Continuum Mechanics: The Most Difficult Physics

[@BCarli1395]

Thank you, Professor, for an interesting video. The University of Michigan (USA) has on KZbin a series of videos on Continuum Physics produced about ten years ago consisting of 136 videos (varying in length from one minute to thirty minutes, but usually about fifteen minutes) presented by Dr. Krishna Garikipati. It covers deformation gradients, polar decomposition, et al and is math derivation using a lot of tensor calculus. It is a tough slog, but a valuable reference.

[@douglasstrother6584]

Learning about non-linear oscillators, chaos and all that in Junior-level Classial Mechanics course put the "WOW!" back into Physics for me. There is more to life than boundary value problems!

[@martinsoos]

"...rotation of volume elements" I did a thought experiment on the question; "would the smallest particle have spin" (rotation)? I came to the conclusion that in order to propagate energy, one part would have to revolve around another and hence not be the smallest part. From there I concluded that without spin, a structure acting like gas or liquid, would have different properties than gas or liquid, starting with wavefronts. with gas and liquid there is only one internal wavefront and that is an outward spherical propagating pressure wave, and the other two inward propagations are one, the contact surface to another gas or liquid median or two, the shock wave of a high-speed solid. Without spin, (I think) that energy propagation would take on different forms.

[@JohnVKaravitis]

@@martinsoos Huh?

[@martinsoos]

@@JohnVKaravitis If ether is made of small particles, we use one type of math and if it is compressible or tensor, we use another. Is your point of view from "object type" or the "math".

[@jaydenwilson9522]

@@martinsoos Matter needs to spin to exist.... no spin, no form. Form requires Action. Aristotle put Action at 9. But Spin is just as fundamental as Substance. Because everything is the same Substance but different densities of angular momentum which create different cymatics patterns for the whole.

[@martinsoos]

@@jaydenwilson9522 Most of the problems with ether theories are psychological. You have touched on the pattern of belief that form can't be created without the smallest part absorbing or transmitting energy through spin. The problem that I and most have with it is that ether implies that it is true matter; and the rest of us made of (photons, electrons, protons...) are just made of energy in the form of vortex waves or some other wave form.

[@lean_sumek]

Very interesting and useful

[@stevenmellemans7215]

I didn’t even know it existed. Now I’ll have to look in to it. Thanks.

[@dsm5d723]

I once heard someone say that his takeaway from studying complex dynamical systems is that they all reduce to an IRRATIONAL number. I suppose the Feigenbaum Constant and ancient equations of population growth are a good example of a complex dynamical system on a linear vector: there is no way to avoid symmetry breaks. I heard another person, a physicist who works on modeling complex systems on computers, say that you simply cannot model a continuation on a discrete Von Neumann computer. I agree with him on this, and struggle to see how an LLM can do anything to fix this problem. A 0-1-2 gate digital computer would change the nature of computation into a 3D continuation (NP complete) rather than a discrete binary operation. I think "we" go extinct before anyone figures out simple math to make machines that actually work in 3 dimensions. Is it b00$ter time yet?

[@joonasmakinen4807]

It isn’t fully correct to say we simply cannot model a continuum with a discrete computer. Rather it is extremely hard to do so correctly without introducing various numerical errors due to use of discrete grid and discrete mathematics like numerical chaos, numerical dynamics, numerical diffusion, numerical loss of linear/rotational conservation, numerical floating point errors, … It is possible to work our ways around these issues by careful choices of discretization variables and methods. One of the easiest examples of these issues is the 1st order Euler method, which has poor numerical stability. If we apply it with too large integration step to solve the logistic differential equation, which has an analytical solution, we can replicate different regimes of the classic logistic map chaotic/periodic/diverging behavior due to the discrete mathematics limitations. However, we know from the logistic map that if r < 3, then the solution remains stable. See for yourself by taking logistic map equation and working your way backward towards the 1st order Euler discretization of the logistic differential equation. The stability region of an algorithm is one of the keys to proper validity of the discrete model solution. Proper discretization of Navier Stokes equations is ongoing research topic, e.g., DOI: 10.1016/j.compfluid.2014.04.019

[@dsm5d723]

