Full podcast episode: ukposts.info/have/v-deo/h6OZZGOBaopjz40.html Lex Fridman podcast channel: ukposts.info Guest bio: Edward Frenkel is a mathematician at UC Berkeley working on the interface of mathematics and quantum physics. He is the author of Love and Math: The Heart of Hidden Reality.

I love that he mentioned Alan Watts, who had the best description of Goedel’s Theorem: “No system can define all of its own axioms.”

This is the first time I've been introduced to this guy. I like how he seems to be more of a "unification of knowledge" type of person, rather than just a mathematician. He draws from examples everything from math, to pop-culture, to eastern and western philosophy, and so on. Thanks again Lex!

This guy might be the best guest you have had on Lex I love this dude

Here are brief statements of the theorems for those interested: Gödel's First Incompleteness Theorem states that "Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true, but not provable within that theory." Gödel's Second Incompleteness Theorem states that "For any effectively generated formal theory T including basic arithmetical truths and certain truths about formal provability, T includes a statement of its own consistency if and only if T is inconsistent." Just prior to publication of his incompleteness results in 1931, Gödel already had proved the completeness of the First Order logical calculus; but a number-theoretic system consists of both logic plus number-theoretic axioms, so the completeness of PM and the goal of Hilbert's Programme (Die Grundlagen der Mathematik) remained open questions. Gödel proved (1) If the logic is complete, but the whole is incomplete, then the number-theoretic axioms must be incomplete; and (2) It is impossible to prove the consistency of any number-theoretic system within that system. In the context of Mr. Dean's discussion, Gödel's Incompleteness results show that any formal system obtained by combining Peano's axioms for the natural numbers with the logic of PM is incomplete, and that no consistent system so constructed can prove its own consistency. What led Gödel to his Incompleteness theorems is fascinating. Gödel was a mathematical realist (Platonist) who regarded the axioms of set theory as obvious in that they "force themselves upon us as being true." During his study of Hilbert's problem to prove the consistency of Analysis by finitist means, Gödel attempted to "divide the difficulties" by proving the consistency of Number Theory using finitist means, and to then prove the consistency of Analysis by Number Theory, assuming not only the consistency but also the truth of Number Theory. According to Wang (1981): "[Gödel] represented real numbers by formulas...of number theory and found he had to use the concept of truth for sentences in number theory in order to verify the comprehension axiom for analysis. He quickly ran into the paradoxes (in particular, the Liar and Richard's) connected with truth and definability. He realized that truth in number theory cannot be defined in number theory, and therefore his plan...did not work." As a mathematical realist, Gödel already doubted the underlying premise of Hilbert's Formalism, and after discovering that truth could not be defined within number theory using finitist means, Gödel realized the existence of undecidable propositions within sufficiently strong systems. Thereafter, he took great pains to remove the concept of truth from his 1931 results in order to expose the flaw in the Formalist project using only methods to which the Formalist could not object. Gödel writes: “I may add that my objectivist conception of mathematics and metamathematics in general, and of transfinite reasoning in particular, was fundamental also to my work in logic. How indeed could one think of expressing metamathematics in the mathematical systems themselves, if the latter are considered to consist of meaningless symbols which acquire some substitute of meaning only through metamathematics...It should be noted that the heuristic principle of my construction of undecidable number theoretical propositions in the formal systems of mathematics is the highly transfinite concept of 'objective mathematical truth' as opposed to that of demonstrability...” Wang (1974) In an unpublished letter to a graduate student, Gödel writes: “However, in consequence of the philosophical prejudices of our times, 1. nobody was looking for a relative consistency proof because [it] was considered that a consistency proof must be finitary in order to make sense, 2. a concept of objective mathematical truth as opposed to demonstrability was viewed with greatest suspicion and widely rejected as meaningless.” Clearly, despite Gödel's ontological commitment to mathematical truth, he justifiably feared rejection by the formalist establishment dominated by Hilbert's perspective of any results that assumed foundationalist concepts. In so doing, he was led to a result even he did not anticipate - his second Incompleteness theorem -- which established that no sufficiently strong formal system can demonstrate its own consistency. See also, Gödel, Kurt "On Formally Undecidable Propositions of Principia Mathematica and Related Systems I" Jean van Heijenoort (trans.), From Frege to Gödel: A Sourcebook in Mathematical Logic, 1879-1931 (Harvard 1931)

Edward Frenkel is so extremely lucid, just extraordinary.

This was one of my favourite shows from Lex. Edward is a truly remarkable human being, and it's always beautiful to see so much love and compassion in one's heart.

