[Fis] Limits of Formal Systems

[Pedro C. Marijuán]

/(This message has general interest, and eliminating its attached docs 
it fits well into the list. Interested parties in these attachments may 
ask Lou directly--Pedro)/

El 13/02/2024 a las 9:05, Louis Kauffman escribió:

Dear Folks,

This message was too long to reach fis. I cannot reduce it and keep the 
content. So I am sending it for a private discussion.

I am interested in your notion Stu of jury rigging. You are interested 
in structures that have a global interrelationship and are not very 
likely to be deduced from some axioms although they may indeed occur in 
processes constrained by some axioms.  I am interested in structures 
that can occur under such constraints. This is what mathematics is often 
about for me. For example we have the Borommean Rings in topology (And 
of course a long history from the Borommean family to Ballentine been 
ads). Yellow surrounds Blue surrounds Red surrounds Yellow...

Below I show you an example of generalized Borommean rings. They are a 
kind of topological autocatalytic structure. Remove any ring and the 
whole thing comes undone, but together they are topologically linked. 
Now we topologists do not deduce these things. We invent them or 
discover them. And if you were to look around you could make more of 
them.Once we know about them we can talk learnedly about the SET OF ALL 
BOROMMEAN TYPE RINGS.  It can be a pastime to to try to find things like 
this. And then we enter a host of mathematical problems such as — how do 
you prove from topological axioms that these rings are indeed linked? 
That is something that can be deduced if you are clever enough. So we go 
back and forth between invention/discovery and deductive challenges. The 
prolixity of topological jury rigging is huge and I imagine that

If you wonder how we might make proofs about such linkedness, this paper 
will give you an introduction. High school students can understand the 
techniques and in fact I wrote this paper with Devika and Claudia when 
they were inhigh school in Chicago. But lets go over to biology and in 
particular to recombination and knotting, this kind of topology suitably 
generalized will be important in biology in the future.

DNA molecules can be knotted and this was proved by the molecular 
biologists Cozzarelli, Spengler and Stasiak by coating DNA with protein 
and using electron microscopy. This allowed one to SEE the weaving and 
to map the DNA via electron microscopy to the diagrammatic formal 
systems of the knot theory. You can draw a knot diagram corresponding to 
the self-entangled DNA. Then it is possible to use topological analysis 
consisting as I said of invention/discovery coupled with deduction to 
find about about the topology of this micro molecular world. 
Representing the DNA as a diagram, we can describe recombination as 
indicated below.

It is a GUESS that the replacement is a right-handed crossover as shown 
above. How can one tell if this guess is right? (See the articles 
above.) Experiments in processive recombination can be performed. If you 
did it with this form recombination the diagrams tell us that the 
results will be like this. A specific “spectrum” of knot types should 
emerge. One can look for these with the electron microscopy and in many 
cases the form of the recombination can be found by this topological 
analysis.

Now you should think about this. I have described one example of how 
people have managed to map the unruly biology to a formal diagrammatic 
topological system and how this could be used to understand and even to 
find out about basically unobservable processes in the biology (the form 
of the recombination). The mathematical formalism into which things are 
mapped contains all sorts of possibilities of its own for improbable and 
not deducible structures. And the formalism admits to deductive analysis 
if we are clever enough to find the ways to do it. There is a lot of 
this kind of topological work going on in molecular biology and in the 
study of protein folding. I see this as an exemplar of how in the future 
it will be possible to flexibly interface biological, topological and 
mathematical problems. I

In relation to the theme of the limits of formal systems, we see here 
that diagrammatic formal systems allow a very wide interaction among 
these different modes of working and if you were to look at the 
practices of topologists, graph theorists, combinatorialists, category 
theorists etc you would see that the sort of strait-laced limitations of 
the era of Russelian and Goedelian worries has been enfolded in a 
growing structure of expanding and interrelated formal systems that in 
their prolixity escape some of the problems of incompleteness. (But not 
all. See my previous email).

