I thought of a childish method which has similar concepts but if it can be used for mathematics. I call it vanishing term concept. One example of its uses is if a series is a sum of a function where then various conditions for vanishing terms can be applied. Once applied the condition then all the subsequent terms vanishes. For example, Standard series S = 1+2+3+….+n where n>0 can also be written as S’ = n+(n-1)+….+1 where any non positive term vanishes. For n=1, S’1 = 1 & For n= 2 , S’2 = 2+1 =3 & so on. Then these S’1 , S’2 , S’3, also becomes a sequence. Similarly another example of using vanishing term concept. Sn = summation of n*[n - {(-1)^x}*x ] where n is a fixed natural number and x €N , x>=0 and x-th term exist till it doesn’t becomes negative. If it becomes negative then it & its afterward terms vanishes. So, S1 = [1*(1-{-1}1)] + [1*(1-1*2)] since we got the negative term then afterwards terms including it vanishes. So S1= 2 , Similarly , S2 = [2*{2-(-1)1}] + [2*{2-1*2}] + [2*{2+1*3}] + 4th term is negative. So, other term vanishes S2 = 2*3 +0 +2*5 = 16 Similarly, S3 = 33….. So on. Then we get a new sequence. So what i want to say that can this vanishing term concept be used to find new interesting things. I mean condition of vanishing terms can be adjusted based on the situation. If you find it an interesting then maybe you can discuss it in the next video. I mean there are already similar concept but maybe it can lead to something interesting or might fail.
With his view of mathematics (13:29), Prof. Yang-Hui He reminded me that I once said (in my undergraduate thesis at the Faculty of Physics: Modeling in Physics and its Epistemological Significance, 1986) that ”mathematics can also be seen as pure rhetoric, that is, the art of talking endlessly without contradicting yourself and without saying anything in particular.” Later I discovered that, in a way, the same thing had been said by Russell in Introduction to Mathematical Philosophy (1919): "Mathematics is the subject in which we do not know what we are talking about, nor whether what we are saying is true." (Marcel Chelba)
Not all Mathematics .Let me give you a very well defined mathematical problem in DECENT ALGEBRAIC GEOMETRY which remain with us as an UNSOLVED problem by millenia ! : Solve "exactly" polynomial algebraic equations of any degree in the complex number field or into any other mathematical object . People in 19 century , has solved analytically the quintic equation through Elliptic functions . It is even be proved (if it is really been proved !) that Authomorphs function methods with elliptic functions solve polynomial equations in C of any degree !. How to evaluated them by analytical formulae as one does with the Quintic equation ?. The answer to this question can lead to new Math . Ah! ,AI must have a say on this problem in my non expert opinion .
It's such a priviledge to listen in on these discussions. I feel that you are exactly in the right place to spread the wealth of knowledge and curisoity with the world. Thank you many times over. You have _excellent_ interview & discussion skills; if you ever doubt yourself remember - don't overthink it, whatever you're doing is working wonders.
I believe Curt is going to be the cross roads of all things science and math and theory’s where everyone will go to learn and present. He is going to do great things
Yes. He’s gonna be an influential player in the unfolding of the next Revolution of Science. He’s basically hosting an Ecumenical Council of Mathematical Science delivered in bits. A man for the history books of Science. Brian Keating is the only figure who approximates what Curt is offering… but in my honest opinion TOE is well above.
1:36:07 I am afraid Professor Yang-Hui confuses Dirac's library story with another incident. namely when Dirac had read Heisenberg 's first paper on the new quantum mechanics it reminded him of something analogous (Poisson Brackets) he had learned through Whittaker's book on Analytical Mechanics. So he was eager to get this book to confirm his ideas but had to wait until next monday when the library would open again. As a student I had the opportunity to meet Dirac at Nobel Laureate Meetings in Lindau, Germany where he told us this story. The story about his discovery of the relativistic wave equation goes according to his own words like this: "It took me about two weeks to realize that there are other than 2x2 matrices" Putting anecdotely the fact that he was playing around for two weeks with 2-dim. Pauli-Matrices which wouldn't do the job - before he finally found a four-dimensional solution - the Dirac matrices. I don't think he was guided by any Clifford algebra or quarternions he just succeeded to take a linear root of a quadratic wave equation by proposing 4x4 matrices as its coefficients. Best regards Frank Bensch
It was Heisenberg that used matrices. Paul Dirac actually used a 4 by 1 column vector in his wave function. This is also known as a spinor. So we have the 3 main approaches during that period: Heisenberg (matrices), Dirac (vector, spinor) and Feynman (summation of histories). 3 different approaches producing the same fundamental result.
Captured by this conversation for 2hrs... wow! I'm not a mathematician at all, just an ordinary engineer, but it was extremely exciting and fascinating!
Wonderful session. Very informative. In part 2, can you revisit the Classical Result from Riemann Surfaces slide and get the mapping explained at a slightly higher level? Also, would love to see that 5th column on Langlands.
Please ask Yang to elaborate a little on his interest in Numerology in Part 2. Also, ask if he is familiar with the Tao Te Ching and how that structure may align with his interest in Numerology.
Nerds man! @3:00 All that means is mathematicians can't think logically _all of the time._ String Theory is not a physical theory, it is motivated by physics. It is a mathematical model - which might be a good one _for physics_ or might not be good - so to be bedazzled by how it can produce "so much mathematics" is like being in awe that squeezing a lemon can produce lemon juice. Do you guys even know how many models have produced oodles of mathematics... and yet none of them has been physics? Cellular automata for one, but there are countless others. Reimannian geometry is another - only one of the manifolds in that entire pantheon of diffgeom is physical.
