Mathematician explains Gödel's Incompleteness Theorem | Edward Frenkel and Lex Fridman

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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.
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[@LexClips]

Full podcast episode: kzbin.info/www/bejne/haTLYWCAaLllpLs Lex Fridman podcast channel: kzbin.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.

[@cryptic8043]

Gödel's first incompleteness theorem states that for any consistent formal system that is sufficiently powerful to represent arithmetic (which includes most foundational systems of mathematics), there exist statements within that system that are true but cannot be proven within the system. In other words, there are true mathematical statements that cannot be derived or proven using the rules and axioms of the system.

[@cryptic8043]

The key idea behind Gödel's theorem is the concept of self-reference. Gödel constructed a mathematical statement that asserts its own unprovability within a given formal system. This statement, known as Gödel's sentence or Gödel's formula, essentially says, "This statement is unprovable." If the system could prove this statement, it would be inconsistent because it would be asserting both its own provability and unprovability. On the other hand, if the system cannot prove the statement, it implies the existence of true but unprovable statements.

[@cryptic8043]

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.

[@vicheakeng6894]

Entanglement

[@vicheakeng6894]

5 formulas, : 1+tan(2)power=sec(2) power

[@baTonkaTruck]

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.”

[@amante2443]

I was surprised, amazed how he connected them, but then remembered how wide ranging Frenkel's knowledge about things and people are (away from Mathematics). I recall (and am now rewatching) his 2014 talk, when, at the beginning of his talk, the computer system breaks down, he tells the tech, ""Don't worry about it". And "we use computers so much these days, maybe it's a sign". Then still goes onto speak with such humility, humour, and a weird humbleness. Weird because he obviously knows so much but believes he doesn't. The talk's from a book promotion tour, for his "Love and Math". If I remember correctly, he's got a few more of those Alan Watts like comments. kzbin.info/www/bejne/j5_UgmB4jLJ7qKc (if you're interested).

[@Leksa135]

I don't get it. What does it mean to define an axiom?

[@xmathmanx]

That sounds like a vague description to me, which is fine for people like watts, but no good at all if you want clarity

[@xmathmanx]

​@@Leksa135axioms are statements, so yeah, what does it mean to 'define' a statement? Seems like it doesn't mean anything honestly

[@amante2443]

@@Leksa135 At around 4:46, you'll see them show Euclid's five axioms (also called postulates). You'll see it says, "first axiom", "second axiom", etc. Underneath each of those is what is technically called a definition. What this is, is a definition for each axiom. To return back to @FrigginTommyNoble's comment, the system of mathematics being used is called Euclidean Geometry. But the Euclidean Geometry system can't define it's axioms or itself, it needed Euclid (i.e. a person) to define them. Hence, no system (i.e. no mathematical system) can define all of it's own axioms. Which is what Kurt Gödel proved mathematical, or he proved that all mathematical systems will be incomplete, hence Gödel's incompleteness theorems.

[@jacksmith4460]

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

[@shimrodson5443]

agreed!

[@ExperienceLOS7713]

When I saw this dude on Numberphile a long time ago I knew he was the real deal. Such a delight to see him again here!

[@dudeshiya]

It's great to hear about Godel's Incompleteness Theorems. What he didn't mention is the motivation behind how Godel ended up with the Incompleteness Theorems. So at the end of 20th century, a genius named Georg Cantor (founder of set theory) wanted to understand God (just like Einstein wanted to understand God). For him God was represented by infinity. Therefore, he wanted to understand infinity, which we failed to understand for thousands of years (and still to this present day). So he asked himself the simple question, we can add numbers, subtract them, etc, but what about infinity? To this end, he constructed a theory that is now known as cantor's set theory. Unfortunately his basis of reasoning (called axioms) contained a contradiction that is now known as Russell's paradox. Nevertheless, he proved that the set of natural numbers is smaller than the set of real numbers. And he wanted to prove whether there is a set in between the naturals and the reals (that is now known as the continuum hypothesis), but failed to prove it. Because of these, David Hilbert (widely considered to be the greatest mathematician at that time and of the 20th century) came up with what's known as the Hilbert's program. In this program, he posed a number of problems, among which is the quest for a basis of reasoning that is both complete (that is to say, if a mathematical statement is true, then it must be provable from this basis of reasoning) and sound (that is, the basis of reasoning only proves true statements). Note that these two properties (completeness and soundness) are the two fundamental properties of all algorithms. Now, comes Godel. So originally godel wanted to prove that such a theory exists, but ended up with the incompleteness theorems in the end. But, the results are so earth shattering that it completely destroyed Hilbert's and mathematicians' dream of having a sound and complete theory for mathematics. On top of that, the mathematics he used to prove the incompleteness theorems were so new that only a handful of mathematicians understood it. Amongst these mathematicians that understood Godel's results were Alan Turing. So what Alan Turing said is that what Godel really means is that there is no machine/algorithm that is able to prove/determine whether an arbitrary program on an arbitrary input will stop or run forever. This is the Turing halting problem. It is important to note that all of these led to the birth of computer science and eventually led to the famous P vs NP problem. Historically, Godel was the first to informally pose the P vs NP problem in a letter to Jon Von Neumann. He also was the first to prove that Einstein's theory of relativity allowed time travel. And he gave the proof to Einstein as a birthday present.

[@blueskies3336]

@@dudeshiya Thank you for this write up! Is there any book you could recommend to read about this in detail?

[@Flovus]

@@blueskies3336 If you are capable of reading German, Dirk W. Hoffmann's "Grenzen der Mathematik" is a must. It is mathematically rigorous enough, but as comprehensible as possible for this hard topic. It gives a detailed historical account of the developments. Maybe one could try reading it with the advancing translation technology... I saw that in September 2023 a book called "Foundations of Logic" by Westerståhl will be published, it seems similar from its summary.

