[Fis] Can We Go Beyond the Limits of Formal Systems?

Wed Feb 7 05:25:30 CET 2024

Dear Karl,
This will be my last communication this week. I was not criticizing your structures. I was speaking of the use of formalism quite generally.
Goedelian limitations happen when the system contains its own logic and is rich enough to do number theory.
In mathematical practice we work with simpler systems that still embody limitations and may embody them even when embedded in larger systems with that logic.

A good example of this hands-on approach to limitations is the following argument.
Suppose that F_{1},F_{2},F_{3},… is any countable list of halting algorithms.
We can assume they are all written in some given programming language and each one can receive a natural number n 
and produce the result F_{k}(n) in a finite amount of time.
With this, I can define a new algorithm by the formula F(n) = F_{n}(n) + 1 that also halts and F is not equal to any F_{k}.
Thus any list of halting algorithms is incomplete.

This result applies in any formal system or computer language that can have such algorithms and is what lies in back of Turing’s Theorem that the 
Halting Problem is not decidable.

All these situations are part and parcel of the fact that formal systems show more than they can prove.
I do not expect to get beyond formal systems since they are just ways of writing down rules and assumptions that we intend to follow.
But we do need to realize that in actual mathematical and scientific practice one can be ready to not only work within certain formal systems, but one can
make new formal systems that are appropriate to the problems at hand.

There is no magic key to solving hard problems or predicting how structures might combine to form new structures.
At every junction we wait for new ideas or new distinctions that may occur through our experience, practice, observation or inspiration.
Best,
Lou
P.S. I will respond further next week.
PP.S. One of the things that we like to do is compare very different situations. Thus the incompleteness of infinite lists of algorithms resonates with the Euclid’s proof 
of the incompleteness of finite lists of primes. Let p1,…, pk be a finite list of prime numbers . Let N = p1x p2 x … x pk +1. Then N is not divisible by and pi and so N is
either a new prime or a product of new primes. The Euclid argument does not apply to infinite products. But Euler gave a different proof of the infinitude of the primes via 
the divergence of Sum_{n}(1/n) by writing it as an infinite product: Sum_{n}(1/n)  = Product_{p}(1-1/p)^{-1} where the product runs over all the primes. The product cannot diverge unless it is an infinite product and so there must be infinitely many primes. I imagine that one of Euler’s thoughts in coming to his new proof over 2000 years after Euclid
was wondering how the finite products of Euclid’s proof might be related to the infinite structure of series and convergence and divergence. Mathematical imagination works
both outside and inside given formalities, looking for relationships. The formal systems are the fallout of such imaginings and give us platforms from which to make further 
explorations. We live and think outside the formalities and we use the logic and formalities to structure, criticize and launch our own thought.

> On Feb 6, 2024, at 8:54 AM, Karl Javorszky <karl.javorszky at gmail.com> wrote:
> 
> Dear Lou,
> 
> (count = and ≠ differently, 2024 01 06)
> 
>  
> Thank you for the serious and elaborate restatement of principles we agree on. Your politely implied question was: What use tabulating numbers, we only see that as a result what we have set as rules?
> 
> The sceptics is well-reasoned. It is on me to show that this time it is different, that my shuffling of numbers is really, really something new and produces spectacles like Nature does.
> 
>  
> I specifically enjoyed your sentence:
> 
> it helps to look at the simpler examples where one does not have the ideal that the formal system might be rich enough to express everything including the logical structure of proofs and demonstrations.
> 
> because your ideas: a: start from simple, from bottom up, b. how rich (flexible) is the logical language to express complicated interdependencies (eg in AI or genetic) deal with central points of what we discuss here.
> 
>  
> Ad 1. Start from scratch
> 
> We address the conflict that (1,3) is on 3rd rank in the sequence (1,1),(1,2),(1,3),… but 4th rank in sequence (1,1),(1,2),(2,2),(1,3),…(which are sorts of a,b). Eq. 1
> 
> Like Pythagoras, we draw two axes in the sand and point out place x: 3, y: 4 as a logical dissolution of the conflict by giving it a solution in the next higher dimension.
> 
> Change in attitude required. So far, we use the Sumerian concept of Unit, comparable to a playing card one side saying “1” and the other side uniform. Up to today, we count the height of the stack of elementary units as the content of the message. The invention turns over the Sumerian playing cards and finds 136 different symbols etched on the reverse side. We play a completely different game, within the general game, based on secret signs that signify belonging to cooperators, if circumstances arise.
> 
> Do it your own way, don’t depend on others. Use only deictic definitions. Create a cohort of pairs of (a,b); a ≤ b;  a,b ≤ 16. We listen to chamber music performed by 136 individuals.
> 
> Ad 2. Flexibility of the language
> 
> We live in the correct impression that the world can be described by a language that uses unform units. The language itself does not hinder us to express differences and contradictions. The inner controversy demonstrated by Eq. 1. can easily be thematized and narrated.
> 
> Seen as a planar coordinate, the contradiction of Eq. 1 dissolves. The language is flexible enough to express the observed facts, if only we would observe the facts.
> 
> My point in answering you:
> 
> This is no joke. Neurology makes use of a numeric fact, which lies in fine details of how we count (perceive) ≠ before a background of =, and = before a background of ≠. If Nature uses two basic descriptive properties of a heap of input, we should try to do likewise. The argument that 0,1 are also two reference systems, is void, because 0,1 are symbols of an abstract nature, while ≠, = describe material differences that are gradated and within limits and thresholds.
> 
> It is nebbich a fact that one has more variants of space available to accommodate that many material variants that n objects can be incorporating, if and only if the objects number < 32 or > 97. We may like or not like the fact, but it remains true that the possibly existing variants of material properties of the objects will not fit in the number of available spatial segments which the objects can produce, if and only if the objects number 32 < n < 97.
> 
> We do not invent anything Nature would not have discovered. The syntax of the DNA is the best argument. Let your logical primitives exercise and the first thing they do is building sequenced threesomes of one of among four (restricted to one of two of two pairs),
> 
>  
> Dear Lou, thank you for your efforts to build bridges for me. The relative isolation of this new family of algorithms lies in the requirements you pointed out at the beginning. If something is that new, that an update on a+b=c is necessary, it is necessarily self-contained and uses deictic definitions. It is pure chance that the Eddington constants and the DNA support the hair-raising ideas transmitted here.
> 
>  
> We have a complicated rhetorical situation here. Let us grow to the challenge while we discuss that the central idea of information is that something is otherwise, that in the Sumerian system nothing ever can be otherwise, therefore the rational language appears to be inadequate to describe, discuss or express something that can not exist by its nature.
> 
> No reason to be overly pessimistic, Some nephews of Pythagoras have surely played with diverse toy soldiers, exercising them, and some say to have heard Pythagoras curse under his beard: what a horrible spell dampens my ability to tabulate and memorize! Elementary logical symbols allow recognizing elementary logical patterns if we perform elementary logical operations on them! I wish I could live in some 2600 years hence. I’d have computers and I’d show them what is a pattern.
> 
>  
> Friendly greetings:
> 
> Karl
> 
>  
>  
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