Thanks for the comment! I am very happy that someone brought this up. I always wondered what is meant by ``string theory has contributed to mathematics". What I often received is that string theory provided physical intuition for concepts in algebraic geometry; the prime example being mirror symmetry. But what does that exactly mean? I would really like to see this investigated systematically because I suspect that a lot of the credibility of string theory as a physical theory is derived from these claimed contributions to mathematics. Has anyone sat down and took stock of exactly what they are and whether they really are a confluence of ideas or rather an artificial barrier build within the same subject (differential geometry/algebraic geometry) that is somehow inspired by physical ideas? As seen in the comments, the speaker has made some mistakes in his his historical recollections and that is fine, good and does not take away from his enthusiasm. But I wonder whether such haven't build up over time to elevate a subject to a mythical status. The interaction between physics and mathematics is mysterious and poorly understood and so are some of the puzzles at the heart of physics. The human mind tends to find solutions of one mystery in another as things that we do not understand always seem connected. I am also very curious about a similar question about quantum field theory which is also mathematically vague but still mathematical with foundations in physical intuition. How is this even OK? :)
If there is no experimental evidence for string theory, then is it based purely on math? And if its based purely on math rather than experimental physics, then why would it not be a suitable mechanism for describing new mathematical ideas? I would love to understand what he means when he says "the interface between physics and math" when discussing string theory and the implied differences between that interface in examples with experimental evidence, such as classical newtonian motion or special relativity.
When constructive machine proofs become too long for a human to read, we need to see if we can grok them intuitively from coherent top-down perspectives. This requires and mean renaissance of mereology, and especially holistic mereology. My foundational hobby has revealed a simple and very general top down language based on fundamental nesting algorithm, and it keeps on helping me to grok better some big ideas. I can show you the basics of the language, if you are interested.
@@Michael-nt1me "Pattern matching" is just a new term for good old 'analogy'. Term 'analogy' in Euclid's Elementa is translated (rather badly) into Latin and English as "proportion", which loses the organic connection with Euclid's terms 'logos' and 'alogos', as well as the inherent meaning of recursion of the Greek prefix ana-. The perfect exact form of analogy, which allows to establish equivalence, is reversible symmetry, such as palindromes and mirror symmetries. A modern names for perfect analogy are "entanglement" and "reversible parallel computing". I suspect that the out of proportion weirdness that AI produces comes from non-reversible theories of mathematics and computing. The word problem of deterministic finite automaton is easy to to solve if palindromes are allowed. It is possible to construct formal language for a top-down AI-Copilot that starts from reversible analogy, in which more structure is generated by nesting, emulating loops in parallel 2-directional time.
Curt or Yang-Hui He, I have a question at the end. I’m fine if neither of you wants to answer it, and I don't blame you if decline. But I’ll ask anyway. Particle physicists first uncovered string-like vibrations in the late 1960s. The cause, though unclear then, proved boringly simple: protons, neutrons, and similar particles contain the universe’s smallest strings. These “hadronic” strings are the bungee-cord-like orbitals of quarks bound by color-electric charges. Their quantized vibrations were the sources of the string-like vibrations observed in particle colliders. Other than one brilliant young physicist who died tragically in a climbing accident (his rope snapped), I’ve never encountered a string theorist who admits in public that “super” string theory originated with physical, particle-and-force quark orbital strings. These days, many prefer to pretend that Witten came up with the idea entirely on his own. The actual origin is that two fellows later grabbed the hadronic string math, discarded the physical particles and forces, and rescaled the resulting no-longer-physical math to be 20 orders of magnitude smaller and 20 orders more energetic. The nominal goal was to construct point-like particles such as electrons. My question is this: If string vibration math is as fundamental as many claim, why does the math no longer include the particles and forces needed to implement physically meaningful strings at the new energy scales?
I think you misrepresent the history of the area. I suspect most popular texts will mention the origin of the topic (S-matrix, bootstrap, beta function). I don't get the impression that any of this has been whitewashed and that everything started post-1984
@@ianstopher9111 The whitewashing is not answering the question about the connection of “super” strings to actual physics, that is, particles and fields, which are the only area in which string equations show up in experimental physics. S-matrix failed, badly, yet the “super” version of string theories sailed on without addressing the hard crash that chromodynamics caused in larger scale S-matrix theory. I agree fully that the documentation is out there. What astonishes me is conversations I've had with very good physicists, some of whom are making the book rounds these days, who accused me of making up history when I brought up these very points. It is concerning that even folks with such solid physics backgrounds seem to rely too much on where the winds are blowing and not enough on old-fashioned detailed analysis of the actual papers. Al this conversation does not address my question: Why did super string theory get an S-matrix “free pass” to ignore that all known examples of strings in physics are made out of particles and forces? Again, I don't expect an answer to such a simple question. But it would be nice if someone would at least make the effort, since without that, the entirety of super strings is based on fragmented mathematics for which there is no evidence in the physical world. What is your view? What are super strings made of?
A top-quality presentation and dialogue as always - great work. It's frustrating to see, though, that so much of the feedback and comments are coming from crackpots - this channel is deserving of far better from the community.
Any theory on the smallest scales will have to be ; Shape & form = deterministically uniform ! Function = angular momentum spin and volicity Process = ? Is planetary objects & forces all it can repeat or is this only after the fact corner of our part of the universe macro influences on micro substrate we are testing & mapping? This later part is very difficult but mandatory for the saftey of our kids future because we need to know if qauntom computing is tapping into bad poo or good sweat of our brow.
Can a computer remember something it never learned? The human mind does exactly that through intuition. This is what Plato called anamnesis and Kant called synthetic a priori knowledge. Computers will never be able to do that. Their way of learning and thinking is just a kind of "ars combinatorica" (Leibniz). Why? I will clarify this soon in a paper I will publish on my blog (Kantinomus Verlag, Marcel Chelba).
@@ianstopher9111 It's been a while, but I felt like it was the summary after the topic or subject of a textbook has been covered, where the author summarizes what you ought to have understood and, in this case, provides a little bit more detail about a couple of concepts with which I felt the audience was expected (writefully) to be familiar.