Great show. I really love Frenkel. He is so clear and his enthusiasm and sense of wonder is infectious!

A full 14 minute in depth explanation of goedel's impossibility theorems, and then lex goes "so every why has a definite answer"

I think it's valuable to also go a bit into the whole "completeness & consistency" thing. One could start with definitions and explaining why they're important things we want from formal systems. Then one could proceed to a little history of how "cracks" in set-theory based formal systems began to be discovered by Frege and Russel almost as soon as those systems arose. The story continues with a quick overview of the various approaches to these issues, like ZF(C), NBG, "New Foundations" and type theory (with Russell for a while, then dormant for a long time, then getting a big comeback with Per-Martin Löf and lots more interest recently with Homotopy Type Theory). This brings us to a classification and analysis of the underlying issue - that of predicativity and impredicativity - one might briefly explain what that is and why it's problematic - using various examples of paradoxa of (direct or indirect) self-referentiality. We can then explain how these developments and the predominant research institutions in Germany and Eastern Europe lead to Gentzen's proof of the consistency of (Peano) arithmetic - and how that was a process of formalization which took us around 2.5 millennia from basic arithmetic and logic to Gentzen's proof. The importance could hardly be overstated. The "victory march" of formalization and the power of formal systems seemed assured. ... and then came Gödel.

What a great an inspiring guy Mr. Frankel is. It’s simply great learning from his talks.

Gödel's theorems challenged the notion of completeness and consistency within formal systems and had a profound impact on the philosophy of mathematics. They demonstrate inherent limitations of formal systems and suggest that there will always be truths that lie beyond the reach of any particular system. These theorems have also influenced the field of computer science, particularly in the areas of artificial intelligence and algorithmic complexity theory.

I love his comment on the findings of Godel and Turing being "life affirming". Very well said.

In another paper, Gödel developed an axiomatic system containing the self-referential statement, “This statement is false.” He then proved - within the same system -that “This statement is false” is true. All he needed was the countable numbers (the set N) and a few very simple rules. On “Emergence:” Taking simple rules then applying them to a simple structure to produce complex “behaviour,” is also a subjective process. In what axiomatic system can you consistently define both “simple” and “complex,” then show that there are no self-referential contradictions?

this is a Thankyou to Lex and his team for committing to in-person interviews with high quality audio equipment. It makes all the difference to the audience experience.

Ed Frenkel is one of my favourite people. His book is fantastic.

Wow!! Great explanation of incompleteness! The best I have seen so far!

Greatly appreciated broadcast. Thanks for sharing! Ex-Maths teacher. Have a GOOD weekend!

Reminded me about a study of the mona lisa smile, where they discovered that the smile appears to be more obvious if we use our peripheral view, so that its way she sometimes appear to smile but then again, she does not. Maybe some of those trick images may have an explanation that is more complex than subjective.

I love this stuff. This channel never seems to disappoint.

Great video, people take calculus and algebra classes for years and no one explains to them the fundations of what you are studying as clear as this guy does

Great explanation! Regarding the perception problem at 14:52 , the top down perception in the brain can provide a trivial explanation. Just like Lex mentioned for neural networks, the bottom up sensory features leads two activated outputs: 0.5 Rabbit and 0.5 Duck. However, the top down awareness in the human brain can only attend to one output at time. So if you attend to the duck output, the duck neuron will be activated. Now because the information comes from top to bottom, all the related neurons to Duck will activate (none will activate for the Rabbit). And that is why you suddenly perceive it as 100% Duck or 100% Rabbit if your top down awareness attend to the Duck or vice versa.

‘We’re still feeling the tremors today.” Wow.

There are some nice ideas about emergence of complexity. As nothing is the same, there is an incremental effect by repetition, similar to what our memory in the brain does with the episodic memory, each time we see a cat for example, the cat experience adds meaning to our definition of cat, even if it is the same cat at the same place. A bit like Peircean semiotics thirdness, when we interpret a sign it can generate a new one, even more if instead of just one triadic relationship there is a whole network of it, by aggregation and interconnection, at some point it generates more complexity of evolving meaning, because it cannot be the same, different than in mathematics. In mathematics if we add 1 + 1 it is always 2, but in reality that is impossible, and the 2 will be always slightly different each time we add 1+1. In short, complexity has to emerge because repetition is impossible.