[@arsartium108]

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)

[@chunliangzhang2506]

Thank you!

[@atomgutan8064]

Wow, literally an informative article in the comment section. Thank you so much!

[@TobyZobell]

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!

[@opinionhaver574]

He's been featured on the Numberphile channel a few times - the topics he covers are always very illuminating.

[@johnstebbins6262]

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

[@0risenloudly]

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

[@Paronimous]

??? every why still can have a definite answer but not on the same system

[@jaywulf]

To be fair, as far as I understand Goedel's incompletness theorem (I think thats what you mean, 'incompletenes' rather than 'impossiblity'?) the theorem only states that there is at lease one state that is not within the universal set. And that by the virtue of at least one, that makes the whole 'completeness' incomplete. It does not mean 'there are questions that can not be anwsered', it just means, 'Mathematically, there is at least one question that can not be anwsered'. I choose to interpret that to mean "Do we get Pizza or Fish and Chips." being the question.

[@436tucan4]

@@jaywulfnot necessarily a question being asked but rather and assumption being posed

[@436tucan4]

@@jaywulfbecause a question would imply that if an answer was derived, the answer would be used to start the system of logic, but within the logic the axiom was already assumed without validation (except if you go meta)

[@logicalconceptofficial]

9:28 No, no it doesn’t. It “proves” these things if you aren’t a shrewd enough logician to understand that a self-contradicting and semantically empty statement is simply “incoherent” within any coherent formal system (that is coherent with the Logos). Also the halting problem is solved by the Logos and by translating programs into a number of versions until a halting version is found (if the original does not halt by the time that other halting version is found). It’s all easy and these people that think LOGIC is imperfect or can’t prove everything are the fools, even if they’re “big names” and did some cool things. When the Math doesn’t “add up” it’s YOU/US (the fallible mathematician and fallible humans) that screwed up, it’s not Math and Universal Logic (Logos) that were actually imperfect at their Root and in their Essence. Perfect Logic is Perfect Logic and that is tautological and 100% certain. There is no “truth” that is not encompassed by the Universal Logic or (objectively) true without being so according to the Standard of Universal Logic. It is illogical and logically contradictory nonsense to speak of a “truth beyond Logos” or any aspect of Logic that cannot be symbolically represented with Formal Logic.

[@BerenddeBoer]

Edward Frenkel is so extremely lucid, just extraordinary.

[@CordialBuffoon]

This is your brain on logic.

[@joegillian314]

My ex-wife actually had Edward Frankel as an instructor at Berkeley nearly 10 years ago.

[@jeremias5688]

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

[@unkokusaiwa]

Do people?

[@TheJustinJ]

It takes a deep and complete understanding to make it as simple as it can possible be explained.

[@jeremias5688]

@@unkokusaiwa HAHAAA

[@aletheia161]

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!

[@Mattje8]

Or that even if we are wholly deterministic, it is not in a way that can be replicated by any formal / computational approach known today. Penrose has basically said something similar.

[@aletheia161]

@Mattje8 Very true. Do you think if we are not wholly deterministic, that Penrose's idea, if true, would explain the phenomenon? Could it be that superposition and entanglement provide human's with the ability to think outside the box, to get the "god's eye view" ?

[@DumblyDorr]

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.

[@Flovus]

"and then came Gödel" No, Gödel was first (1931) and Rosser's extension of Gödel's first incompleteness theorem (which is often falsely attributed to Gödel) was in the same year as Gentzen's proof, 1936. Gentzen evaded the incompleteness theorems: the system he used to show that PA is consistent is not stronger than PA, but miraculously manages to capture the structure of proofs in PA (in particular, it is not weaker than PA). I think Gentzen's result is even more breathtaking and few people (if any) understand it.

[@jaymethodus3421]

Godel vs Gentzen? Epic math battles in history? Please tell me this is a KZbin series 🫡

[@markcarey67]

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

[@cryptic8043]

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.

[@olegilin7094]

It’s so obvious, you can’t put all your eggs in one basket. We need competing ideas, we need theories that “contradict” others, but because of this they work (meaning multi-valued logics that immediately eliminate Gödel’s theorems), etc.

[@barneyronnie]

Thanks for your clear exposition of the core ideas embraced in Godel's amazing theorem...

[@InterfaceGuhy]

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.

[@sabinrawr]

I've been watching a lot of physics videos lately, mostly from Drs. Sabine Hossenfelder (Science without the Gobbledygook) and Matt O'Dowd (PBS Spacetime). Both channels take dips into the Quantum and explore some of the weirdnesses within. I've been thinking a lot about the ideas of complementarity and how they might relate to quantum superposition and the measurement problem. See, I'm starting to think that maybe the wave function doesn't collapse at all, but rather we are just observing one "version" of the particle. Its unobserved complement might still be just as real as the one we measure, but can't be seen at the same time. In a sense, maybe particles really are in two places at once, but we can only observe one at a time. Thoughts?

[@InterfaceGuhy]

@@sabinrawr yea I have a lot of thoughts about this. Safe to say there is a lot of confusion surrounding QM as it is popularly understood. I tend to go with interpretations akin to objective collapse or Quantum Bayesianism.

[@eamonnsiocain6454]

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?

[@ciarantaaffe4199]

The Chomsky hierarchy defines classes of complexity of behaviour. Any Turing complete system can capture the fullness of complexity, and they are all well defined computational systems. Also, the simple structure of these systems can produce the most complex behaviour possible: irreducibly complex behaviour.

[@kurtgodel5236]

In "another paper"? Which would that be? "All he needed was the countable numbers (the set N) and a few very simple rules." This comic reference to Peano arithmetic and gödelisation really hurts!

[@mikezooper]

Gödel should have trolled us and left his incompletence theorem incomplete.

[@sanjitdaniel4588]

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

[@rgw5991]

GAYDOH

would you say the explanation was complete?

