There was a rendering error and you can listen on any of the audio platforms to hear the the missing audio at 1:22:30. For example, Spotify (open.spotify.com/show/6yIO9TophRyStq6abbzOjl) and iTunes (podcasts.apple.com/us/podcast/id1521758802). Since I can't re-render on UA-cam, here's a summary of the missing audio. "CURT: Sometimes I hear this from the finitists (or the ultrafinists): it doesn't matter if each plank length cubed volume of the universe was covered in a transistor, we still wouldn't be able to carry out a certain calculation before the heat death of the universe, therefore it's "meaningless" -- however..."
hello curt, i really enjoyed your interview with prof wildberger. I have been following his channel since 2014-2015. One more person you may find interesting to talk to on the topic of formal math is prof ck raju. Would love to see you interview him. thanks.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
Of course numbers are not real. They are just symbols of concepts or patterns. They have certain rules they follow and we created those rules and we selected those numbers(patterns). This is because it is convenient for us to manipulate these concepts by symbolizing into simple one letter or number. Math and reality: Language is mixture of conceptual words. Long sentences carry big concept. Reality is much bigger complex than language and or math. Math is nothing but interplay of concepts and jump from one concept of problem to another concept solution. It is conversion of objectivity into subjectivity back to objectivity. It is mind play with objects in the universe. When you say: F=MA Each letter is a concept E=MC2 -- each letter is concept and we chose those concept and put them into play and make another concept!!!!!!!!!!! Concepts exist in our mind subjectively. Saying math exists discovered or invented is rubbish Math is invented my our mind and body invented mind. Hahahahahahahaha!!! I win 🏆 🥇 All these big heads are just big !!!
A big thanks to Curt for a fun interview, steering us toward lots of interesting and important topics. And thanks also to the audience members that contributed nice questions too.
I've already commented on how wonderful I found this conversation, but wanted to let you know that I have since subscribed to your own channel and was astonished at the sheer volume of output there. I'm very much looking forward to working my way through. Thank you Prof. Wildberger and Curt for a brilliant conversation. Its much appreciated by so many people from all walks of life.
Professor Wilderberger's series of lectures on the history of mathematics is really great: check it out. I was delighted to see him being interviewed on this channel. Looking forward to the second interview.
I am infinitely grateful to all the knowledge I have gained in math by watching Norman's videos with his drumstick pointer and crisp and clean wall blackboards. He almost makes it too easy. I did buy his book as a means of supporting his efforts and I admire his cavalier attitude toward making only rational decisions in this one life we have.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
So happy to see you in the wild Norm, wishing you the best and want to say that I absolutely love your channel. I have learned so much from your teachings. Thank you Curt for inviting our guest today.
Always really interesting watching your videos Curt, but for some reason, this was very easy to follow and understand. Maybe it’s because you edited or possibly because you guys know how to communicate effectively. Either way, love you man!
Yeah, what does it mean "to do"? NJW says you can't do an infinite number of things as if we are talking about some sort of time-dependent manual process.
I am in awe. I confess that this is the first time I have heard of Prof. Wildberger but was blown away by his clear ability to explain to a non mathemetician such as I. I will be looking up his channel for sure. Prof. Wildberger showed such a deep appreciation for being included in the stellar 2022 Theories of Everything lineup. Thank you Curt for giving me the opportunity to be exposed to such accomplished thinkers. This was a great episode, already watched it twice. 👍👍👍🏴
This man is a mathematician in name only. He completely misunderstands what Mathematics is at the most fundamental level. He is basically a flat earther for Mathematics. Flat earthers say, "The world looks flat, so it must be flat." He says, "I can't see infinity, so it must not exist." But just as we can measure the radius of the earth using the shadows cast by measured sticks, we can also describe and understand some properties about infinite objects, even if we don't have any tangible examples on hand. We know it's real because we can infer its existence. We know that we can add one to every natural number and get a new natural number, so we can infer that there must be infinitely many natural numbers, without needing to list them all out. That's the whole point of Mathematics, which he completely misses.
He has a series on his channel called something like Banking in the Wild West (probably butchering the title, but should be easy to find the playlist), where he doesn't (yet) talk about the future of banking/economics, but he does an interesting job of giving a mathematician's view on how the banking system has evolved over time from being based on actual 'coin'/metals to being more and more based on loans on top of loans and other more abstract and dubious systems. I've enjoyed watching it. Might not be exactly what you're looking for though, not sure.
If not addressed in part one, I would like to hear what his thoughts are on finitary statements that require the notion of infinite sets to prove, such as the Paris-Harrington theorem.
A prophet is what thats called we have become to etymologically corrupted to recognize that. Esoteric prophetic predictions based upon a man made language of symbols we agree upon. No different that using alphabet and recorded history of society then you make future predictions based upon it
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
@@hyperduality2838pagans and the occults have believed it was an esoteric theory of everything, written in the stars for many millinia. From ancient zigurats to it's modern eq.colleges. from the ancient Chaldeans prophets to the modern day priestky physics professor. Search continues lol
@@dadsonworldwide3238 It is very easy to understand:- en.wikipedia.org/wiki/Gaussian_curvature Negative Gaussian curvature is derived using two separate or distinct points, origins where as positive curvature is defined using only one point as origin. k1, k2 = reciprocal of the respective radii which are perpendicular to each other and in negative curvature the origins have to be separate at all times -- non null homotopic. Dark energy = repulsive gravity as galaxies are accelerating away from us at faster and faster velocities, and this is consistent with negative curvature or hyperbolic space. The big bang is an explosion or repulsive, divergent (inflation) = negative curvature singularity (a point). Positive curvature is dual to negative curvature -- Gauss, Riemann geometry. If the big bang is a negative curvature singularity then it requires a minimum of two points or origins as the definition of negative curvature according to Gauss has two dual origins. The big bang = duality (isomorphism) -- non null homotopic. The physicist Julian Barbour has written a book about Janus points/holes:- ua-cam.com/video/FMD8B7VRX0w/v-deo.html The big bang is a Janus hole/point (two faces = duality). Topological holes cannot be shrunk down to zero -- non null homotopic. The Necker cube is a good example of duality and you can watch the spinning dancer here:- www.medicaldaily.com/right-your-eyes-science-behind-famous-spinning-dancer-optical-illusion-336122 Clockwise is dual to anti-clockwise -- duality. Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality). Gravity is dual to acceleration -- Einstein. Einstein did not believe in black holes so he did not take his equations about negative curvature seriously, so negative curvature singularities do not appear in his theory of General Relativity. Dark energy or negative curvature must be added to his equations. Energy is dual to mass -- Einstein Dark energy is dual to dark matter. "Always two there are" -- Yoda.
As as a theoretical physicist myself, I cannot thank Professor Wildberger enough for his contribution to teaching mathematics in a more pragmatic and intuitive way to students. I do not think mathematics education should be simpler, but clearer, with many computable examples. As the Professor pointed out in this interview, we do not need brilliance, but clarity. Brilliance is a by-product of the latter, not the other way around. Nobody is required to show off or impress to be regarded as a genius, unless we suffer from some ultra-ego complex. If you cannot explain any subject, being math, physics or wood-working, using day-to-day intuition, then you are fooling yourself and your audience in presuming you have understood anything at all. Regardless of whether the real numbers do exist or not, it is mandatory for any educator out there to be able to convince their students that your theory is solid. From the very foundations. If you are a student, never assume what you are taught is correct, always ask questions and require precise answers. This is neither presumption nor arrogance, it is the true nature of what makes us humans, therefore curious about Mother Nature.
You may disagree with Norman's conclusions, but you have to agree that the notion of infinity as a complete extension (i.e. the axiom of infinity in set theory) is a very strange notion. I appreciate mathematicians actually thinking about the foundations of mathematics instead of just leaving that to the philosophers.
I see no problem with infinite sets. If we assume there are finitely many natural numbers, then we must accept that not every natural number has a successor. So it becomes possible to say "there exists a natural number N such that N+1 is not a natural number, or rather, is undefined" . I find that statement more ridiculous than the notion there are infinitely many natural numbers
This was great! I've been a follower of Prof. Wildberger's channel for about 7 years, and I'm glad I got to know your project thanks to him. I would love to suggest mathematician Gregory Chaitin as a next guest. His work is amazing and too little known, and touches many of Wildberger's points and many others (including his work on Leibniz), critically Chaitin's Omega Number, or Chaitin's constant. Kudos Curt!
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
Okay, I'm not a mathematician, but I've listened in on Brian Greene's master classes in quantum physics for hours without understanding how to do the mathematics, yet the concepts he talks about are fascinating. So, without having a deep understanding of his math, I listened to this conversation from two levels: 1) Since I work with meditation and the multi-dimensionality of consciousness-and how to live that on a practical level at the grocery store, for example-I was curious to see how a mathematician would view infinity. 2) His philosophy of mathematics was very interesting and, oddly enough, connected a few dots that I hadn't connected before. That being said, it would be an amazing interview/conversation if you could get a physicist together with Norman Wildberger and asked them to play "What if..." on switching over to the older mathematics system Prof. Wildberger spoke of. If it was ever actually tried, the ramifications and ripple effect is mind boggling. But it would be an interesting thought experiment to listen in on, you know? One of your strengths is you're a good listener, so you ask good questions that feed the curiosity of the listener. If they're curious and have an open mind, that is. Anyway, thank you for giving me a glimpse into the point of view of someone who I wouldn't meet in my daily life.
I am a finitist and think the concept of infinity should be eliminated completely from mathematics (and from physics, philosophy, and religion too while you're at it). The answer to the question of an implied largest integer is that it is context dependent. It depends on the capacity of your computing device. If you fill the display on your pocket calculator with 9s, you get a large number but you can't add 1 to it. As Norman suggested, if the entire universe acted as a computer with planck-length components, you would have a limit on the size of numbers that could exist, and thus establish a largest integer. In the context of an abacus, the largest integer would be determined by the number of beaded wires on the abacus.
@@quantumbubbles7977 If so, the process of "turning out" will be interesting to examine. It should be evidence of someone, or something, doing an infinite number of things. I don't think that will ever be done.
I’m not a finitist but I think the concept of infinity is not well founded. The mathematicians just stumbled along without having properly defined or understood it, which is terrible.
This reminds me of how observation itself effects the state of things on a quantum level, because as soon as you’ve “grasped” infinity it escapes you and becomes something greater. So it’s an imaginary thing.
I notice Norman didn't once uttered the name Cantor, rather he used a word demeaning him. Thinking about infinity we are soon drawn to a one dimensional natural numbers, when infinity remains what they call 'undefined'. What Cantor said is simple, he divided the one dimension into two, but then infinity isn't manageable but infinity can be accommodated in what turns out to be finite. So since the 19th century when Cantor discovered how to side step infinity, mathematical logicians pounced on Cantor's mistake (which faced much objections, right from the beginning) when Godel (using symbols of PEANO's natural numbers) arrived at 'undecidability' even though these numbers suffer from the same nature of 'undecidability', termed by Wildberger as 'arrogance', its like pouring milk into a new bottle. Cantor cardinality/ordinality helps logicians like Hilbert to approximate infinity with finite, that has some mathematical usefulness. But in essence neither Godel nor Cantor can be said to be right. They remain 'arrogant'.