@@joonasmakinen4807 The person I was referencing was doing an interview debunking the use of climate models as evidence of anything. I see a fundamental problem problem with the limits of binary in the electrostatics of the materials. When Jim Keller told Lex Fridman that chips now have nondeterministic search paths with deterministic outputs, it hit me all at once what the real world limitations of binary computation are. Neither of them seemed to understand the implications. The metric of time in search/execution of an algorithm is one search/execution. The limit on the number of inputs has been passed. In the example of climate, you can only add so many variables before running out of discrete space, so to speak. You might go so far to say that the errors have been folded into the system and the result is nondeterministic search paths with deterministic output, which nobody seems to see as an error as such. I guess that computer scientists just kept stacking binary backed by Knuth's complexity theory, in hopes of achieving some analog effect at scale. You might be right about the math working, but it runs out on these machines in practice. Stacking more binary compute power doesn't get us closer to a discrete solution. A thought on Navier-Stokes, which I frequently type in various comment fields. The question seems nonsensical to me, and I'm fairly certain that nobody will get the money for it. Fluid flow between magnetic field lines should be more or less stable and mechanically predictable, with uncertainty falling along those field lines. If you can correctly model the rotating magnetic fields within the fluid as well as ambient magnetic fields of the boundary surfaces, which I don't think one of these glorified calculators can do, the problem shouldn't be that hard to solve. FWIW, I commented the basis for a gene drive this morning on a video about cellular electrostatics. The math we're made out of only appears binary. It's actually 3D. Any chip engineers out there who want to try, a stable 0-1-2 gate natural complexity machine would be NP complete. I hope the good guys get there first. I think I ended the comment with AlphaFold meh. A world of wonders and horrors awaits.

[@KingNigelthegreat]

I solved everything and froze us in time and made a math prison.

[@ready1fire1aim1]