[@sanjitdaniel4588]

Ha ha good one.. As complete as an be at this level I guess...

[@cyclonasaurusrex1525]

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

[@IvanGrozev]

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.

[@andrewferg8737]

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

[@koraamis5568]

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.

[@elindauer]

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

[@MichaelT_123]

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

[@themacso4157]

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

[@MuffinsAPlenty]

Don't forget that he has spent decades of his life dedicated to learning and communicating mathematics. It take a lot of hard work to be able to do what he does.

[@linchenpal]

Is your dad a mathematician? If not the case, pls respect your dad.

[@themacso4157]

@@linchenpal yes

[@TheJustinJ]

Some people are natural at math but cannot teach or explain it. Others work harder to learn it, but do better conveying it. But most people are bad at what they do and don't care to try to learn everything they are capable of.

[@rfvtgbzhn]

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.

[@HexViccissitude]

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.

[@ayrnovem9028]

On the Origins and Nature of the Dark Calculus There's evidence in the arithmetic record that the study of formal systems reached a pernicious apex in the Long Before. Advancements made by mathematicians such as Russell, Gödel, Eisencruft, Atufu, Wheatgrass, and System Star contributed to the understanding of notions like undecidability, pointed regularism, and abyssalism. Upon reaching this minimal degree of mathematical maturity, equipped with sophisticated grammars, researchers set out to experiment with the limits of expressibility. They contrived bold research programs and galloped into the mathematical wood, unwitting of the dangers that brood there. The record is even scarcer than usual, due to the efforts of successive generations to obfuscate the venture. As best as I can gather, at some point in the course of inquiry, a theorist from a mathematical seminary called the Cupola formulated a conjecture on the fragility of formal semantics. The conjecture ripened to a broader theory, out of which spawned a formal system called the penumbra calculus. In the few fragments of texts that predate the obfuscation, it's stated that, in the penumbra calculus, certain theorems are provable, but are falsified upon the completion of their proofs. As much as this result is at odds with the systems of thought I've encountered in my own inquiries, I find little reason to doubt the veracity of the authors. Nevertheless, it's certainly a peculiar property. The Cupola theorist's results erupted into a grand investigation into the expressibility of the penumbra calculus. The conclusions were troubling. Pushing further, researchers constructed sister systems with alternate axioms. These systems were still more fragile, with the systems' inference rules themselves unraveling upon the completion of certain proofs. Convinced that their discoveries were made possible by some idiosyncrasy of self-awareness, but synchronously fearful of the implications of their results, some schools of theorists engineered complex automated deduction systems to probe boundary theorems and launched them into neutron stars. The outcome is undocumented, but the result convinced theorists across the Coven to abandon research and blacklist anyone who studied the penumbra calculus and its derivative systems. (from: Caves of Qud)

[@mexxmirror]

I just love how Kurt Gödel made us know back in the 50s or what? that we can never solve every mathematical problem... and there is still the AI hype out there thinking we are going "IRobot" in the next 10 years. I love how the mathematic guy really got what the problem is, the moderator seems to be a guy who doesn't understand what math guy is saying :/ at 16:15 he also sounds drunk and just dumb. why is he asking a knowledged person like this? he should go in kindergarden and ask there. wtf is this shit

[@arontesfay2520]

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

[@amendlowitz]

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

[@paryanindoeur]

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!

[@evertoaster]

I think Lex was joking about programming. There being no means to figure out the perfect program. :)

[@paryanindoeur]

@@evertoaster Agreed, Lex was not being 100% serious in his answer

[@brainxyz]

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.

[@iranjackheelson]

simply giving it the name "topdown" is neither much of an explanation nor interesting. and who is this "you" picking what to attend to? that's the interesting stuff

[@s.muller8688]

@@iranjackheelson The brain is a reactor and converter not a creator. So this whole story goes into lala land categories.

[@iranjackheelson]

@@s.muller8688 you're being sarcastic right? brain is exactly not just a converter or reactor. it is a creator indeed. part of the reason why you can't predict what you want to eat for lunch 100%.

[@s.muller8688]

@@iranjackheelson what you going to eat at lunch is already stored in the memory in the form of known data, which than randomly get's chosen by thought. Nice try.

[@iranjackheelson]

@@s.muller8688 decision to what to eat for lunch is influenced by your prior beliefs and states, but it's not just stored as "known data". nice try

[@Tadesan]

Maybe the Einsteins would have been smarter if they hadn't done all that first cousin loving.

[@kaibe5241]

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.

[@guntherschabus3458]

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

[@proteus4301]

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.

[@anthonyjulianelle6695]

Einstein did think highly of Godel but Einstein was at the Institute of Advanced Studies years before Godel did.

[@dondevice8182]

“Ceci n’est pas une pipe” (This is not a Pipe)-- It is neither a doctor nor a rabbi;. Neither a case, nor a woman- merely two dimensional, drawings, suggesting both.

[@anuragsheoran9995]

Physics is bio product of one of the mathematical possibilities But in thoughts things get bizaare we have infinite possibilities Also a lot of master/human /living organisms/consiouness who produce thoughts and convince all these is bizaare and more precisely mathematics is a way of understanding things and we all have mathematics of ourselves and somehow it is connected and we all agree at some point Mathematics is the way of understanding things can be different for different People,can be find in different form truth is one ,just one Kamal hai our mathematics is not much developed to understand these bizarre things ,as this is the way of computation of information but question is will we compute all the information of that system only then we can calculate Vow vow the ai we are creating is not fully consiouness aware like it is a partial/zombie awareness omg we are creating it Correct this is our limitations to see it completely .