I’ll edit this comment as I listen to this podcast, if my questions are already answered in it. (up to 38:40 now) Ive watched a handful of Dr Wildberger’s lectures and always wanted to ask questions like these: 1. A common refrain of his is that infinity is not suitable as a formal concept because actual infinities of various sorts are not realizable. For instance, one cannot ever claim to have “completed” an infinite task, or represent most real numbers as they’d require infinitely large computers. In this sense he claims infinity (and thus most real numbers) do not exist, and also, we cannot do formal reasoning with them as concepts. I fail to see how the non-realizability of some X (in this universe) relates to our ability to formally reason about it. What is "realizable" is largely a contingent truth. We can still reason about entities and properties in logically possible universes, even if those entities / properties are never realized in this one. When he states that a particular infinite (or large) sum "could never be calculated in this universe", Dr. Wildeberger is demonstrating this fact (he is in fact reasoning about such a non-realizable entity!). Formally speaking, we can define a symbol X in set theory / number theory, which, by virtue of its relationships to other symbols in that system, is a suitable representation of infinity, and reason about it. We can even check the validity of those acts of reasoning with computers. Can't we? 2. The usefulness of infinity, real numbers in physics Q: Why are mathematical systems involving real numbers and grounded in infinities so consistently useful for expanding our scientific knowledge and representing the world, if theres a deep problem with them? 3. Putting infinity to the test Q. If there are logical holes in the ground of modern mathematics, can Dr Wildberger demonstrate this formally, by doing something like deriving a contradiction from them? Can he perhaps show that the system is incomplete (other than the incompleteness arising from godel’s theorems?) I look forward to listening to this in full, and hopefully seeing some of these questions resolved :)
I really like these questions and would love to have them answered! As someone writing numerical algorithms for living, I need to understand what these algorithms do and how they behave on a computers. The understanding usually comes in two steps. First, you proof certain properties of an algorithm imagining it is running on a computer that can do real number arithmetics. Second, you proof that the real computer with finite(floating point) arithmetics gives you a good approximation. As a formalist, mathematics is nothing more then consistent manipulation of formal statements, I do not care if the idealised computer with real arithmetics "exists" or not. It is just a formal tool (a very useful one) to at the end prove something about the algorithm I run on my computer in front of me. So I still fail to see why should one give up the notion of infinity/real numbers. As the saying, the shortest path between two points goes through the complex plain. We could easily say that the shortest path between computations with large numbers goes through infinity.
I would've liked to have seen thorough follow-up questions in this interview, e.g. after a weak response to a question about e in physics at 42:50. Wildberger answered by saying physicists only have in mind approximations to e, which isn't a fair characterization of how physicists work. e as an exact transcendental number occurs prominently in physical theory, e.g. in solutions to differential equations. Much of what Wildberger says deserves tougher questioning and criticism than it gets here.
One of the points is, you don't *need* the formalism of infinity. You don't need to take a limit to infinity, there is a way to do it that is entirely constructive and finite
You can get around Norman's objections to infinite processes by the same way of saying, there is no largest number. The epsilon/delta method works like this - for all epsilon, I can specify a delta. Limits work like this as do iterative processes - give me a level of accuracy, and I can specify a number of iterations. No mention of infinity. The notion 'as N tends to infinity' is tagged onto the proofs and definitions of Real Analysis.
_"Mr. Owl, how many licks does it take to get to the Tootsie Roll center of a Tootsie Pop?"_ I feel so deeply vindicated by this man's thinking. I know I have some kind of brain damage. So, when it came to math, I must have engaged a work-around so as to avoid memorizing an ever increasing number of axioms. He articulated this perfectly. The work around ended up going backwards every time a new principle was introduced and figuring out *why* it was true. I managed thru grade school and some of Jr. High, but then this method no longer worked because HS math is where they require you to "just do the calculation". I wanted to understand, but no one would take the journey with me to find out why something worked. The factory model in schools destroys so much value. Several years ago online I was in a disagreement over some of these items because I wanted him to defend the concept of "real" in a way that connected with the lived-in world. He refused and insisted that I wasn't smart enuf to do this esoteric math because I couldn't understand the ground that these axioms stood on. Thank you for all this confirmation. Bless you both. 🤗 P.S. Can you tape that convo about UFO's and give yourselves a chance to change your mind later about publishing it? I think he'd be very surprised by the positive reception it would get. Even from peers. You might even be in the majority. 👽🖖
47:07 I think in quantum mechanics you cannot avoid irrational numbers. You can always rotate a light polarizer some degrees (let's assume the universe is discrete as so the degrees are rational) that are not of the form p = q^2 for some rational number q. Therefore, the amplitude for the state of the light being in any of the two orthogonal basis will be related to sqrt(p) which isn't rational.
Any moment in which we say that we understand what is essentially pure Mystery, we are losing our place. Infinite, God, Consciousness, Unconditional Love, all those mysteries cant be understood, and that is fine. We are the childs of the Universe, lets play.
I don’t understand why he thinks referring to an infinite set, say the natural numbers, involves assuming that one can complete an infinite number of tasks. I don’t need to have to write a list of every natural number in order to speak about the set as a whole.
it goes to his latter comments. it might not be a problem for pure math (where you can just use certain symbols to hand-wave away what you actually explicitly mean), but if you're using a math system (counting, enumerating etc) to help model truths about "physical reality", you want to be more grounded that you can do the things you claim to be able to do, at least that's how i see the argument, which i agree with. at least, as he further argues, mathematicians should have an assortment of systems to choose from, so there is value to explore one without the concept of infinity, especially if our tools which we use to measure the "physical reality" cannot compute/enumerate with infinite concepts. infinity problems turn up in quantum physics with the current tools we use. so ultimately i think its a pragmatic argument that could help.
@@julsius I don’t really see how that responds to my comment. I was trying to say that Wildberger’s criticism of standard pure math is that it wrongly assumes that we can complete an infinite number of tasks. But I don’t think anyone actually assumes that, so he’s making a straw-man argument. More generally, he seems to be confused about the relation between pure and applied math. Any pure mathematician will happily grant that applications involve approximations. But that doesn’t mean that they are simply “hand waving” or doing anything illegitimate when they use symbols to stand for real numbers, infinite sets, etc. It’s as if he wants to eliminate anything in pure math that doesn’t directly correspond to applications or concrete computations. But, even after watching this video and some of Wildberger’s videos, I don’t see any real value in his project.
@@synaestheziac im not fully sold on his argument that a system without infitinities will be more true/correct/sound, but im ultimately a pragmatist (closer to applied math), if it proves to be useful then that was time and energy well worth spent developing his system. i think there are already some potentially useful ideas in his videos. For example, its another valid question to ask, "what if your system of infinities leads to false assumptions about limits of approximation methods?"
“Real numbers aren’t real” is a philosophy of nihilistic mathematics, because in order to maintain a consistent philosophical commitment to this claim, you must also commit to a denial of the existence of the number 1. I’m not claiming that there’s a mathematical proof of an inconsistency in systems that do not posit the existence of the real number system. Nor do I contend that the denial of the existence of real numbers like the square root of 2 implies that there are no numbers like zero, one or two; I only contend that the philosophical reasons given for choosing to commit to a metaphysical claim that there is no square root of two are not more convincing than reasons given for objecting to metaphysical claims that there is a number such as zero or that there is a number with the properties ascribed to 1, or, even accepting its existence, or for objecting to metaphysical claims that there is a way to add it to itself to obtain the number 2. In fact, if we adapt the contention that committing to existence of specific numbers requires providing algorithms to construct or compute them, then in that system of philosophical commitment, we can produce such an algorithm to construct a square root of 2, precisely by taking as a square root of two the algorithm obtained by applying to the equation x^2-2=0 the bisection method, in the interval (0,2). If we declare that to be the (metaphysical) square root of 2, then we have met your requirement for justifying the existence of that specific number, because we can define addition and multiplication of algorithms so that this algorithm has all of the properties required of a positive number whose square is 2, and we need not resort to equivalence classes of Cauchy sequences for the definition of a (positive) square root of two. We can then, however, use the notion of a Cauchy sequence, the notion of convergence of (Cauchy) sequences, and the notion of an equivalence class of Cauchy sequences to explain how this particular algorithm could have been taken as merely a representative of such a class, so that someone else, having used a different algorithm as a definition of a (positive) square root of two, and therefore adopting a different metaphysics, is guaranteed to have a metaphysical interpretation of the problem and solution that is naturally isomorphic to our metaphysics. The traditional development of mathematics using the theory of Cauchy sequences and defining real numbers as equivalence classes of Cauchy sequences has provided the foundation for writing such a proof already.
Curt you continue to be one of, if not THE, best casters on the net today, thanks for sticking with it! In Part 2 I'd love to hear the Professor's take on Eric Weinstein's statement that the hopf fibration is the most important construct in the universe. Secondly, that we can derive basically the entirety of physics from it and with changing only a few parameters can construct alternative physics (if I understand his contention correctly). As I'm sure you know, Eric mirrors the Professor's intuition that physicists can benefit from mathematicians understandings as they are fluent in the language that physicists use to express their domain.
Surely, one of the favourites which I’ll watch again and again, like the Richard borcherds interview. There are more mathematicians you would want to interview in this space, like Joel David Hamkins, (Mathematician, philosophy of mathematics and set theory, same name yt channel), Terry Tao, Timothy Gowers, (same name yt channel, Cambridge courses), also himself a fields medalist. (Both Borcherds and Gowers won in 1998).
The guy whose main project is to develop a theoretical strategy in a theoretical game of infinite chess... Yes he's really advancing the subject in important ways
@@TheRosyCodex Well I mentioned him because he was on UA-cam as well, and set theory, computability are his main area, he has multiverse view in set theory, philosophy and nature of mathematics matters a good deal, and that maybe he and Norman could debate in future. Chess seems to be one his things as well.
Professor Wildberger you have been a huge influence on me and have completely transformed by understanding of the subject of mathematics. I strongly recommend for everyone to go through your Math Foundations and Math History series, and to sign up for Algebraic Calculus! Sometimes though, I feel you go too easy on the physicists. In your video "object-oriented vs expression-oriented mathematics", you encourage a point of view where the expression should be primary, and we ought to be careful and flexible about objective interpretations of an expression. How would you respond to the criticism that theoretical physics is neither careful nor flexible about metaphysical interpretations of successful mathematical models?
@Cody Rose I believe that while the foundational problems with modern pure mathematics are almost self-evident, there are also a range of foundational difficulties with modern physics, but since this is not my area of expertise I am less clear about how to lay this out. But perhaps it comes down to similar issues-- the need to define terms very clearly and precisely BEFORE we get into long discussions involving them. In physics I think there are a whole range of "obvious" terms that really need to be defined much more clearly -- especially as we delve into the first few nanoseconds of the big bang or the supposed interior of a black hole etc. But perhaps most obviously when abstract discussions of "alternate universes" or "many worlds" abound: what the Dickens are we actually talking about here?
I think the problem with real numbers he mentions is actually a consequence of the limitations of our language, not a limit of mathemetics itself. Our language is countable and thus it can only represent countably infinite objects, while the real numbers are uncomputably infinite by nature, so it is naturally impossible to acurately represent the total information of real number with language as we usually do with rational numbers. But even so we can make abstractions about real numbers, the reason for that is because our thought go beyond the limitations of language, language is just an instrument used for communication, it does not limit our abstractions themselvs. Our abstractions can indeed reach uncountable infinite objects. In other words, we dont reason in terms of language, we reason using something more powerful than natural language, the limitation only exists in when trying to describe the continuum with language.
Ask Dr. Wildberger: Can you ascertain that there might be some validity in the following statement: "A Rubik's cube is a mathematical manipulative that allows a student to study trigonometry through the process of solving using a mixture of the concepts and ideas you put forth in your video that reveals the magic and mystery of chromogeometry, UHG, multiset theory, rational/finite trigonometry, etc?"... If you can get him talking about that then maybe you can convince him to learn to solve one...
Curt , man ! In 1:00:18 you asked the same question I was thinking about ! Is complex analysis affected by this? And honestly extracted the nugget of this podcast out, in a very humble and valuable way and thank you for that!
