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

Mon Feb 5 23:30:40 CET 2024

Dear Karl,
Well put. Indeed formal systems are built into our language and culture.
Before we start talking about incompleteness in the Goedelian form 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. Consider the formality of ordinary arithmetic as we learned it in school.
You could take as the base formality for this the numerals |,||,|||,||||,… and the definitions of operations such as a + b = ab (juxtaposition of strings of bars) and 
a x ||…||| = aa…aaa (multiplication as duplication of one string by copies of another. This is (without the logic and various abbreviations) the formalization of elementary arithmetic. Note that we can prove things. E.g  || + || = |||| is the proof that 2+ 2 = 4, and || x ||| = || || ||  is the proof that 2 x 3 = 6.  After arithmetic we have algebra where a sign can stand for a numeral. And we can show algebraic identities such as a x ( b + c ) = a x b + a x c and prove theorems such as (a + b) x (a + b) = a x a + a x b + b x a + b x b.
And if you went down into these beginnings you might wonder just how you would prove that a x b = b x a (it is a good exercise). 

After enough language and concept has evolved you get good systems of numerals like the Arabic system that we use, and you can explore problems. 
A good problem, one that any grade school student can understand, is the Collatz Problem. 

Start with a number n,
If n is even, replace it by n/2.
If n is odd, replace it by 3n + 1.
Keep going unless you reach the number 1.
If you reach 1, STOP.

The problem is: Does the Collatz process always stop?

For example 7 —> 22 —> 11 —> 34 —> 17—> 52 —> 26 —> 13 —> 40 —> 20 —> 10 —> 5 —> 16 —> 8 —> 4 —> 2 —> 1.
Try 27. It does stop but goes quite high and for many terms.

Collatz Conjecture: The Collatz process always stops for any positive integer input.

No one has proved this yet and since we have defined it in the minimal formalism of elementary arithmetic, we have not said that it cannot be proved by some specific formal system. It might be just plain unprovable without making some assumption that is equivalent to the conjecture.

Before thinking about the Goedelian limitations, I suggest that one should look at actual mathematical problems like the Collatz. We face problems like this all the time where we see a phenomenon happen (in a certain formalism such as elementary arithmetic) and we do not have a proof, a demonstration or any understanding why this phenomenon happens. We would like to explain it in terms of how the system is built, in terms of basic ideas about the system. But we are perplexed.

It is this kind of perplex that drives mathematical investigation. We act just like scientists looking at the real world, but we are looking at what happens in the formalities of our 
calculations and patterns that follow certain rules. We find patterns that happen and we do not know how to explain them.

Sometimes we find an explanation. For example Euclid proved that there are an unlimited number of prime numbers back in 500 BC. It is easy to conjecture this because as you go counting up through the numbers you find that new primes keep occurring. Recall that a number is prime if it has no factorization into smaller numbers and is not equal to 1. Thus the first few primes are 2,3,5,7,11,13,17,19,23, … Euclid’s proof that there are an unlimited number of primes is very simple. Suppose that p1,p2,…,pk are primes
then the number N = 1 + p1 x p2 x … x pk leaves a remainder of 1 on division by p1or p2 or … or pk.. Thus N is either a new prime, or N has new prime factors. 
Play with Euclid’s argument. Use examples. After you do that you can say that you really do understand why there must be an unlimited number of primes. But many questions remain about just where the new primes occur and how often. There is a whole world to explore and the simple generating principles of arithmetic seem to be unable to explain everything. We explore the mathematics to find new principles just as much as to solve old problems.

If you have followed what I have said you realize that making up or discovering formalisms and then exploring what can be done with them is part of the depths of our human activity of language, sign symbol and calculation. Whatever we call thinking, indicating and communicating is tied up with this, and indeed it would not be easy to see what it
could mean to go beyond formalism.

On the other hand it is possible to go beneath formalism to those creative activities that generate new formalisms. This is also what mathematicians, theorists and poets and artists do. In mathematics itself there are many examples both contemporary and from the past. Good ones to start with are the invention/discovery of the number 0, the invention/discovery of the number i with i^2 = -1, the invention/discovery of non-Euclidean and differential geometry, the discovery of the quaternions and their relationship with and discovery of vector calculus. As you look at this prolixity you see how applications flourished in relation to these new departures.