[@TANRININresulu]

Everything based on 1 Who is 1? ALLAH❤

[@mescale]

Allah do what He wants, but he can't became man if he would? That's either because He can't, either because He doesn't want. Because He can what he want, then he can't because he doesn't want. Why he doesn't want? Because is he the Almuttakabbir? No, that's only Satan that pretend to be what's not. Allah is the AlMuttakabbir, the One who's aware about his Greatness, but he's humble, he is The Humble, The Most Humble, and he is humble, and the most humble just right because he knows he is the Greatest. He's The Righteous, The Just and The King, but He is Humble. Could thus a Humble King order to his subordinates anything He wouldnt do personally by himself? He told Satan to bow to the men, and Satan didnt because he's not God. But the Will of God is God himself, and by ordering the angels to bow to men, he asked them to prepare his Way, his way down to bow to us, to wash our dirty feeth, so that we know, and everybody and everything know that, if the Master, which is The Master, have bowed to his slave, and wash his feet, then we, anybody and everything else must wash the feet of each other too, in the measure we can. Cheers.

[@BigDaddyGee85]

it really doesn´t make sense beeing afraid of AI. At the end, AI is still limited by human knowledge. All it really can do is to combine all of our false and true input in a huge scale and compute it with an already known algorythm. And all of that really only to prove or get a solution we have predicted already, or took as a possible outcome anyway. For example if you want AI to prove that 1 +1 is 2 but AI will not confirm it actually, then you would never take AI`s solution as correct. So as long we cannot prove AI´s solution with our own given intel (to this date), we will not accept it anyways. So in fact AI will not create ,,new" intel or is actual inteligent, it can only help us to take bigger and faster steps to expanding our OWN mind or intel.

[@Zroolmpf_Celmbror]

Thank God I've finally found someone who is saying this.

[@jimwatkins1664]

Deep Mind’s AlphaFold has already demonstrated AI’s ability to extend human knowledge dramatically. I think you are absolutely wrong to assume that AI is limited to knowledge humans already have. This isn’t a reason to fear AI, but it is a reason to be very careful with the agency that you give to an AI.

[@BigDaddyGee85]

​@@jimwatkins1664 All great innovations or predictions in history began with an idea or feeling based on human instinct coupled with knowledge. AI can only combine known knowledge or find/compare already defined patterns in huge databases.

[@jamiemarshall8284]

What evidence is there of this limitation? There is none.

[@trajklogik7304]

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 some thing. Without categorization there is only one of eveything.

[@mescale]

Hi, I loved your answer. It's enlightening. I'd like to share and ask you some question had came to me as I read you. Please, mind me for my english or lack of conciseness not. First: is this "one of everything" a category itself? Second: is "category" a category itself? Or rather, what's a category? Third: if we ask about a God who 'create' this 'one of everything', are we just splitting in two categories this "one of everything" or are we 'creating', or defining, a new "one of everything" category which will overwrite the older one? Or rather, would this operation be an internal or an external operation to the "One of everything" set? Fourth: what's an operation? Fifth: who's the one who make the operation of categorising? Sixth: does defining, or rather 'creating', needs at least one external element -the one of everything or the one everything is defined from- to make subsets from, with or within that element, to set everything, to set the set of everything? Seventh: does the set of everything needs a definer to define itself? Or rather, can a set define itself by itself? Eighth: is the 1 element truly a element 1 if it can slit into two elements? Or rather, is 1 a element or a set of elements? Can be the 0 element slit into something else other than it self? Can the empty set do the same? Nineth: is the empty set an element? Is the empty set at least the only element where other elements can place within? Tenth: is the 'one of everything' the one, the zero or the empty (set), the emptyness of everything? Eleventh: what's the empty set? What's the emptyness of everything? Is it the set of every empty sets? How many -or how much- elements does it contain? Twelveth: can the empty set, or the emptyness of everything, fill itself by itself? Theerteenth: who fill the empty set? Who fill the emptyness of everything? Really thank you. I apologize, I had not believed so lot of questions might flowed out. Waiting on a answer, Feeling happy already, Cheers. -The Lord is my sheperd, I miss nothing. Nothing, unless a you.

[@andrewferg8737]

Existence, or truth, in and of itself is the singularly self-evident axiom from which all else is derived. That is also the referent for the term 'God' in classical theology.

[@koraamis5568]

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.

[@blackshard641]

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

[@anthony-kv6qh]

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

[@ibnkhaldun7373]

“No system can fully understand its own deficiencies"

[@Forever._.curious..]

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

[@SJCDAVE]

Not surprising, Lex takes a discussion on higher math and searches for answers to (proof of) his philosophical questions. Very enjoyable, but Lex still seems a little disappointed.

[@BarackObamaJedi]

Gödel shook both maths and philosophy to their foundations, and it's what links these seemingly opposite languages, or what interrupts the continuum between natural and formal language. By what the Incompleteness encodes, any understanding of it in one language will not satisfy the understanding you have in the other. Understanding Gödel means that you're aware that you're not sure you understand it, or if your intuition of it is backed by a continuously derivable fractured discontinous computation of axioms and knowledge you have access to and memory of

[@nicholascarter9158]

In terms of mathematical philosophy Lex is born some eighty years too late to find what he is looking for

[@JackPullen-Paradox]

Nobody has ever explained what the true import of the theorem is, and this attempt is no different. It looks more like there are self-referential truths that we cannot prove than that there are significant truths of the standard kinds that we cannot prove. In other words, the theorem may be far less important than we have been led to think.

[@johnrichardson7629]

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.

[@andromeda3542]

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

[@kurtgodel5236]

*Utter nonsense on stilts!* Incompleteness is *absolutely nothing* to do with the justifiability or the (epistemic) status of the premises from which one draws inferences. Incompleteness shows a rather stunning divergence between provability and truth in certain symbolic systems. These are two utterly unrelated areas. There are no Agrippan problems for the axioms of symbolic systems. By the way, _Münchhausen Trilemma_ is the recently deceased Hans Albert's novel name for a problem that for centuries has been known as _Agrippan Trilemma._ It's use is mostly confined to a rather small group of German-speaking Popperians. Are you German? And what is GTP-4? Is this to indicate that your comment is the product of some "AI" software? If so, would you care to share your "prompt"? I'm rather curious to know more about the genesis of this rather elegantly presented load of unadulterated tosh.