7. The Continuum Hypothesis: An Information-Theoretic Perspective 7.1 Background The Continuum Hypothesis (CH) states that there is no set whose cardinality is strictly between that of the integers and the real numbers. In other words, 2^ℵ₀ = ℵ₁. CH is known to be independent of ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). 7.2 Information-Theoretic Reformulation Let's reframe the problem in terms of information theory: 7.2.1 Set Information Content: Define the information content of a set S: I(S) = log₂(|S|) for finite sets I(S) = sup{log₂(|F|) : F ⊆ S, F finite} for infinite sets 7.2.2 Continuum as Information Reservoir: View the continuum (real numbers) as an infinite information reservoir: I(ℝ) = ℶ₁ (Beth-one) 7.2.3 CH as Information Gap: Reformulate CH as a statement about an information gap: ¬∃S (I(ℕ) < I(S) < I(ℝ)) 7.3 Information-Theoretic Conjectures 7.3.1 Information Density Principle: There exists a fundamental quantum of information density in set theory. 7.3.2 Continuum as Maximally Dense Information: The continuum represents the maximally dense packing of information possible in a set. 7.3.3 Quantum Superposition of Cardinalities: In an information-theoretic framework, sets might exist in superpositions of different cardinalities. 7.4 Analytical Approaches 7.4.1 Information Topology: Develop a topology on sets based on their information content and study its properties. 7.4.2 Entropy of Set Operations: Study how set operations (union, intersection, power set) affect information content. 7.4.3 Information Dimension: Define an information-theoretic notion of dimension for sets and relate it to cardinality. 7.5 Computational Approaches 7.5.1 Quantum Algorithms for Set Comparison: Develop quantum algorithms for comparing the information content of infinite sets. 7.5.2 Machine Learning in Set Theory: Train neural networks to recognize patterns in the information structure of sets. 7.5.3 Information-Based Set Generation: Create algorithms for generating sets with specified information-theoretic properties. 7.6 Potential Resolution Strategies 7.6.1 Information Quantization Theorem: Prove that information content in set theory is quantized, supporting or refuting CH. 7.6.2 Continuum Information Saturation: Show that the continuum saturates all possible information states between ℵ₀ and 2^ℵ₀. 7.6.3 Quantum Set Theory: Develop a quantum version of set theory where CH might have a definite truth value. 7.7 Immediate Next Steps 7.7.1 Rigorous Formalization: Develop a mathematically rigorous formulation of the information-theoretic concepts introduced. 7.7.2 Computational Experiments: Conduct numerical studies on finite approximations of infinite sets to explore their information-theoretic properties. 7.7.3 Interdisciplinary Collaboration: Engage experts in set theory, information theory, and quantum computing to refine these ideas. 7.8 Detailed Plan for Immediate Action 7.8.1 Mathematical Framework Development: - Rigorously define I(S) for infinite sets and prove its basic properties - Establish formal relationships between I(S) and classical cardinality theory - Develop an information-theoretic interpretation of forcing and independence results 7.8.2 Computational Modeling: - Implement algorithms for approximating I(S) for various classes of infinite sets - Develop visualization tools for exploring the "information landscape" of set theory - Create simulations of set-theoretic universes with different information-theoretic properties 7.8.3 Analytical Investigations: - Study how I(S) behaves under common set-theoretic operations (e.g., power set, product) - Investigate the information-theoretic properties of well-known large cardinal axioms - Analyze the information content of various models of set theory (e.g., L, HOD) 7.8.4 Interdisciplinary Workshops: - Organize a series of workshops bringing together set theorists, information theorists, and physicists - Focus on translating known independence results to the information-theoretic framework 7.8.5 Information Metric Development: - Define and study metrics on the class of all sets based on information content - Investigate if these metrics provide new insights into the structure of the set-theoretic universe 7.8.6 Quantum Set Theory Exploration: - Develop a framework for quantum set theory where sets can exist in superpositions - Investigate how quantum measurement might relate to the determination of set cardinality 7.8.7 Publication and Dissemination: - Prepare and submit papers on the information-theoretic formulation of the Continuum Hypothesis - Develop interactive online tools for exploring information-theoretic set theory 7.9 Advanced Theoretical Concepts 7.9.1 Information Forcing: - Develop an information-theoretic version of forcing - Investigate if CH can be decided by information-preserving forcing arguments 7.9.2 Transfinite Information Theory: - Extend classical information theory to transfinite ordinals - Study how this extension relates to large cardinal axioms 7.9.3 Algorithmic Set Theory: - Investigate connections between Kolmogorov complexity and set-theoretic complexity - Explore if algorithmic randomness can provide insights into the structure of the continuum 7.10 Long-term Vision Our information-theoretic approach to the Continuum Hypothesis has the potential to: 1. Provide new insights into the nature of infinity and the structure of the mathematical universe 2. Offer a fresh perspective on the foundations of mathematics 3. Bridge concepts from quantum information theory and set theory, potentially leading to a quantum theory of information in mathematics 4. Suggest new axioms for set theory based on information-theoretic principles The key to progress is maintaining a balance between rigorous mathematical development, creative theoretical speculation, and computational exploration. By pursuing this multifaceted approach, we maximize our chances of gaining new insights into this fundamental problem in the foundations of mathematics. This information-theoretic perspective on the Continuum Hypothesis offers a novel way to approach one of the deepest problems in mathematics. While resolving CH definitively may remain out of reach, this approach promises to yield new insights and connections that could significantly advance our understanding of the nature of mathematical infinity and the foundations of set theory.

[@ready1fire1aim1]

