine, and if they
survive a digital substitution at some level of description, they
their soul is not a machine, once we admit to identify the soul with
the knower, and translate the Theaetetus' definition of knowing (true
opinion, []p & p) in arithmetical terms. If my body is a machine, then
my soul is not (says Peano Arithmetic!).
I explain the main things in my two JPMB contributions. The key idea,
related to Gödel, is that incompleteness separates clearly what is
true about the machine and what is justifiable by the machine on one
part, and on the other part, it separates clearly the justifiability
([]p), knowability ([]p & p), "observability" ([]p & <>t), sensibility
([]p & <>t & p) with []p for Gödel's probability predicate, and <>t =
~[]~t = ~[]f = consistency (t = constant true, f = constant false). I
predicted that []p & p, []p & <>p and []p & <>t & p should give a
quantum logic (when p is "computably accessible, or Sigma_1), which I
manage to prove after 30 years of research. The whole things leads to
a many dream interpretations of arithmetic, from which a derivation of
quantum mechanics should be possible (and is partially done).
All this has been made possible by Gödel's theorem and its many
generalizations in many directions: Löb's theorem, Kleene's theorems
and mainly Solovay's completeness theorem for some modal logics, which
capture everything in the undecidability field in two modal logics (G
and G*). I will just refer everyone interested to my papers.
The main point relevant here is that incompleteness saves machine from
all reductionism. It shows that a machine (or any effective theory)
like Peano Arithmetic (say) is already quite clever. PA can already
refute all reductionist theories about its soul, and indeed can
already derive physics from Mechanism + Arithmetic.
Unfortunately, this uses mathematical logic, which is not well know by
non-logicians. Mathematical logic contains important sub-branches,
like computation theory, computability, theory, proof theory, and
model theory. Model theory is the study of meanings and semantics of
formal theories. Physicists used the term "model" for theory, and that
often lead to a "dialog of deaf".
I don't want be too long so I will stop here. I will reply to possible
comments next week, because this is my second post of the week.
Best regards, have a good week-end,
Bruno
I found Minds, Machines and Godel very useful to read since it
seemed to confirm aspects of my offering to you all.
On 3 May 2016 at 07:49, Francesco Rizzo
<13francesco.ri...@gmail.com> wrote:
Cari Terrence, Louis, Maxine e Tutti,
I state that I am a "exponential poor" who do not claim to lead any
claim whatsoever. But I think I have understood from the
triangulation of three colleagues, I do not think it is concluded,
that:
- Rosario Strano, a mathematician at the University of Catania,
lecturing on "Goodel, Tarski and liar" in the end he said: "In
closing, we conclude with a remark 'philosophical' suggested during
the conference by fellow F. Rizzo : a result that we can draw from
the theorems of exposed is that the search for truth, both in
mathematics and in other sciences, can not be caged by mechanical
rules, nor reduced to a formal calculation, but it requires
inspiration, intuition and genius, all features own human intellect
( "Bulletin Mathesis" section of Catania, Year V, n. 2, April 28,
2000);
- Also it happened to me, to defend the economic science from
encroachment or domain of the infinitesimal calculus, to declare
that the mathematical models resemble simulacra, partly true
(according to the logic) and partly false (according to reality) :
see. lately, F. Rizzo, ... "Bursars (c) to" Arachne publishing,
Rome, April 2016;
-in theory and in economic practice the capitalization rate "r" of
the capitalization formula V = Rn. 1 / r can be determined or
resorting to "qualitative quantity" of Hegel ( "The Science of
Logic") or to complex or imaginary numbers that, among other things,
allowed the Polish mathematician Minkowski, master of A. Einstein,
to adjust the general theory of relativity, so much so that I wrote:
"the imaginary and / or complex numbers used to conceive the
'Minkowiski of' universe that transforms time into space, making it
clearer and more explicit the isomorphic influence that space-time
exercises the capitalization formula and equation of special
relativity, maybe they can illuminate with new lights the function
of the concept of co-efficient of capitalization "(F. Rizzo," from
the Keynesian revolution to the new economy ", Franco Angeli, Milan,
2002, p . 35).
How you see the world looks great but basically it is up to you and
even to poor people like me to make it, or reduce it to the
appropriate size to understand (and be understood) by all.
Thank you for the opportunity you have given me, I greet you with
intellectual and human friendship.
Francesco
2016-05-03 5:28 GMT+02:00 Louis H Kauffman <lou...@gmail.com>:
Dear Folks
I realize in replying to this I surely reach the end of possible
comments that I can make for a week. But nevertheless …
I want to comment on Terrence Deacon’s remarks below and also on
Professor Johnstone’s remark from another email:
"This may look like a silly peculiarity of spoken language, one best
ignored in formal logic, but it is ultimately what is wrong with the
Gödel sentence that plays a key role in Gödel’s Incompleteness
Theorem. That sentence is a string of symbols deemed well-formed
according to the formation rules of the system used by Gödel, but
which, on the intended interpretation of the system, is ambiguous:
the sentence has two different interpretations, a self-referential
truth-evaluation that is neither true nor false or a true statement
about that self-referential statement. In such a system, Gödel’s
conclusion holds. However, it is a mistake to conclude that no
possible formalization of Arithmetic can be complete. In a formal
system that distinguishes between the two possible readings of the
Gödel sentence (an operation that would considerably complicate the
system), such would no longer be the case.