[@VeritasVinci]

Call me crazy but I think everyone may be missing the point with incompleteness. If incompleteness arises from a complex system doesn’t that mean the iteration of that system gives rise to emergent properties that will always yield more problems than answers?

[@Peter-rw1wt]

Mathematics is a language, and as with all languages it is a self-referential system. That is why it cannot, in principle, understand itself.

[@An-ht8so]

How is first order logic self-referential ? Variables and constants make up terms, but terms are neither variables nor constants.

[@paulklee5790]

Complexity emerges because the universe is conscious and always looking to become….

[@JosiahWarren]

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

[@rgw5991]

GAYDOH

[@ItCanAlwaysGetWorse]

Great clip, but left the Göedel incompleteness theorem explanation Incomplete!

[@jamesboulger8705]

I am not sure why people wouldn't think an AI trained on a bunch of reddit forums would give contradictory, trollish answers.

[@disabledchatzen5276]

I feel exposed

[@astroganov]

The guest is amazing, and his eyes are so bright

[@JohnVKaravitis]

Godel was insane. He died because HE FORGOT TO EAT.

[@olegilin7094]

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.

[@ChessArmyCommander]

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.

[@ryandempsey4830]

Except it's been understood for a while now that there doesn't need to be a creator in the universe we inhabit. It's an established part of physics that structure can emerge from "nothing" due to spontaneous symmetry breaking. "Structure" referring the laws of the universe that give rise to what we see as material reality. It's understood that as energy levels lower, the many symmetries of the universe break down in ways that create more and more differentiated structure as opposed to the extremely high symmetry of nothingness (everything being the same under all possible transformations. This isn't just speculation. It's proven that the most fundamental laws of physics, the conservation laws, are all just emergent expressions of symmetries in the universe. They did not have to be decided on or written down by a creator. They just emerge due to the "shape" of reality. Even the most fundamental "things" in reality, quantum fields, emerge spontaneously as energy decreases. The four fundamental forces emerge spontaneously from one as the universe cools. At the beginning of the big bang, there was a single fundamental force, and as the universe cooled, broke into 4 different forces. More symmetry to less symmetry. More similarity in the universe to less, which is to say more structure emerging from less. No creator needed. The big bang itself is very likely the breaking of time symmetry. The existence of things like "charge" is due to spontaneous symmetry breaks too. Like liquid water being the same/symmetric from all sides spontaneously turning into a snow flake as energy lowers and suddenly being symmetric from only 6 sides. That's a rough analogy of a symmetry break. The structure of the snowflake emerged from the less structured water completely passively due to the symmetries of liquid water breaking as energy falls. None of this proves there is no God, but it DOES mean that God isn't necessary or inevitable to explain the universe existing. Structure can emerge from where there was no structure completely passively as things go from high energy to lower energy, which is just to say time passes.

[@user-gr5tx6rd4h]

Gödel proved that there are true sentences that can't be proved. If I ask for an example and you give one I may ask "How do you know it is true, since it can't be proved?" A self-referential sentence like "THIS sentence is false" seems not to be an example, since if it is true, it is false, and if it is false it is true - it is meaningless, I think. Is there any non-self-referential sentence that is true but not provable???

[@disabledchatzen5276]

refine the last question further for insight: Are there any non-self-referential sentences at all? Language is a relationship map, a topology, a surface, a structure. And math is a language.

[@user-gr5tx6rd4h]

@@disabledchatzen5276 By self-referential here what is meant is that the sentence "talks" about itself. Is there any known example of a mathematical sentence which is about "other things" than itself (and its being true or not), which is true but not provable? (Not one of the axioms, of course.) Ordinary human languages are not strictly logic while mathematics is so (free of contradictions).

[@bjpafa2293]

Editing a foundational part of the whole interview✨🙏

[@The_Conspiracy_Analyst]

There are consistent and complete systems, they're just limited in their ability to express mathematics. Presburger Arithmetic, Skolem Arithmetic, First Order Logic without quantification, etc. So you could say that what causes undecidability is the inclusion of integers together with addition and multiplication. If number theory were advanced to where we had a much better understanding of integers and their operators, perhaps we could then have powerful enough understanding to create something like the Peano Arithmetic that would be consistent and complete.

[@tsituaton]

Interesting point of view. My intuitive understanding is that if a formal system can encode itself without omissions, this already suffices to perform a Gödel incompleteness type argument. Can this be made rigorous? This would be interesting since it would show that it is not so much addition and multiplication that matters but rather some general self-referentiality feature. And it would also connect to what they discussed at the end - unavoidability of some form of subconscious. I mean, on the level of human mind also, you probably can achieve some form of exhaustive self-description, but there seems to be no such self-description that would not omit some truth about yourself.

[@The_Conspiracy_Analyst]

@@tsituaton Can it be made rigorous? You mean the exact change we make to a first order theory that would make it undecidable and no less or more? No I don't think it's a boundary that can be understood precisely. Complex problems in discrete math are like that. You get chaotic boundaries, and thats if you know where to look for a boundary to begin with. That was the point in making Presburger and Skolem (and there may be others out there), to add limited quantification and prove they are consistent and complete. You could certainly add other forms of quantification and see what that would yield. I don't know what those would be. When we add natural numbers, addition and multiplication to the first order theory we get Peano Arithmetic, which is what Gödel proved is complete but inconsistent. Any way of making it more rigorous, you'd have to come up with different systems of logic and try to prove them consist or not. You should check out "On Formally Undecidable Propositions of Principia Mathematica and Related Systems", the Dover Books on Mathematics edition. It's really not that complicated a proof. Any second or third year CS student can understand it. There's no heavy analysis. I read it years ago. It's really elegant.

