Re: [Fis] Gödel Discussion

[Louis H Kauffman]

Dear Alex,
Thank you for this very balanced viewpoint about this part of the debate.

You write "Although formal systems are designed to apply to concepts within the 
world of thought i.e. the world of (abstract) phenomenal experience, they are 
not intended to have semantic application, but only syntactical consistency. To 
judge their validity or invalidity from a semantic (or even semiotic) 
perspective of Husserlian phenomena - experience, therefore seems to me to be 
inappropriate. “

From the point of view of a practicing mathematician the matter is a bit more 
complex. For we certainly are interested in the semantics of our formalities.
We do not regard what we do as purely syntactical. There are meanings involved. 
For example one interprets algebra (quite formal) in terms of geometry
(often quite intuitive) and one may think philosophically about the meanings of 
the Goedelian limitations on formal systems. Furthermore the most basic 
mathematical concepts such as the number three are highly conceptual and are 
understood that way (through the appearance of triples in both the mental and 
physical realms).
The point of having good formalisms is sthat the formalism should have the 
least possible necessary interpretation and this leads to the multiplicity of 
interpretations that makes mathematics so fruitful. Think of the number of ways 
to interpret a second order linear differential equation, from fluid flow, to 
electricity to the oscillations of a pendulum. The point of nearly 
uninterpreted formalisms is that they are PATTERNS that can occur in many 
situations and lead us into the unknown including the internal unknown of their 
self-application.

Thank you again,
Lou Kauffman

> On May 11, 2016, at 1:44 PM, Alex Hankey  wrote:
> 
> Dear Colleagues, 
> 
> This discussion is continuing to be very enlightening, I feel, 
> for those aware of, but not intimately familiar with, details of 
> both Husserl's approach and Gödel’s statements / theorems. 
> 
> I suspect that part of the problem lies in the fact that we are dealing with 
> a highly contrasting pair of intellectual discussions, about two entirely 
> different universes of analysis (if that is the right term). I suspect that 
> they may not be compatible, and that that is the real cause for the conflict 
> of perspectives. 
> 
> Husserl was concerned with formulating a philosophically rigorous discussion 
> of the world of experience, from within the world of experience, and set up 
> his criteria on that basis. 
> 
> Gödel on the other hand was operating within the world of formal systems, and 
> showed that if a set of axioms containing arithmetic was consistent, it had 
> to be incomplete - valid statements could be made that are not derivable 
> within the formal system. 
> 
> Although formal systems are designed to apply to concepts within the world of 
> thought i.e. the world of (abstract) phenomenal experience, they are not 
> intended to have semantic application, but only syntactical consistency. To 
> judge their validity or invalidity from a semantic (or even semiotic) 
> perspective of Husserlian phenomena - experience, therefore seems to me to be 
> inappropriate. 
> 
> They are categorically different (linguistic?) structures.
> Or have I snafued?
> 
> Alex
>  
> 
> 
> 
> -- 
> Alex Hankey M.A. (Cantab.) PhD (M.I.T.)
> Distinguished Professor of Yoga and Physical Science,
> SVYASA, Eknath Bhavan, 19 Gavipuram Circle
> Bangalore 560019, Karnataka, India  
> Mobile (Intn'l): +44 7710 534195 
> Mobile (India) +91 900 800 8789
> 
> 
> 2015 JPBMB Special Issue on Integral Biomathics: Life Sciences, Mathematics 
> and Phenomenological Philosophy 
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> Fis mailing list
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[Fis] Gödel Discussion

[Alex Hankey]

Dear Colleagues,

This discussion is continuing to be very enlightening, I feel,
for those aware of, but not intimately familiar with, details of
both Husserl's approach and Gödel’s statements / theorems.

I suspect that part of the problem lies in the fact that we are dealing
with a highly contrasting pair of intellectual discussions, about two
entirely different universes of analysis (if that is the right term). I
suspect that they may not be compatible, and that that is the real cause
for the conflict of perspectives.

Husserl was concerned with formulating a philosophically rigorous
discussion of the world of experience, from within the world of experience,
and set up his criteria on that basis.

Gödel on the other hand was operating within the world of formal systems,
and showed that if a set of axioms containing arithmetic was consistent, it
had to be incomplete - valid statements could be made that are not
derivable within the formal system.

Although formal systems are designed to apply to concepts within the world
of thought i.e. the world of (abstract) phenomenal experience, they are not
intended to have semantic application, but only syntactical consistency. To
judge their validity or invalidity from a semantic (or even semiotic)
perspective of Husserlian phenomena - experience, therefore seems to me to
be inappropriate.

They are categorically different (linguistic?) structures.
Or have I snafued?

