"Everybody knows what a Calabi-Yau manifold is."

I read Yao's book, The Shape of Inner Space, several years ago. It was fascinating!

I love this episode. Thank you for it.

Inspiring and very well conducted. Thanks!!!

Great piece, Robert.

TOTALLY fascinating, but for ME mixed with about 40 percent or less of comprehension, and that was after listening to parts twice.. LOVED it..

Thanks

Mathematics breakthroughs in my humble opinion: Calculus (Newton and Leibniz) XIX Century: Diferential Geometry (Riemman and Co.) And then Einstein. Hamilton (Quantum Physics) Wave equation. Boole (Binarian Logic) Digital World.

a breakthrough is considered as such if it offers a better understanding or better precision of particular natural processes...

Hi. First of all, I'ld like to congratulate you for the constant quality of these contents, aesthetically pleasing as well as functional and concise in their technical explanations. I would like to ask a question, or to expose an argument regarding breakthroughs in science in modern times and to come, so here it is: Setting as a premise the probable (since supported by a statistical convergence in the linearity of the evolutionary path of technology, not only non-ignorable but even reasonably sustainable) creation of a Hard Artificial Intelligence, therefore available of a large Autonomy* capacity - for which, given a limited number X of inputs, it is possible to have a nonpredictable order of computational output Y, such that Y is significantly bigger than X - which allows it to establish progression in its internal operations (autopoietic & self-referential function of a complex system), It is logically possible to hypothesize a partial - if not primary - role of these intelligences in the very advancement of the sciences, especially those that are made in an exact and rigorous manner, that are fundamentally Mathematics and Geometry. In informal logical terms: AI: Artificial Intelligence; (in)x: number of input; (ou)y: number of output; (s.P): property of statistical Predictability; AP: Auto-Poietic function; Au: state of Autonomy; f(M&G): function of Math&Geometry; (HpV): Valid Hypothesis; (Bt in M&G): breaktroughs in M&G; • if {AI: (in)x --> (ou)y} | {(ou)y > (in)x }; • if {(ou)y /not contained in/ (s.P)}; • if {AI --> AP} | {AI = Au}; • if {AI: f(M&G)}; ___________ • Therefore {E! (HpV)} | {(HpV) = (Bt in M&G) c (AI) capability}. I want to underline this idea as I often observe how the considerations of a possible interference of artificial intelligences in human activities are made mainly in artistic and humanistic fields; instead, I fervently think that, if they demonstrate, in their creation, a capacity for autonomy such that they are not statistically predictable, Hard Artificial Intelligences will control hegemonically the revolutions in the mathematical field, almost inevitably. The reason for their preponderance in the field is also the following: for the human mind, a massive and articulated use of orderly and lucid rationality is undoubtedly an effort -for some Very heavy, for others less-; for an Artificial Intelligence constructed in a strict mathematical language and purely computational, and with the provision of independent research, the extreme and ultimate dimension of mathematical knowledge would be directly accessible.

Mathematics is where pure intellect lives.

The invention of digits was a great breakthrough, especially zero. Before that, people used to add up letters ! Duh!

Does this guy have a private jet? How is it possible to interview all of these people?

Mathematics is not only to calculate numbers. It also represents processes like cause and effect. Complex number i is defined as the ratio of effect by cause, when change in y (effect) due to change in x (cause) of a complex number z=x+iy, in VISUAL COMPLEX ANALYSIS by Tristan Needham, page 217.

How do you "test" that 2+2=4? How do you fabricate a Moebius strip without a cut and a twist? How can a math equation open up a 5th dimensional portal to another point in time? There is a sense that breakthroughs in mathematical thinking are mostly like funny puns: plays on words, and sometimes like interesting riddles or eye opening paradoxes. Formulations of metaphorical, substitutions that takes the form of mathematical "objects". Waves are made of sin waves or cos waves in a cartesian coordinate system or in a polar or cylindrical or take the form of an Eigenvector and between this system and that system these formulas are identical or inverse or.... Similarly in physics, matter is made of atoms is made of quarks is made of fermions and bosons is made of.. Are there any breakthroughs in "math", or are all the breakthroughs in metaphor? Real numbers, probabilities, imaginary numbers, Euclidean Geometries, Riemannian Geometries, knots, etcetera. What is probability? What does probability tell you? Isnt it telling you the most absurd thing? It tells you it doesn't know what the next roll of die will be, but it can tell you that from what it does know it "can't" be more than what it tells you it is, but it "can" be more than what it tells you it isn't. There's a 16% chance of rolling a 1, but there's an 84% chance it won't be a 1. Is that rational? It certainly isn't helpful. Who knew reason could be so...smart. Math is simply scaling. Nanoseconds, femtometers, gigavolts, kilograms embedded in a disguise as a 6 dimensional non-linear solution to a 5 twist, moebius knot 🪢 on an inverse toroid. "Why was six nervous? Because seven eight nine." Only if eight means "ate" does this make sense. Get it.