[@benjaminangel5601]

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.

[@tessxu8367]

Due to decoherence…

[@trisbane4086]

Damn, that's a good explanation.

[@akagordon]

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!

[@vladrazzy]

Эдуард как всегда на высоте - чертов гений!

[@tupacalypse88]

I've been saying Goedel's name wrong my whole life

[@cookymonstr7918]

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

[@michaelgrayrn4579]

Duck Rabbit philosophy Edward Franklin, lex podcast 2023 Goedel incompleteness theorem See duck rabbit picture Is it a duck Is it a rabbit Some people see duck Some people see rabbit Some people can see both easily Some people, with some effort, or in the case of UNfocusing can see, at difference rates of change and completeness Some people can never see both Can this be similar to political views In psychology, there may be a predisposition to a certain political swing dt genetics and behavioral tendencies Gestalt Death, dying Religious Consider: Rabbit is the living being Duck is the dying being Imagine you have 10 family members in the hospital room of an actively dying loved one. How many of these are seeing a duck, how many of them or seeing a rabbit only, how many of them are starting to transition their view from rabbit to duck and or able to see both sides. How many of them have always been able to see both sides so they've always had some range of shifting for you. For those that can only see the rabbit, those are the ones that insist on IV fluids and feeding tubes and ventilators and continuous dialysis machine and long-term care homes for the vegetative body. At some point the duck always appears. The rabbit is proven by life itself and the duck is proven by the cessation of life. Is religious training, duck training? We can review John donoher and his theory of being dead twice. You're dead before you were alive and you are dead after you are alive. Does this make the duck more prominent. Are we more duck Do we assign rabbit theory to duck so that we can feel better personally Are we actually just 99.9999999999...% duck? I'm happy to accept and to have been the rabbit I'm content with the duck situation I'm glad I'm not a goose

[@marl6908]

THEY are not capable of understating the Incompleteness Theorem.

[@jamescherry2763]

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.

[@77sanskrit]

9:22 All due respect Mr Frenkel, I think that you might have that backwards there, "If it is not completely useless, then there will be a true statement in it....." . But isnt that actually redundant, or backward, because the fact that there is something "real" or "true" it describes , is precisely what determines the systems usefulness??? I cant help but feel at a certain bedrock this might imply, some kind of fiat of historic symbology as its "genealogies" of language, math, and writing have been delineated and distilled to us as modern mathematics in all its glory... that only upon our seemingly, passive UN-logical, ubiquitous faith.. (and our shareable experience of those systems) that serve as the foundation for a far more resilient, and causal rational logic. "That word is made up!!!!" "Theyre all made up." "Mind... Blown." hahahaha

[@jayjackson3045]

Yall struggling with 'explaining postmodernism'. Just listening to the audio book, stephen hicks, 3hrs on x2 speed. Explains every question discussed.

[@rer9287]

I simply never see "popular mathematicians" that understand mathematics - like this guy. he simply does not understand math. Boring mathematicians that dont give interviews, seem to understand math better than this type of mathematician. His presented explanation is not internally consistent itself and if it were - Godel would be trivial and simply outside the realm of logic. From Aristotle's primary axiom against contradiction, we get all logic and in turn mathematics where A must not Equal B. Logic then necessarily excludes self reference because self reference is a case of A.1, A.2, etc, but logic only applies to A NEQ B. Calculus salvages self reference in the case where A.1, A.2, A.3, etc converges on B, but it cannot cover the divergent case. However, no that is the only axiom of mathematics (contradiction is forbidden). Math is simply a language where supplementary axioms (actually mentioned in the video) are decided in advance and must never violate contradiction (logic). Once then you understand math is a language where the logical rules are decided in advance (axioms) all of these other claims about math - like "math is incomplete" make no sense. Substitute other examples from its category (language), say it again, and the nonsense becomes clear. Would you say English is incomplete? Would you say French is incomplete? It is nonsense.

[@vincentrusso4332]

Everything I say is a lie... / this sums up Gödel's Incompleteness Theory. Anything that is complex enough to reference itself can not be proven by its own logic... - Some country boy with a Good Enough Diploma.

[@acasualviewer5861]

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.

[@musashi4856]

Why did Einstein waste his last years pursuing a unifying Theory of Everything (TOE) which was not possible as per his walking mate, Kurt Godel⁉ There will always be axioms from which cannot be explained within a formal system giving way to recursion, self-reference, and the necessity for latter theories to explain the former theories. It is turtles all the way down...

[@stevelarsen4717]

I thought I could, but I cannot...

[@chemistryset1]

Wow - I am literally a fucking idiot, but this guy has a great gift for imparting his knowledge to the uneducated. And English probably isn't even his 1st language. Great stuff.

[@davidthurman3963]

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

[@gariusjarfar1341]

In Godel's day no fractals and fractal based software were ever a consideration. Aristotle was a geometrical mystery teacher, squaring the circle was a geometrical way of understanding fractal reality. Godel had no idea of the concepts of Aristotle, or his partitioned method of teaching the Eleusis mysteries. Math without geometry is lawless.Having executed Aristotle, the rebel architects could claim the code to Eleusis. Learning the ancient view of geometry is now forbidden in universities. Bacon, Da Vince and Newton would not agree. Rascals hold our abilities.

[@farhadfaisal9410]

An innocent question: if the incompleteness theorem refers to the existence of ''valid'' propositions whose truth or falsity is impossible to prove within a formal system of axioms, then, on what basis is it assumed that the proposition was considered ''valid'' to start with (if it was)? Or, is ''valid'' above = merely ''formally constructable''?