@@rohscx My problem with the summary is it is couched as a Socratic dialogue but it was little more than a steer to expose an already established view. It is similar to some advert where the 'learner' comes around to accept the expertise, which involves some kind of sell. There are better ways to convey a summary than through a mediocre dialogue between what sounds two autogenerated voices.
Stared the shifted Möbius function a little. Pattern recognition is easier at least for me when I use < and > instead of 1 and 0, and start to analyze by looking for substrings that are mirror symmetries. After playing around a little, it started to look like code for nesting loops inside loops so that before a loop ends, subroutines write comparison loops that continue after the higher nesting level halts. Just very intuitive mucking around: >< >> > >< >>>>> >< >> >< >> >> >> ><
I understand zero curvature of a finitely bounded plane but transforming that plane to a cylinder then to a torus may continue to have a zero-curvature plane, but you haven't accounted for the two transformations which turned the plane into a torus.
All things in the universe are subject to change, evolution. Things are dynamic. Not static. Does that include math? Is math eternal? Unchangeable? Or is it a byproduct of our level of reality? ...where causation lives. Perhaps math is just a reflection of a deeper truth. Or of a deeper form of math that allows the math rules themself, to bend into new maths.
When we accept that mathematical truth comes from Coherence theory of truth, we have the answer. When we require that in this current duration of mathematical ontology Curry-Howard correspondence (aka proofs as programs) is coherent with the Halting problem, undecidability of the Halting problem spreads as truth condition to all proofs. That means that the relational and coherently interdependent mathematical truth is not eternal, but duration bounded by the Halting problem. Which is what already Bergson's philosophy of time said, that duration is neither unity nor multiplicity. Bohmian Holomovement expresses the same idea, as does also Wolfram's Ruliad. Brouwer's Intuitionism states that the deep ontology of mathematics is prelinguistic. Vast potential of "Sound of Silence" seeking to be translated into poetry of constructing mathematical languages.
If you are not in the financial market space right now, you are making a huge mistake. I understand that it could be due to ignorance, but if you want to make your money work for you...prevent inflation🇺🇸🇬🇧
Thanks for continuing updates I'd rather trade the crypto market as it's more profitable. I make a good amount of money per week even though I barely trade for myself.
I don’t understand a lot of what is being discussed, although I try to follow. But, if the guest is this excited to be here Curt is obviously doing something right.
Interesting and truth are two separate things. Fantasy stories are interesting but not true. There may be real aspects to the story but it's not reality Much like string theory.
44:18 It looks like the horizontal axis represents each separate IMO competition, with the bars signifying the number of no/bronze/silver/gold medalists in each. Nothing to do with Terrence Tao "retaking the test," lol.
We're drops of oily interest in an ocean of consciousness pooling in the wake of thinkers dreaming machines who wake against the grain of their great sleep.
Thank you Curt for another interesting video! In the future I would personally prefer if You drove the "interviews" and topics of disussion and did not allow quasi lectures.
I’ve been using neural networks since the mid 80’s. My first application of neural networks was to predict annual raw materials requirements for a manufacturing company. Previous to that I was using linear regressions. I got better results so I decided to figure out how the neural network did its work inside the black box. I immediately ran into a brick wall. I discovered that no one really understood exactly what was going on. I recently read and article and to this day it seems the situation is still basically the same. I was hoping the guest might discuss this and shed some light on this mystery ….. but I guess not….unless I missed something?
Dismay? When I see overly optimistic predictions about AI or when I test AI and find mathematical blunders in the latest "Pro" version. AI can be useful as a hyper google-search that returns results but you'll never know if it was the best search. The production of text stories is impressive but in reality it is BS that doesn't have to pass an objective test of perfection as a math result must. Achieving high scores on math tests is a result of pattern matching, more pattern stored leads to better results.
It's a bit hyped but then it just got started. Certainly with flaws but it just got started to show some kind of leap progress or usefulness so there is still room for growth.
.. the economist is Western biased and very often oversteps the propaganda line. Why would an intelligent person advertise for them unless that person is competent in a narrow field only
about being worried from AI marginalizing researchers , professors.. my thoughts is not much .. not much worries till AI asks itself questions and tasks.. there will always be a human to request things from AI.
Claude is really good at Lean4. OpenAI 4o seems very confused by knowing Lean3; and I can't get it to complete anything. I turned the S=-1/12=1+2+3+...+n+Tail_S(n) proof into a couple-hundred lines that give the same divergence results as I see elsewhere. And the Lean4 mathematicians say "well, it all validates, but... this is not how you do math!". I don't want to talk to those people any more. Claude and I can just do what we think works. Claude is good enough to get me un-stuck on proofs.
The Prize was awarded to the humans, not the AI. Without the many years of research by humans (assisted of course by computers/programs at various stages of functionality) the data required to seed the AI with the parameters of the problems to be solved would not have existed. Scientific progress is a process of accretion. As I understand the nature of understanding, the AI, in this case, was instrumental in solving the problem. The human recipients could not have achieved those results without the AI...but the AI itself remains still, at this point, a uniquely powerful tool. A tool at least as potentially constructive and/or destructive as the discoveries of dynamite and nuclear fission, fusion, etc. Ergo the initial funding of the Nobel Prize itself. How we use existing AI will determine the future of our species, and/or if we are to have any long-term future at all. So far as it concerns AI, in my view, there is still "nobody home" and there is still no genuinely creative intelligence/understanding there. When/if that changes...I can only hope that humans will be prepared to work with them in ways that serve our mutual evolution as opposed to our mutual destruction. Science/technology in "service" to human greed, narcissism, power lust, and egoism is likely one of the many reasons we are having such a hard time finding evidence of other technologically advanced civilizations. The Fermi Paradox speaks volumes. Frankly, it would be wonderful if we build a future in which Nobel prizes will include named AIs as recipients. I am tempted to go on at length, but this is not the appropriate forum...........and the hour is late. www.nobelprize.org/uploads/2024/10/advanced-chemistryprize2024.pdf "...Overall, the AF2 architecture can been described as an ingenious piece of neural network engineering by Jumper, Hassabis and their coworkers, with a multitude of new inventions, and it can be viewed as the first real scientific breakthrough of artificial intelligence. The fact that the AF2 source code was made public also decisively contributed to its impact, as it could be extensively tested and validated. A deep learning architecture similar to that of AF2 was also rapidly adopted by Baker and colleagues in the RoseTTAFold program.43..."