Honesty, clarity and carefulness (awareness) 1:33:40 are traits (or objects) that become infinite if we strive to accomplish them daily! Actually, for some of us, death will be the end of the time window for these objects, however our post-mortum presence in the memory of others, as well as the content we create for others (literature, media, what else?) will extend this time window. Plato, Homer, Cave Paintings may still have traits like honesty to be considered infinite.. until a finite end 😂
People who say that 0 and infinity are the same don't understand what they talk about when they say infinity. Wonder how many enlightened people have checked Brouwer or Cantor's logic, or looked at the zenith of the Riemann sphere which apparently is infinite
Thank you Curt. I was so psyched when I saw you were interviewing ProfessorNJ Wildberger, one of my favorite educators in the world (along with Joscha Bach). Now if you get Nima Arkani-Hamed on your podcast, my life will be complete!! BTW, where did you get that thumbnail? I barely recognized him;)
Infinity in the Hegelian/yinyang sense: is there void, or more infinity beyond infinity. The choice sounds like a Halting problem (stretching it a lot here), or at least a split in dimensionality with that choice, like a crossroad where one number line splits in two diverging ones. It depends on the function you use, which kind of infinity you will get: a limit or no limit (like Brouwer). My knowledge stops here I am much confused by the discovery of the projectively extended real line. This should explain it all, or all should be explained in here I guess Does modulo exist in this real line?
@@0001nika Do you know any names? If no keep your comments for yourself. Name dropping = methods of others before us, instead of 'inventing random things and private names on the spot' like most people do, which is useless The problem with the introduction of the variable of infinity, is that you basically cut it off and introduce a void: you turn it into a set. And this should not always be correct or appropriate in my view (and the interviewee's). I just learned this is also kind of what happens with Dedekind cuts. Look it up; this stuff is very interesting. I'm using UA-cam as a notebook basically, I thought this was live and was responding to the interviewee. Didn't really think it through, also I don't really care
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
Instead of truncating infinite sequences, an alternative is to turn them into recursive equations and solve them as algebra. You can algebraically expand -1 = S into S=1+2S=1+2+4+8+...2^n+2^{n}S=(2^n - 1)+(2^n)S, where it works whatever n is. But since it's recursive, you don't need to speak of infinity at all. Then people then object to seeing -1=S=1+2+4+8S. But you can show it with exact and finite algebra! It's not saying that "1 + 2 + 4 + 8 + ..." = -1. It's saying that S=-1, and that S expands like: S=1+2S. S = Head_S + Tail_S = (2^n-1) + 2^n S. The trick is that (2^n - 1) and (2^n S) are both "large" when n is "large",. The "limit" isn't S. It's (S - Tail_S), because Head_S = Limit_S, for every n, including large n. ie: S =(2^n - 1) + 2^n. S(1 - 2^n) = (2^n - 1) = -1(1 - 2^n) = (2^n-1). My favorite example: -(1/12) = S = 1 + 13S. When you expand it like: S = 13^0 + 13^1 + 13^3 + ... 13^n S, it's very clear why it sums to -(1/12). Recursion seems to define "..." well, but infinite iteration is impossible, so there is no paradox. Especially because the sequence is S. But Limit_S = S - Head_S.
the great limitation of mathematics is that much more is possible than can be written down . in other words, there are concepts for which there is no possible grammar to express them
I once watched a lecture on real numbers given at the Institute for Advanced Studies, and distinctly heard Ed Witten asking if real numbers existed or were real, or something to that effect. Kinda surprised me, such a basic and fundamental question, asked by such a titan in his field of physics.
Excellent content Kurt. Would be great if you were on Rumble, or any of the other platforms that do not censor/suppress comments. A lot of us are trying to move away from UA-cam for this reason. Keep up the good work 👍
Has anyone on here heard of a lattice called Post's Lattice? Technically speaking, it's the "lattice of clones on a two element set". The lattice is a beautiful piece of mathematics that contains countably infinite parts but has a finite global structure (the lattice has a top and a bottom). Can we design an algorithm that generates this lattice? Also, to get an idea of complexity of Clone Theory, "the lattice of clones on a three element set" is unknown and its accurate description would earn you at least a masters, if not a doctorate... What's my point? To disregard "an infinite set of objects", countable or otherwise, particularly in the computational example above, would be completely erroneous. P.S. Great convseration guys!
This simple algorithm of concatenating mediants has many similarities with Post's Lattice: < > < > < > < > etc. The nice feature is that when we interprete second iteration as three different countable objects, < or > as numerator and as denominator (if < or > part of , not counted separately), then the generated words can be numerically interpreted by unary count, and form Stern-Brocot type mereological theory of ordered co-prime rationals. The same generative algorithm (as well as counting algorithm) can be applied to full combinatorical range of palindromic seeds consisting of symbols < and > and blank white space for mediant space and word separability, And of course the generation can be done upwards or downwards from the seed string, or both. Semantically, the symbols can be interpreted as continued directed processes < 'increases', > 'decreases', 'both and' and >< as 'neither nor', as well as arrows of time, which can also form L R type paths e.g. along the binary tree of blanks between the words with rational values, corresponding with "irrationals". E.g. construction type "one symbol pair, top to bottom outwards, bottom to top inwards" with Boolean NOT symmetry up and down < > < > < > ::: > >>< >< >>< < > >< < > < has nothing to do with "infinite set" as such, there's only potential to keep on computing as many rows of increasing resolution as we bother. To avoid the trap of imagining that a Zeno machine is actually possible, it could be advisable to define the relational operators < and > as bounded by the Halting problem.
Such a shame that we wouldn't be able to witness a conversation between the two of you about the UFO phenomenon! Would love to just watch two incredibly intelligent people speculate. Thank you for yet another interview with a fascinating guest!
Excellent topic. Before I watch the video I wanted to say that N J Wildberger has said that infinity as something complete doesn't exist. That made me think that there is only one absolute infinity, but later I realized that Wildberger is correct! Infinity is an inexhaustible potential, meaning one that can never be manifested in a completed form.
Haven't seen the video just yet, but that's sounds like it could have familiarities with Henri Bergson's idea of 'Duration'. I'll be interested going in
@@phillt436 Yes, potential infinity as process corresponds well with Bergson-duration - which is neither unity nor multiplicity. Especially when you further define general process by Undecidability of the Halting problem - of which Gödel's incompleteness theorems are special cases.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
I run into lots of rounding questions when performing QR decompression as part of the PSLQ algorithm when input is really large (e.g., 100 digits). What works in practice is using a precision of 1500 bits and considering a number to be zero if its first 500 or so binary digits after the decimal point are zero. But a solid theoretical foundation of the kind Dr. Wildberger is suggesting could potentially allow me to use far less precision and still identify zeroes reliably. I'm all for someone else figuring this out and telling me about it!
Personally I am convinced the issues with math presented in this video are (at least in part) the fundamental reasons why we don't (and may never be able to) understand gravity on very large and very small scales.
WIldberger is incorrect when he says that computers are incapable of doing real-number arithmetic. There is a "Type-2 Theory of Effectivity" that allows you to do actual real-number arithmetic on a computer. And, of course, for the reasons Wildberger outlines, it does not use infinite decimals to represent reals. See Weihrauch's book, "Computable Analysis." The price is that even simple discontinuous functions, such as "sign of x", are not computable; you have to remove the point(s) of discontinuity from the domain to make them computable.
At 43:20 Norman is saying that physicists are fine because they're not thinking about e as an infinite decimal, but then he also says that "you can sort of choose the resolution at which you view e". But that to my minds totally implies that there is some real number e that you're simply taking a look at, you're kind of summoning it out of the Platonic space! It seems to me a contradiction to say that you can choose the resolution at which you define or use e for an application, and then say that there is nothing there that you're approximating, that it's just a magic spell or something (which does sound cool I admit). Maybe it's like Josha was saying in his conversation with Donald Hoffman, that maybe continuity doesn't exist and that you're just kind of using ingenious tools to estimate certain ratios that exist in reality, but these relationships are finite and perhaps variable, while the perfect imaginary world has infinite but unreachable potential (or at least that's my take on it). Now that I think about it that may be what Norman is saying also. Thanks for these most interesting interviews. I would like to have my own channel at some time. Cheers!
The Potential (with capital letter) is not unreachable. Sometimes psychadelics take you there for a visit. Continuity and infinity should not be confused as the same. You can have a continuous line segment which is finite in the sense that it has two endpoints. It's not too complicated, if you don't make it too complicated and absurdly absurd with Cantorian 'actual infinity'.
@@santerisatama5409 Continuity is an infinity of sorts, it's an infinity of detail, not of reach, you could say. And it's not because you think or feel like you've been there, that it's really true. Crucial distinction to make there. I've had some very strange experiences too but I wouldn't say that they were necessarily as I interpreted them. I don't think you've really answered my question. But I didn't phrase it really well either, for that I would have to watch these interviews again, which I probably will sometime in the future.
@@kjekelle96 The mistake is to treat infinity as an object, as a noun. Continuous processes - such as generative algorithms - are verbs, and can continue and grow both internally (more and more details, as you say, or more resolution) and externally. In linguistics this distinction between internal and external corresponds roughly with intransitive and transitive verbs. In my own foundational hobby I start from the concept of 'open interval', which is open both internally and externally: both increases and decreases; both intransitively and transitively. The processes of increasing and decreasing are further qualified by the undecidability of the Halting problem, which I take as foundational - the continuity of e.g. generative algorithms to ad infinitum is thus only potential, not actual.
@@santerisatama5409 I have to admit that I don't understand much of what you're saying here. I'm quite unfamiliar with linguistics and have just googled the distinction between intransitive and transitive verbs, which seems straightforward but I don't really see the connection immediately with internal and external infinities. Nor am I very familiar with computer science. I'm currently an undergraduate in math so I'll probably look deeper into these problems later on, but for now I'm quite sure I don't have enough knowledge to properly respond to this. But thanks for your time :)
@@kjekelle96 A good example of "intransitive" growth of internal resolution is Stern-Brocot tree, the generative algorithm of concatenating mediant. Standard example of external growth could be the successor function of Peano axioms.
Curt asked if it isn't circular to claim that mathematics leads to physics and that physics leads to mathematics. Penrose suggests adding a third "world", the mental world, to the mix. He then observes the paradox that the mental world leads to the mathematical world which leads to the physical world which leads back to the mental world (via the evolution of biology and brains.) I think that in the grandest picture, the rock-paper-scissors paradox is broken by, instead of closing the loop, ascending to a new level, or dimension, with each revolution to form an immense helix. Unlike Tegmark, who would claim that the starting point is mathematics, I prefer it to be the mental world (consciousness being the most basic). I also prefer to name Penrose's three worlds by crediting the originators: I call them the Cartesian World, the Platonic World, and the Aristotelian World.
Claim: Since Pi = C / D, and Pi is irrational, a perfect circle cannot exist in reality. But rotate one blob of matter around a second blob of matter. Assume the blobs have the same shape in 2 dimensions. Then by the Intermediate Value Theorem, we might expect there to be a point in each that remains equidistant for the rotation. Now if the points are in empty space so much the better. If the points are within a particle, though, we might count ourselves lucky. If space is quantized, we might imagine that we have a continuous deletion of points. Hence, it would appear reasonable to assume that a perfect circle can exist in reality in principle.
Curt, imagine that wave functions have complex values that are actually rational in their “real” and “imaginary” parts. Then the magnitudes squared (and the probabilities) would also be rational. So, no problems doing the calculation in that direction. But what if you’re doing a different calculation that starts with a probability of 50% and you want to know the magnitude of that eigenstate? It CAN’T be “one over root two” because the “rational” wave function does not admit irrational values. It must be some rational value that is VERY close to that value, so that we don’t notice when we do experiments, but it would be impossible to say exactly what the value might be! The consequence for physics would be that a probability of EXACTLY 50% for an outcome would be considered non-physical.
@AdrianBoyko Thanks for that nice explanation: better I think than the one I gave. In physics we should not be thinking about any functions -- especially wave functions from QM as given with "infinite precision". That is just way too big an assumption to impose on the world, in my view.