********”
I will begin with the paragraph above.
Many mathematicians felt on first seeing Goedel’s argument that it
was a trick, a sentence like the Liar Sentence that had no real
mathematical relevance.
This however is not true, but would require a lot more work than I
would take in this email to be convincing. Actually the crux of the
Goedel Theorem is in the fact that a formal system that
can handle basic number theory and is based on a finite alphabet,
has only a countable number of facts about the integers that it can
produce. One can convince oneself on general grounds that there are
indeed an uncountable number of true facts about the integers. A
given formal system can only produce a countable number of such
facts and so is incomplete. This is the short version of Goedel’s
Theorem. Goedel worked hard to produce a specific statement that
could not be proved by the given formal system, but the
incompleteness actually follows from the deep richness of the
integers as opposed to the more superficial reach of any given
formal system.
Mathematicians should not be perturbed by this incompleteness.
Mathematics is paved with many formal systems.
In my previous email I point to the Goldstein sequence.
https://en.wikipedia.org/wiki/Goodstein%27s_theorem
This is an easily understood recursive sequence of numbers that no
matter how you start it, always ends at zero after some number of
iterations.
This Theorem about the Goodstein recursion is not provable in Peano
Arithmetic, the usual formalization of integer arithmetic, using
standard mathematical induction.
This is a good example of a theorem that is not just a “Liar
Paradox” and shows that Peano Arithmetic is incomplete.
And by the way, the Goodstein sequence CAN be proved to terminate by
using ‘imaginary values’ as Professor Deacon describes (with a tip
of the hat to Spencer-Brown).
In this case the imaginary values are a segment of Cantor’s
transfinite ordinals. Once these transfinite numbers are admitted
into the discussion there is an elegant proof available for the
termination of the Goodstein sequence. Spencer-Brown liked to talk
about the possibility of proofs by using “imaginary Boolean values”.
Well, the Goodstein proof is an excellent example of this philosophy.
A further comment, thinking about i (i^2 = -1) as an oscillation is
very very fruitful from my point of view and I could bend your ear
on that for a long time. Here is a recent paper of mine on that
subject. Start in Section 2 if you want to start with the
mathematics of the matter.
http://arxiv.org/pdf/1406.1929.pdf
And here is an older venture on the same theme.
http://homepages.math.uic.edu/~kauffman/SignAndSpace.pdf
More generally, the idea is that one significant way to move out of
paradox is to move into a state of time.
I feel that this is philosophically a deep remark on the nature of
time and that i as an oscillation is the right underlying
mathematical metaphor for time.
It is, in this regard, not an accident that the Minkowski metric is
X^2 + Y^2 + Z^2 + (iT)^2.
TIME = iT
This is an equation with double meaning.
Time is measured oscillation.
Time is rotated ninety degrees from Space.
And one can wonder: How does i come to multiply itself and return -1?
Try finding your own answers before you try mine or all the previous
stories!
Best,
Lou
(See you next week.)
On May 2, 2016, at 9:31 PM, Terrence W. DEACON
<dea...@berkeley.edu> wrote:
A number of commentators, including the philosopher-logician
G. Spencer Brown and the anthropologist-systems theorist Gregory
Bateson, reframed variants of the Liar’s paradox as it might apply
to real world phenomena. Instead of being stymied by the
undecidability of the logic or the semantic ambiguity, they focused
on the very process of analyzing these relationships. The reason
these forms lead to undecidable results is that each time they are
interpreted it changes the context in which they must be
interpreted, and so one must inevitably alternate between true and
false, included and excluded, consistent and inconsistent, etc. So,
although there is no fixed logical, thus synchronic, status of the
matter, the process of following these implicit injunctions results
in a predictable pattern across time. In logic, the statement “if
true, then false” is a contradiction. In space and time, “if on,
then off” is an oscillation. Gregory Bateson likened this to a
simple electric buzzer, such as the bell in old ringer telephones.
The basic design involves a circuit that includes an electromagnet
which when supplied with current attracts a metal bar which pulls
it away from an electric contact that thereby breaks the circuit
cutting off the electricity to the electromagnet which allows the
metal bar to spring back into position where the electric contact
re-closes the circuit re-energizing the electromagnet, and so on.
The resulting on-off-on-off activity is what produces a buzzing
sound, or if attached to a small mallet can repeatedly ring a bell.
Consider another variant of incompletability: the concept of
imaginary number. The classic formulation involves trying to
determine the square root of a negative number. The relationship of
this to the liar’s paradox and the buzzer can be illustrated by
stepping through stages of solving the equation i x i = -1.
Dividing both sides by i produces i = -1/i, and then substituting
the value of i one gets i = -1/-1/i and then again i = -1/-1/-1/i
and so forth, indefinitely. With each substitution the value
alternates from negative to positive and cannot be resolved (like
the true/false of the liar’s paradox and the on/off of the buzzer).