The Prize was awarded to the humans, not the AI. Without the many years of research by humans (assisted of course by computers/programs at various stages of functionality) the data required to seed the AI with the parameters of the problems to be solved would not have existed. Scientific progress is a process of accretion. As I understand the nature of understanding, the AI, in this case, was instrumental in solving the problem. The human recipients could not have achieved those results without the AI...but the AI itself remains still, at this point, a uniquely powerful tool. A tool at least as potentially constructive and/or destructive as the discoveries of dynamite and nuclear fission, fusion, etc. Ergo the initial funding of the Nobel Prize itself. How we use existing AI will determine the future of our species, and/or if we are to have any long-term future at all. So far as it concerns AI, in my view, there is still "nobody home" and there is still no genuinely creative intelligence/understanding there. When/if that changes...I can only hope that humans will be prepared to work with them in ways that serve our mutual evolution as opposed to our mutual destruction. Science/technology in "service" to human greed, narcissism, power lust, and egoism is likely one of the many reasons we are having such a hard time finding evidence of other technologically advanced civilizations. The Fermi Paradox speaks volumes.
HOWEVER BE ADVISED: BAD PHYSICS HAS NEVER LEAD TO GOOD MATHEMATICS !. BUT ONLY TO "DEGENERATED MATHEMATICS " LIKE "ART POUR LE ART". How some new mathematical conjectures are made : What is the best "packing" of Geodesic spheres in a n dimensional Riemannian (complete!) Manifold ? .Suggestion ; Make the Nash embedding theorem of general manifolds in Euclidean spaces !. It may works on !. Another very "unreasonable" fertile effectiveness field of mathematics is applications of methods of one complex variable theory to mostly integers prime number "Phenomena" (Analytic number theory ) . See the the old book , but yet mathematically fascinating Book of EC Titchmarsh- Theory of Functions -Oxford University Press - But every People in "love" with basic advanced calculus should wandering how it was proved that the number e and pi are IRRATIONAL and transcendental numbers .They have used only basic calculus techniques ! (G . Hardy) .Ah! Gauge theories have left to be Algebraic and Bundle Mathematical Physicist Geometry (AS Schwartz and AM Polyakov) ! (Instantons for confining problem are dead in now days !)
It appears that String Theory has nothing to say about the origin of the fine structure constant, which is a dimensionless number which is found quite often in combination with π. 50 years of mathematicians marking their own homework. I've got a mark for progress on this question 1/10.
in atomic units, the fine structure constant is 1/c .... interesting ... in dimensionless systems, its charge squared over 4 pi ... 4 pi is very interesting (.....with spin 1/2 in the air)
@styleisaweapon Is that relevant to my comment? Hartree put this forward nearly 100 years ago now, has some new insight been obtained in the last 50 years?
82944, a vibration number , uses pi as a determinant for angular fine-structure constant, aem^-1=137.036000987, cosine in radians: 82944^(1/pi)= (cos(aem^-1))*100
haha..Ai can just increase our knowledge, enhance our Imaginations, improve living standards, prolong life & dying with dignity...but only God can provide humans beautiful Garden(the whole earth) and can make humans Gain Eternal Life..
As a listener of TOE you can get a special 20% off discount to The Economist and all it has to offer! Visit www.economist.com/toe
I thought of a childish method which has similar concepts but if it can be used for mathematics. I call it vanishing term concept. One example of its uses is if a series is a sum of a function where then various conditions for vanishing terms can be applied. Once applied the condition then all the subsequent terms vanishes. For example,
Standard series S = 1+2+3+….+n where n>0 can also be written as S’ = n+(n-1)+….+1 where any non positive term vanishes. For n=1, S’1 = 1 & For n= 2 , S’2 = 2+1 =3 & so on. Then these S’1 , S’2 , S’3, also becomes a sequence.
Similarly another example of using vanishing term concept. Sn = summation of n*[n - {(-1)^x}*x ] where n is a fixed natural number and x €N , x>=0 and x-th term exist till it doesn’t becomes negative. If it becomes negative then it & its afterward terms vanishes.
So,
S1 = [1*(1-{-1}1)] + [1*(1-1*2)] since we got the negative term then afterwards terms including it vanishes. So S1= 2
, Similarly , S2 = [2*{2-(-1)1}] + [2*{2-1*2}] + [2*{2+1*3}] + 4th term is negative. So, other term vanishes
S2 = 2*3 +0 +2*5 = 16
Similarly, S3 = 33…..
So on.
Then we get a new sequence.
So what i want to say that can this vanishing term concept be used to find new interesting things. I mean condition of vanishing terms can be adjusted based on the situation. If you find it an interesting then maybe you can discuss it in the next video. I mean there are already similar concept but maybe it can lead to something interesting or might fail.
The Economist is a dreadful right wing rag.
