In this art book(let) I want to collect a lot of information about dynamical systems and chaotic processes - and explain iterations by using methods of iteration - with the help of transparent foils. [It explains itself in the way/kind it explains itself, so to speak. [Self-referring recursive loop - I might leap into the topic of Gödel's Incompleteness Theorems and non-linear axiomatic systems occassionally... ] ]
I also want to portray cell division as concrete example - and I also want to include some ideas about Feynman diagrams and explain them as interaction patterns - step by step using transparent foils. --- to render a better understanding of this complex topic about dynamical systems and their axiomatization - and to explain my analogies and neologisms of my concept of "information weaving"
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When it comes to my really unexact approach of thinking and non-linear reasoning, I came to understand that I used the concept of Bayesian Inference since I can remember my own thinking processes. To validate or falsify assumptions in a probabilistic, yet logically consistent manner. - The difference between non-linear logic and irrational illogic is that non-linear logic's assumptions follow a logically consistent route when further investigation of thought is made. Irrational illogic is rather fallacy-based, statements are arbitrarily defined as true or wrong, no sufficient logical consistency can be found. - In non-linear logical reasoning many parallel paths of reasoning not just occur, but also interfere, hence, only a probabilstic description of truthness is possible, which changes with new data/measured evidence. It's neat how such statistical methods can help with some of my ADHD-induced fallacies and wrong application of generalizations upon other special cases, using some sort of metacognition.
[I think the autism side may be the part that helps me with sufficient pedantry in linearizing/concretizing and de-tangling the parallel reasoning processes into more logically consistent-appearing chunks of more linear thought paths. - making me return to certain details and dissecting the details as if they were big pictures themselves - to find logical consistencies inside that information clot...)}
We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mat
Abstract:
We study the logic of dynamical systems, that is, logics and proof principles for properties of dynamical systems. Dynamical systems are mathematical models describing how the state of a system evolves over time. They are important in modeling and understanding many applications, including embedded systems and cyber-physical systems. In discrete dynamical systems, the state evolves in discrete steps, one step at a time, as described by a difference equation or discrete state transition relation. In continuous dynamical systems, the state evolves continuously along a function, typically described by a differential equation. Hybrid dynamical systems or hybrid systems combine both discrete and continuous dynamics. This is a brief survey of differential dynamic logic for specifying and verifying properties of hybrid systems. We explain hybrid system models, differential dynamic logic, its semantics, and its axiomatization for proving logical formulas about hybrid systems. We study differential invariants, i.e., induction principles for differential equations. We briefly survey theoretical results, including soundness and completeness and deductive power. Differential dynamic logic has been implemented in automatic and interactive theorem provers and has been used successfully to verify safety-critical applications in automotive, aviation, railway, robotics, and analogue electrical circuits.
Published in: 2012 27th Annual IEEE Symposium on Logic in Computer Science
That is really interesting.
Just before I googled it, I tried to summarize my special interests - and it took a few hours, when I gave them the title axiomatization of dynamical systems. (Writing about how I came from my passion for polymath stuff and chaos theory to that would exceed the margin right now.)
Please bare with me, the next stated thoughts might be very ill-formulated gibberish at this stage of thought ...primordial soup or such heeh...
Axiomatization: definition in merriam-webster: the process of reducing down to a system of basic truths or axioms
Let's reduce that idea to basic language using some analogies: (It's very simplified and reduced, many details are omitted.)
Axioms are basic building blocks. Building blocks can be arranged and combined in all sort of ways, like a mosaic. You can chunk them and re-arrange them.
A dynamical system is a chaotic system that changes over time. Its compounds interact. These interactions correlate to the alterations in the dynamical system (compound = subordinate or detail, system = superordinate or whole/big picture)
-> (quantum) contextuality: [Quantum contextuality means the whole alters the details, as well as the details alter the whole. The system alters the compounds, and the compounds alter the system.] [Also: to understand quantum contextuality you can use the example of translating language: One word often has multiple meanings. It is ambigious. Which word is meant in that sentence? The right meaning of the word can be found when the context is analyzed. That is basically the principle behind quantum contextuality: (Also: Good read: https://www.quantamagazine.org/the-spooky-quantum-phenomenon-youve-never-heard-of-20220622/ )] {Further idea in my brain: What is the role of quantum contextuality in quantum error correction algorithms such as Bayesian Optimization?}
Well, back to axiomatization:
A system is a collection of interwoven building blocks. Axiomatization is a process of segmentation, of chunking, and of de-tangling the whole into its basic building blocks.
Dynamical systems are emergent, which means their working mechanisms alter themselves and emerge collectively.