Best,

Lou

On Feb 12, 2024, at 1:16 PM, Louis Kauffman <loukau at gmail.com 
<mailto:loukau at gmail.com>> wrote:

>> Dear Folks,
>> Please examine the Kleene argument that you cannot list all 
>> algorithms (that halt) because you can form
>> F(n) = F_{n}(n) + 1.
>> This is exactly Cantor diagonal transposed to algorithms.
>> It is the core of incompleteness.
>> It is what it is.
>> You cannot sweep it under the rug.
>>
>> This argument fits into many different formal systems and once you 
>> place it there,
>> then the fact that all algorithms in a given formal system (not 
>> necessarily halting) can be listed (it is routine to check if some 
>> text is an algorithm, not routine to see if it halts),
>> shows that there can be no way in the formal system to decide whether 
>> algorithms halt. If you could do that, you could list all the halting 
>> algorithms and run into
>> a contradiction from the above.
>>
>> Thus (Turing)  the halting problem is undecideable in a wide class of 
>> formal systems.
>>
>> This part of the limitations of formal systems is just what it is.
>>
>> There is something else however and I would like to illustrate it 
>> with the Goldbach problem.
>> Try writing even numbers > 4 as a sum of two odd primes.
>>
>> 6 = 3 + 2
>> 8 = 5 + 3
>> 10 = 7 + 3 = 5 + 5
>> 12 = 7 + 5
>> 14 = 11 + 3 = 7 + 7
>> 16 = 13 + 3 = 11 + 5
>> 18 = 13 + 5 = 11 + 7
>> 20 = 17 + 3 = 13 + 7
>> 22 = 19 + 3 = 17 + 5 = 11 + 11
>> 24 = 19 + 5 = 17 + 7 = 13 + 11
>> 26 = 23 + 3 = 19 + 5 = 13 + 13
>> 28 = 23 + 5 = 17 + 11
>> 30 = 23 + 7 = 19 + 11 = 17 + 13
>> 32 = 29 + 3 = 19 + 13
>> …
>> These decomposition go up and down but the number of decompositions 
>> grows as the even numbers get larger.
>> There is some principle here that we are missing about how numbers 
>> get constructed.
>> The problem to prove that there is at least ONE way to write any even
>> number greater than 4 as a sum of two odd primes is completely open.
>> That is the Goldbach Conjecture.
>> It bet that 12 is the last time you get only one decomposition for an 
>> even number into two odd primes.
>>
>> It is possible that by thinking about how numbers are made
>> we will find new principles by which to reason about them and hence 
>> new formal systems, unknown at this time.
>> Any number theorist worth his or her salt is going to think about 
>> this. This means that the number theorist is thinking outside of the
>> known formal systems, trying to find new and better ways to work. 
>> This is normal. We need to promote the fact that creative thinking 
>> may use formal systems, but is
>> not limited to using only systems that already exist.
>> Best,
>> Lou
>>
>>
>>> On Feb 9, 2024, at 9:44 AM, eric werner <eric.werner at oarf.org 
>>> <mailto:eric.werner at oarf.org>> wrote:
>>>
>>> Dear Carlos,
>>>
>>> Notice you contradicted yourself:
>>>
>>> On 2/5/2024 4:43 PM, Carlos Gershenson wrote:
>>>> It is clear that models/descriptions will never be as rich as the 
>>>> modeled/phenomena, and that is the way it should be. As Arbib 
>>>> wrote, “a model that simply duplicates the brain is no more 
>>>> illuminating than the brain itself”. [1]
>>>
>>> On the one hand, you state that a model/description will never be as 
>>> rich as the model/phenomena it describes. And, then in the next 
>>> sentence you quote Arbib who presupposes that there could be a model 
>>> that duplicates the brain.  Duplicating the brain presupposes that 
>>> that model is as rich in structure and information as the thing it 
>>> models, namely the brain.
>>>
>>> Of course, Arbib is wrong. If we did have model that duplicates the 
>>> brain, then given the model is something like an LLM residing on my 
>>> laptop, that model would not only be as rich but richer in many ways 
>>> than the brain it modeled.  It would give us unprecedented insight 
>>> into the organization and function of its architecture.
>>>
>>> My point is that models are often richer than the object that is 