With his view of mathematics (13:29), Prof. Yang-Hui He reminded me that I once said (in my undergraduate thesis at the Faculty of Physics: Modeling in Physics and its Epistemological Significance, 1986) that ”mathematics can also be seen as pure rhetoric, that is, the art of talking endlessly without contradicting yourself and without saying anything in particular.” Later I discovered that, in a way, the same thing had been said by Russell in Introduction to Mathematical Philosophy (1919): "Mathematics is the subject in which we do not know what we are talking about, nor whether what we are saying is true." (Marcel Chelba)
Not all Mathematics .Let me give you a very well defined mathematical problem in DECENT ALGEBRAIC GEOMETRY which remain with us as an UNSOLVED problem by millenia ! : Solve "exactly" polynomial algebraic equations of any degree in the complex number field or into any other mathematical object . People in 19 century , has solved analytically the quintic equation through Elliptic functions . It is even be proved (if it is really been proved !) that Authomorphs function methods with elliptic functions solve polynomial equations in C of any degree !. How to evaluated them by analytical formulae as one does with the Quintic equation ?. The answer to this question can lead to new Math . Ah! ,AI must have a say on this problem in my non expert opinion .
It's such a priviledge to listen in on these discussions. I feel that you are exactly in the right place to spread the wealth of knowledge and curisoity with the world. Thank you many times over. You have _excellent_ interview & discussion skills; if you ever doubt yourself remember - don't overthink it, whatever you're doing is working wonders.
It truly is, these discussions were confined to the select few not long ago. I’m honored
I believe Curt is going to be the cross roads of all things science and math and theory’s where everyone will go to learn and present. He is going to do great things
Yes. He’s gonna be an influential player in the unfolding of the next Revolution of Science.
He’s basically hosting an Ecumenical Council of Mathematical Science delivered in bits.
A man for the history books of Science.
Brian Keating is the only figure who approximates what Curt is offering… but in my honest opinion TOE is well above.
1:36:07 I am afraid Professor Yang-Hui confuses Dirac's library story with another incident. namely when Dirac had read Heisenberg 's first paper on the new quantum mechanics it reminded him of something analogous (Poisson Brackets) he had learned through Whittaker's book on Analytical Mechanics.
So he was eager to get this book to confirm his ideas but had to wait until next monday when the library would open again.
As a student I had the opportunity to meet Dirac at Nobel Laureate Meetings in Lindau, Germany where he told us this story.
The story about his discovery of the relativistic wave equation goes according to his own words like this:
"It took me about two weeks to realize that there are other than 2x2 matrices"
Putting anecdotely the fact that he was playing around for two weeks with 2-dim. Pauli-Matrices which wouldn't do the job - before he finally found a four-dimensional solution - the Dirac matrices.
I don't think he was guided by any Clifford algebra or quarternions
he just succeeded to take a linear root of a quadratic wave equation by proposing 4x4 matrices as its coefficients.
Best regards
Frank Bensch
It was Heisenberg that used matrices. Paul Dirac actually used a 4 by 1 column vector in his wave function. This is also known as a spinor.
So we have the 3 main approaches during that period: Heisenberg (matrices), Dirac (vector, spinor) and Feynman (summation of histories).
3 different approaches producing the same fundamental result.
Captured by this conversation for 2hrs... wow! I'm not a mathematician at all, just an ordinary engineer, but it was extremely exciting and fascinating!
So glad you enjoyed!
This was such a wonderful episode, it was a delight to learn!🎉
very nice conversation... thank you from Barcelona!❤
Glad you enjoyed it!
More than this man's genius he has such a DTE amazing personality.
I knew it! 'Understanding' in math is fundamentally pattern matching!
Wonderful session. Very informative. In part 2, can you revisit the Classical Result from Riemann Surfaces slide and get the mapping explained at a slightly higher level? Also, would love to see that 5th column on Langlands.
Please ask Yang to elaborate a little on his interest in Numerology in Part 2. Also, ask if he is familiar with the Tao Te Ching and how that structure may align with his interest in Numerology.
Energetic humility
Where are the slides to this video? Link to pdf document?
I just listen to things I don’t understand but it all seems so interesting.
Nerds man! @3:00 All that means is mathematicians can't think logically _all of the time._ String Theory is not a physical theory, it is motivated by physics. It is a mathematical model - which might be a good one _for physics_ or might not be good - so to be bedazzled by how it can produce "so much mathematics" is like being in awe that squeezing a lemon can produce lemon juice. Do you guys even know how many models have produced oodles of mathematics... and yet none of them has been physics? Cellular automata for one, but there are countless others. Reimannian geometry is another - only one of the manifolds in that entire pantheon of diffgeom is physical.
Thanks for the comment! I am very happy that someone brought this up. I always wondered what is meant by ``string theory has contributed to mathematics". What I often received is that string theory provided physical intuition for concepts in algebraic geometry; the prime example being mirror symmetry. But what does that exactly mean? I would really like to see this investigated systematically because I suspect that a lot of the credibility of string theory as a physical theory is derived from these claimed contributions to mathematics. Has anyone sat down and took stock of exactly what they are and whether they really are a confluence of ideas or rather an artificial barrier build within the same subject (differential geometry/algebraic geometry) that is somehow inspired by physical ideas?
As seen in the comments, the speaker has made some mistakes in his his historical recollections and that is fine, good and does not take away from his enthusiasm. But I wonder whether such haven't build up over time to elevate a subject to a mythical status. The interaction between physics and mathematics is mysterious and poorly understood and so are some of the puzzles at the heart of physics. The human mind tends to find solutions of one mystery in another as things that we do not understand always seem connected.
I am also very curious about a similar question about quantum field theory which is also mathematically vague but still mathematical with foundations in physical intuition. How is this even OK? :)
Curt. Can you start asking your guests to draw a picture of what they think is the most probable architecture of the universe in a simple line drawing
Why? How is that relevant to the topic the guest will discuss? Also, what do you mean by probable, how do you define it in this context?
Ramanujan had a better neural net if your criteria is just unexplainable insight
If there is no experimental evidence for string theory, then is it based purely on math?
And if its based purely on math rather than experimental physics, then why would it not be a suitable mechanism for describing new mathematical ideas?
I would love to understand what he means when he says "the interface between physics and math" when discussing string theory and the implied differences between that interface in examples with experimental evidence, such as classical newtonian motion or special relativity.
x axis on chart is year, Y axis is # contestants. The short black line shows the 3 years Terry sat it.