I believe Paul Davies is the one.

Can anyone tell me why we don't use sextigezimmal math like the Sumerians were taught and used. The math systems were changed solely on who was is in power on Earth then. That's a whole different topic but I want to know about the math. The more I look into the past the more that I see that we are using a substandard math system. Or so it seems. Thank you for your responses in advance

I think comparing Steve Jobs inventing iPhone to science innovations is a bad one. Even though iPhone was a leap forward technically, we would have had smartphones anyways without it and devices reaching that level of technology would have been there anyways.

It seems that very few people, mathematicians and physicists alike, want to consider the idea that the base in which we do math is significant. There IS a difference in the bases, and it lies in the size, or length, of the decimal place. There is a foundational geometric pattern which exists on the base twelve cartesian plane which has been invisible to us because we explore solely in base ten. The discovery of this geometric pattern will be a breakthrough for both the mathematics and physics communities, as it has to do with our understanding of the geometry of the circle, and therefore our understanding of Pi, which is pervasive, and from a physics standpoint is actually the ratio of energy and mass to empty space. Among other things, it will help shed light on dark matter ;-) and do away with the necessity of invoking infinity when calculating the size of the very, very smallest of the small. So, stated concisely: An upcoming breakthrough will be arriving soon as the result of someones deep explorations of base twelve mathematics.

The next biggest step, or the biggest step for whole human culture will be: using AI to help human to prove all those difficult mathematic hypotheses.

3:42 ... but I don't think anybody knew what would come out of the calabi conjecture once he solved it. 3:49

The Iphone analogy must have been a sarcastic joke. What is Iphone - essentially? An ordinary phone at the time, but with the screen covering it, right? Jobs didn't come up with this idea - not even Wozniac. A screen covering the whole side was "invented" by somebody who had nothing to do with Apple - look it up. The whole idea is also pretty simple half the population of the globe probably thought of this after seeing the limitation with a small, 240*320 pix screen. Calling it an invention is a huge stretch.

19:15 -·¹ 22:05 ¹·-

Good try !

Steve Jobs did not create iPhone. It was designed by the engineers under his leadership. He has no knowledge of engineering, neither he first proposed to develop. He might have had asthetics sense, but most ordinary people do have that, the difficulty was realising it. Do not forget many company builds different models, some gain popularity some don't. Some other company would have created similar device anyway within a few years. On the other hand, Andrew Wiles solved one of the greatest problem almost single handedly, when noone came even close. Comparing them was a very silly way (if not insulting) to start the topic.

Discovering 0 was a breakthrough

multiplicative areas had in algebraic geometry... perhaps in the shrinking and the enlargement new advances in renormilization can be seen, the sequencing around the nodes when enlarged may give some novel new uniqueness in terms of the consrevation represented there.

Mathematics should be taught to physicists in such a way that familiarise them with the problems of mathematics: physicists can also make breakthroughs in mathematics. Similarly, physics should be taught to mathematicians in such a way that familiarise them with the problems of physics: mathematicians can also make breakthroughs in physics. In this case, we will witness more breakthroughs both in mathematics and physics.

the poiynting vector, and the stress energy tensor, the step of movement in or around such compact places, will be both continuuous and non continuuous. Fractal energy storage.