It has been a topic in the discussion about the limitations of describing how biological systems move into new forms, evolve and become. We have little in the way of formal models for that, but we do see how intelligent human activity in the face of choices and possibility for invention is also prolix and not predictable by simple models.
It is so.

I say that the formalisms that we make are not meant to become worlds in which we restrict ourselves to live. But we have to understand that these worlds often exert 
a sufficient fascination on persons to lead them to devote all their energies in such directions. Thus a person with the talent to be a grandmaster at chess may well confine all her energy to the study of what goes on in that formalism for a 64 square board. A physicist may imagine that the world is all composed of mathematics. A mathematician may assert that mathematical structures are more real than everyday reality. And so it goes. Perhaps at this juncture it is indeed the biologists who can remind us that 
formalisms are cultural artifacts of the communication patterns of certain animals, based on their capacity for living.
Best,
Lou

> On Feb 5, 2024, at 10:23 AM, Karl Javorszky <karl.javorszky at gmail.com> wrote:
> 
> The formal systems are tools to structure perception with. Our counting, reasoning, self=certifying rational model of the world is like a headset into a virtual reality the rules of which we have learnt since childhood.
> 
> The formal systems we use are our culture. We cannot simply take off the viewer apparatus and see the world as a newborn. That experiment would hit the cns too hard. 
> 
> Being completely discultured allows looking completely new structures into the world. 
> (example : depraved child squints in order to maintain neuronal necessity of processing anything. Questions : a. Are the 2 pictures different, b. Is it information how and how much the 2 pictures caused by squinting are different?)
> 
> The paradigm change relates indeed to the limits of formal systems. That one which was introduced by the Sumerians has reached the limits of its utility. Everything that can be said by using 1 unit and multiples thereof has been already said or could be said easily. 
> 
> The general idea is in peril. We have to admit that it is useless to fabricate more models that use 1 degree of similarity and diversity in monocaustic, monoperspectivist assemblies.
> 
> The duality comes back to the intellectual arena. We have to learn to count in stereo. 
> 
> The end of the formal systems comes as we say:
> 1. Learn to conceptually accept that the 2 squinting representations of a picture are PRIOR to the unified one, 
> 2. Take off your helmet. What it channels are two strands of one stream. That is not stereo. Stereo is when you perceive by two sensory organs and you do the integrating. 
> 
> Duality is more archaic than uniformity. It has its own rules. Neurology processes signals based on contrasts, similarity and diversity. She does well so, because the basic mathematics of counting '=' before a background of '/=' is different to the counting of '/=' before a background of '='. 
> 
> As it happens, there is a paper of 26 pages called Update on a + b = c.
> 
>  https://urldefense.com/v3/__https://www.qeios.com/read/TRPD96__;!!D9dNQwwGXtA!SK1-cxWFdb98AopSRffPzPwKAwVt3UbGm5U-xNi8g9ajPUcdB0z7J3ZPqctRXqNm_SiOjsLDGGiebbOA$  <https://urldefense.com/v3/__https://www.qeios.com/read/TRPD96__;!!D9dNQwwGXtA!QuX6RS0q2172JFRJzaQVbrfBM5litLk-IQjCAPkuQdkUpRwbQp-NCQ8P-hmJP6PEezrJScXGgZEbOcEjkEBDwgvRBcI$>
> 
> The paper gives a down to earth tour d'horizon about formal systems and how to use the same old formal system in non-traditional ways. It's a piece of cake, really.
> 
> There is nothing non-respectful in making dots on planes from ranks in sequences, and from planes the construction of Descartes spaces is also an activity for a Sunday School. 
> 
> Yet, the murmuring voice in the headset "this is the correct reality and there is only one reality and it is completely and absolutely congruent within itself" is throwing a deep punch against the ability to think.
> 
> We approach and in fact present the idea of being otherwise and diverse. 
> 
> If something goes against the grains of Sumerians, then such ideas. 
> 
> That sits extremely deep. Good luck with unsolving childhood associations 
> 
> Try it by brain power. Force yourself by insight to reorder 12 books on your table, even if it is a viscerally revolting task. Unlearning childhood certainties is a complicated business. 
> 
> Karl 
> 
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