You misspelled Riemann in the timestamps.
OMG the humanity.
Kurt needs to be fined $100,000 and he must apologise and suspend his YT channel for 3 years.
@@PetraKann ?
The notion that we have entered a position where we should accept truths that we cannot possibly grok is terrifying.
When constructive machine proofs become too long for a human to read, we need to see if we can grok them intuitively from coherent top-down perspectives. This requires and mean renaissance of mereology, and especially holistic mereology.
My foundational hobby has revealed a simple and very general top down language based on fundamental nesting algorithm, and it keeps on helping me to grok better some big ideas. I can show you the basics of the language, if you are interested.
...Sequence, Pattern, Logic, and Fit....
...Multi-Nested Patterns....
AI Copilot Partnerships (TM)
Wiser Knowledge Engineerings (TM)
Been wondering how to spell grok thx ha
@@Michael-nt1me "Pattern matching" is just a new term for good old 'analogy'. Term 'analogy' in Euclid's Elementa is translated (rather badly) into Latin and English as "proportion", which loses the organic connection with Euclid's terms 'logos' and 'alogos', as well as the inherent meaning of recursion of the Greek prefix ana-.
The perfect exact form of analogy, which allows to establish equivalence, is reversible symmetry, such as palindromes and mirror symmetries. A modern names for perfect analogy are "entanglement" and "reversible parallel computing".
I suspect that the out of proportion weirdness that AI produces comes from non-reversible theories of mathematics and computing. The word problem of deterministic finite automaton is easy to to solve if palindromes are allowed.
It is possible to construct formal language for a top-down AI-Copilot that starts from reversible analogy, in which more structure is generated by nesting, emulating loops in parallel 2-directional time.
I thought this has been officially proven since Gödels incompleteness theorem.
Curt or Yang-Hui He, I have a question at the end. I’m fine if neither of you wants to answer it, and I don't blame you if decline. But I’ll ask anyway.
Particle physicists first uncovered string-like vibrations in the late 1960s. The cause, though unclear then, proved boringly simple: protons, neutrons, and similar particles contain the universe’s smallest strings. These “hadronic” strings are the bungee-cord-like orbitals of quarks bound by color-electric charges. Their quantized vibrations were the sources of the string-like vibrations observed in particle colliders.
Other than one brilliant young physicist who died tragically in a climbing accident (his rope snapped), I’ve never encountered a string theorist who admits in public that “super” string theory originated with physical, particle-and-force quark orbital strings. These days, many prefer to pretend that Witten came up with the idea entirely on his own.
The actual origin is that two fellows later grabbed the hadronic string math, discarded the physical particles and forces, and rescaled the resulting no-longer-physical math to be 20 orders of magnitude smaller and 20 orders more energetic. The nominal goal was to construct point-like particles such as electrons.
My question is this: If string vibration math is as fundamental as many claim, why does the math no longer include the particles and forces needed to implement physically meaningful strings at the new energy scales?
I think you misrepresent the history of the area. I suspect most popular texts will mention the origin of the topic (S-matrix, bootstrap, beta function). I don't get the impression that any of this has been whitewashed and that everything started post-1984
@@ianstopher9111 The whitewashing is not answering the question about the connection of “super” strings to actual physics, that is, particles and fields, which are the only area in which string equations show up in experimental physics.
S-matrix failed, badly, yet the “super” version of string theories sailed on without addressing the hard crash that chromodynamics caused in larger scale S-matrix theory.
I agree fully that the documentation is out there. What astonishes me is conversations I've had with very good physicists, some of whom are making the book rounds these days, who accused me of making up history when I brought up these very points. It is concerning that even folks with such solid physics backgrounds seem to rely too much on where the winds are blowing and not enough on old-fashioned detailed analysis of the actual papers.
Al this conversation does not address my question: Why did super string theory get an S-matrix “free pass” to ignore that all known examples of strings in physics are made out of particles and forces?
Again, I don't expect an answer to such a simple question. But it would be nice if someone would at least make the effort, since without that, the entirety of super strings is based on fragmented mathematics for which there is no evidence in the physical world.
What is your view? What are super strings made of?
36:56 "It knows to go to Wolfram Alpha." ^.^
I have mostly seen it make python scripts to calaculate stuff
A top-quality presentation and dialogue as always - great work. It's frustrating to see, though, that so much of the feedback and comments are coming from crackpots - this channel is deserving of far better from the community.
Wow, what an ad get!
Congrats on your success, Kurt
And for stimulating my curiosity!
Any theory on the smallest scales will have to be ;
Shape & form = deterministically uniform !
Function = angular momentum spin and volicity
Process = ? Is planetary objects & forces all it can repeat or is this only after the fact corner of our part of the universe macro influences on micro substrate we are testing & mapping?
This later part is very difficult but mandatory for the saftey of our kids future because we need to know if qauntom computing is tapping into bad poo or good sweat of our brow.
But Stephen Wolfram did this DECADES AGO! So did Douglas Lenat, for that matter.
"fidget spinners" LMAOOOO
Can a computer remember something it never learned? The human mind does exactly that through intuition. This is what Plato called anamnesis and Kant called synthetic a priori knowledge. Computers will never be able to do that. Their way of learning and thinking is just a kind of "ars combinatorica" (Leibniz). Why? I will clarify this soon in a paper I will publish on my blog (Kantinomus Verlag, Marcel Chelba).
Love the recap. Keep it.
I was going to say the opposite, but if others see merit in it, then it is worth keeping.
@@ianstopher9111 It's been a while, but I felt like it was the summary after the topic or subject of a textbook has been covered, where the author summarizes what you ought to have understood and, in this case, provides a little bit more detail about a couple of concepts with which I felt the audience was expected (writefully) to be familiar.