Reposting and slight editing of recent mathematical ideas into one post: Split-complex numbers relate to the diagonality (like how it's expressed on Anakin's lightsaber) of ring/cylindrical singularities and to why the 6 corner/cusp singularities in dark matter must alternate. The so-called triplex numbers deal with how energy is transferred between particles and bodies and how an increase in energy also increases the apparent mass. Dual numbers relate to Euler's Identity, where the thin mass is cancelling most of the attractive and repulsive forces. The imaginary number is mass in stable particles of any conformation. In Big Bounce physics, dual numbers relate to how the attractive and repulsive forces work together to turn the matter that we normally think of into dark matter. The natural logarithm of the imaginary number is pi divided by 2 radians times i. This means that, at whatever point of stable matter other than at a singularity, the attractive or repulsive force being emitted is perpendicular to the "plane" of mass. In Big Bounce physics, this corresponds to how particles "crystalize" into stacks where a central particle is greatly pressured to degenerate by another particle that is in front, another behind, another to the left, another to the right, another on top, and another below. Dark matter is formed quickly afterwards. Ramanujan Infinite Sum (of the natural numbers): during a Big Crunch, the smaller particles, not the dominating black holes, are about a twelfth of the total mass involved. This also relates to 12 stacked surrounding universes that are similar to our own "observable universe" - an infinite number of stacked universes that bleed into each other and maintain an equilibrium of Big Bounce events. i to the i power: the "Big Bang mass", somewhat reminiscent of Swiss cheese, has dark matter flaking off, exerting a spin that mostly cancels out, leaving potential energy, and necessarily in a tangential fashion. This is closely related to what the natural logarithm of the imaginary number represents. Mediants are important to understanding the Big Crunch side of a Big Bounce event. Matter has locked up, with particles surrounding and pressuring each other. The smaller matter, not the black holes, gets broken up into fractions of what it was and then gets added together with flattened black holes to form the dark matter known from our Inflationary Epoch. Sectrices are inversely related, as they deal with all stable conformations of matter being broken up, not added like the implosive "shrapnel" of mediants. Ford circles relate to mediants. Tangential circles, tethered to a line. Sectrices: the families of curves deal with black holes and dark matter. (The Fibonacci spiral deals with how dark matter is degenerated/broken up and with supernovae. The Golden spiral deals with how the normal matter, that we usually think of, degenerates, forming black holes.) The Archimedean spiral deals with dark matter spinning too fast and breaking into primordial black holes, smaller dark matter, and regular matter. The Dinostratus quadratrix deals with the laminar flow of dark matter being broken up by lingering black holes and the dark matter also being broken up. Delanges sectrices (family of curves): dark matter has its "bubbles" force a rapid flaking off - the main driving force of the Big Bang. Ceva sectrices (family of curves): spun up dark matter breaks into primordial black holes and smaller, galactic-sized dark matter and other, typically thought of matter. Maclaurin sectrices (family of curves): newly formed dark matter gets slowed down, unstable, and broken up by black holes. Jimi Hendrix's "Little Wing". Little wing = Maclaurin sectrix. Butterflies = Ceva sectrix. Zebras = Dinostratus quadratrix. Moonbeams = Delanges sectrix. Jimi was experienced and "tricky". Jimi was commenting on dark matter. How it could be destabilized by being slowed down, spun up, broken up by lingering black holes, or flaked off. (The Delanges trisectrix also corresponds to stable atomic nuclei.) Dark matter, on the stellar scale, are broken up by supernovae. Our solar system was seeded with the heavier elements from a supernova. I'm happily surprised to figure out sectrices. Trisectrices are another thing. More complex and I don't know if I have all the curves available to use in analyzing them. But, I can see Fibonacci and Golden spirals relating to the trisectrices. The Clausen function of order 2: black holes start spinning up just before being crunched. Methods of arbitrary angle trisection: Neusis construction relates to how dark matter has its corner/cusp singularities create "bubbles", driving a Big Bang event. Repetitious bisection relates to dark matter spinning so violently that it breaks, leaving smaller dark matter, primordial black holes, and other more familiar matter, and to how black holes can orbit other black holes and then merge. Belows method (similar to Sylvester's Link Fan) relates to dark matter being broken up by lingering black holes that came late to the Big Bounce event. General relativity: 8 shapes, as dictated by the equation? 4 general shapes, but with a variation of membranous or a filament? Dark matter mostly flat, with its 6 alternating corner/cusp edge singularities. Neutrons like if a balloon had two ends, for blowing it up. Protons with aligned singularities, and electrons with just a lone cylindrical singularity? Prime numbers in polar coordinates: note the missing arms and the missing radials. Matter spiraling in, degenerating? Matter radiating out - the laminar flow of dark matter in an Inflationary Epoch? Connection to Big Bounce theory? "Operation -- Annihilate!", from the first season of the original Star Trek: was that all about dark matter and the cosmic microwave background radiation? Anakin Skywalker connection?

Also the axes may be weakly coupled, or up to some degree taken to be close enough to orthogonality.

Mathematics is one of humankind’s gifts to itself. To make up for teaching it religion.

The problem with math as a model for the natural world is that it inevitably leads us to unnatural conclusions and dead ends we can’t make sense of: infinity

What does this have to do with Jesus?

Huh?

Mathematics is a Intelectual tool of counting and dusintigusing Second, third person till infinity and so on its a kind of second backup of mind activity and a directional accuracy creating to manifest reality from the weirdness of particle behavior, but there are more powerful theorem such a Quantum wave function which mathematic cant define , flipping effect of fundamental reality is far to calculate with mathematic, it changes at the moment awareness strike

They were all there together, they were hear, there and everywhere, connected, got old started growing hair? We weren't meant to be held back like this, the weight of each was too little and it collapsed, idk but I understand the non binary anymore, so too much, or so much nothing it was forced to start something, delete , where did the no thing come from idk, I'm sorry

Yay now they know the earth is flat .

This would have been a lot better if you took the third interviewee's push-back a bit more seriously; this is a philosophy of mathematics question, and you only talked with practitioners. 'Breakthroughs', many would say, are a misnomer and a much less singular affair.