Alex




-- 
Alex Hankey M.A. (Cantab.) PhD (M.I.T.)
Distinguished Professor of Yoga and Physical Science,
SVYASA, Eknath Bhavan, 19 Gavipuram Circle
Bangalore 560019, Karnataka, India
Mobile (Intn'l): +44 7710 534195
Mobile (India) +91 900 800 8789


2015 JPBMB Special Issue on Integral Biomathics: Life Sciences, Mathematics
and Phenomenological Philosophy

___
Fis mailing list
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Re: [Fis] Gödel discussion

[Bruno Marchal]

Dear Albert,


On 07 May 2016, at 06:57, Albert A Johnstone wrote:


Greetings everyone,
I’d like to say a few words about Smullyan’s thought experiment and  
its relevance to Gödel’s Theorem in the hope of putting an end to  
discussion of a topic somewhat tangential to the main one. Before  
doing so, I am forwarding an email from Lou Kauffman which gives a  
very clear account of Smullyan’s reasoning.


 Original Message 
Subject: Re: [Fis] _ FIS discussion
Date: 2016-05-04 12:30
From: Louis H Kauffman 
To: Maxine Sheets-Johnstone 

Dear Maxine,
I am writing privately to you since I have used up my quota of forum  
comments for this week.

I am going to discuss a Smullyan puzzle in detail with you.
I call this the Smullyan Machine.

THE SMULLYAN MACHINE
The machine has a button on the top and when you press that button,  
it prints a string of symbols using the following three letter  
alphabet.

{ P, ~ ,R}
Thus the machine might print P~~~NRRP.
I shall designate an unknown string of symbols by X or Y.
Strings that begin with P, ~P, PR or ~PR are INTERPRETED (given  
meaning) as follows:


Meaningful Strings
(When I say “X can be printed by the Machine” I mean that when you  
press the button the machine will print exactly X and nothing else.)


Actually Smullyan meant "printable" or "printed soon or later by a  
machine which is programmed to print all what she can print (it can be  
shown easily that this is always possible by a dovetailing technic).


The point will be that if the machine is correct, then the set of what  
is printable will be included properly in the set of what is true.






PX:  X can be printed by the Machine.
~PX: X cannot be printed by the Machine.
PRX: XX can be printed by the Machine.
~PRX: XX can not be printed by the Machine.

Thus it is possible that the machine might print
~PPR
This has meaning and it states that the machine cannot bring PR all  
by itself when the button is pressed.


AXIOM OF THE MACHINE
The Smullyan Machine always tells the truth when it prints a  
meaningful string.


THEOREM. There is a meaningful string that is true but not printable  
by the Smullyan Machine.


PROOF. Let S = ~PR~PR. This string is meaningful since it starts  
with ~PR.
Note that S = ~PRX where X = ~PR. Thus by the definition (above) of  
the meaning of S,  “XX is not printable by the Machine.”
We note however that XX = ~PR~PR = S. Thus S has the meaning that “S  
is not printable by the Machine.”
Since the Machine always tells the truth, it would be in a  
contradiction if it printed S. Therefore the Machine cannot print S.
But this is exactly the meaning of S, and so S is true. S is a true  
but not printable string. The completes the proof.

—

Now I have an assignment for you.
Please criticize the Smullyan Machine from your phenomenological  
point of view.
If you wish you could include my description of the Machine and make  
a statement about it on FIS.
My point and Smullyan’s point in his Oxford University Press Book on  
Godel’s Theorem, is that the Machine is an accurate depiction of the  
Godel argument, with
Printabilty replacing Provablity. The way that self-reference works  
here, and the way the semantics and syntax are controlled is very  
much like the way these things happen in the
full Godel theorem. The Machine provides a microcosm for the  
discussion of Godel and self-reference.

Yours truly,
Lou Kauffman
P.S. “This sentence has thirty-three letters.”
is a fully meaningful and true English sentence.
Self-referential sentence can have meaning and reference.


Johnstone again:

	In response to the above assessment, let us first distinguish  
syntactic self-reference which is reference to the words or sentence  
that one is using, from semantic self-reference, which is reference  
to the MEANING of the words or sentences one is using. There is  
nothing wrong with syntactic self-reference but semantic self- 
reference invariably generates vacuity and sometimes paradox.


Now Smullyan’s sentence ‘~PR~PR’ is often interpreted (as by Lou,  
Bruno, and by myself earlier) as making a syntactically self- 
referential statement that says that the sentence expressing that  
statement is not printable. On the supposition that such is the  
case, the statement it makes must also be semantically self- 
referential for the following reason. In Smullyan’s scenario, the  
printing machine prints only true statements. As a result, a  
sentence is printable if and only if the statement it makes is true.