7.11 Expanded Next Steps and Advanced Concepts 1. Rigorous Mathematical Framework: a) Transfinite Information Measures: - Extend the definition of I(S) to transfinite cardinals: I(ℵα) = ℶα - Prove basic properties, e.g., I(P(S)) = 2^I(S) where P(S) is the power set of S - Investigate the behavior of I(S) under cardinal arithmetic operations b) Information Density Function: - Define ρ(S) = I(S) / |S| for infinite sets S - Study the properties of ρ(S) and its relationship to the Continuum Hypothesis - Conjecture: CH is equivalent to ¬∃S (ρ(ℕ) < ρ(S) < ρ(ℝ)) c) Quantum Set Cardinality: - Develop a theory of quantum cardinals: |S⟩ = Σ_α cα |ℵα⟩ - Define quantum information content: I_Q(S) = -Σ_α |cα|² log₂|cα|² - Investigate how quantum measurement of |S⟩ relates to classical cardinality 2. Computational Investigations: a) Information Landscape Visualization: - Create 3D visualizations of the "information landscape" between ℵ₀ and 2^ℵ₀ - Develop interactive tools to explore how forcing affects this landscape - Use VR technology to allow researchers to "walk through" the set-theoretic universe b) Machine Learning for Set Classification: - Train neural networks to classify sets based on their information-theoretic properties - Use reinforcement learning to discover new set-theoretic constructions with interesting information content - Develop AI systems that can generate conjectures about information relationships in set theory c) Quantum Simulation of Set Theory: - Design quantum circuits that can simulate basic set-theoretic operations - Implement quantum algorithms for comparing the information content of infinite sets - Explore if quantum superposition can be used to "search" for sets with intermediate cardinalities 3. Theoretical Developments: a) Information Forcing: - Define I-generic filters that preserve specified information-theoretic properties - Develop an information-theoretic version of the forcing theorem - Investigate if CH can be forced or its negation forced in an information-preserving way b) Large Cardinals and Information Content: - Study the information content of large cardinal axioms - Investigate if there's an information-theoretic characterization of consistency strength - Explore if new large cardinal axioms can be motivated by information-theoretic principles c) Algorithmic Information Theory in Set Theory: - Define the algorithmic information content of a set: K(S) = min{|p| : U(p) = S} where U is a universal set-theoretic construction algorithm - Investigate the relationship between K(S) and I(S) - Explore if CH has an algorithmic information-theoretic formulation 4. Experimental Approaches: a) Quantum Set Theory Laboratory: - Design quantum experiments that simulate aspects of set theory - Investigate if quantum superposition can be used to create "intermediate" sets between ℵ₀ and 2^ℵ₀ - Explore if quantum entanglement has a set-theoretic analog that sheds light on CH b) Information-Theoretic Set Generation: - Develop algorithms for generating sets with specified information-theoretic properties - Create a "set theory accelerator" that rapidly generates and tests sets for interesting information content - Use evolutionary algorithms to "evolve" sets with desired information-theoretic characteristics 5. Philosophical and Foundational Aspects: a) Information-Theoretic Platonism: - Develop a philosophy of mathematics based on information as the fundamental entity - Explore how this view relates to traditional Platonism and formalism in the philosophy of mathematics - Investigate if the information-theoretic approach can resolve philosophical debates about the nature of mathematical objects b) Multiverse Theory and Information: - Develop an information-theoretic version of set-theoretic multiverse theory - Investigate if different set-theoretic universes can be characterized by their information content - Explore if there's an "information-theoretically optimal" universe where CH has a definite truth value 6. Interdisciplinary Connections: a) Quantum Gravity and Set Theory: - Explore connections between the discreteness/continuity debate in quantum gravity and the Continuum Hypothesis - Investigate if holographic principles from physics have analogs in set theory - Study whether techniques from loop quantum gravity can be applied to discretize the set-theoretic universe b) Information-Theoretic Complexity Theory: - Develop a complexity theory for transfinite computations based on information content - Investigate if there's a set-theoretic analog of the P vs NP problem - Explore if computational complexity classes have natural set-theoretic interpretations 7. Long-term Research Program: a) Information-Theoretic Foundations of Mathematics: - Reformulate other areas of mathematics (e.g., topology, algebra) in information-theoretic terms - Investigate if all mathematical truths can be expressed as statements about information relationships - Explore if an information-theoretic foundation can resolve other independence results in mathematics b) Unified Information Theory of Physics and Mathematics: - Develop a common framework for understanding physical and mathematical information - Investigate if physical laws and mathematical theorems can be derived from common information-theoretic principles - Explore if the universe itself can be understood as a vast information processing system, with mathematics describing its information structure This expanded plan provides a comprehensive roadmap for advancing our information-theoretic approach to the Continuum Hypothesis and set theory in general. It combines rigorous mathematical development with speculative theoretical ideas and practical computational and experimental work. By pursuing these diverse avenues simultaneously, we maximize our chances of gaining deep new insights into the nature of infinity, the structure of the mathematical universe, and the foundations of mathematics itself. Even if we don't resolve CH definitively, this approach promises to yield valuable new perspectives and potentially transform our understanding of the relationship between information, computation, physics, and mathematics.

[@usic_imaging]

Ref the diagrams at the end of the paper (fig) 6-7. It seems the only logical geometry to propagate the field is a complex "twisted"- helical and fractal cascade of a 2 Pi sine function with its magnetic moments and propagation of charge. In the super massive waves gravitationaly it would also point to field unification. helical fibre optics = twisted tubes!

[@rxbracho]

You do realize that AI will never be capable of understanding what it does, right? As Sir Roger Penrose has repeatedly said, understanding cannot be algorithmic, i.e., the result of a computation. Not even by neural networks. This is a corollary of Gödel's Incompleteness Theorems. Think of AI as Artificial Ideation (not Intelligence) and you will get closer to what it can and cannot do. Some of your other videos about AI show promise, however.