Unfortunately, with all the maths available, most "scientists" cannot even calculate the curvature of the Earth and have to use bullying tactics against anybody who dares to question the retarded spinning ball Earth.

What was insane Trump rant advertisement?

Mathematics does not represent reality because it cannot handle Infinity which is possibly the reality . Right now, maths regards infinity as a error of sorts

The following formulae is the so-called “THEORY OF EVERYTHING”, much sought-after by theoretical physicists for the past century: S+O = ∞BCP (The Subject and all objective reality is Infinite Being-Consciousness-Peace [“satyam jnañam anantam brahma”, in Sanskrit]). Alternatively, and more parsimoniously (as well as somewhat more elegantly) expressed as: E = A͚ (Everything, including all potential and actual objects, plus The Subject, is Infinite Awareness [“sarvam khalvidam brahma”, in Sanskrit]). For a thorough explanation of the above equations, refer to Chapters 05 and 06.

Realizing nature, learning the physics of aviation, engineering, constructing towers/bridges, developing tech for health care & medicine, testing and manipulating so fine tuning for efficieny is a marvel.... amidst it all, it never satiates that, which is a longing, a very deep-rooted void in all our centers. Merely are we beating around the bush. The break through is never really the 'tool' or 'method', but the very realization or recollection within us. I don't know what breakthroughs are in math or science or theology - quality of life, education, technology certainly make life less painful. When i think about math, i see that we're talking relativity and is limited to mind & senses; in retroduction such method reaches a threshold and can not go on further. Whereas the ancients, particularly the Pythagoreans and those before them, acknowledged Arithmetics, which accounts for the Principle - therefore is a superior system. The obvious to me is that mathematics points to Metaphysics. There are persons who believe that Principle and attribute are two entirely different things i.e, metaphysics and physics - well, it's a demonstrable fact that they're not. You can't cut illumination off from Light. 'A wave isn't what something is, it's what something does' and is it's nature. Modern man: quantum, physical, information, mathematics, empiricism. Ancient man: Metaphysics, nature, Knowledge, arithmetic, Wisdom. ^ the latter comprehends the ONE, the good, the Divine, beauty, being, form, essence, substrata, Principle, Reason, nature, Wisdom, Emanation. The former acknowledges limited physics(nature), quantity(quantum), information, and the ability to control, manipulate and exploit. Regardless, what we see is that man has inherent(of Soul) capabilities and in a way cooperates with God in tending to the earth and animals so working with nature maintaing balance and protecting, 'all life', and can apply his own creative & intellectual faculties in utilizing the laws, much like God, or rather more properly put, the God within. The breakthrough: the acknowledhement that something miraculous is 'ALL of this', and that we all are part of it, in fact are actually one with it, but we deal with the tribulations and quarrels of matter and the senses, impulses and desire, pain and suffering, managing the little time we have, managing the limited energy we have. Don't wait for your "friends". Don't wait for a mathematician to tell you where to go or what to do. 'Know thyself' - the ancient adage. Knowing thyself, is to come Knowing of God. Seek to Know thyself. ----------------------------------- Here is a list of the best content in the world today. These men knew the Intellect, inquiry, critical thinking, dialectic, Reason, Logic, retroduction, via negativa, arithmetic, science, evidence, the very nature of... Periphyseon, by Eriugena, translation by O'Meara. Plotinus Enneads, select works translated by Thomas Taylor and complete translation by Lyyod. Plato complete works. Proclus books. Iamblichus books. Syrianus books. Bhagavad Gita Upanishads translated by Nikhilananda 4 vol. set, and the 18 principal Upanishads translated by Radhakrisnan. Upadesa sahashria by sankara, translated by jagadananda. Vivekacudamani by sankara, translated by Madhavananda. Philosophy as a rite of Rebirth by Algis U. Meister Eckhart complete works. The Unknown God, by D. Carabine. Mystical languages of unsaying, by M. Sells. Plotinus: Road to Reality, by JM Rist. Bible - KJV translation only. archaic is very important here with mysticism. Jacob Bohme books - a German mystics Emmanuel Swedenborg books - a scientist turned mystic and metaphysics. Coomaraswamy books. The presocratic Philosopher's - book. Sweet touches of harmony - book; Pythagorean influence. Lore and science in ancient pythagoreanism - book. The Universal One, by Walter Russel. The gods of field theory: Henri Poincare Tesla Steinmetz Maxwell Dollard

Formulation of Mathematics relies solely on understanding causes and effects that this Physical Universe can possibly provide, but NOT beyond, as intended by the Holy Spirit and, so, its breakthroughs are limited... ...this is the reason why there is no mathematics or equation to recreate, copy, or understand Consciousness because Consciousness is beyond physical... Consciousness is part of th essence of the Holy Spirit who has no beginning and no end. We are Conscious because our free immortal souls are free splits of the Holy Spirit.