[Fis] Fis Digest, Vol 118, Issue 38

[Louis Kauffman]

See my previous email. I assert that human consciousness cannot be encompassed by any single formal system.
This goes beyond set theory. I assert the validity of arguments such as those given in Penrose books Emporer’s New Mind, but state these arguments in my way
And without speculation about what kind of physics goes beyond Turing.

As I said before, such arguments are hard for some people to take. The assertion really is that if you accept the original Goedelian argument, then it tells you that a human cognizer reasoning about 
a formal system can do more than the formal system on its own. If you accept this, then you cannot be such a formal system without being inconsistent. I do think that people find this annoying.
But there it is. And maybe you find it annoying because it is proving what you already knew.

NotTuring
LK

1. We prove Goedel’s Theorem as follows: 
Let T be a formal system that is consistent 
and contains at least the Peano axioms for number theory.
I examine T as a mathematical object and produce (via Goedel coding) 
a sentence G that declares its own unprovability in T. 
This declaration has an external meaning and it is 
devised so that a proof of G in T would lead to a contradiction. 

Thus, since T is consistent, G cannot be proved in T. 
But G states the non-provability of G in T. 
Thus G is true but not provable in T. 
We have proved, from outside T, that G is true. 
This proof is a mathematical proof of the statement G 
and it does not contradict T’s unprovability inside T, 
since we work in the larger system of 
reasoning about formal systems, including T.

2. Could I be identical with T as above? 
Certainly not. 
For I have proved G. 
So if I = T, then T has proved G. 
I have shown that T cannot prove G.
Thus if I = T, then T is inconsistent. 
We have assumed that T is consistent. 
Therefore I am not identical with T as a mathematical reasoner.

3. Could I be a Turing machine T, 
consistent and rich enough to contain Peano Arithmetic? 
Suppose it is so and 
go to 1. and 2. above 
to arrive at the conclusion that 
this is not possible.

4. Go back to 1. 
and note that I have the capacity to take T as an object of study. 
The discussion in 2. and 3. leads to the 
ancient questions about whether a person can know themselves. 

In the mathematical context, 
if I do stand outside my own processes of reasoning 
and then reason about these processes, 
this is a practical capacity that I have.

The history of mathematics and logic is 
a long spiral of such self-examination. 
In order for it to spiral as it does, 
the whole process can not be encompassed in a single formal system. 

This is the import of Goedel’s theorem 
and it actually applies to the entities 
that we call persons, 
individual reasoners with understanding. 
The individual reasoners are not single formal systems 
(to the extent that they are consistent).


> On Jan 15, 2025, at 7:09 AM, Stuart Kauffman <stukauffman at gmail.com> wrote:
> 
> Hello to All, 
> 
> in support of Lou, I attach two references that say the becoming of the world, including, presumably, human consciousness, is beyond any mathematical formulation based on set theory.
> 
> Kind wishes, 
> 
> Stu
> 
> Kauffman, S. and Roli, A. (2021) The World Is Not A Theorem” Entropy vol 23, issue 11 
> 
> Kauffman, S. and Roli, A. (2022), What is Consciousness? Biological Journal of the Linnean Society,_ _2022
>    
> 
> 
> 
>> On Jan 15, 2025, at 3:38 AM, Marcus Abundis <55mrcs at gmail.com> wrote:
>> 
>> <  I am sympathetic with mathematical and formal modeling of “cognitive processes” but feel that it should be clear that formal models will not capture the whole phenomenon. >
>> 
>> For *myself*, while I accept an essential truth lies in this statement . . . I am ALSO inclined to think 'surrendering' prematurely is a lack of scientific imagination ('heavy lifting') – where 'science' is SUPPOSED to be in the business of continually reinventing itself. That said, I also accept that many do not see science as an actual/active creative process. For me, it is different. I think the core issue here is “cognitive processes = psychology”, a notoriously . . . .uhhh, I am not sure of the best word to use here, so I will just say 'difficult topic'.
>> 
>> And thanks for the lovely taoist imagery . . . taoism being the last word in Natural Psychology.
>> 
>> Marcus
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