It follows only that all the sentences printed by the machine will be  
true. It does not entail that the machine will print all true  
sentences. It is "only if", not "if and only if" when you say "As a  
result, a sentence is printable if and only if the statement it makes  
is true.".





Consequently, the two predicates ‘is not printable’

Re: [Fis] Gödel discussion

[Louis H Kauffman]

Dear Albert,
I see that I do want to say a bit more.
 
Lets turn to the Godel Theorem. We have that there is a coding method that 
assigns to each formula F a Godel number g that can be algorithmically decoded 
into that formula.
For this I write g ——> F. Colloquially this means “g is the Godel number of the 
formula F” but remember that there is an algorithm that allows this to happen 
in the formal system.
Furthermore, if F(u) is a formula with a free variable u, then we would have a 
g such that g ——> F(u). But u is an arithmetical variable and so we can define 
a function that takes Godel numbers of this type to new Godel numbers via the 
pattern
g ——> F(u)
then 
#g ———> F(g).
That is, #g is the Godel number of the result of substituting g for the free 
variable of F(u).
The function g to #g is arithmetically defined and can be computed by the 
formal system.
Thus we can take # as an abbreviation for a special text in the formal system 
and consider functions with one free variable u of the form F(#u).
Then we have
g ——> F(#u)
#g ———> F(#g).
The formula F(#g) has the Godel number #g.
I shall refrain from saying that F(#g) “talks about itself”. This is just a way 
of saying that inside the formula F(#g) appears a representative for the Godel 
number of the formula as a whole.
But we also have a very rich formal system, and so there is a predicate in the 
system that has the form ~B(u) which is the statement that there is no proof in 
the system of the formula whose Godel number is u. This is a statement that can 
be made inside the formal system. Inside the system, if confronted by the 
statement ~B(g) where g ——> F, the system can decode g into F and than ~B(g) 
states that there is no proof of F. The formal system is equipped with enough 
internal language so that such a statement is meaningful. The statement that 
there is no proof of F is a statement about sequences of numbers since a proof 
is a sequence of statements of a certain kind ending in F. But now, Godel uses 
the # function and examines the Godel number g of ~B(#u).
We have
g ———> ~B(#u)
#g ———> ~B(#g).
So ~B(#g) asserts that there is no proof of the formula with Godel number #g 
and this is ~B(#g).
The result is this:
If decoding of h is provable, then B(h) is also provable in the formal system.
For B(h) says that the decoding of h is provable.
If the formal system were to have a proof of the decoding of h then it would 
also have a proof of B(h).
If the formal system were to have a proof of the decoding of #g then it would 
also have a proof of B(#g).
But the decoding of #g is ~B(#g). So if there were a proof of the decoding of 
#g then there would be proofs of 
~B(#g) and B(#g). Hence the formal system would be inconsistent.
We assume that the formal system is consistent, and we conclude that it is 
incomplete.

The point about these constructions is that they would happen all the way to 
inconsistency INSIDE the formal system. It is independent of our talk from the 
outside.
The Godel argument shows that if you have a sufficiently rich formal system 
that includes arithmetic then there are statements in it, so that if the system 
could prove them, then the system would be inconsistent. We can only conclude 
that if the system is consistent then these statements have no proof in the 
system. 

Now finally I get to the point. We CAN interpret the statement ~B(#g) as 
asserting its own
unprovability. You can unwind what it says about arithmetic. It says that there 
is no valid proof in the sense of a sequence of statements in the formal system 
(all encoded by numbers) so that the sequence starts with givens in the formal 
system and ends with the formula
~B(#g). That means that it says that there is no proof of ~B(#g) in the system. 
And we have proved that. So we have proved, from the outside that ~B(#g) is 
indeed true. But only by assuming the consistency of the formal system.

Best,
Lou



> On May 9, 2016, at 1:16 AM, Albert Johnstone  wrote:
> 
> Greetings Lou,
> 
> If I understand you correctly, 'P' means 'an empty string is
> printable', and so is a sentence rather than merely a predicate.
> Actually the point has no effect on Smullyan's argument, a simpler
> version of which is the following:
> 
> The sentence, ''c'  is not printable', is not printable by a printer
> that prints only sentences making true statements, because it contains
> the letter 'c' which, if the sentence were true, would not be
> printable.
> 
> Smullyan couches his argument in a more complex form that echoes the
> situation in Gödel's theorem, and in which Smullyan's sentence echoes
> the Gödel sentence. He does so in order to suggest that the two
> sentences are essentially similar. They are not, however, because the
> Gödel sentence is supposed to express a statement in arithmetic, not a
> statement about words. As a result, Smullyan's engaging thought
> experiment ends up muddying the water more than clarifying Gödel's
> theorem.
> 
> The Godel sentenc

[Fis] Gödel discussion

[Albert A Johnstone]

Greetings everyone,
I’d like to say a few words about Smullyan’s thought experiment and its 
relevance to Gödel’s Theorem in the hope of putting an end to discussion 
of a topic somewhat tangential to the main one. Before doing so, I am 
forwarding an email from Lou Kauffman which gives a very clear account 
of Smullyan’s reasoning.