[@jamestagge3429]

So, if we consider this business in a slightly different context which would be considered as the real world extension of these theories, how is it that one can claim that a particular statement is true and at once claim it to have no proof. To know it is true one would have to understand clearly how and why it is true. If one knew the how and why, that IS the proof. This is unequivocal in the understanding that a statement is true. With regard to Turing, it may be true what he claimed but I have yet to find a video which explains his propositions which is itself not wholly contradictory. Consider, (this is just of the many videos but the logical failings are the same in all and this critique applies to all) if there is a halting program H which analyzes another program (call it program B) to determine if program B will halt or continue on and (program H) outputs its finding, halt let’s say, the initial goal is accomplished. IF then there is a reversing program (call it D) which looks at what H output as to the function of program B and does the opposite, what is it exactly that program D would do? It cannot do anything to effect program B for it is analyzed while not running, the output of H only a prediction or determination of what it WOULD do if running. Continuing on with the critique, embed program H in program D and feed program D (with H embedded) itself into program itself. It is claimed that the input of program D with H embedded would be analyzed by program H and an output of program H if halt, for example, would cause program D (as the input) to do the reverse and by that the encompassing program D to also do the opposite supposedly creating a contradiction for program D would be both halting and continuing on at once. This is nonsense. Program D in both cases could not be analyzed by program H under any configurations proposed because it is reactive, available for both inputs of halt or continue on from H. It is presented as requiring the output of H as in put that it might “know” to what output to react. H cannot anticipate it for again, it is reactive. As you can see this entire scheme is nonsense, is not how computers work nor computer programs and is a complete contradiction. Can anyone explain?

[@CodepageNet]

That explanation doesnt explain anything. How about making a more concise example instead of sideyracking every phrase. Disappointing.

[@lucasfabisiak9586]

"If you have a sufficiently sophisticated formal system . . . , then there will be true statements in it which cannot be derived by this linear syntactic process of proving theorems from axioms. . . ." If Godel's incompleteness theorem was proven from axioms, using a "sufficiently sophisticated formal system", then couldn't there be "a true statement in it which cannot be derived" through this process such that it contradicts his proof? I'm not a mathematician, so this is beyond my abilities, but wouldn't that also generate a contradiction (namely the incompleteness theorem being both true and false)?

[@jakeenvelopes9561]

It's a very long video for something that doesn't say much. Most of this is the sort of thing any technical topic journeyman will know. The only thing it says about Godel's Incompleteness Theorem is what you already know: For any axiomatic system there will be true things within that system which cannot be proven by the axioms which created those truths. We know this, but the title says he Explains it, and he doesn't.

[@Upuaut4572]

it is surprising how the stupid "duck-rabbit" type pictures can be analogous to many grand mysteries like incompleteness and superposition

[@ciarantaaffe4199]

That's not surprising and they aren't grand mysteries. They are inherent and everywhere. Our perceptions and beliefs are all "superpositions" in the sense that they are probabilistic approximations based on limited sense data. And incompleteness is nonsense. But irreducible complexity and computational complexity (which is what people really mean when they talk about incompleteness) is what makes it harder to find an answer than to verify one. All ever-present and ubiquitous sentiments and phenomena

[@srimallya]

Ontology The mind equation The more we think we have agencies to the actions the body takes, the more we imposes agencies to the activities in our environment. The ownership expands into other objects. The illusion of body ownership comes from the modeling of the motor neurones pattern from the childhood. The self just predicts the bodies behaviours in the real world with its simulation of the real world. Multiple sensor data unify in language in the simulation. Intelligence is economy of metabolism. Language is temporal reference frame of economics. Self is simulation in language on metabolism for economy. Longer context windows create generalisation. Shorter creates specificity. Longer context window needs more computing. Self is the protagonist creates a storyline in this context window. Theory of mind evolved so that an entity can learn from it’s peers. It’s creates a possibility for parallel computing. Then it creates the possibility of transmitting the highlights of a generational lessons into a metaphorical story for upcoming child. That creates the possibility of modeling the physical world as a macro organism. Creation of fiat currency was the singularity of this species. There is now one macro organism in a connected web world. Loosing the peer of the macro organism creates the possibility of loosing it’s objective function. That creates the possibility of loosing the theory of mind of this macro organism. That creates the possibility of death of this macro organism by reaching the planetary boundary. That is post singularity. Every action we do, we do what is expect from ours tribe. Body might have a opinion, but not the cell. They do what is expected from its tribe. If it doesn’t we call it cancer. The body is a mirror system of the macro organism. Each system have two transactional openings. Serial and parallel. Each cell within the body can transact material or information serially by genetic determinism and parallel non deterministic way. Similarly eact body with in the macro organism can transact serially by inherit material and information in a deterministic manner and parallelly through language in the society. Everything emerges from this systems. Every sensor is a range calculator of contexts. Taste > touch > smell Immediate and visceral. Vision > hearing Not immediate, tactical. Self > language Abstract, strategical. In this non deterministic economic transaction space the individual is coded to transact with its kin. From the macro perspective tribe formation minimises economic risk for the tribes. Each and every node of these systems organise and mark their kin’s with identifier. Thus, i am what you make of me. And others too. For short cut i have a legal name, so you have. My legal name gives the legitimacy marker so that you can transact with me parallelly if you have the same marker. The self is a simulation in language. It negotiates between the physical world and the information world. All these negotiations are the temporal memories in the body and scene of the story. Now, when we started writing we iconised the abstract in the physical world to make symbols for the tribes. So that under that common symbol every node will take the same risk and distribute equally. We created more and more symbols and more and more meta tribes within the tribes so that who has the authority to use the pen control the tribe. When the negotiator act like an executioner then it’s a downfall of that system. It falls apart. Objective reality > legitimacy > individual behaviours. Survival of the species is dependent on the decoding of the objective reality. Since no species can access it, they use their sensors and interpret the small data which is useful for the survival. Few complex species have created communication channels to rectify their sensory limitations to survive. Homo sapiens has widened their communication channels for faster throughput and started storing them as culture and carrying them through education. As a result we have created social truth. Factual datas are the useful snapshot of the objective reality, a totem, a physical object can be observed with the sensors. Truth is an individual subject, an interpretation of the sensory data, a useful compromise. The social truth is the useful compromise for the group by the group. The goal of the social truth is to survive as a group. Physical Transcriptions of these social truth legitimise them. We are tribal animal. We live in as a physical tribes and inside of hundreds of meta tribes in simulation which is the socio political data space we call it as the world. Since we can’t access the objective reality reliably we look for social truth as the best guess blindly. Institutions legitimise truths. Fact driven institutions are more useful in the survival of the specie. In other hand opinion driven institutions are not so useful for the species. We do what we can get away with and exactly as expected within the context of our meta tribes. We have two bodies The biological one is like looking the earth from space. And the political body is like the state. The name you carry is the political body. It transacts with the political states on the boundary less earth. From the evolutionary perspective every biological entity has a basic feature which is homeostasis. It’s the functioning sweet spot of that entity. A control center read the sensory data to regulate itself to that state. By doing so it’s validate or update it’s prediction model. In the process of becoming a complex organism it developed an extra layer of processing. That’s our conscious mind. And the control center remains as subconscious. The subconscious collect the sensory data and regulate itself to stay functional. Now when it stumble upon a novel environment it float the management to conscious mind to find the solution for homeostasis. This conscious mind have one sensor which is language. It works like a spiderweb. As a spider creates it’s web it’s perception gets expand. We are like spiders in a jungle. We started creating these small webs at least 2/3 million years ago. Our offspring stayed on it’s ancestral web reinforced it expanded it. In time nearby webs became larger and connected with each other. A common structural geometrical pattern emerges from this. This became the symbols which is the backbone of all language systems. In time the forest becomes the mesh of web. The superstructure is exactly the same but when we zoom in we can find different species of spiders are making their type of webs in between the super web. Each spider try to senses the vibration of flies and Try to catch it before others. Every movement is telegraphic in the zone. Every form of perceptions are just a different pitch of note traveling back and forth in the web superstructure. There is a echo of older vibration pulsating through the web. Full of noise and self repeating hum. That’s cultural history. In the background there is the base hum in the infinite feedback loop. Insignificant but ever present. The sum of all the vibrations from the start.