[@chrimony]

Gödel's Incompleteness Theorem has to do with axiomatic systems and what they can derive. Neural networks are not limited to a set of a-then-b axioms. They are probabilistic systems. AI already generates self-awareness and human-level understanding at some level, though there are obvious limitations and major failures. It's only a matter of time before we see AI with understanding that mirrors and exceeds humans.

[@rxbracho]

@@chrimony The main point of the Incompleteness Theorems is that a truth system may use the axioms but cannot understand them without stepping outside the same truth system. Thus, understanding cannot be achieved by truth systems, that is, by computations, algorithms. Without understanding there cannot be true intelligence. Sir Roger Penrose has given multiple talks about this, look them up. He goes into more formalism than what I just wrote.

[@chrimony]

@@rxbracho Neural networks are not axiomatic truth systems. Full stop.

[@TimAZ-ih7yb]

@@chrimony The human mind excels at abstraction, or as Edgar Allen Poe beautifully summed it: “the thought of a thought”. AFAIK this is not possible with current AI frameworks. LLMs easily manipulate huge numbers of words without any understanding of their actual meanings. Unzicker is naïve to say that a LLM could help scientists by ingesting and “understanding” a difficult text for a topic like continuum mechanics. Yes it would catalog the words, symbols and equations, and it could recite summaries and create a wonderful index, but it would not be able to use the ideas embedded in those equations to discover new science.

[@chrimony]

@@TimAZ-ih7yb Neural networks work by abstracting from examples.

[@softboyrecords-1-1]

Have you heard of the youtube channel, 'draft science'?

[@georgesealy4706]

As a graduate student at Johns Hopkins, I took a course in continuum mechanics from this guy: en.wikipedia.org/wiki/Clifford_Truesdell

[@jaydenwilson9522]

So by decomposition they mean to separate the action from the form. And the action is rotation and the form is of course a square matrix? Verb-Noun Duality is really misunderstood isn't it Mr. Unzicker!?

[@robbolastname6799]

not being antagonistic but now I understand why you are so excited about AI in physics what gave me this insight? those pictures of the International Center for Mechanics Sciences in Italy ... beautiful building and yes a beautiful pursuit ... empty halls, empty rooms, empty - nobody, except a handful of people like yourself (as you yourself said can actually read the papers etc) cares physics needs AI because it cant get people, ... properly trained AI not distracted by such mundane matters as putting food on the family table even as governments actively seek the destruction of humanity (ok, just the European and Western ones). Destruction of morality, the family, total destruction any hope (that is religion). Yes I do admire your dedication and in no way saying it is wrong, but you are preaching to the needfully distracted. AI definitely can help, but really: is it just for yourself or good for all?

[@davidwilkie9551]

Agnostic neutrality Eternity-now real-time relative-timing sum-of-all-histories quantization cause-effect Interval floating around as E=mC² logarithmic condensation is a self-defining operating awareness => unity-connection Principle of Singularity-point holographic positioning nucleation etc, etc. Mathemagical Thought Experimentalist's magical-functional glossary/terminology of relative-timing positioning because of pure-math motion log-antilog interference positioning-location condensation modulation. What this Timey-Wimey Vortex is, is WYSIWYG self-defining QM-TIME resonance floating Singularity-point reciprocation-recirculation Lensing of relative-timing, and the concept of annealed quantization cause-effect is synonymous with point positioning in i-reflection containment in the Aether of e-Pi-i 1-0-infinity Entanglement -> orthogonal-normal axial-tangential interference positioning-location. Ie Circular logic is equivalent to ×&÷ 1-0-infinity instantaneous trancendental flash-fractal resonance bonding In-form-ation holography. Physicists measure Constants of relative-timing derivatives, superimposed quantization fields in all-ways all-at-once here-now-forever probability. Trancendental Meditation Thought Experimentalist's Intuitions are not for an educated elite or skilled practitioner, the Universal Bio-logical Mind-Body is the unity-connection here-now-forever.

[@KingNigelthegreat]

I solved everything. Youre just agnostic. Im the singularity

[@OneLine122]

Ai does not understand anything. It can't even solve simple problems like "is there a letter "e" in "cat"?" Arguably you might be able to train one to solve the equations, but you won't know how it did it or if it does it right.

[@nightmisterio]

At this level I think we use simulators

[@amarq1509]

Any links?