[Fis] A new discussion session

Louis Kauffman loukau at gmail.com
Tue Nov 8 01:12:50 CET 2022

Dear Pedro,
Here are some comments about Goedel numbering and coding.

It is interesting to think about Goedel numbering in a biological context.
Actually we are talking about how a given entity has semantics that can vary from context to context.
It is not simply a matter of assigning a code number. If g —> F is the relation of a Goedel number g to a statement F, then we have two contexts for F.
1. F as a well formed formula in a formal system S.
2. g as a number in either a number system for an observer of S or g as a number in S, but g, as a representative for F can be regarded in the system S with the meanings so assigned.
Thus we have produced by the assignment of Goedel numbers a way for a statement F to exist in the semantics of more than one context.
This is the key to the references and self-references of the Goedelian situations.

Lets look at this more carefully. Recall that there is a formal system S and that to every well formed formula in S, there is a code number g = g(S). The code number can be produced in many ways.
For example, one can assign different index numbers n(X) to each distinct generating symbol in S. Then with an expression F regarded as an ordered string of symbols, one can assign to F the product of
the prime numbers, in their standard order, with exponents the indices of the sequence of characters that compose F. For example, 
g(~ x^2 = 2) = 2^{n(~)} 3^{n(x)}5^{n(^)}7^{n(2)}11^{n(=)}13^{n(2)}. From such a code, one can retrieve the original formula in a unique way.
The system S is a logical system that is assumed to be able to handle logic and basic number theory. Thus it is assumed that S can encode the function g: WFFS(S) —> N where N denotes the natural numbers.
And S can decode a number to find the corresponding expression as well. It is assumed that S as a logical system, is consistent.

With this backgound, let g —> F denote the condition that g = g(F). Thus I write a reference g —> F for a mathematical discussion of S, to indicate that g is the Goedel number of F.

Now suppose that F(x) is a formula in S with a free variable x. Free variables refer to numbers. Thus if I write x^2 = 4 then this statement can be specialized to 2^2 = 4 with x =2 and the specialization is true.
Or I can write 3^2 = 4 and this is a false statement. Given F(x) and some number n, I can make a new sentence F(n). 
Now suppose that
g —> F(x).
Then we can form F(g) and this new statement has a Goedel number. Let #g denote the Goedel number of F(g).
#g —> F(g).
This # is a new function on Goedel numbers and also can be encoded in the system S. I will abbreviate the encoding into S by writing #n for appropriate numbers n handled by S.
Then we can consider 
F(#x) and it has a Goedel number
h —> F(#x)
And we can shift that to 
#h —> F(#h).
This is the key point.
Now we have constructed a number #h so that F(#h) discusses its own Goedel number.

This construction allows the proof of the Goedel Incompleteness Theorem via the sentence B(x) that states
B(x) =  “The statement with Goedel number x is provable in S.” (This can also be encoded in S.)

We then construct 
h—> ~B(#x)
#h —> ~B(#h)
and obtain the statement 
G= ~B(#h).
G states the unprovability of the Goedel decoding of #h.
But the Goedel decoding of #h is the statement G itself.
Thus G states its own unprovability.
Therefore, S being consistent, cannot prove G. 

By making these arguments we have have proved that G cannot be proved by S.
Thus we have shown that G is in fact true.
We have shown that there are true statements in number theory unprovable by system S..

The above is a very concise summary of the proof of Goedel’s Incompleteness Theorem, using Goedel number encoding.

It is a very interesting question whether such encoding or such multiple relationships to context occur in biology. Here are some remarks.

1. In biology is is NORMALLY the case that certain key structures have multiple interpretations and uses in various contexts.
The understanding of such multiple uses and the naming of them requires an observer of the biology. Thus we see the action of a cell membrane and we see the 
action of mitosis, and so on.

2. There are implicit encodings in biology such as the sequence codes in DNA and RNA and their unfoldment. To what extent do they partake of the properties of Goedel coding?

3. The use of the Goedel coding in the Incompleteness theorem depends crucially on the relationship of syntax and semantic in the formal system and in the mathematician’s interpretation 
of the workings of that system. The Goedel argument depends upon the formal system S being seen as a mathematical object that itself can be studied for its properties and behavior.
When we speak of the truth of G, we are speaking of our assessment of the possible behaviour of S, given its consistency. We are reasoning about S just as Euclid reasons about the structure of 
right triangle.

4. In examining biological structures we take a similar position and reason about what we know about them. Sufficiently complex biological structures can be seen as modeled by certain logical formal systems.
And then Goedelian reasoning can be applied to them. This can even be extended to ourselves. Suppose that I am modeled correctly in my mathematical reasoning by a SINGLE CONSISTENT FORMAL SYSTEM S.
Then “I” can apply the above proof of Goedel’s Therem to S and deduce that G cannot be proven by S. Thus “I” have exceeded the capabilities of S. Therefore it is erroneous to assume that my 
mathematical reasoning is encapsulated by a single formal system S. If I am a formal system, that system must be allowed to grow in time. Such reasoning as this is subtle, but the semantics of the relationship
of mathematicians and the formal systems that they study is subtle and when biology is brought in the whole matter becomes exceedingly interesting.

5. We man not need numbers to have these kinds of relationships. And example is the Smullyan Machine that prints sequences of symbols from the alphabet {~,P,R} on a tape. Sequences that begin with 
P,~P,PR and ~PR are regarded as meaningful, with the meanings:
PX: X can be printed.
~PX: X cannot be printed.
PRX: XX can be printed.
~PRX: XX cannot be printed.
Here X is any string of the symbols {~,P,R}.
Thus PR~~P means that XX can be printed where X = ~~P. Thus PR~~P means that ~~P~~P can be printed.
By printed we mean on one press of the button on the Machine, a string of characters is printed.
Then we have the 

Theorem. There are meaningful true strings that the Smullyan Machine cannot print.

This is a non-numerical analog of the Goedel Theorem. And the string that cannot be printed is
G = ~PR~PR.
For you see that G is meaningful and since G = ~PRX, G says that XX cannot be printed. But X = ~PR and XX = ~PR~PR = G. So G says that G cannot be printed.
If the machine were to print G, it would lie. And the machine does not lie.
Therefore G is unprintable.
But this is what G says.
So we have established the truth of G and proved the Theorem.

6. Examine this last paragraph 5. The Machine is like an organism with a limitation. This limitation goes through the semantics of reference. ~PRX refers to XX and so can refer to itself if we take 
X = ~PR. ~PX refers to X and cannot refer to itself since it is longer than X. In biological coding the DNA code is fundamentally smaller or equal to the structure to which it refers.
Thus the self-reproduction of the DNA is possible since DNA = W+C the convention of the Watson and Crick strand and each of W and C can by themselves engage in an action to encode, refer to,
the other strand. W can produce a copy of C in the form W+C and C can produce a copy of W in the form W+C each by using the larger environment. Thus W+C refers to itself, reproduces itself by 
a method of encoding quite similar to the self reference of the Smullyan Machine.

7. Von Neuman devised a machine that can build itself. B is the von Neuman machine and B.x —> X,x where x is the plan or blueprint or code for and entity X. B builds X with given the blueprint x.
Then we have B,b —> B,b where b is the blueprint for B. B builds itself from its own blueprint. I hope you see the analogy with the Goedel code.

8. I will stop here. The relationships with biology are very worth discussing.
Before stopping it is worth remarking that the Maturana Uribe Varela autopoeisis is an example of a system arising into a form of self-reference that has a lifetime due to the probabilisitic dynamics of its

Very best,
Lou Kauffman

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