Re: [Fis] _ Re: _ Re: re Gödel discussion

[Francesco Rizzo]

Cari Terrence, Louis, Maxine e Tutti,
premetto che sono un "poverino esponenziale" che non ha la pretesa di
menare alcun vanto. Ma mi pare di aver capito dalla  triangolazione dei tre
colleghi,che non credo si sia conclusa,  quello che:
- Rosario Strano, un valente matematico dell'Università di Catania, tenendo
una conferenza su "Goodel, Tarski e il mentitore" alla fine ha affermato:
"In chiusura concludiamo con un'osservazione 'filosofica' suggerita durante
la conferenza dal collega F. Rizzo: una conseguenza che possiamo trarre dai
teoremi su esposti è che la ricerca della verità, sia nella matematica che
nelle altre scienze, non può essere ingabbiata da regole meccaniche, nè
ridursi a un calcolo formale, ma richiede estro, intuizione e genialità,
tutte caratteristiche proprie dell'intelletto umano ("Bollettino Mathesis"
della sezione di Catania, Anno V, n. 2, 28 aprile 2000);
- anche  a me è capitato, per difendere la scienza economica dall'invadenza
o dominio del calcolo infinitesimale, di dichiarare che i modelli
matematici assomigliano a dei simulacri, in parte veri (secondo la logica)
e in in parte falsi (secondo la realtà): cfr. ultimamente, Rizzo F.,
..."Economi(c)a", Aracne editrice, Roma, aprile 2016;
-nella teoria e nella pratica economica il saggio di capitalizzazione "r"
della formula di capitalizzazione V = Rn. 1/r si può determinare o
ricorrendo alla "quantità qualitativa" di hegel ("La scienza della logica")
oppure ai numeri complessi o immaginari che, fra l'altro consentirono al
matematico polacco Minkowski, maestro di A. Einstein, di aggiustare la
teoria della relatività generale, tanto che ho scritto: "I numeri
immaginari e/o complessi usati per concepire l''universo di Minkowiski' che
trasforma il tempo in spazio, rendendo più chiara ed esplicita l'influenza
isomorfica che lo spazio-tempo esercita sulla formula di capitalizzazione e
sull'equazione della relatività ristretta, forse possono illuminare di luce
nuova la funzione del concetto di co-efficiente di capitalizzazione" (Rizzo
F., "Dalla rivoluzione keynesiana alla nuova economia", FrancoAngeli,
Milano, 2002, p. 35).
Come vedete il mondo sembra grande ma in fondo spetta a Voi e, anche ai
poverini come me di renderlo o ridurlo alla dimensione adeguata per
comprendere (ed essere compreso) da tutti.
Nel ringraziarVi per l'opportunità che mi avete dato, Vi saluto con
amicizia intellettuale ed umana.
Francesco

2016-05-03 5:28 GMT+02:00 Louis H Kauffman :

> 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
> 

[Fis] _ Re: _ Re: re Gödel discussion

[Louis H Kauffman]

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  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. 
>