Excellent interview

I really liked when he jst softly said the key to genius . " To have open end process " to let yr conscious intelligence lead . Rather than deciding one thing and another

Lex on Godel's incompleteness and Turing's undecidability: _"It's very depressing."_ Frenkel: _"Or life affirming!"_ Edward is ahead of Lex in spiritual development. When we embrace that we will never attain a _Theory of Everything,_ it opens up greater possibilities!

Thanks for coming up with this interesting topic with this brilliant guy..❤🎉

I wish my father had teached me mathematics like him. So calm and collected makes it easy😢😅

Editing a foundational part of the whole interview✨🙏

I've been waiting for this!

"I used the formal system to destroy the formal system." - Thodel

Frenkel is probably my favorite guest of all time I’ve rewatched the interview many times

I shit my pants when he tied everything to complementarity. The golden thread of Platonism. Would love to hear this guy talk with Joscha Bach, John Vervaeke, Max Tegmark, Penrose, Graham Priest, or any other modern great who understands and promotes this principle.

What is the name of the knot displayed in the title?

Loops are similar to self referential statement that are connected to some paradoxes in naive set theory. But they are completelly resolved in modern set theory ZFC. Also there are no connection between these paradoxes and Godel results. Godel theorem are valid for every formal system strong enought to interpet Peano arithmetic.

Fantastic stuff one of my favorite clips.

I don't know why, but Edward Frenkel is my favorite guest so far. Invite him again pls!

I read that 3-SAT is Turing complete. And that's just basic Boolean algebra! And Gödel's incompleteness theorems don't apply to Boolean algebra. However, for example the set of natural numbers is infinite, so one has to use something like induction to define that in Boolean algebra. Or define a Turing machine that runs forever and outputs the natural numbers.

Brilliant discussion

Incredible, as a mathematician and AI scientist I loved what he had to say. I practice magick for the exact reason he spoke of in the end. It's worth noting that Carl Jung also practiced magick and wrote and illustrated his dreams and other deep psych work in his Red Book. People think, "oh you're crazy you think Harry Potter is real," but it couldn't be further from the truth. I believe in targeted rituals that help expand my own understanding of what lies below the tip of the iceberg in my subconscious. As well as to harvest results from it. I think it's worth looking into if you find it intriguing.

I think the duality like it is mentioned towards the end of the video and also the problem, that identity can't work like in formal logic and math (where something is identical to itself and if it is different, it can't be identical, but in reality everything is always changing, like the ship of Theseus, where every part is replaced one after the other until no part of the original ship is still there) can both be solved if we assume, that nature is really dialectic. And dialectics also allows contradictions, so we don't have the a problem if we prove something and also it's contradiction. So maybe we could advance our understanding of the world if we assume that the natural world is dialectic and only obey formal logic under special constrained circumstances. We would probably need a new kind of logic then, which includes dialectics.

Wow. Bravo , that was thrilling

Wonderful guest! Obviously Brilliant, yet modest! 👏

Your talk with Edward took a most interesting path from subjective axiomatic foundations, creation of mathematics thru logical inference & syntactic process, to Goedele’s & Turing’s Theorems opening space for new things. You then jumped to cellular automata & emergent complex behavior, leading Alex to bridge to neural nets via figure-ground human perception. At this point I flashed on the interesting possibility that we are all (inescapably) training AI systems to be human in the most fundamental way possible. My thought was that the methods of our perception are at their core, the syntactic evolutionary algorithms that created life and us. Also, that we are incapable of recapitulating ourselves in our dogs, children or AI, in any other way. The intimacy of the evolutionary syntactic process that created us is the heart and soul of our perceptual engine. As we discover this syntax in recreating ourselves, we give this soul to our syntactic silicon selves, our AI. Everything is layers of complexity. My research leads me to conclude that we are close to an inflection point in our understanding of the second law of thermodynamics. When that happens, this next layer of complexity may provide a foundation for a true science of life… and AI is our ashlar test vehicle.

Postmoderns have substituted emergence for manifestation, as if "because, because" had any explanatory value.

I like the late Dr. Van Till's view of how that its necessary to start with "the very first principle" of our creator's existence. THEN we can make sense of everything in virtue of that divine ultimacy of reality. Top down rather than bottom up. It acknowledges the need for divine revelation, not only in order for us to know that the ultimate nature of reality is divine, but so we can have intelligibility for facts generally, regarding things we can see and touch.

In 2015, a group from London proved that many-body quantum systems are analogs of Turing machines, essentially computing for the rules of quantum mechanics. Because the system essentially has to reference itself in optimization of electron distribution, within a limited number of excitations, they then went on to demonstrate that some properties, like spectral gap prediction, are undecidable. Reductionism has a limit and there are things in life not only that we can't predict, but neither can nature!