Another possible take on this: if we are willing to work projectively then we don't need to renormalize to get unit complex numbers values, that is we can work with the vector (1,1) rather than its "irrational" normalization "(1/sqrt(2), 1/sqrt(2)".
Real is dual to imaginary. Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
I love his lectures for things that are unrelated to foundational topics. His view, that I disagree with, makes for some really interesting new perspectives on some mathematical objects, and in the end, that's one of the most beautiful and powerful things about mathematics, how you can consider the same thing from multiple different angles that are all equally correct.
At 39:01 we have, "suppose that you are given two computers that will output [a real to the nth digit]... and you're asked to provide another program that will output the sum [of these two outputs]. Okay, if you want to leave it all in the hands of a computer, you have a point. But why suppose we're stuck only with computers? And why, for that matter even assume that ordered operations are ordered _in time?_
What I basically got from his argumentation can be simplified to “affine”, “elementary/digital”, and without openly saying it “integer/whole”. Even with these supposed complete personal views/perspectives/systems I feel like he is granting more power than such a notion could either allow or even handle. He seemingly means to recreate the power of the “real” numbers by means of a more clarified “rational” symbolic system… without openly calling it such. He essentially describes what math already is but one that would satisfy his perceived misgivings.
The analog of this problem that also exists in the inner world is that every aspect of consciousness is always illusion because it's not what precedes the list of 12 dependent originations. This is resolved in experience whenever experts in meditation return to the same self, body and world each time they achieve complete dissolution of self. By analogy, the problem you identify in the outer world is resolved in experience too. Truncated decimals to chosen degrees of resolution provides predictable results, and this gives evidence to prove the existence of real numbers you are looking for. Thus, the proof youarelookingfor doesn'texist, but there is enough empirical evidence to prove it. This is the same situation for much of epistemology in every field of physics and math and science.
The UFO topic should call into question certain assumptions about laws of physics, inertial reference frames (and what they're made of), is there a better interpretation of renormalization of wave functions, etc...
While i very much enjoyed the interview and it led me down the rabbit hole of hypergroups and helped me find a new algebraic structure based on my rho-calculus, i felt you were too star struck to ask probing questions. For example, a huge number of physical processes that are relevant to humans are iterative. Biological processes certainly are. But, even before we reach the complexity of biological phenomena processes like diffusion as NW studies in hypergroups are also iterative. Iterative processes are extremely susceptible to chaos. This means that even very tiny differences in input result in arbitrarily large differences in output. This means that arithmetic that depends on some finite precision can be very unreliable for a wide range of physical processes. It would have been interesting to get NW’s take on the relationship between chaos and his ultra finitist view of the reals. Likewise, Conway’s conception of numbers as games offers a very compelling and computationally grounded notion of quantity. It would have been interesting to get NW’s perspective on Conway’s construction.
Thank you. I have "known" Prof. Wildberger for several years, from his youtube contributions. I like especially his videos on the history of mathematics. Having graduated (masters) in mathematics myself (back in 1994) and having taken set theoretical, logic, history, analysis, numerical and applied mathematics and physics classes, I do not see what point prof Wildberger is trying to make that is not already covered by the known subjects in mathematics, numerical mathematics and physics.
@31:23 he makes the point there is a distinction between "things that are intrinsically approximate and things that are intrinsically exact." i would like to point out that is also distinction between inductive logic and deductive logic. induction being the approximate/probably, and the deductive being exact/certain.
Thank you Kurt for talking about how some people believe that a wave function is real. When I say PSI is real, I mean that it's real in the sense that whatever makes physics constants real, uses wave functions. In other words, psi is the carrier of physics constants. I can't help it that physicists won't understand what I'm telling you. Maybe they should meditate on what I say.
I kind of agree with him about the computing power of this universe running out before you run out of real numbers in fact I completely agree with him about that if the computing power is solely determined on the energy or mass or some physical property. What about when you throw fractals into the mix? These are just some things to ponder as well as the fact that they're more than likely other universes to add their energy to the computational power although I daresay there's something that's computing what those universes are in the very first place
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
Great convo, Curt! I am right there with Wildberger and his integer math, which reflects the reality of our discrete and finite universe and of course, is part of my TOE. You should ask Wildberger about which math TOE he uses most, set theory, category theory, group theory, ...? Also, what does he think about the sporadics and in particular, the monster groups?
Thanks Steve. I believe he's formulating his own, outside of set / category / group theory. I will be talking to him about this new approach to unify mathematics next time.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
We are using an infinite ortagonal grid as our primary coordinate system. This grid is useful but apparently is only an approximation of reality, and not the best one when it comes to very big and very small numbers. Both, relativity and quantum mechanics, don't use this Euclidean ortagonal grid. Do you think that using a different coordinate system that better reflects on the nature of reality will help us to grasp the reality better and eliminate the "infinity" problem?
Excellent discussion. On the question of why there is no clear contradiction in spite of mathematics being flawed, I consider the Banach-Tarski "Theorem" to be just such a contradiction.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective. Infinite or affine projections are dual to finite projections or projective hyperbolic geometry. Generalization is dual to localization. All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic). From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics! Negative curvature is dual to positive curvature -- Gauss, Riemann geometry. Curvature or gravitation is dual, gravitational energy is dual. The definition of negative curvature requires a minimum of two dual points -- Gauss. A negative curvature singularity therefore requires two dual points -- non null homotopic. The big bang is a negative curvature singularity -- non null homotopic. The big bang is therefore dual -- non null homotopic. Points are dual to lines -- the principle of duality in geometry. The point duality theorem is dual to the line duality theorem. Topological holes cannot be shrunk down to zero -- non null homotopic. Questions are dual to answers. Physics question:- Is the big bang a negative curvature singularity, a white hole? "Always two there are" -- Yoda.
@@hyperduality2838 It is commonly reported by people who have experienced NDE, or other supposed glimpses into a greater reality, that their experience was ineffable and impossible to express in human language. Your response, Hyperduality, seems to be an attempt to do exactly that.
@@Qdogsman I am using concepts and ideas from physics, mathematics & philosophy. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Making predictions to track targets, goals & objectives is a syntropic process -- teleological. Teleological physics (syntropy) is dual to non-teleological physics (entropy). There is a pattern of duality hardwired into physics, mathematics & philosophy. Normal gravity is positive curvature or attractive (elliptic, spherical) so repulsive gravity would be negative curvature or hyperbolic. Action is dual to reaction -- Sir Isaac Newton (the duality of force). Attraction is dual to repulsion, push is dual to pull -- forces are dual. Space is dual to time -- Einstein. You cannot have 3 dimensions of space in physics as you always need to include time as the 4th dimension, it is incorrect to think in terms of 3 dimensions and mathematicians make this mistake all the time. A negative curvature singularity is by definition repulsive from a point or divergent -- the big bang an explosion (inflation). If physicist want a big bang then simple logic dictates that it must be divergent from a point source. Sources (white hole) are dual to sinks (black hole). Divergence (entropy) is dual to convergence (syntropy). Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! A white hole is opposite, opposame or dual (isomorphic) to a black hole. Dark energy (divergent, repulsive) is dual to dark matter (convergent, attractive) -- a dipole. A magnet has a dipole structure (north, pole, south pole) -- duality! "May the force (duality) be with you" -- Jedi teaching. "The force (duality) is strong in this one" -- Jedi teaching.
@@hyperduality2838 It seems that the only thing you are missing is Descartes' mind-body duality. I think we should revive that notion and take it seriously. I also agree with you that 3 dimensions are not enough. But instead of just adding time, we should seriously consider the reality of multiple extra astronomically large spatial dimensions. The mathematical notion of manifolds explains how those extra dimensions are inaccessible to us 3D critters limited by our 3D tools and equipment.
@@Qdogsman Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes. According to Descartes the mind/soul is dual! "The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts)" -- The mind duality of Thomas Aquinas. Concepts are dual to percepts -- the mind duality of Immanuel Kant. Your mind converts perceptions or measurements into conceptions or ideas -- rational thinking or thought forms -- a syntropic process, teleological. Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics! Teleological physics (syntropy) is dual to non-teleological physics (entropy). Making predictions to track targets, goals or objectives is a syntropic (converging) process. From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics. All observers have a syntropic perspective. "Always two there are" -- Yoda.
I've really enjoyed this discussion on the problem of infinity. Your guest puts it into a quite sensible framework. I've been concerned for a while now about natural numbers and just how unnatural they are. They are a very abstract digitisation of our phenomenological impression of "oneness, twoness ..." which are quite fuzzy in practise. This digitisation creates the expectation that reality is a computation. Perhaps we need some form of analogical arithmetic. That would certainly remove infinities. I'm curious if I'm alone in these doubts.
You are not alone doubting 'number'. An Amazonian tribe consciously refuses to leard and adopt discrete number theory. Analogical arithmetic sounds very cool. I've been playing with the idea of building 'analogical' formal language of indefinite continua, starting from relational operators < and > interpreted as verbs.
"plus 1 whatever you say" sounds like infinity to me. Thinking simply seems to confound and escape the academic mindset. I feel sorry for people that actually take discussions like this seriously.
Are you a mathematician? Have you researched Cantor's proofs of infinities in set theory and Brouwers remarks? The reason this guy (who is professor) has this discussion' which you don't take seriously, is because mathematicians actually don't think so simply as you suggest. So you should applaud this guest actually...
@Gennady Arshad Notowidigdo I'm replying to the guy saying he feels sorry for people who take discussion like this (about infinity and mathematics) seriously. What do you think about this?
There was a rendering error and you can listen on any of the audio platforms to hear the the missing audio at 1:22:30. For example, Spotify (open.spotify.com/show/6yIO9TophRyStq6abbzOjl) and iTunes (podcasts.apple.com/us/podcast/id1521758802). Since I can't re-render on UA-cam, here's a summary of the missing audio. "CURT: Sometimes I hear this from the finitists (or the ultrafinists): it doesn't matter if each plank length cubed volume of the universe was covered in a transistor, we still wouldn't be able to carry out a certain calculation before the heat death of the universe, therefore it's "meaningless" -- however..."
hello curt, i really enjoyed your interview with prof wildberger. I have been following his channel since 2014-2015. One more person you may find interesting to talk to on the topic of formal math is prof ck raju. Would love to see you interview him. thanks.
@@pvmanson Thank you Pavan, I'll look into this.
Than you
Thank you
Thank you 🙏
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
Of course numbers are not real. They are just symbols of concepts or patterns. They have certain rules they follow and we created those rules and we selected those numbers(patterns). This is because it is convenient for us to manipulate these concepts by symbolizing into simple one letter or number.
Math and reality:
Language is mixture of conceptual words. Long sentences carry big concept. Reality is much bigger complex than language and or math.
Math is nothing but interplay of concepts and jump from one concept of problem to another concept solution. It is conversion of objectivity into subjectivity back to objectivity. It is mind play with objects in the universe.
When you say: F=MA
Each letter is a concept
E=MC2 -- each letter is concept and we chose those concept and put them into play and make another concept!!!!!!!!!!!
Concepts exist in our mind subjectively.
Saying math exists discovered or invented is rubbish
Math is invented my our mind and body invented mind. Hahahahahahahaha!!!
I win 🏆 🥇
All these big heads are just big !!!
A big thanks to Curt for a fun interview, steering us toward lots of interesting and important topics. And thanks also to the audience members that contributed nice questions too.
Dug your chan from the get-go. 👍
So glad you did this interview with Curt! Been a long time viewer and supporter of your channel. Thank you 🙏🏼
@@DoctorCalabria Thanks Frank!
I've already commented on how wonderful I found this conversation, but wanted to let you know that I have since subscribed to your own channel and was astonished at the sheer volume of output there.
I'm very much looking forward to working my way through.
Thank you Prof. Wildberger and Curt for a brilliant conversation. Its much appreciated by so many people from all walks of life.
Been tracking you for a little longer than Curt. A great surprise interview!