 Original Message 
 Subject: Re: [Fis] _ FIS discussion
 Date: 2016-05-04 12:30
 From: Louis H Kauffman 
 To: Maxine Sheets-Johnstone 

 Dear Maxine,
 I am writing privately to you since I have used up my quota of forum 
comments for this week.

 I am going to discuss a Smullyan puzzle in detail with you.
 I call this the Smullyan Machine.

 THE SMULLYAN MACHINE
 The machine has a button on the top and when you press that button, it 
prints a string of symbols using the following three letter alphabet.

 { P, ~ ,R}
 Thus the machine might print P~~~NRRP.
 I shall designate an unknown string of symbols by X or Y.
 Strings that begin with P, ~P, PR or ~PR are INTERPRETED (given 
meaning) as follows:


 Meaningful Strings
 (When I say “X can be printed by the Machine” I mean that when you 
press the button the machine will print exactly X and nothing else.)


 PX:  X can be printed by the Machine.
 ~PX: X cannot be printed by the Machine.
 PRX: XX can be printed by the Machine.
 ~PRX: XX can not be printed by the Machine.

 Thus it is possible that the machine might print
 ~PPR
 This has meaning and it states that the machine cannot bring PR all by 
itself when the button is pressed.


 AXIOM OF THE MACHINE
 The Smullyan Machine always tells the truth when it prints a meaningful 
string.


 THEOREM. There is a meaningful string that is true but not printable by 
the Smullyan Machine.


 PROOF. Let S = ~PR~PR. This string is meaningful since it starts with 
~PR.
 Note that S = ~PRX where X = ~PR. Thus by the definition (above) of the 
meaning of S,  “XX is not printable by the Machine.”
We note however that XX = ~PR~PR = S. Thus S has the meaning that “S is 
not printable by the Machine.”
Since the Machine always tells the truth, it would be in a contradiction 
if it printed S. Therefore the Machine cannot print S.
 But this is exactly the meaning of S, and so S is true. S is a true but 
not printable string. The completes the proof.

—

Now I have an assignment for you.
 Please criticize the Smullyan Machine from your phenomenological point 
of view.
 If you wish you could include my description of the Machine and make a 
statement about it on FIS.
 My point and Smullyan’s point in his Oxford University Press Book on 
Godel’s Theorem, is that the Machine is an accurate depiction of the 
Godel argument, with
 Printabilty replacing Provablity. The way that self-reference works 
here, and the way the semantics and syntax are controlled is very much 
like the way these things happen in the
 full Godel theorem. The Machine provides a microcosm for the discussion 
of Godel and self-reference.

 Yours truly,
 Lou Kauffman
 P.S. “This sentence has thirty-three letters.”
is a fully meaningful and true English sentence.
 Self-referential sentence can have meaning and reference.


Johnstone again:

	In response to the above assessment, let us first distinguish syntactic 
self-reference which is reference to the words or sentence that one is 
using, from semantic self-reference, which is reference to the MEANING 
of the words or sentences one is using. There is nothing wrong with 
syntactic self-reference but semantic self-reference invariably 
generates vacuity and sometimes paradox.


Now Smullyan’s sentence ‘~PR~PR’ is often interpreted (as by Lou, Bruno, 
and by myself earlier) as making a syntactically self-referential 
statement that says that the sentence expressing that statement is not 
printable. On the supposition that such is the case, the statement it 
makes must also be semantically self-referential for the following 
reason. In Smullyan’s scenario, the printing machine prints only true 
statements. As a result, a sentence is printable if and only if the 
statement it makes is true. Consequently, the two predicates ‘is not 
printable’ and ‘is not true’ are logically equivalent. A sentence that 
says of itself that it is not printable is consequently logically 
equivalent (each entails the other) to a statement that says of itself 
that it is not true, that is, it is equivalent to a Liar statement. As 
such, it is semantically incomplete or vacuous; it does not make a 
statement, and hence is neither true nor false, and so cannot possibly 
be an unprintable true statement.
The equivalence of the two predicates has the result that ‘~PR~PR’ is 
both syntactically AND semantically self-referential.


	On reflection, however, I suspect that the sentence ‘~PR~PR’ has been 
incorrectly interpreted. The second expression ‘~PR’ at the end of 
Smullyan