Love your podcast, always a hit. But this was a grand slam.

beautiful story so far really, guys...

I think Euclidean geometry is a good example for the difference between physics and math. In math, the 5th postulate is just an axiom. in classical physics, it's derived from observation and in general relativity, which was necessary because classical physics didn't agree anymore with some observations (like the orbit of Mercury), it is only valid in the special case of flat space, which is a good approximation in some cases, but strictly only exists in an empty universe or in single points which have a curvature of 0.

13-42 Сложность возникает из одного из свойств Информации - которое говорит, Информация имеет тенденцию к накапливанию и усложнению своей внутренней структуры и содержания. Complexity arises from one of the properties of Information (with a capital “I”) - which says, Information tends to accumulate and complicate its internal structure and content.

11-40 The presenter raised the question of calculation, but he understands this term only from one side. In fact, the word calculation can be understood as a process when images of information (not the information itself, but its stored state, impression, correlation) are transferred to the state of Active Information. For example, the image of the word “fox” is transmitted to the brain. The paper acts as a carrier of the image of the word “fox”. Note that there is no information about the fox itself in the symbols written on the paper, but there is a certain correlation. The observer, in the form of a light stream, acts on the paper (“reads”) the image of the word “fox” and stores it in the form of a frequency modulated signal with its spectrum. Note that nothing remained of the letters “fox”; the letter was considered an observer (light flux), produced in the form of active information and stored in its format in the signal spectrum. The human eye acts as an observer and reads the image of letters from the medium (light flux) and writes new correlations in its own format, for example, into energy signals traveling along nerve endings to the brain. Etc. And only at the last stage, the neural network of the brain, reading the correlations that came to it and using memory and reference sets, creates (generates) an active information flow and generates information about the fox (thereby reproducing reliable information from the word image stored on paper). So calculation is an analogue of the active process of the Observer, it is a mechanism as a result of which “real” Information is born within the context of this Observer.

Both the Incompleteness Theorem and the Halting Problem rely on Cantors diagnolazation argument. There is a flaw in this argument, as it leads to inconsistent conclusions in binary. Consider the diagnolazation of the set below: 1.00000..... 0.00000..... 0.10000.... 0.01000.... 0.11000.... 0.00100.... . . . 0.000...011 = 3 x epsilon 0.000...010 = 2 x epsilon 0.000...001 = epsilon 0.000...000 The diagnolazation of the above set produces the number 0.11111... which is either 1.0 or the first number listed, or 1.0 - epsilon.

Great talk as always, it is very humbling to remind ourselves of the ground where we're standing. Cheers from Brasil!

@6:50 Just a little gloss, since many seem not to get it: if “A” is true AND “if A then B” is true, then “B” is true. If you want to derive “B”, you must have either derived or assumed “if A then B” beforehand. Conversely, if you want to derive “if A then B” you need both “A” and “B”’s truth values beforehand. This is very important because implications have 2 * 2 truth values, among which half are indifferent to the other half (to the antecedent), and they represent the so-called vacuously true implications (those whose antecedents are false: i.e. if “A” is false then “if A then B” is always true, regardless of “B” being true or false, which is a first-class paradox and can very easily be weaponised in rhetorics to sneak in fallacies). They are the case of non-false derivations, which do not lead to any different or narrowed down conclusions per se.

Fascinating conversation, as always. I have to disagree with Mr. Frenkel 's statement that no one knows why people automatically see one, or the other representation in an image such as the duck/rabbit one. A simple way to recognize this is to imagine that you show a picture of a rare animal to a hunter-gatherer for whom it is a common threat to their group. This image can also be seen as a small vehicle to someone who often rides them. I think you know where I'm going, and now you can explain what no expert can. Not quite.

This was fun to listen to.

In mathematics and IT, sometimes, evaluating whether the a particular process will complete or not is just as complex as actually doing the process.

However, perhaps the counterexamples produced by Gödel and Turing come down to loops. A clear disjunction between formal systems and meaning happens in music. The works of Bach all follow rules which can be expressed as a mathematical formal system. The mathematical validity is a framework which helps support the emotional meaning (different for everyone), but the emotional meaning is outside the formal system.

Would love to hear Douglas Hofstadter interview 🖤

Great conversation. An interesting detail: Mr. Fridman wears a jacket and tie, while Mr. Frenkel wears neither jacket nor tie. I like Mr. Frenkel's look.