Curt, you are as earnest and sincere as it gets. How you remain so kind while still being fiercely passionate is very inspiring.
Well that's kind of *you*. Thank you Justin.
Professor Wilderberger's series of lectures on the history of mathematics is really great: check it out. I was delighted to see him being interviewed on this channel. Looking forward to the second interview.
I am infinitely grateful to all the knowledge I have gained in math by watching Norman's videos with his drumstick pointer and crisp and clean wall blackboards. He almost makes it too easy. I did buy his book as a means of supporting his efforts and I admire his cavalier attitude toward making only rational decisions in this one life we have.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
This Professor seems like the nicest person and as a non-math-guy I found this conversation highly interesting.
Great interview. I've been a fan of Professor Wildberger for years now. You're both awesome.
So happy to see you in the wild Norm, wishing you the best and want to say that I absolutely love your channel. I have learned so much from your teachings. Thank you Curt for inviting our guest today.
@@ReverendDr.Thomas thanks for that.
Always really interesting watching your videos Curt, but for some reason, this was very easy to follow and understand. Maybe it’s because you edited or possibly because you guys know how to communicate effectively. Either way, love you man!
Great to hear
Yeah, what does it mean "to do"? NJW says you can't do an infinite number of things as if we are talking about some sort of time-dependent manual process.
I am in awe. I confess that this is the first time I have heard of Prof. Wildberger but was blown away by his clear ability to explain to a non mathemetician such as I.
I will be looking up his channel for sure. Prof. Wildberger showed such a deep appreciation for being included in the stellar 2022 Theories of Everything lineup.
Thank you Curt for giving me the opportunity to be exposed to such accomplished thinkers. This was a great episode, already watched it twice. 👍👍👍🏴
I've watched maybe 150 of his videos and it's totally transformed and leveled up my understanding of the subject
This man is a mathematician in name only. He completely misunderstands what Mathematics is at the most fundamental level.
He is basically a flat earther for Mathematics. Flat earthers say, "The world looks flat, so it must be flat." He says, "I can't see infinity, so it must not exist." But just as we can measure the radius of the earth using the shadows cast by measured sticks, we can also describe and understand some properties about infinite objects, even if we don't have any tangible examples on hand.
We know it's real because we can infer its existence. We know that we can add one to every natural number and get a new natural number, so we can infer that there must be infinitely many natural numbers, without needing to list them all out. That's the whole point of Mathematics, which he completely misses.
I would like to hear Prof. Norman's take on the future of the global economic/financial system.
He has a series on his channel called something like Banking in the Wild West (probably butchering the title, but should be easy to find the playlist), where he doesn't (yet) talk about the future of banking/economics, but he does an interesting job of giving a mathematician's view on how the banking system has evolved over time from being based on actual 'coin'/metals to being more and more based on loans on top of loans and other more abstract and dubious systems. I've enjoyed watching it. Might not be exactly what you're looking for though, not sure.
If not addressed in part one, I would like to hear what his thoughts are on finitary statements that require the notion of infinite sets to prove, such as the Paris-Harrington theorem.
Physics is finitary; metaphysics is infinitary. A finitist mathematician is a mathematical physicist in disguise 😌💭
Metaphysics isn't infinitary tho?
A prophet is what thats called we have become to etymologically corrupted to recognize that.
Esoteric prophetic predictions based upon a man made language of symbols we agree upon.
No different that using alphabet and recorded history of society then you make future predictions based upon it
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
@@hyperduality2838pagans and the occults have believed it was an esoteric theory of everything, written in the stars for many millinia. From ancient zigurats to it's modern eq.colleges. from the ancient Chaldeans prophets to the modern day priestky physics professor.
Search continues lol
@@dadsonworldwide3238 It is very easy to understand:-
en.wikipedia.org/wiki/Gaussian_curvature
Negative Gaussian curvature is derived using two separate or distinct points, origins where as positive curvature is defined using only one point as origin.
k1, k2 = reciprocal of the respective radii which are perpendicular to each other and in negative curvature the origins have to be separate at all times -- non null homotopic.
Dark energy = repulsive gravity as galaxies are accelerating away from us at faster and faster velocities, and this is consistent with negative curvature or hyperbolic space.
The big bang is an explosion or repulsive, divergent (inflation) = negative curvature singularity (a point).
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
If the big bang is a negative curvature singularity then it requires a minimum of two points or origins as the definition of negative curvature according to Gauss has two dual origins.
The big bang = duality (isomorphism) -- non null homotopic.
The physicist Julian Barbour has written a book about Janus points/holes:-
ua-cam.com/video/FMD8B7VRX0w/v-deo.html
The big bang is a Janus hole/point (two faces = duality).
Topological holes cannot be shrunk down to zero -- non null homotopic.
The Necker cube is a good example of duality and you can watch the spinning dancer here:-
www.medicaldaily.com/right-your-eyes-science-behind-famous-spinning-dancer-optical-illusion-336122
Clockwise is dual to anti-clockwise -- duality.
Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Gravity is dual to acceleration -- Einstein.
Einstein did not believe in black holes so he did not take his equations about negative curvature seriously, so negative curvature singularities do not appear in his theory of General Relativity.
Dark energy or negative curvature must be added to his equations.
Energy is dual to mass -- Einstein
Dark energy is dual to dark matter.
"Always two there are" -- Yoda.
As as a theoretical physicist myself, I cannot thank Professor Wildberger enough for his contribution to teaching mathematics in a more pragmatic and intuitive way to students. I do not think mathematics education should be simpler, but clearer, with many computable examples.
As the Professor pointed out in this interview, we do not need brilliance, but clarity. Brilliance is a by-product of the latter, not the other way around. Nobody is required to show off or impress to be regarded as a genius, unless we suffer from some ultra-ego complex.
If you cannot explain any subject, being math, physics or wood-working, using day-to-day intuition, then you are fooling yourself and your audience in presuming you have understood anything at all.
Regardless of whether the real numbers do exist or not, it is mandatory for any educator out there to be able to convince their students that your theory is solid. From the very foundations. If you are a student, never assume what you are taught is correct, always ask questions and require precise answers. This is neither presumption nor arrogance, it is the true nature of what makes us humans, therefore curious about Mother Nature.
I have literally no idea what they're talking about, but I'm listening anyway.
Prof. Wildberger is a huge discovery for me. Thank you very much for this interview!
*Curt got nervous when asked about being interviewed. Priceless expression.* 😊
Agreed
You may disagree with Norman's conclusions, but you have to agree that the notion of infinity as a complete extension (i.e. the axiom of infinity in set theory) is a very strange notion. I appreciate mathematicians actually thinking about the foundations of mathematics instead of just leaving that to the philosophers.
I see no problem with infinite sets.
If we assume there are finitely many natural numbers, then we must accept that not every natural number has a successor. So it becomes possible to say "there exists a natural number N such that N+1 is not a natural number, or rather, is undefined" .
I find that statement more ridiculous than the notion there are infinitely many natural numbers
This was great! I've been a follower of Prof. Wildberger's channel for about 7 years, and I'm glad I got to know your project thanks to him. I would love to suggest mathematician Gregory Chaitin as a next guest. His work is amazing and too little known, and touches many of Wildberger's points and many others (including his work on Leibniz), critically Chaitin's Omega Number, or Chaitin's constant. Kudos Curt!
You're in luck. Chaitin and I spoke a couple weeks ago and we're both excited to speak to each other on TOE. It will occur in a month or two. :)
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
Just finished this episode on apple podcast. Was very interesting. Thanks curt!
Okay, I'm not a mathematician, but I've listened in on Brian Greene's master classes in quantum physics for hours without understanding how to do the mathematics, yet the concepts he talks about are fascinating. So, without having a deep understanding of his math, I listened to this conversation from two levels: 1) Since I work with meditation and the multi-dimensionality of consciousness-and how to live that on a practical level at the grocery store, for example-I was curious to see how a mathematician would view infinity. 2) His philosophy of mathematics was very interesting and, oddly enough, connected a few dots that I hadn't connected before.
That being said, it would be an amazing interview/conversation if you could get a physicist together with Norman Wildberger and asked them to play "What if..." on switching over to the older mathematics system Prof. Wildberger spoke of. If it was ever actually tried, the ramifications and ripple effect is mind boggling. But it would be an interesting thought experiment to listen in on, you know? One of your strengths is you're a good listener, so you ask good questions that feed the curiosity of the listener. If they're curious and have an open mind, that is.
Anyway, thank you for giving me a glimpse into the point of view of someone who I wouldn't meet in my daily life.
Thank you Glenn
I am a finitist and think the concept of infinity should be eliminated completely from mathematics (and from physics, philosophy, and religion too while you're at it). The answer to the question of an implied largest integer is that it is context dependent. It depends on the capacity of your computing device. If you fill the display on your pocket calculator with 9s, you get a large number but you can't add 1 to it. As Norman suggested, if the entire universe acted as a computer with planck-length components, you would have a limit on the size of numbers that could exist, and thus establish a largest integer. In the context of an abacus, the largest integer would be determined by the number of beaded wires on the abacus.
@@quantumbubbles7977 If so, the process of "turning out" will be interesting to examine. It should be evidence of someone, or something, doing an infinite number of things. I don't think that will ever be done.
I’m not a finitist but I think the concept of infinity is not well founded. The mathematicians just stumbled along without having properly defined or understood it, which is terrible.
@@maxwang2537yeah... That's just not true
This reminds me of how observation itself effects the state of things on a quantum level, because as soon as you’ve “grasped” infinity it escapes you and becomes something greater. So it’s an imaginary thing.
It makes me think that we can only _count_ , but we cannot _measure_ (except approximately)
FAR OUT, Man SOLID Thanks Dr Wildberger, and you,too, Curt.
I notice Norman didn't once uttered the name Cantor, rather he used a word demeaning him. Thinking about infinity we are soon drawn to a one dimensional natural numbers, when infinity remains what they call 'undefined'. What Cantor said is simple, he divided the one dimension into two, but then infinity isn't manageable but infinity can be accommodated in what turns out to be finite. So since the 19th century when Cantor discovered how to side step infinity, mathematical logicians pounced on Cantor's mistake (which faced much objections, right from the beginning) when Godel (using symbols of PEANO's natural numbers) arrived at 'undecidability' even though these numbers suffer from the same nature of 'undecidability', termed by Wildberger as 'arrogance', its like pouring milk into a new bottle.
Cantor cardinality/ordinality helps logicians like Hilbert to approximate infinity with finite, that has some mathematical usefulness. But in essence neither Godel nor Cantor can be said to be right. They remain 'arrogant'.
I’ll edit this comment as I listen to this podcast, if my questions are already answered in it. (up to 38:40 now)
Ive watched a handful of Dr Wildberger’s lectures and always wanted to ask questions like these:
1. A common refrain of his is that infinity is not suitable as a formal concept because actual infinities of various sorts are not realizable. For instance, one cannot ever claim to have “completed” an infinite task, or represent most real numbers as they’d require infinitely large computers. In this sense he claims infinity (and thus most real numbers) do not exist, and also, we cannot do formal reasoning with them as concepts.
I fail to see how the non-realizability of some X (in this universe) relates to our ability to formally reason about it. What is "realizable" is largely a contingent truth. We can still reason about entities and properties in logically possible universes, even if those entities / properties are never realized in this one. When he states that a particular infinite (or large) sum "could never be calculated in this universe", Dr. Wildeberger is demonstrating this fact (he is in fact reasoning about such a non-realizable entity!).
Formally speaking, we can define a symbol X in set theory / number theory, which, by virtue of its relationships to other symbols in that system, is a suitable representation of infinity, and reason about it. We can even check the validity of those acts of reasoning with computers. Can't we?
2. The usefulness of infinity, real numbers in physics
Q: Why are mathematical systems involving real numbers and grounded in infinities so consistently useful for expanding our scientific knowledge and representing the world, if theres a deep problem with them?