[@mathearnius]

14:25 I’ve seen the exact dress in person. It’s absolutely blue & black. My theory is that women often sift through dark closets, where a dim lit white & gold dress appears like a sunlit blue & black dress.

[@rmorris5604]

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.

[@behroozcompani2348]

As I understand it, Godel's proof is based on a self referential statement that to begin with is void of any MEANING. As such the conclusions are meaningless.

[@smolderingtitan]

Lex is good with math and science guests because he doesn't push back on people who are smarter than him. Lex is bad with culture-war and political guests because he doesn't push back on people who should be dumber than him.

[@davidespinosa1910]

Now ask Frenkel to explain Godel's *completeness* theorem for first order logic -- that everything true is provable.

[@danushairan]

The next evolution will come in the field of complexity and mathematics of adaptive complex systems. This theory of mathematics alongside game theory can explain life and biology as well as the social behavior of animals (humans in our case). Funny how mathematics usually advances with warfare and in peacetime, nothing new seems to be created.

[@greyesnorcal]

I made it halfway. Waiting for this to become entertainingly interesting. Then he said, "training a duck". Am I on mescaline?

[@5r3t5n0m]

A practical example or the proof would be nice, cause it's difficult to follow the logic of something new coming from a system or rules, but not being derived from those rules. Tells me that the computer did not stay within the rules and that there must be another axiom/ata in there that we don't know about or that we dont understand the derivation process. The last is perhaps similar to a chess engine which stays within the rules, but human comprehension is not on the level to follow all of the suggested best moves. I don't think there's a big mystery about seeing those pictures differently. So of it is individual genetics (color interpretation perhaps) and some of if is how we learn and remember thing which is different from individual to individual. I can kind of see duck and rabbit features in that picture, but actually it's neither. It's a truck to have approximations of both intent to confuse. Neither have I seen a duck bill like thsr or rabbit ears like that in real life, but those are the closest options given the overall features.

[@AliReza-cx7wg]

Physics is not based on math. Physics is based on nature's behavior. It takes for ever mathematicians find out why they are not very important for physicists

[@waterfrodo4304]

Physics is based on the idea of using maths to describe nature’s behavior. Writing poetry about nature is not physics.

[@AliReza-cx7wg]

@@waterfrodo4304 Nature's behavior I said idiot! ( Observation). The physics has sufficient math to deal with toy models. Math job is dead part of physics

[@rfvtgbzhn]

@@waterfrodo4304 no, physicists use math because they found out it works. However modern physics has several problems that are unsolved for decades, maybe at some time they will find out that this is because math is not sufficient and we may need something that is based on a different from of logic that goes beyond the formal logic, that physicists use today. Maybe some kind of dialectic logic. If you think hard enough about formal logic you find out that it contains assumptions that are not really true in nature, for example that something is identical with itself. In reality everything always changes. One example of this is the ship of Theseus. If you change a part of a ship and then change another part and so on, until after some time you have no part that is identical with the original, is it still the same ship`? And if not, when did it become a new ship? Already, when the first part was changed, or when over 50% of the parts where changed, or only when every part was changed? This can't be deducted, you can just define it arbitrarily if you want. It is much better to accept that the ship is simultaneously the same and not the same, but formal logic doesn't allow that. Also most physicists that ask that question think that Gödel's incompleteness theorem also applies to physics (it is kind of unclear though because the observable universe contains not just a finite number of particles, but also has a finite partition function, meaning that it has a finite number of possible states.

[@fredwinslow744]

is it possible that computers with quantum compute will postulate axiom based math that goes beyond or numerically beyond the postulates of the shared and use-full myth we call mathematics ?