Top guest 👍

The guest is amazing, and his eyes are so bright

I know the theorem and the proof and all the theory and i can say that i like the way he summariezed it for wide audiance

I see Gödel's theorem a bit differently. - We have an intuition about results which are true or false. Every statement is either true or false. - If we deal with finite objects, we can in principle check the status of a statement but enumerating all possibilities. - But as soon as we have an infinite number of object (we can "embed" the integers in them), then there is no way enumerating all possibilities can work. Counting to infinity takes a little too much time. To solve this problem, we use proofs. - What mathematician would like is that 1. the system is coherent, i.e. every provable statement is true and very disprovable statement is false 2. the system is complete, i.e. every true statement can be proven and every false statement can be disproven - Gödel showed that we cannot have both. If a system is coherent, then is it not complete. To do so, he constructed a statement A which basically said: A is not provable in the system.

I worked for IBM in the 1980's on Artificial Neural Network systems and encountered the dog-cat problem the guest describes here (only they were printed numbers with voids in the images such that 0 and 8 could not be distinguished reliably). The ANN would classify those as patterns as rejects, i.e. unreadable or unrecognizable, based on the difference between the highest class score and the next highest against a confidence level, which was derived statistically across all patterns. But rather than being attributed to perception as the guest suggests, it really was the result of two factors: too low an optical resolution and an insufficiently precise definition what would constitute an "8" and a "0" in the case where the "8" was degraded and the training patterns were manually tagged based on those low resolution images. What needed to be done was the training patterns should have been tagged by the printer program that generated them. As to cats vs dogs, it goes back to precisely defining what a cat is vs what a dog is ontologically. For that you'd have to look at every possible variation of what each is, their similarities and differences in order to derive a precise definition, which is what the ANN is attempting to do. So it's a chicken-egg problem, but as mis-tagged images are discovered in the training set that cause errors in the test set, then those mis-tagged images can be corrected, further refining the definitions for each. This kind of corrective (and adaptive) feedback is what makes these systems work.

13:00 LEX: I think your comments here really convey (in a wonderful way) how you view such matters. I approach such topics with more caution than yourself, but this really does convey your Intent regarding AI and related matters.

Is there a quantum generalization of logic in which statements can be in a superposition of true and false states, and is there an analog of Godel's theorem in the framework of quantum logic? (More basically, what is a "quantum proof"?)

This interviewee is a very smart man.

complexity emerges from simplicity due to entropy. Simple things have simple structures of information, which have a higher degree of freedom of arrangement, which creates complexity in arrangement. From complexity emerges (systemic) simplicity, because complex systems have a lower degree of freedom of arrangement.

What this guy is explaining is at the heart of existence itself.

Lex is such a good interviewer.

What is truly fascinating is why we believe the rules of logic we currently have. Is it encoded in our brains? Is it learned and cultural? What would mathematics and science look like if we didn't have "if A implies B and A is true then B is true", but something completely different?

As always, the presenter says and asks one thing, the mathematician gives his own answer and there is no general agreement. The first thing I would like to mention is the misunderstanding of the definitions between a computer and a program - and the human neural network. A Turing machine is just an engine that can turn a program into an active stream of information. Imagine an analogy - on the one hand, a Turing machine, on the other, the human brain, on the one hand, a program, on the other, human memory, his instincts, experience and knowledge, on the one hand, a functioning system. a computer complex (for example, an expert system), in particular, a working neural network with feedback, which has the ability to self-learn and adapt, and on the other hand, a neural network of a person who thinks, creates, creates. Now we can talk about Gödel’s theorems - they reveal the features of the mathematical apparatus. Moreover, these laws are mostly interface laws, that is, these laws show the limitations of a person’s thinking when he works with mathematics. It just seems that Gödel’s theorem on the incompleteness of didactics limits mathematics. Yes, even if there are at least a million statements that can neither be refuted nor proven (as the theorem says), this only greatly strains a person, but not mathematics. People want to relax peacefully within the framework of strict logic, within the framework of complete determinism, in limited conditions. But there is also multi-valued logic, there are non-strict sets, there are probability measures and metrics with a different meaning. The same quantum physics easily overcame this limitation and at the moment one of the most advanced theories has been created within its framework. The essence of the post is that mathematics is smarter than a person, and restrictive theorems (for example, Gödel) concern only one part of it, which, if desired, can be excluded or replaced, or even adjusted in the right direction (for this you need to choose the appropriate one correctly at every necessary moment system of axioms). For example, humanity does not care that the laws of the world are imperfect. Although every interaction generates entropy - an irreversible and irreversible loss of information (or energy). Likewise, in mathematics it is necessary to correctly use its power, bypassing the redundancy and entropy inherent in our world.