3. Putting infinity to the test
Q. If there are logical holes in the ground of modern mathematics, can Dr Wildberger demonstrate this formally, by doing something like deriving a contradiction from them? Can he perhaps show that the system is incomplete (other than the incompleteness arising from godel’s theorems?)
I look forward to listening to this in full, and hopefully seeing some of these questions resolved :)
I really like these questions and would love to have them answered!
As someone writing numerical algorithms for living, I need to understand what these algorithms do and how they behave on a computers. The understanding usually comes in two steps. First, you proof certain properties of an algorithm imagining it is running on a computer that can do real number arithmetics. Second, you proof that the real computer with finite(floating point) arithmetics gives you a good approximation.
As a formalist, mathematics is nothing more then consistent manipulation of formal statements, I do not care if the idealised computer with real arithmetics "exists" or not. It is just a formal tool (a very useful one) to at the end prove something about the algorithm I run on my computer in front of me. So I still fail to see why should one give up the notion of infinity/real numbers.
As the saying, the shortest path between two points goes through the complex plain. We could easily say that the shortest path between computations with large numbers goes through infinity.
I have messed up the quote, Jacques Hadamard:
"The shortest path between two truths in the real domain passes through the complex domain."
I would've liked to have seen thorough follow-up questions in this interview, e.g. after a weak response to a question about e in physics at 42:50. Wildberger answered by saying physicists only have in mind approximations to e, which isn't a fair characterization of how physicists work. e as an exact transcendental number occurs prominently in physical theory, e.g. in solutions to differential equations.
Much of what Wildberger says deserves tougher questioning and criticism than it gets here.
One of the points is, you don't *need* the formalism of infinity. You don't need to take a limit to infinity, there is a way to do it that is entirely constructive and finite
Looking forward to this . Thanks Curt . Will you try to interview dr. Neil Turock .
You can get around Norman's objections to infinite processes by the same way of saying, there is no largest number. The epsilon/delta method works like this - for all epsilon, I can specify a delta. Limits work like this as do iterative processes - give me a level of accuracy, and I can specify a number of iterations. No mention of infinity. The notion 'as N tends to infinity' is tagged onto the proofs and definitions of Real Analysis.
_"Mr. Owl, how many licks does it take to get to the Tootsie Roll center of a Tootsie Pop?"_
I feel so deeply vindicated by this man's thinking.
I know I have some kind of brain damage. So, when it came to math, I must have engaged a work-around so as to avoid memorizing an ever increasing number of axioms. He articulated this perfectly. The work around ended up going backwards every time a new principle was introduced and figuring out *why* it was true. I managed thru grade school and some of Jr. High, but then this method no longer worked because HS math is where they require you to "just do the calculation". I wanted to understand, but no one would take the journey with me to find out why something worked. The factory model in schools destroys so much value.
Several years ago online I was in a disagreement over some of these items because I wanted him to defend the concept of "real" in a way that connected with the lived-in world. He refused and insisted that I wasn't smart enuf to do this esoteric math because I couldn't understand the ground that these axioms stood on.
Thank you for all this confirmation.
Bless you both.
🤗
P.S. Can you tape that convo about UFO's and give yourselves a chance to change your mind later about publishing it? I think he'd be very surprised by the positive reception it would get. Even from peers. You might even be in the majority.
👽🖖
47:07 I think in quantum mechanics you cannot avoid irrational numbers.
You can always rotate a light polarizer some degrees (let's assume the universe is discrete as so the degrees are rational) that are not of the form p = q^2 for some rational number q. Therefore, the amplitude for the state of the light being in any of the two orthogonal basis will be related to sqrt(p) which isn't rational.
Any moment in which we say that we understand what is essentially pure Mystery, we are losing our place.
Infinite, God, Consciousness, Unconditional Love, all those mysteries cant be understood, and that is fine. We are the childs of the Universe, lets play.
I don’t understand why he thinks referring to an infinite set, say the natural numbers, involves assuming that one can complete an infinite number of tasks. I don’t need to have to write a list of every natural number in order to speak about the set as a whole.
it goes to his latter comments. it might not be a problem for pure math (where you can just use certain symbols to hand-wave away what you actually explicitly mean), but if you're using a math system (counting, enumerating etc) to help model truths about "physical reality", you want to be more grounded that you can do the things you claim to be able to do, at least that's how i see the argument, which i agree with. at least, as he further argues, mathematicians should have an assortment of systems to choose from, so there is value to explore one without the concept of infinity, especially if our tools which we use to measure the "physical reality" cannot compute/enumerate with infinite concepts. infinity problems turn up in quantum physics with the current tools we use. so ultimately i think its a pragmatic argument that could help.
@@julsius I don’t really see how that responds to my comment. I was trying to say that Wildberger’s criticism of standard pure math is that it wrongly assumes that we can complete an infinite number of tasks. But I don’t think anyone actually assumes that, so he’s making a straw-man argument. More generally, he seems to be confused about the relation between pure and applied math. Any pure mathematician will happily grant that applications involve approximations. But that doesn’t mean that they are simply “hand waving” or doing anything illegitimate when they use symbols to stand for real numbers, infinite sets, etc. It’s as if he wants to eliminate anything in pure math that doesn’t directly correspond to applications or concrete computations. But, even after watching this video and some of Wildberger’s videos, I don’t see any real value in his project.
@@synaestheziac im not fully sold on his argument that a system without infitinities will be more true/correct/sound, but im ultimately a pragmatist (closer to applied math), if it proves to be useful then that was time and energy well worth spent developing his system. i think there are already some potentially useful ideas in his videos. For example, its another valid question to ask, "what if your system of infinities leads to false assumptions about limits of approximation methods?"
@Gennady Arshad Notowidigdo as I suspected. Thanks for the inside scoop!
@Gennady Arshad Notowidigdo yes, I’d be interested in that, thanks. I found your channel but I don’t see your email
“Real numbers aren’t real” is a philosophy of nihilistic mathematics, because in order to maintain a consistent philosophical commitment to this claim, you must also commit to a denial of the existence of the number 1.
I’m not claiming that there’s a mathematical proof of an inconsistency in systems that do not posit the existence of the real number system. Nor do I contend that the denial of the existence of real numbers like the square root of 2 implies that there are no numbers like zero, one or two; I only contend that the philosophical reasons given for choosing to commit to a metaphysical claim that there is no square root of two are not more convincing than reasons given for objecting to metaphysical claims that there is a number such as zero or that there is a number with the properties ascribed to 1, or, even accepting its existence, or for objecting to metaphysical claims that there is a way to add it to itself to obtain the number 2.
In fact, if we adapt the contention that committing to existence of specific numbers requires providing algorithms to construct or compute them, then in that system of philosophical commitment, we can produce such an algorithm to construct a square root of 2, precisely by taking as a square root of two the algorithm obtained by applying to the equation x^2-2=0 the bisection method, in the interval (0,2). If we declare that to be the (metaphysical) square root of 2, then we have met your requirement for justifying the existence of that specific number, because we can define addition and multiplication of algorithms so that this algorithm has all of the properties required of a positive number whose square is 2, and we need not resort to equivalence classes of Cauchy sequences for the definition of a (positive) square root of two. We can then, however, use the notion of a Cauchy sequence, the notion of convergence of (Cauchy) sequences, and the notion of an equivalence class of Cauchy sequences to explain how this particular algorithm could have been taken as merely a representative of such a class, so that someone else, having used a different algorithm as a definition of a (positive) square root of two, and therefore adopting a different metaphysics, is guaranteed to have a metaphysical interpretation of the problem and solution that is naturally isomorphic to our metaphysics. The traditional development of mathematics using the theory of Cauchy sequences and defining real numbers as equivalence classes of Cauchy sequences has provided the foundation for writing such a proof already.
There is never a limited amount of numbers there's just a limited amount of names you have given them
Curt you continue to be one of, if not THE, best casters on the net today, thanks for sticking with it! In Part 2 I'd love to hear the Professor's take on Eric Weinstein's statement that the hopf fibration is the most important construct in the universe. Secondly, that we can derive basically the entirety of physics from it and with changing only a few parameters can construct alternative physics (if I understand his contention correctly). As I'm sure you know, Eric mirrors the Professor's intuition that physicists can benefit from mathematicians understandings as they are fluent in the language that physicists use to express their domain.
Great conversation!! I've often struggled with the concept of "infinity" and this was a great exploration of that topic 👏
I'm so glad!
Surely, one of the favourites which I’ll watch again and again, like the Richard borcherds interview. There are more mathematicians you would want to interview in this space, like Joel David Hamkins, (Mathematician, philosophy of mathematics and set theory, same name yt channel), Terry Tao, Timothy Gowers, (same name yt channel, Cambridge courses), also himself a fields medalist. (Both Borcherds and Gowers won in 1998).
The guy whose main project is to develop a theoretical strategy in a theoretical game of infinite chess... Yes he's really advancing the subject in important ways
@@TheRosyCodex Well I mentioned him because he was on UA-cam as well, and set theory, computability are his main area, he has multiverse view in set theory, philosophy and nature of mathematics matters a good deal, and that maybe he and Norman could debate in future. Chess seems to be one his things as well.
@@thea.igamer3958 sorry I was being snarky
@@TheRosyCodex
and misleading
Hamkins’ main work is definitely not infinite chess.
Professor Wildberger you have been a huge influence on me and have completely transformed by understanding of the subject of mathematics. I strongly recommend for everyone to go through your Math Foundations and Math History series, and to sign up for Algebraic Calculus! Sometimes though, I feel you go too easy on the physicists. In your video "object-oriented vs expression-oriented mathematics", you encourage a point of view where the expression should be primary, and we ought to be careful and flexible about objective interpretations of an expression. How would you respond to the criticism that theoretical physics is neither careful nor flexible about metaphysical interpretations of successful mathematical models?
@Cody Rose I believe that while the foundational problems with modern pure mathematics are almost self-evident, there are also a range of foundational difficulties with modern physics, but since this is not my area of expertise I am less clear about how to lay this out. But perhaps it comes down to similar issues-- the need to define terms very clearly and precisely BEFORE we get into long discussions involving them. In physics I think there are a whole range of "obvious" terms that really need to be defined much more clearly -- especially as we delve into the first few nanoseconds of the big bang or the supposed interior of a black hole etc. But perhaps most obviously when abstract discussions of "alternate universes" or "many worlds" abound: what the Dickens are we actually talking about here?
How can infinity be bounded into a finite space? There is only one way, as a fractal.
Without infinity negative numbers make no sense and thus complex numbers aren't necessary
I think the problem with real numbers he mentions is actually a consequence of the limitations of our language, not a limit of mathemetics itself.
Our language is countable and thus it can only represent countably infinite objects, while the real numbers are uncomputably infinite by nature, so it is naturally impossible to acurately represent the total information of real number with language as we usually do with rational numbers.
But even so we can make abstractions about real numbers, the reason for that is because our thought go beyond the limitations of language, language is just an instrument used for communication, it does not limit our abstractions themselvs. Our abstractions can indeed reach uncountable infinite objects.
In other words, we dont reason in terms of language, we reason using something more powerful than natural language, the limitation only exists in when trying to describe the continuum with language.
So much beauty. Thanks Curt !
Ask Dr. Wildberger: Can you ascertain that there might be some validity in the following statement: "A Rubik's cube is a mathematical manipulative that allows a student to study trigonometry through the process of solving using a mixture of the concepts and ideas you put forth in your video that reveals the magic and mystery of chromogeometry, UHG, multiset theory, rational/finite trigonometry, etc?"...
If you can get him talking about that then maybe you can convince him to learn to solve one...
Curt , man ! In 1:00:18 you asked the same question I was thinking about ! Is complex analysis affected by this? And honestly extracted the nugget of this podcast out, in a very humble and valuable way and thank you for that!