>>> modeled. They often have further dimensions that go beyond the 
>>> object modeled. This extra-dimensionality and richness enable us to 
>>> understand that object and utilize it.
>>>
>>> To be fair, you do state that computers can go beyond axiomatic 
>>> systems, implying perhaps that they can model phenomena that 
>>> axiomatic systems cannot. But I am skeptical of what appears to be 
>>> your wish to throw all axiomatic systems together and then get 
>>> meaning out of such a hodgepodge.
>>>
>>> I must admit I long for the beauty of mathematics and logic, the 
>>> crystalline world of truth, even if Goedel and other's seem to have 
>>> made a mess of it.
>>>
>>> And yet if you actually read Goedel's proof, reading as I did while 
>>> a student of Kleene (who was a student of Goedel's at Princeton) and 
>>> later teaching it as I did to undergrads, that proof is itself a 
>>> thing of beauty even if a bit messy.
>>>
>>> But Goedel did steal the core idea from Cantor's diagonal method. 
>>> And, someday we may find that Cantor's method is flawed, in yet 
>>> another higher dimensional mathematical space.  Which will bring us 
>>> back to the Greeks!
>>>
>>> Thank you for your contribution, Carlos, and remember this is all in 
>>> good fun,
>>>
>>> -Eric
>>>
>>> ****
>>> Dr. Eric Werner, FLS
>>>
>>>
>>>
>>> On 2/5/2024 4:43 PM, Carlos Gershenson wrote:
>>>> In the 1920s, David Hilbert's program attempted to get rid once and 
>>>> for all from the paradoxes in mathematics that had arisen from the 
>>>> work of Cantor, Russell, and others. Even when Hilbert’s PhD 
>>>> student — John von Neumann — was working avidly on demonstrating 
>>>> that mathematics were complete, consistent, and decidable, Kurt 
>>>> Gödel proved in the early 1930s that formal systems are incomplete 
>>>> and inconsistent, while Alan Turing proved in 1936 their 
>>>> undecidability (for which he proposed the "Turing Machine", laying 
>>>> the theoretical basis for computer science).
>>>>
>>>> Digital computers have enabled us to study concepts and phenomena 
>>>> for which we did not have the proper tools beforehand, as they 
>>>> process much more information than the one our limited brains can 
>>>> manipulate. These include intelligence, life, and complexity.
>>>>
>>>> Even when computers have served us greatly as "telescopes for 
>>>> complexity", the limits of formal systems are becoming even more 
>>>> evident, as we attempt to model and simulate complex phenomena in 
>>>> all their richness, which implies emergence, self-organization, 
>>>> downward causality, adaptation, multiple scales, semantics, and more.
>>>>
>>>> Can we go beyond the limits of formal systems? Well, we actually do 
>>>> it somehow. It is natural to adapt to changing circumstances, so we 
>>>> can say that our "axioms" are flexible. Moreover, we are able to 
>>>> simulate this process in computers. Similar to an interpreter or a 
>>>> compiler, we can define a formal system where some aspects of it 
>>>> can be modified/adapted. And if we need more adaptation, we can 
>>>> generalize the system so that a constant becomes a variable 
>>>> (similar to oracles in Turing Machines). Certainly, this has its 
>>>> limits, but our adaptation is also limited: we cannot change our 
>>>> physics or our chemistry, although we have changed our biology with 
>>>> culture and technology.
>>>>
>>>> Could it be that the problem lies not in the models we have, but in 
>>>> the modeling itself? We tend to forget the difference between our 
>>>> models and the modeled, between the map and the territory, between 
>>>> epistemology and ontology; simply because our language does not 
>>>> make a distinction between phenomena and our perceptions of them. 
>>>> When we say "this system is complex/alive/intelligent", we assume 
>>>> that these are inherent properties of the phenomenon we describe, 