But if we ignore this irresolvability and just explore the
properties of this representation of an irresolvable value, as have
mathematicians for centuries, it can be shown that i can be treated
as a form of unity and subject to all the same mathematical
principles as can 1 and all the real numbers derived from it. So i
+ i = 2i and i - 2i = -i and so on. Interestingly, 0 x i = 0 X 1 =
0, so we can conceive of the real number line and the imaginary
number line as two dimensions intersecting at 0, the origin.
Ignoring the many uses of such a relationship (such as the use of
complex numbers with a real and imaginary component) we can see
that this also has an open-ended consequence. This is because the
very same logic can be used with respect to the imaginary number
line. We can thus assign j x j = -i to generate a third dimension
that is orthogonal to the first two and also intersecting at the
origin. Indeed, this can be done again and again, without
completion; increasing dimensionality without end (though by
convention we can at any point restrict this operation in order to
use multiple levels of imaginaries for a particular application,
there is no intrinsic principal forcing such a restriction).
One could, of course, introduce a rule that simply restricts
such operations altogether, somewhat parallel to Bertrand Russell’s
proposed restriction on logical type violation. But mathematicians
have discovered that the concept of imaginary number is remarkably
useful, without which some of the most powerful mathematical tools
would never have been discovered. And, similarly, we could
discount Gödel’s discovery because we can’t see how it makes sense
in some interpretations of semiosis. On the other hand, like G.
Spencer Brown, Doug Hofstadter, and many others, thinking outside
of the box a bit when considering these apparent dilemmas might
lead to other useful insights. So I’m not so willing to brand the
Liar, Gödel, and all of their kin as useless nonsense. It’s not a
bug, it’s a feature.
On Mon, May 2, 2016 at 2:19 PM, Maxine Sheets-Johnstone <m...@uoregon.edu
> wrote:
Many thanks for your comments, Lou and Bruno. I read and pondered,
and finally concluded that the paths taken by each of you exceed
my competencies. I subsequently sent your comments to Professor
Johnstone—-I trust this is acceptable—asking him if he would care to
respond with a brief sketch of the reasoning undergirding his
critique,
which remains anchored in Gödel’s theorem, not in the writings of
others
about Gödel’s theorem. Herewith his reply: ******** Since no one
commented on the reasoning supporting the conclusions reached in
the two cited articles, let me attempt to sketch the crux of the
case presented. The Liar Paradox contains an important lesson about
meaning. A statement that says of itself that it is false, gives
rise to a paradox: if true, it must be false, and if false, it must
be true. Something has to be amiss here. In fact, what is wrong is
the statement in question is not a statement at all; it is a pseudo-
statement, something that looks like a statement but is incomplete
or vacuous. Like the pseudo-statement that merely says of itself
that it is true, it says nothing. Since such self-referential truth-
evaluations say nothing, they are neither true nor false. Indeed,
the predicates ‘true’ and ‘false’ can only be meaningfully applied
to what is already a meaningful whole, one that already says
something. The so-called Strengthened Liar Paradox features a
pseudo-statement that says of itself that it is neither true nor
false. It is paradoxical in that it apparently says something that
is true while saying that what it says it is not true. However, the
paradox dissolves when one realizes that it says something that is
apparently true only because it is neither true nor false. However,
if it is neither true nor false, it is consequently not a
statement, and hence it says nothing. Since it says nothing, it
cannot say something that is true. The reason why it appears to say
something true is that one and the same string of words may be used
to make either of two declarations, one a pseudo-statement, the
other a true statement, depending on how the words refer. Consider
the following example. Suppose we give the name ‘Joe’ to what I am
saying, and what I am saying is that Joe is neither true nor false.
When I say it, it is a pseudo-statement that is neither true nor
false; when you say it, it is a statement that is true. The
sentence leads a double life, as it were, in that it may be used to
make two different statements depending on who says it. A similar
situation can also arise with a Liar sentence: if the liar says
that what he says is false, then he is saying nothing; if I say
that what he says is false, then I am making a false statement
about his pseudo-statement. This may look like a silly peculiarity
of spoken language, one best ignored in formal logic, but it is
ultimately what is wrong with the Gödel sentence that plays a key
role in Gödel’s Incompleteness Theorem. That sentence is a string
of symbols deemed well-formed according to the formation rules of
the system used by Gödel, but which, on the intended interpretation
of the system, is ambiguous: the sentence has two different
interpretations, a self-referential truth-evaluation that is
neither true nor false or a true statement about that self-
referential statement. In such a system, Gödel’s conclusion holds.
However, it is a mistake to conclude that no possible formalization
of Arithmetic can be complete. In a formal system that
distinguishes between the two possible readings of the Gödel
sentence (an operation that would considerably complicate the
system), such would no longer be the case. ******** Cheers, Maxine
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University of California, Berkeley
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Distinguished Professor of Yoga and Physical Science,
SVYASA, Eknath Bhavan, 19 Gavipuram Circle
Bangalore 560019, Karnataka, India
Mobile (Intn'l): +44 7710 534195
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