Honesty, clarity and carefulness (awareness) 1:33:40 are traits (or objects) that become infinite if we strive to accomplish them daily! Actually, for some of us, death will be the end of the time window for these objects, however our post-mortum presence in the memory of others, as well as the content we create for others (literature, media, what else?) will extend this time window. Plato, Homer, Cave Paintings may still have traits like honesty to be considered infinite.. until a finite end 😂
Chris Langan seemed like a reasonable chap when I watched that video, at the least an entertaining character
People who say that 0 and infinity are the same don't understand what they talk about when they say infinity. Wonder how many enlightened people have checked Brouwer or Cantor's logic, or looked at the zenith of the Riemann sphere which apparently is infinite
Thank you Curt. I was so psyched when I saw you were interviewing ProfessorNJ Wildberger, one of my favorite educators in the world (along with Joscha Bach). Now if you get Nima Arkani-Hamed on your podcast, my life will be complete!! BTW, where did you get that thumbnail? I barely recognized him;)
A share your last sentence man.
Infinity in the Hegelian/yinyang sense: is there void, or more infinity beyond infinity. The choice sounds like a Halting problem (stretching it a lot here), or at least a split in dimensionality with that choice, like a crossroad where one number line splits in two diverging ones. It depends on the function you use, which kind of infinity you will get: a limit or no limit (like Brouwer). My knowledge stops here
I am much confused by the discovery of the projectively extended real line. This should explain it all, or all should be explained in here I guess
Does modulo exist in this real line?
What language are you speaking? Is this some sort of name dropping exercise?
@@0001nika Do you know any names? If no keep your comments for yourself. Name dropping = methods of others before us, instead of 'inventing random things and private names on the spot' like most people do, which is useless
The problem with the introduction of the variable of infinity, is that you basically cut it off and introduce a void: you turn it into a set. And this should not always be correct or appropriate in my view (and the interviewee's).
I just learned this is also kind of what happens with Dedekind cuts. Look it up; this stuff is very interesting.
I'm using UA-cam as a notebook basically, I thought this was live and was responding to the interviewee. Didn't really think it through, also I don't really care
This was brilliant! Thank you to both of you.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
The quality of infinity ♾️ is inconsistent with logic: ♾️+1 element=♾️ elements. Can't be true. Compare with 0+1=0.
*Curt is genuinely honest. Good interview.* 👍
to my mind, this is the best of the best, Today. Thank You.
I'm glad you enjoyed it!
Instead of truncating infinite sequences, an alternative is to turn them into recursive equations and solve them as algebra. You can algebraically expand -1 = S into S=1+2S=1+2+4+8+...2^n+2^{n}S=(2^n - 1)+(2^n)S, where it works whatever n is. But since it's recursive, you don't need to speak of infinity at all. Then people then object to seeing -1=S=1+2+4+8S. But you can show it with exact and finite algebra! It's not saying that "1 + 2 + 4 + 8 + ..." = -1. It's saying that S=-1, and that S expands like: S=1+2S.
S = Head_S + Tail_S = (2^n-1) + 2^n S. The trick is that (2^n - 1) and (2^n S) are both "large" when n is "large",. The "limit" isn't S. It's (S - Tail_S), because Head_S = Limit_S, for every n, including large n. ie: S =(2^n - 1) + 2^n. S(1 - 2^n) = (2^n - 1) = -1(1 - 2^n) = (2^n-1).
My favorite example: -(1/12) = S = 1 + 13S. When you expand it like: S = 13^0 + 13^1 + 13^3 + ... 13^n S, it's very clear why it sums to -(1/12). Recursion seems to define "..." well, but infinite iteration is impossible, so there is no paradox. Especially because the sequence is S. But Limit_S = S - Head_S.
the great limitation of mathematics is that much more is possible than can be written down . in other words, there are concepts for which there is no possible grammar to express them
Very much looking forward to this one.
Superb! Eagerly waiting!
I once watched a lecture on real numbers given at the Institute for Advanced Studies, and distinctly heard Ed Witten asking if real numbers existed or were real, or something to that effect. Kinda surprised me, such a basic and fundamental question, asked by such a titan in his field of physics.
Excellent content Kurt. Would be great if you were on Rumble, or any of the other platforms that do not censor/suppress comments.
A lot of us are trying to move away from UA-cam for this reason.
Keep up the good work 👍
Has anyone on here heard of a lattice called Post's Lattice? Technically speaking, it's the "lattice of clones on a two element set". The lattice is a beautiful piece of mathematics that contains countably infinite parts but has a finite global structure (the lattice has a top and a bottom). Can we design an algorithm that generates this lattice?
Also, to get an idea of complexity of Clone Theory, "the lattice of clones on a three element set" is unknown and its accurate description would earn you at least a masters, if not a doctorate...
What's my point? To disregard "an infinite set of objects", countable or otherwise, particularly in the computational example above, would be completely erroneous.
P.S. Great convseration guys!
This simple algorithm of concatenating mediants has many similarities with Post's Lattice:
< >
< >
< >
< >
etc.
The nice feature is that when we interprete second iteration as three different countable objects, < or > as numerator and as denominator (if < or > part of , not counted separately), then the generated words can be numerically interpreted by unary count, and form Stern-Brocot type mereological theory of ordered co-prime rationals.
The same generative algorithm (as well as counting algorithm) can be applied to full combinatorical range of palindromic seeds consisting of symbols < and > and blank white space for mediant space and word separability, And of course the generation can be done upwards or downwards from the seed string, or both.
Semantically, the symbols can be interpreted as continued directed processes < 'increases', > 'decreases', 'both and' and >< as 'neither nor', as well as arrows of time, which can also form L R type paths e.g. along the binary tree of blanks between the words with rational values, corresponding with "irrationals".
E.g. construction type "one symbol pair, top to bottom outwards, bottom to top inwards" with Boolean NOT symmetry up and down
< >
< >
< >
:::
> >>< >< >>< <
> >< <
> <
has nothing to do with "infinite set" as such, there's only potential to keep on computing as many rows of increasing resolution as we bother. To avoid the trap of imagining that a Zeno machine is actually possible, it could be advisable to define the relational operators < and > as bounded by the Halting problem.
Such a shame that we wouldn't be able to witness a conversation between the two of you about the UFO phenomenon! Would love to just watch two incredibly intelligent people speculate. Thank you for yet another interview with a fascinating guest!
Excellent topic. Before I watch the video I wanted to say that N J Wildberger has said that infinity as something complete doesn't exist. That made me think that there is only one absolute infinity, but later I realized that Wildberger is correct! Infinity is an inexhaustible potential, meaning one that can never be manifested in a completed form.
Haven't seen the video just yet, but that's sounds like it could have familiarities with Henri Bergson's idea of 'Duration'. I'll be interested going in
@@phillt436 Yes, potential infinity as process corresponds well with Bergson-duration - which is neither unity nor multiplicity. Especially when you further define general process by Undecidability of the Halting problem - of which Gödel's incompleteness theorems are special cases.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
I run into lots of rounding questions when performing QR decompression as part of the PSLQ algorithm when input is really large (e.g., 100 digits). What works in practice is using a precision of 1500 bits and considering a number to be zero if its first 500 or so binary digits after the decimal point are zero. But a solid theoretical foundation of the kind Dr. Wildberger is suggesting could potentially allow me to use far less precision and still identify zeroes reliably. I'm all for someone else figuring this out and telling me about it!
Thought I'd let you know that the audio cuts off at 1:22:30 (on my end in any case).
I left a note about this as the pinned comment. Hope this helps.
Personally I am convinced the issues with math presented in this video are (at least in part) the fundamental reasons why we don't (and may never be able to) understand gravity on very large and very small scales.
WIldberger is incorrect when he says that computers are incapable of doing real-number arithmetic. There is a "Type-2 Theory of Effectivity" that allows you to do actual real-number arithmetic on a computer. And, of course, for the reasons Wildberger outlines, it does not use infinite decimals to represent reals. See Weihrauch's book, "Computable Analysis." The price is that even simple discontinuous functions, such as "sign of x", are not computable; you have to remove the point(s) of discontinuity from the domain to make them computable.
Don't we define e^x as the limit of a summation, rather than the total of a summation?
At 43:20 Norman is saying that physicists are fine because they're not thinking about e as an infinite decimal, but then he also says that "you can sort of choose the resolution at which you view e". But that to my minds totally implies that there is some real number e that you're simply taking a look at, you're kind of summoning it out of the Platonic space! It seems to me a contradiction to say that you can choose the resolution at which you define or use e for an application, and then say that there is nothing there that you're approximating, that it's just a magic spell or something (which does sound cool I admit). Maybe it's like Josha was saying in his conversation with Donald Hoffman, that maybe continuity doesn't exist and that you're just kind of using ingenious tools to estimate certain ratios that exist in reality, but these relationships are finite and perhaps variable, while the perfect imaginary world has infinite but unreachable potential (or at least that's my take on it). Now that I think about it that may be what Norman is saying also.
Thanks for these most interesting interviews. I would like to have my own channel at some time. Cheers!
The Potential (with capital letter) is not unreachable. Sometimes psychadelics take you there for a visit.
Continuity and infinity should not be confused as the same. You can have a continuous line segment which is finite in the sense that it has two endpoints. It's not too complicated, if you don't make it too complicated and absurdly absurd with Cantorian 'actual infinity'.
@@santerisatama5409 Continuity is an infinity of sorts, it's an infinity of detail, not of reach, you could say. And it's not because you think or feel like you've been there, that it's really true. Crucial distinction to make there. I've had some very strange experiences too but I wouldn't say that they were necessarily as I interpreted them.
I don't think you've really answered my question. But I didn't phrase it really well either, for that I would have to watch these interviews again, which I probably will sometime in the future.
@@kjekelle96 The mistake is to treat infinity as an object, as a noun.
Continuous processes - such as generative algorithms - are verbs, and can continue and grow both internally (more and more details, as you say, or more resolution) and externally. In linguistics this distinction between internal and external corresponds roughly with intransitive and transitive verbs.
In my own foundational hobby I start from the concept of 'open interval', which is open both internally and externally: both increases and decreases; both intransitively and transitively. The processes of increasing and decreasing are further qualified by the undecidability of the Halting problem, which I take as foundational - the continuity of e.g. generative algorithms to ad infinitum is thus only potential, not actual.
@@santerisatama5409 I have to admit that I don't understand much of what you're saying here. I'm quite unfamiliar with linguistics and have just googled the distinction between intransitive and transitive verbs, which seems straightforward but I don't really see the connection immediately with internal and external infinities. Nor am I very familiar with computer science. I'm currently an undergraduate in math so I'll probably look deeper into these problems later on, but for now I'm quite sure I don't have enough knowledge to properly respond to this. But thanks for your time :)
@@kjekelle96 A good example of "intransitive" growth of internal resolution is Stern-Brocot tree, the generative algorithm of concatenating mediant.
Standard example of external growth could be the successor function of Peano axioms.
Thanks for this. Great questions. I look forward to part two.
Curt asked if it isn't circular to claim that mathematics leads to physics and that physics leads to mathematics. Penrose suggests adding a third "world", the mental world, to the mix. He then observes the paradox that the mental world leads to the mathematical world which leads to the physical world which leads back to the mental world (via the evolution of biology and brains.) I think that in the grandest picture, the rock-paper-scissors paradox is broken by, instead of closing the loop, ascending to a new level, or dimension, with each revolution to form an immense helix. Unlike Tegmark, who would claim that the starting point is mathematics, I prefer it to be the mental world (consciousness being the most basic). I also prefer to name Penrose's three worlds by crediting the originators: I call them the Cartesian World, the Platonic World, and the Aristotelian World.
Please discuss the apparent infinity as ascribed to black hole singularity in equations of GR.