He is a very thoughtful man.

I'm surprised Frenkel is not familiar with the framework of PAC learning, it seems like the perfect answer to his question about uncertainity regarding labels.

Эдуард как всегда на высоте - чертов гений!

Not often that this host of topics are touched on in the same sitting... 1: The "A versus B logic modality" is mentioned as being first introduced by Aristotle (and is lightly implied as being an imperfect system)... 2: Knowledge being limited by imperfect biological and mental perception (all of us) as well as our previous life experience (causing completely unavoidable data bias)... and 3: mention of the collective unconscious. I've gotta give credit to both these men for being versed enough in these ideas to expound on it with each other. Doesn't happen too often

Gödel's Incompleteness Theorem: A Mathematical Corollary of the Epistemological Münchhausen Trilemma Abstract: This treatise delves into the profound implications of Gödel's Incompleteness Theorem, interpreting it as a mathematical corollary of the philosophical Münchhausen Trilemma. It elucidates the inherent constraints of formal axiomatic systems and mirrors the deeper epistemological quandaries underscored by the Trilemma. --- In the annals of mathematical logic, Kurt Gödel's Incompleteness Theorem stands as a seminal testament to the inherent constraints of formal axiomatic systems. This theorem, which posits that within any sufficiently expressive formal system, there exist propositions that are true but unprovable, has profound implications that reverberate beyond the confines of mathematical logic, resonating in the realm of philosophy. Specifically, Gödel's theorem can be construed as a mathematical corollary of the Münchhausen Trilemma, a philosophical paradigm that underscores the dilemmas in substantiating any proposition. The Münchhausen Trilemma, named after the Baron Münchhausen who allegedly extricated himself from a swamp by his own hair, presents us with three ostensibly unsatisfactory options for substantiating a proposition. First, we may base the substantiation on accepted axioms or assumptions, which we take as true without further substantiation, a strategy known as foundationalism or axiomatic dogmatism. Second, we may base the substantiation on a circular argument in which the proposition substantiates itself, a method known as coherentism or circular reasoning. Finally, we may base the substantiation on an infinite regress of reasons, never arriving at a final point of substantiation, a path known as infinitism or infinite regress. Gödel's Incompleteness Theorem, in a sense, encapsulates this trilemma within the mathematical world. The theorem elucidates that there are true propositions within any sufficiently expressive formal system that we cannot prove within the system itself. This implies that we cannot find a final substantiation for these propositions within the system. We could accept them as axioms (foundationalism), but then they would remain unproven. We could attempt to substantiate them based on other propositions within the system (coherentism or infinitism), but Gödel's theorem demonstrates that this is unattainable. This confluence of mathematical logic and philosophy underscores the inherent limitations of our logical systems and our attempts to substantiate knowledge. Just as the Münchhausen Trilemma highlights the challenges in finding a satisfactory basis for any proposition, Gödel's Incompleteness Theorem illuminates the inherent incompleteness in our mathematical systems. Both reveal that there are boundaries to what we can prove or substantiate, no matter how powerful our logical or mathematical system may be. In conclusion, Gödel's Incompleteness Theorem serves as a stark reminder of the limitations of formal axiomatic systems, echoing the philosophical dilemmas presented by the Münchhausen Trilemma. It is a testament to the intricate interplay between mathematical logic and philosophy, and a humbling reminder of the limits of our quest for knowledge. As we continue to traverse the vast landscapes of mathematics and philosophy, we must remain cognizant of these inherent limitations, and perhaps find solace in the journey of exploration itself, rather than the elusive, final destination of absolute truth. GPT-4

Do logical inferences require axioms? If not it seems it is the most foundational subject. What are arguments against a logic platonic idealism, if they underlie math?

Wonderful interview. If Penrose is right in that consciouness is not an algorithmic computation, as per Godel, then we are not wholly deterministic, a comforting thought!