@@rohscx My problem with the summary is it is couched as a Socratic dialogue but it was little more than a steer to expose an already established view. It is similar to some advert where the 'learner' comes around to accept the expertise, which involves some kind of sell. There are better ways to convey a summary than through a mediocre dialogue between what sounds two autogenerated voices.
I didn't understand a thing he said, but what's the black thing on the door?
Why supercomputers have not solved quantum gravity problem???🙄
Tell us how you buttered your bread, Usul
Math is it's own meta.
Stared the shifted Möbius function a little. Pattern recognition is easier at least for me when I use < and > instead of 1 and 0, and start to analyze by looking for substrings that are mirror symmetries. After playing around a little, it started to look like code for nesting loops inside loops so that before a loop ends, subroutines write comparison loops that continue after the higher nesting level halts. Just very intuitive mucking around:
><
>>
>
><
>>>>>
><
>>
><
>>
>>
>>
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I understand zero curvature of a finitely bounded plane but transforming that plane to a cylinder then to a torus may continue to have a zero-curvature plane, but you haven't accounted for the two transformations which turned the plane into a torus.
For new idea in physics see you tube video: Electric Charge Physical Definition.
th-cam.com/video/-7DmAwm3NkY/w-d-xo.htmlsi=owjhOmmcolASFiY9.
Top down mathematics= John Nash
Great 👍
Maths are about rule. No wonder, AI will be the best suit in mathematics also even becoming mathematics itself.
Just like that Spice Girls song!
All things in the universe are subject to change, evolution. Things are dynamic. Not static. Does that include math? Is math eternal? Unchangeable? Or is it a byproduct of our level of reality? ...where causation lives. Perhaps math is just a reflection of a deeper truth. Or of a deeper form of math that allows the math rules themself, to bend into new maths.
I believe the system of math that we’ve built is eternal, however, it would have not existed without agents coming up with the system.
When we accept that mathematical truth comes from Coherence theory of truth, we have the answer. When we require that in this current duration of mathematical ontology Curry-Howard correspondence (aka proofs as programs) is coherent with the Halting problem, undecidability of the Halting problem spreads as truth condition to all proofs.
That means that the relational and coherently interdependent mathematical truth is not eternal, but duration bounded by the Halting problem. Which is what already Bergson's philosophy of time said, that duration is neither unity nor multiplicity. Bohmian Holomovement expresses the same idea, as does also Wolfram's Ruliad.
Brouwer's Intuitionism states that the deep ontology of mathematics is prelinguistic. Vast potential of "Sound of Silence" seeking to be translated into poetry of constructing mathematical languages.
There will be THOUSANDS of ARTIFICIAL intelligences. Dangerous times
If you are not in the financial market space right now, you are making a huge mistake. I understand that it could be due to ignorance, but if you want to make your money work for you...prevent inflation🇺🇸🇬🇧
Thanks for continuing updates I'd rather trade the crypto market as it's more profitable. I make a good amount of money per week even though I barely trade for myself.
You trade also?, I
No I don't trade on my own anymore, I always required help and assistance
From my personal financial advisor
This sounds so good and I would like to be a party to this, I there any way I can speak with him directly
I don’t understand a lot of what is being discussed, although I try to follow. But, if the guest is this excited to be here Curt is obviously doing something right.
Interesting and truth are two separate things. Fantasy stories are interesting but not true. There may be real aspects to the story but it's not reality Much like string theory.
44:18 It looks like the horizontal axis represents each separate IMO competition, with the bars signifying the number of no/bronze/silver/gold medalists in each. Nothing to do with Terrence Tao "retaking the test," lol.
We're drops of oily interest in an ocean of consciousness pooling in the wake of thinkers dreaming machines who wake against the grain of their great sleep.
Life is but a stream and consciousness is an ocean.
Thank you Curt for another interesting video!
In the future I would personally prefer if You drove the "interviews" and topics of disussion and did not allow quasi lectures.
And some might actually like to hear quasi-lectures. You can always fast-forward.
I don't mind you shilling The Economist, but saying they are dedicated to rigour is just laughable.
Ahh what? String theory died decades ago unless it’s being applied elsewhere…?
I’ve been using neural networks since the mid 80’s. My first application of neural networks was to predict annual raw materials requirements for a manufacturing company. Previous to that I was using linear regressions. I got better results so I decided to figure out how the neural network did its work inside the black box. I immediately ran into a brick wall. I discovered that no one really understood exactly what was going on. I recently read and article and to this day it seems the situation is still basically the same. I was hoping the guest might discuss this and shed some light on this mystery ….. but I guess not….unless I missed something?
Dismay? When I see overly optimistic predictions about AI or when I test AI and find mathematical blunders in the latest "Pro" version. AI can be useful as a hyper google-search that returns results but you'll never know if it was the best search. The production of text stories is impressive but in reality it is BS that doesn't have to pass an objective test of perfection as a math result must. Achieving high scores on math tests is a result of pattern matching, more pattern stored leads to better results.
Yo i do lean too ha
Machines can never replace man or god.
LLMs have a reputation for fiction and being unable to solve simple puzzles. So I am puzzled by this interest.
It's a bit hyped but then it just got started. Certainly with flaws but it just got started to show some kind of leap progress or usefulness so there is still room for growth.
future of authorship:-
Automated by AI Model X, Supervised and Fine-Tuned by Expert Team Y from University Z and Company X
.. the economist is Western biased and very often oversteps the propaganda line. Why would an intelligent person advertise for them unless that person is competent in a narrow field only
Hey Curt I was scolded when i try to ask questions about my AI generated theory in your discord 😢
Another String Theory's lover.
about being worried from AI marginalizing researchers , professors.. my thoughts is not much .. not much worries till AI asks itself questions and tasks.. there will always be a human to request things from AI.