Claim: Since Pi = C / D, and Pi is irrational, a perfect circle cannot exist in reality. But rotate one blob of matter around a second blob of matter. Assume the blobs have the same shape in 2 dimensions. Then by the Intermediate Value Theorem, we might expect there to be a point in each that remains equidistant for the rotation. Now if the points are in empty space so much the better. If the points are within a particle, though, we might count ourselves lucky. If space is quantized, we might imagine that we have a continuous deletion of points. Hence, it would appear reasonable to assume that a perfect circle can exist in reality in principle.
Curt, imagine that wave functions have complex values that are actually rational in their “real” and “imaginary” parts. Then the magnitudes squared (and the probabilities) would also be rational. So, no problems doing the calculation in that direction.
But what if you’re doing a different calculation that starts with a probability of 50% and you want to know the magnitude of that eigenstate? It CAN’T be “one over root two” because the “rational” wave function does not admit irrational values. It must be some rational value that is VERY close to that value, so that we don’t notice when we do experiments, but it would be impossible to say exactly what the value might be!
The consequence for physics would be that a probability of EXACTLY 50% for an outcome would be considered non-physical.
@AdrianBoyko Thanks for that nice explanation: better I think than the one I gave. In physics we should not be thinking about any functions -- especially wave functions from QM as given with "infinite precision". That is just way too big an assumption to impose on the world, in my view.
@@njwildberger I'm not sure there is anything in physics that has infinite precision, with the possible exception of the number of digits in the pi.
Another possible take on this: if we are willing to work projectively then we don't need to renormalize to get unit complex numbers values, that is we can work with the vector (1,1) rather than its "irrational" normalization "(1/sqrt(2), 1/sqrt(2)".
@@njwildberger Does this mean I should learn projective geometry?
Real is dual to imaginary.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
We Always enjoy the Love Stream!
I love his lectures for things that are unrelated to foundational topics. His view, that I disagree with, makes for some really interesting new perspectives on some mathematical objects, and in the end, that's one of the most beautiful and powerful things about mathematics, how you can consider the same thing from multiple different angles that are all equally correct.
Gromov reads Archimedes, Scott Aaronson reads Archimedes, perhaps Borcherds as well given his profile pick
At 39:01 we have, "suppose that you are given two computers that will output [a real to the nth digit]... and you're asked to provide another program that will output the sum [of these two outputs]. Okay, if you want to leave it all in the hands of a computer, you have a point. But why suppose we're stuck only with computers? And why, for that matter even assume that ordered operations are ordered _in time?_
Thanks for another great video.
I have always been fascinated by mathematics.
Glad you enjoy it!
What I basically got from his argumentation can be simplified to “affine”, “elementary/digital”, and without openly saying it “integer/whole”. Even with these supposed complete personal views/perspectives/systems I feel like he is granting more power than such a notion could either allow or even handle. He seemingly means to recreate the power of the “real” numbers by means of a more clarified “rational” symbolic system… without openly calling it such. He essentially describes what math already is but one that would satisfy his perceived misgivings.
The analog of this problem that also exists in the inner world is that every aspect of consciousness is always illusion because it's not what precedes the list of 12 dependent originations. This is resolved in experience whenever experts in meditation return to the same self, body and world each time they achieve complete dissolution of self. By analogy, the problem you identify in the outer world is resolved in experience too. Truncated decimals to chosen degrees of resolution provides predictable results, and this gives evidence to prove the existence of real numbers you are looking for. Thus, the proof youarelookingfor doesn'texist, but there is enough empirical evidence to prove it. This is the same situation for much of epistemology in every field of physics and math and science.
The UFO topic should call into question certain assumptions about laws of physics, inertial reference frames (and what they're made of), is there a better interpretation of renormalization of wave functions, etc...
While i very much enjoyed the interview and it led me down the rabbit hole of hypergroups and helped me find a new algebraic structure based on my rho-calculus, i felt you were too star struck to ask probing questions. For example, a huge number of physical processes that are relevant to humans are iterative. Biological processes certainly are. But, even before we reach the complexity of biological phenomena processes like diffusion as NW studies in hypergroups are also iterative. Iterative processes are extremely susceptible to chaos. This means that even very tiny differences in input result in arbitrarily large differences in output. This means that arithmetic that depends on some finite precision can be very unreliable for a wide range of physical processes. It would have been interesting to get NW’s take on the relationship between chaos and his ultra finitist view of the reals. Likewise, Conway’s conception of numbers as games offers a very compelling and computationally grounded notion of quantity. It would have been interesting to get NW’s perspective on Conway’s construction.
Thank you. I have "known" Prof. Wildberger for several years, from his youtube contributions. I like especially his videos on the history of mathematics. Having graduated (masters) in mathematics myself (back in 1994) and having taken set theoretical, logic, history, analysis, numerical and applied mathematics and physics classes, I do not see what point prof Wildberger is trying to make that is not already covered by the known subjects in mathematics, numerical mathematics and physics.
Ahhh the Wildburger... my 2nd favorite meme mathematician after Mochizuki! This interview needs more drum sticks
;)
what do you think of John Gabriel's new calculus?
@31:23 he makes the point there is a distinction between "things that are intrinsically approximate and things that are intrinsically exact." i would like to point out that is also distinction between inductive logic and deductive logic. induction being the approximate/probably, and the deductive being exact/certain.
Thank you Kurt for talking about how some people believe that a wave function is real.
When I say PSI is real, I mean that it's real in the sense that whatever makes physics constants real, uses wave functions. In other words, psi is the carrier of physics constants.
I can't help it that physicists won't understand what I'm telling you. Maybe they should meditate on what I say.
I kind of agree with him about the computing power of this universe running out before you run out of real numbers in fact I completely agree with him about that if the computing power is solely determined on the energy or mass or some physical property. What about when you throw fractals into the mix? These are just some things to ponder as well as the fact that they're more than likely other universes to add their energy to the computational power although I daresay there's something that's computing what those universes are in the very first place
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
Great convo, Curt! I am right there with Wildberger and his integer math, which reflects the reality of our discrete and finite universe and of course, is part of my TOE.
You should ask Wildberger about which math TOE he uses most, set theory, category theory, group theory, ...? Also, what does he think about the sporadics and in particular, the monster groups?
Thanks Steve. I believe he's formulating his own, outside of set / category / group theory. I will be talking to him about this new approach to unify mathematics next time.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
Until you have that chat on what it means to “do” something.
I DO Mathematics, but I am not a Mathematician.
Brother you are amazing. You and lex are joe rogans of the scientific world
We are using an infinite ortagonal grid as our primary coordinate system. This grid is useful but apparently is only an approximation of reality, and not the best one when it comes to very big and very small numbers. Both, relativity and quantum mechanics, don't use this Euclidean ortagonal grid. Do you think that using a different coordinate system that better reflects on the nature of reality will help us to grasp the reality better and eliminate the "infinity" problem?
Good interview.
Well done to both of you.
Kurt, you are an excellent interviewer, one of a kind.
Excellent discussion. On the question of why there is no clear contradiction in spite of mathematics being flawed, I consider the Banach-Tarski "Theorem" to be just such a contradiction.
Infinity is dual to the finite, absolute is dual to relative, objective is dual to subjective.
Infinite or affine projections are dual to finite projections or projective hyperbolic geometry.
Generalization is dual to localization.
All observers have a finite or localized perspective -- projective hyperbolic perspective (syntropic).
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics!
Negative curvature is dual to positive curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual, gravitational energy is dual.
The definition of negative curvature requires a minimum of two dual points -- Gauss.
A negative curvature singularity therefore requires two dual points -- non null homotopic.
The big bang is a negative curvature singularity -- non null homotopic.
The big bang is therefore dual -- non null homotopic.
Points are dual to lines -- the principle of duality in geometry.
The point duality theorem is dual to the line duality theorem.
Topological holes cannot be shrunk down to zero -- non null homotopic.
Questions are dual to answers.
Physics question:- Is the big bang a negative curvature singularity, a white hole?
"Always two there are" -- Yoda.
@@hyperduality2838 It is commonly reported by people who have experienced NDE, or other supposed glimpses into a greater reality, that their experience was ineffable and impossible to express in human language. Your response, Hyperduality, seems to be an attempt to do exactly that.
@@Qdogsman I am using concepts and ideas from physics, mathematics & philosophy.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Making predictions to track targets, goals & objectives is a syntropic process -- teleological.
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
There is a pattern of duality hardwired into physics, mathematics & philosophy.
Normal gravity is positive curvature or attractive (elliptic, spherical) so repulsive gravity would be negative curvature or hyperbolic.
Action is dual to reaction -- Sir Isaac Newton (the duality of force).
Attraction is dual to repulsion, push is dual to pull -- forces are dual.
Space is dual to time -- Einstein.
You cannot have 3 dimensions of space in physics as you always need to include time as the 4th dimension, it is incorrect to think in terms of 3 dimensions and mathematicians make this mistake all the time.
A negative curvature singularity is by definition repulsive from a point or divergent -- the big bang an explosion (inflation).
If physicist want a big bang then simple logic dictates that it must be divergent from a point source.
Sources (white hole) are dual to sinks (black hole).
Divergence (entropy) is dual to convergence (syntropy).
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
A white hole is opposite, opposame or dual (isomorphic) to a black hole.
Dark energy (divergent, repulsive) is dual to dark matter (convergent, attractive) -- a dipole.
A magnet has a dipole structure (north, pole, south pole) -- duality!
"May the force (duality) be with you" -- Jedi teaching.
"The force (duality) is strong in this one" -- Jedi teaching.
@@hyperduality2838 It seems that the only thing you are missing is Descartes' mind-body duality. I think we should revive that notion and take it seriously. I also agree with you that 3 dimensions are not enough. But instead of just adding time, we should seriously consider the reality of multiple extra astronomically large spatial dimensions. The mathematical notion of manifolds explains how those extra dimensions are inaccessible to us 3D critters limited by our 3D tools and equipment.
@@Qdogsman Mind (the internal soul, syntropy) is dual to matter (the external soul, entropy) -- Descartes.
According to Descartes the mind/soul is dual!
"The intellectual mind/soul (concepts) is dual to the sensory mind/soul (percepts)" -- The mind duality of Thomas Aquinas.
Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Your mind converts perceptions or measurements into conceptions or ideas -- rational thinking or thought forms -- a syntropic process, teleological.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non-teleological physics (entropy).
Making predictions to track targets, goals or objectives is a syntropic (converging) process.
From a converging, convex or syntropic perspective everything looks divergent, concave or entropic -- the 2nd law of thermodynamics.
All observers have a syntropic perspective.
"Always two there are" -- Yoda.
The real question here should be: what if the computational properties of our local universe....change?!
I've really enjoyed this discussion on the problem of infinity. Your guest puts it into a quite sensible framework. I've been concerned for a while now about natural numbers and just how unnatural they are. They are a very abstract digitisation of our phenomenological impression of "oneness, twoness ..." which are quite fuzzy in practise. This digitisation creates the expectation that reality is a computation. Perhaps we need some form of analogical arithmetic. That would certainly remove infinities. I'm curious if I'm alone in these doubts.
You are not alone doubting 'number'. An Amazonian tribe consciously refuses to leard and adopt discrete number theory.
Analogical arithmetic sounds very cool. I've been playing with the idea of building 'analogical' formal language of indefinite continua, starting from relational operators < and > interpreted as verbs.
"plus 1 whatever you say" sounds like infinity to me. Thinking simply seems to confound and escape the academic mindset. I feel sorry for people that actually take discussions like this seriously.
Are you a mathematician? Have you researched Cantor's proofs of infinities in set theory and Brouwers remarks?
The reason this guy (who is professor) has this discussion' which you don't take seriously, is because mathematicians actually don't think so simply as you suggest. So you should applaud this guest actually...
@Gennady Arshad Notowidigdo I'm replying to the guy saying he feels sorry for people who take discussion like this (about infinity and mathematics) seriously. What do you think about this?
Let's go bros.