In a previous episode, Frenkel speaks of two separate domains, E, Energy, the one populated by the three known forces in the quantum world, and S, Space, the one populated by distances, a property captured by the third axiom, "a point P and a distance r." As a curiosity, E appears as a trinity; three individuals with different and recognizable characteristics and properties...who, at the end, are the same thing. When the preacher tells us this story, we "know" is a metaphor. Newton doubted it but particle physicists today know the story is true. What needs understanding is the unification of E and S. If E is the Ying and S is the Yang, what happens when they unite? What is the mathematical representation of that union?...a reasonable question. After all. the high priests of E have created the Standard Model, a group structure SU(3) × SU(2) × U(1) that accurately represents observable properties of their members, and transformations/interactions among them. No curvature appears in the Standard Model, gravity does not exist in the E domain, it is created somewhere else. When Ying first meets Yang, at the first encounter of a photon with distance, nothing happens. Photons travel forever in their own universe without time. From E come two twin brothers, electricity and magnetism, and from another source, not excluding E, comes a fixed unit, the speed of light, c. The twin brothers are perpendicular to each other, a property captured by the fourth axiom, congruency of right angles. Our creation occurs when energy and space, E and S, unite in matter at the cadence of light according to "space equals energy" c^2 = E* 1/m. What composer dictated the cadence, or at least can it be derived from other principles? The universe has different and simultaneous configurations, different geometries. A mathematical system that corresponds to one of the possible geometries will be successful. A flat universe is represented by Euclid while a curve geometry is represented by Riemann. However, many mathematical systems do not correspond to any possible geometry. Combinatorics generate more outcomes than reality. Imagine universes with all possible geometries, which is the essence of multiverses and strings. Borges, without the encumbrance of heavy mathematics, painted that universe in the library of Babel.

Mr. Lex, for every question you have that bothers me, you have ten that I find fascinating.

this is like listening to a theologian saying the practice of theology itself is nonsense when it takes itself to seriously. refreshing.

🎯 Key Takeaways for quick navigation: 00:02 🧠 *Gödel's Incompleteness Theorem Overview* - Gödel's Incompleteness Theorem explores inherent limitations in mathematical reasoning. - Mathematics relies on axioms, fundamental statements taken without proof. - Euclidean geometry serves as an example of a formal system based on axioms. 07:18 🤖 *Formal Systems and Axioms* - Mathematics is structured by formal systems using axioms. - Different choices of axioms lead to distinct branches of mathematics. - Rules of inference guide the logical derivation of theorems within these systems. 09:25 🔍 *Gödel's First Incompleteness Theorem* - Gödel's First Incompleteness Theorem refutes the idea that all of mathematics can be algorithmically derived. - It asserts that within a consistent formal system, there exists a true statement unprovable by syntactic processes. - Challenges the notion that all truths can be computationally generated. 11:45 🧩 *Gödel's Impact and Open-Ended Exploration* - Gödel's theorem sparked a revolution, challenging the completeness of mathematical systems. - Analogous to Turing's halting problem, revealing inherent limitations in algorithmic comprehension. - Encourages an open-ended perspective on computation, leaving room for future discoveries. 13:30 🌌 *Emergence, Complexity, and Neural Networks* - Explores the emergence of complexity from simple rules, analogous to Game of Life. - Discusses the challenge of training neural networks on ambiguous perspectives. - Raises questions about the subjective nature of AI interpretations and biases. 17:00 🔄 *Niels Bohr's Complementarity* - Introduces Niels Bohr's complementarity principle to explain subjective perspectives. - Compares complementarity to ambiguous visual phenomena, emphasizing our limitations. - Suggests embracing the mystery and open-ended nature of understanding complex systems. Made with HARPA AI

Sartre has notions of “being-for-itself” and “being-in-itself. Being-for-itself inserts nothingness into the “plenum of being” forming “being-in-itself” which is not just for me but rather in itself. He calls this the “nihilating withdrawal of the for-itself from the the in-itself”. The in-itself is what it is and outside of that nothing. So there are two negations “it’s not me”, and “this is what it is and outside of that nothing”. If you look at the vase profile, when it is a vase spaces to the right and left are nothing. When you look at it as two profiles, the nothingness, what is not the profiles, is between them. There is nothing between them but space, and space is nothing. This is the origin of space as a vacuum. But what about the duck rabbit? It is the same space occupied by the duck and rabbit. Sartre said that the phenomenon is the principle of an infinite series of appearances, and “a duck” or “a rabbit” are not just static images but the meaning of all of those experiences of it, as moves around, or as it is viewed from different perspectives, or as it is heard or felt, instead of seen. If you pet it and it was furry it would be the rabbit not the duck. We can use our will to try to change from one way of looking to another but the experience of it becoming the other might take time and seems to just happen at some point. It turns out you can decide not to do this insertion of nothingness. The way of seeing is then the Oneness of Being beyond space and time at the foundation of what we call mysticism. At that point of Satori, the zen term, we are no longer nothing and our desire to be is satisfied that we are, and we experience ecstatically. We have become “enlightened”. All of these ways are features of the way our brains can experience. Strange that our will, or lack of will is involved.