>>>> forgetting that the moment we name anything, we are already 
>>>> simplifying and limiting it. It is clear that models/descriptions 
>>>> will never be as rich as the modeled/phenomena, and that is the way 
>>>> it should be. As Arbib wrote, “a model that simply duplicates the 
>>>> brain is no more illuminating than the brain itself”. [1]
>>>>
>>>> Still, perhaps we're barking up the wrong tree. We also tend to 
>>>> forget the difference between computability in theory 
>>>> (Church-Turing's) and computability in practice (what digital 
>>>> computers do). There are non-Turing-computable functions which we 
>>>> can compute in practice, while there are Turing-computable 
>>>> functions for which there is not enough time in the universe to 
>>>> compute. So maybe we are focussing on theoretical limits, while we 
>>>> should be concerned more with practical limits.
>>>>
>>>> As you can see, I have many more questions than answers, so I would 
>>>> be very interested in what everyone thinks about these topics.
>>>>
>>>> I'll just share some idea I've been playing with recently, although 
>>>> it might be that it won't lead anywhere. For lack of a better name, 
>>>> let's call them "multi-axiom systems". For example in geometry, we 
>>>> know that if we change the 5th axiom (about intersecting parallel 
>>>> lines), we can go from Euclidean to other geometries. We can define 
>>>> a "multi-axiom geometry", so that we can switch between different 
>>>> versions of the 5th axiom for different purposes. In a similar way, 
>>>> we could define a multi-axiom system that contains several 
>>>> different formal systems. We know we cannot have all at once 
>>>> universal computation and completeness and consistency. But then, 
>>>> in first-order logic, we can have completeness and consistency. In 
>>>> second-order logic we have universal computation but not 
>>>> completeness. In paraconsistent logics we sacrifice consistency but 
>>>> gain other properties. Then, if we consider a multi-axiom system 
>>>> that includes all of these and perhaps more, in theory we could 
>>>> have in the same system all these nice properties, but not at the 
>>>> same time. Would that be useful? Of course, we would need to find 
>>>> rules that would determine when to change the axioms. Just to 
>>>> relate this idea to last month's topic — as it was motivated by 
>>>> Stu's and Andrea's paper [2] — if we want to model evolution, we 
>>>> can have "normal" axioms at short timescales (and thus 
>>>> predictability), but at longer (evolutionary) timescales, we can 
>>>> shift axioms set, and then the "rules" of biological systems could 
>>>> change, towards a new configuration where we can use again "normal" 
>>>> axioms.
>>>>
>>>>
>>>>
>>>> [1] Michael Arbib, The Metaphorical Brain 2. Neural Networks and 
>>>> Beyond (1989)
>>>> [2] Stuart Kauffman, Andrea Roli. Is the Emergence of Life an 
>>>> Expected Phase Transition in the Evolving Universe? 
>>>> https://urldefense.com/v3/__https://arxiv.org/abs/2401.09514v1__;!!D9dNQwwGXtA!TiT7B6YivFo0EXKp-4E3_-zIHBGfXLNDTU9esPVKABtWTiTxCpdDhBtj_JjnZSE5sDe6v4tshwf5JFPdAAwtfwHdZcqo$  
>>>> <https://urldefense.com/v3/__https://arxiv.org/abs/2401.09514v1__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2RoZgOmc$>
>>>>
>>>>
>>>> Carlos Gershenson
>>>> SUNY Empire Innovation Professor
>>>> Department of Systems Science and Industrial Engineering
>>>> Thomas J. Watson College of Engineering and Applied Science
>>>> State University of New York at Binghamton
>>>> Binghamton, New York 13902 USA
>>>> https://urldefense.com/v3/__https://tendrel.binghamton.edu__;!!D9dNQwwGXtA!TiT7B6YivFo0EXKp-4E3_-zIHBGfXLNDTU9esPVKABtWTiTxCpdDhBtj_JjnZSE5sDe6v4tshwf5JFPdAAwtf03onWr-$  
>>>> <https://urldefense.com/v3/__https://tendrel.binghamton.edu__;!!D9dNQwwGXtA!Q9Wf2QzNb33Rbcm_rxf9I_P4EziZ3qwzNM9drNcS2M856SZcvJx6al-U8ZnYt5Fj0OfDWnNsNDd2yTKmSVg$>
>>>>
>>>>
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