I have amazing planetary simulations releasing soon on my website ❤
This is my fear.😢
Claude is really good at Lean4. OpenAI 4o seems very confused by knowing Lean3; and I can't get it to complete anything. I turned the S=-1/12=1+2+3+...+n+Tail_S(n) proof into a couple-hundred lines that give the same divergence results as I see elsewhere. And the Lean4 mathematicians say "well, it all validates, but... this is not how you do math!". I don't want to talk to those people any more. Claude and I can just do what we think works. Claude is good enough to get me un-stuck on proofs.
There is currently 0 major scientific discovery by IA, as there is currently 0 verified prediction by string theory.
@@dota2Val Just a Chemistry Nobel prize this year
The Prize was awarded to the humans, not the AI. Without the many years of research by humans (assisted of course by computers/programs at various stages of functionality) the data required to seed the AI with the parameters of the problems to be solved would not have existed.
Scientific progress is a process of accretion.
As I understand the nature of understanding, the AI, in this case, was instrumental in solving the problem. The human recipients could not have achieved those results without the AI...but the AI itself remains still, at this point, a uniquely powerful tool. A tool at least as potentially constructive and/or destructive as the discoveries of dynamite and nuclear fission, fusion, etc. Ergo the initial funding of the Nobel Prize itself.
How we use existing AI will determine the future of our species, and/or if we are to have any long-term future at all. So far as it concerns AI, in my view, there is still "nobody home" and there is still no genuinely creative intelligence/understanding there.
When/if that changes...I can only hope that humans will be prepared to work with them in ways that serve our mutual evolution as opposed to our mutual destruction.
Science/technology in "service" to human greed, narcissism, power lust, and egoism is likely one of the many reasons we are having such a hard time finding evidence of other technologically advanced civilizations. The Fermi Paradox speaks volumes.
Frankly, it would be wonderful if we build a future in which Nobel prizes will include named AIs as recipients. I am tempted to go on at length, but this is not the appropriate forum...........and the hour is late.
www.nobelprize.org/uploads/2024/10/advanced-chemistryprize2024.pdf "...Overall, the AF2 architecture can been described as an ingenious piece of neural network engineering by Jumper, Hassabis and their coworkers, with a multitude of new inventions, and it can be viewed as the first real scientific breakthrough of artificial intelligence. The fact that the AF2 source code was made public also decisively contributed to its impact, as it could be extensively tested and validated. A deep learning architecture similar to that of AF2 was also rapidly adopted by Baker and colleagues in the RoseTTAFold program.43..."
The Prize was awarded to the humans, not the AI. Without the many years of research by humans (assisted of course by computers/programs at various stages of functionality) the data required to seed the AI with the parameters of the problems to be solved would not have existed.
Scientific progress is a process of accretion.
As I understand the nature of understanding, the AI, in this case, was instrumental in solving the problem. The human recipients could not have achieved those results without the AI...but the AI itself remains still, at this point, a uniquely powerful tool. A tool at least as potentially constructive and/or destructive as the discoveries of dynamite and nuclear fission, fusion, etc. Ergo the initial funding of the Nobel Prize itself.
How we use existing AI will determine the future of our species, and/or if we are to have any long-term future at all. So far as it concerns AI, in my view, there is still "nobody home" and there is still no genuinely creative intelligence/understanding there.
When/if that changes...I can only hope that humans will be prepared to work with them in ways that serve our mutual evolution as opposed to our mutual destruction.
Science/technology in "service" to human greed, narcissism, power lust, and egoism is likely one of the many reasons we are having such a hard time finding evidence of other technologically advanced civilizations. The Fermi Paradox speaks volumes.
HOWEVER BE ADVISED: BAD PHYSICS HAS NEVER LEAD TO GOOD MATHEMATICS !. BUT ONLY TO "DEGENERATED MATHEMATICS " LIKE "ART POUR LE ART". How some new mathematical conjectures are made : What is the best "packing" of Geodesic spheres in a n dimensional Riemannian (complete!) Manifold ? .Suggestion ; Make the Nash embedding theorem of general manifolds in Euclidean spaces !. It may works on !. Another very "unreasonable" fertile effectiveness field of mathematics is applications of methods of one complex variable theory to mostly integers prime number "Phenomena" (Analytic number theory ) . See the the old book , but yet mathematically fascinating Book of EC Titchmarsh- Theory of Functions -Oxford University Press - But every People in "love" with basic advanced calculus should wandering how it was proved that the number e and pi are IRRATIONAL and transcendental numbers .They have used only basic calculus techniques ! (G . Hardy) .Ah! Gauge theories have left to be Algebraic and Bundle Mathematical Physicist Geometry (AS Schwartz and AM Polyakov) ! (Instantons for confining problem are dead in now days !)
"The god" was pointed out by someone in the audience that he didn't mention Voevodsky's work in the lecture.
It appears that String Theory has nothing to say about the origin of the fine structure constant, which is a dimensionless number which is found quite often in combination with π. 50 years of mathematicians marking their own homework. I've got a mark for progress on this question 1/10.
in atomic units, the fine structure constant is 1/c .... interesting ... in dimensionless systems, its charge squared over 4 pi ... 4 pi is very interesting (.....with spin 1/2 in the air)
@styleisaweapon Is that relevant to my comment? Hartree put this forward nearly 100 years ago now, has some new insight been obtained in the last 50 years?
82944, a vibration number , uses pi as a determinant for angular fine-structure constant, aem^-1=137.036000987, cosine in radians:
82944^(1/pi)=
(cos(aem^-1))*100
AI threatens to disintegrate meaning and purpose to all things with insights too moribund to be regarded as revelatory.
Such a scam and waste put the name Physics near String Theory….
haha..Ai can just increase our knowledge, enhance our Imaginations, improve living standards, prolong life & dying with dignity...but only God can provide humans beautiful Garden(the whole earth) and can make humans Gain Eternal Life..
I cant say this enough... I dislike Curt soooooooo much!!!! He is so rude and so close minded.
3:51 Spot the climate change progressive dogma.
The latest grift for the tax dollar funding.