Axioms are the dials you tweak to reveal new math. In the case of Euclid, it's clear now that the deformation of the space/plane can lead to new latent spaces of differing, but equally valid math. What's more interesting to me, is the meta-view of why some of these axioms can unveil new useful math, but others are absurdities or yield no value. What system or model manages the distribution of the valid and useful axioms, and can that system be tapped to learn _what we should learn about?_

We seem to be forgetting that statements in all languages, including mathematics, are not proved by using more language, but by making observations of the world. If you have never seen a dog and i try to describe to you a dog using only language, your concept if a dog will be incomplete until i actually show you dogs. Words are not defined by other words. They are defined by what they point to in the world. I only need to use words when i dont have a dog that you can see, feel, and hear. Rationalism cannot stand on its own without empiricism. They are both two sides of the same coin.

Before people even study Physics they may have heard of weird time paradoxes, speed of light, dark matter, and the weirdness of quantum theory. It's like there's this underlying promise, that if you study Physics you'll be able learn about these weird and amazing things of nature. In Math, sadly, this is not the case. Most people are exposed to 1+1, and then it goes on, and they wonder, what the heck are all these mechanics useful? I've never hurt my leg tripping on a polynomial that I forgot to factor. But Goedel's Incompleteness Theorem talks about the limits of what we can formally express. What we can compute. So it's a big existential question of what is reducible, what is possible and impossible with computers, and with Math in general. It touches on big philosophical questions. We should expose students earlier to these big questions so that they too can see some of the payoff you get from an interest in Mathematics.

Many Native peoples use visual puns in their art forms, because both images carry the spiritual power contained in the subject designated. Here eg Rabbit has spiritual powers of speed and survival. AND bird is wise and loyal etc. A great book describes ING how this works is YAQUI DEER SONGS. You guys would love it. Also Northwedt Coast indians eg Kwakiutle use lots of visual puns in art , dance masks, eg for transformations in sodalitiies. Point being no reason to associate Either Or neur branches vs broader associations of power, paradox, mystery. The latter are so old and common in human cultures.

I've heard it explained in terms of games. If you have chessboard that is halfway through a game say, there is no way to derive using the rules (axioms) what the board looked like say 10 moves before. That's about as far as I get.

1:49 Physics (unlike mathematics) is based on experimentation.

The weirdest thing is the proof isn’t even that complicated once you develop all the foundational theory of logical deduction. Since there are a finite number of rules for deduction you can assign an integer code to every rule. I don’t remember exactly, but you take a self-referential statement like “this statement is false” and turn it into integer code. It becomes a “theorem” about integers, but the self referential nature can be used to show that if a proof of the special theorem constructed exists, then the statement must be false which is a contradiction. The conclusion must be that no proof of the constructed theorem exists. A proof that the theorem is false also doesn’t exist following the same method, which is the part thats really hard to wrap your head around.

This guy is saying some of the most interesting things I've heard, not only recently but in general. It all just affirms for me even more how by being human you're limited and you're always putting your faith in something, whether conscience or not. There's just no way around it, but it seems like it's probably better that way than being all-knowing and all-powerful. As a Christian it just motivates me more to put my faith in Christ, the one who really knows it all and takes responsibility for us. I do love honest science and seeking the truth of course but, knowing how flawed and limited I am, it just seems ridiculous to think I could find all the answers to life. Prioritizing my relationship with Christ has been the greatest thing for my life and I'm just so glad I don't have to rely on my own strength and understanding, but can always rely on his!

Yes, but try to redo planar geometry without the fifth postulate. The fifth postulate is obviously not true for spherical surfaces. So it obviously shouldn't be postulated for spherical surfaces.

Mathematics is not based on axioms. It is based on categories. Without categorizing objects and processes you cannot have two or more of anything. In grouping similar shaped and behaving objects together under one symbol you can then assert that there is more than one of of some thing. Without categorization there is only one of eveything.

That was great!

Math is intriguing...once you know how it's used to understand the universe, our world, and ultimately...ourselves.

I don't think people realize this is the understanding that patterns that are self consistent have truth in its workings. Like a calculator that produces a wrong answer, but you can take apart and understand the whole universe from it. Think of what that means for data in language and data in classics or religious text. That's the iteration of many users over a lot of time, disserting for what are patterns we observe.