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Terence Tao explains 6 essential mathematical concepts [video]
- chrisweekly 15d agoSee also the amazing https://betterexplained.com https://betterexplained.com -- one of the best pedagogical resources I've ever encountered.
- FelixDigga 14d agoI'm happy for you I wish I was like you
- FelixDigga 14d agoWow that's awesome of you
- pixelpoet 16d agoAt least spell his name correctly, my goodness... Terence Tao.
- 129476 16d ago[flagged]
- femiagbabiaka 16d agoIt was expected that people without established credibility would be looked down on for using AI, but it's been weird to see people with all the credibility in the world lose it for embracing AI.
- deleted 16d ago[deleted]
- orf 16d agoWhat credibility has he lost?
- femiagbabiaka 16d agoTo sensible people? None. But the amount of criticism of his work that boils down to "he uses AI" is undeniable.
- orf 16d agoCan you share some of the criticism? I’ve not seen any from anyone real.
- pfdietz 16d agoIt's weird to believe he's lost credibility.
- nutjob2 16d agoTao gives or takes credibility from AI, not the other way around.
- oliculipolicula 16d agoIs Tao the rising angel? Or AI the falling ape? --Another Terry, Perhaps AI-avatar-of-Tao wielded by an internet rando lies to or gets fooled by Tao, not the other way around --Me, attempting to take your Untouchable* joke all the way to a (sociopathic, one-party) joke *https://archive.ph/2023.11.28-152809/https://www.astralcodexten.com/p/book-review-the-gervais-principle#:~:text=So%20Loser%20jokes%20are%203%2B%20people%2C%20and%20Clueless%20jokes%20are%202%20people.%20Continuing%20the%20pattern%2C%20a%20Sociopath%20joke%20must%20be%20for%20one%20person%20%2D%20the%20joker%20amusing%20himself%2C%20totally%20unconcerned%20whether%20anyone%20else%20appreciates%20it. https://archive.ph/2023.11.28-152809/https://www.astralcodex...
- dang 16d agoCan't believe we missed that. Fixed now. Thanks!
- matthewsinclair 16d agoApologies for the typo. I’ve asked mods to fix it.
- matthewsinclair 16d agoLooks like it’s been fixed. Thanks.
- laybak 16d agothis is great foundational content! I also enjoyed his earlier appearance on 3Blue1Brown https://www.youtube.com/watch?v=YdOXS_9_P4U https://www.youtube.com/watch?v=YdOXS_9_P4U
- Agentus 16d agoi blinked and read terrence howard. i was like good god
- stuxnet79 16d agoFor those not in the know, Terrence Howard's dreams of a career in Astrophysics were eviscerated after Neil Degrasse Tyson reviewed his "Theory of Everything" paper [1] [1] https://www.youtube.com/watch?v=1uLi1I3G2N4 https://www.youtube.com/watch?v=1uLi1I3G2N4
- iTokio 16d agoNumbers Algebra Geometry Probability Analysis Dynamics I loved this talk, but these concepts are like an attempt at dimensional reduction of math research, science, the academics knowledge. I would have loved to have his thoughts on the mathematical mind, the process, how to reason, infer vs deduct, abstract, prove.. I don’t really know, what are the primitives, essential concepts of math reasoning?
- Isamu 16d agoSo… numbers, algebra, geometry, probability, analysis, dynamics are not the primitives or essentials ?
- MrbroJangles 16d agoImo it's not numbers at all but linked to our awareness of physical relationships Making it about numbers is like making it about cans when it's more about grasping adding one can to a bag of cans, adding 100 cans (multiplication), or the inverse with subtraction and division Which is why I never liked numbers before algebra which then chucks numbers in the bin more or less. Numbers are just syntax meant to represent $anything; 1.5 can be half a pill and a whole pill or T or A; numbers are euphemism. That they can be infinitely big and yadda yadda isn't that meaningful in and of itself and that little bigness is all due to additive qualities of physical space Geometry is addition or subtraction of shape It's all built on 4 operations we see in daily life all the time
- andsoitis 16d ago> Numbers are just syntax Numerals are syntax; numbers are mathematical objects. “5”, “V”, “101₂”, and “|||||” are different representations of the same number. A variable such as x is closer to what the statement means by something that can stand for arbitrary things. > It’s all built on four operations Elementary arithmetic emphasizes +,-,×,÷, but mathematics isn’t reducible to them. Mathematics studies operations and relations such as composition, exponentiation, differentiation, integration, limits, logical implication, set membership, mappings, probability, topology, symmetry, transformations, equivalence relations, and many others.
- bobajeff 16d agoI really enjoyed this video (watched it the other day). It makes me feel like it's possible for me to understand the math that I'm currently trying to understand.
- esalman 16d agoI respected Terence Tao but since listening to his "Mathematics in the age of AI" talk, I've become a fan. I have had nobody else explain so succinctly what is the purpose of Mathematical research, why it matters, and why it is so important to preserve the ways we do math. Even more importantly, I feel it resonates so well with every other field AI is taking over.
- SMEbooop 16d ago[flagged]
- vismit2000 16d agoI also loved this talk (went through printed version: https://news.ycombinator.com/item?id=49362728 https://news.ycombinator.com/item?id=49362728) Tao talks how this has become even more valuable in maths: understanding, verification, exposition, community judgment, synthesis and canonicalization given how proof generation has become easy (which has historically been considered most valuable). So I just mapped this to coding also in my expereience and broader industry sentiment. Code generation was always the hardest and most valuable part. Now that is the cheapest part with claude code and other AI tools. But taking the candidate output (code) and building harness around it like verification, exposition, human understandability have become all the more important. Not just generate code, but generate code that other engineers can confidently modify and extend. Or even better - generate reusable canonical abstractions that improve codebase.
- not2b 16d agoEven before LLMs were a thing, it wasn't this way: rapidly generating code was not the most valuable skill. As you say, it's much more important that the code can be confidently modified and extended, and reused, not just now, but then. In a mature product, the initial writing of the code will be the least of the work; maintenance is much more expensive. Ideally, design decisions should appear only once in the code when this can be achieved, because then there's one place to fix or one place to modify, instead of dependencies on some detail that appear all over the code. It's too easy with auto-generated code to wind up with redundancy and code duplication, resulting in a brittle mess.
- stillpointlab 16d agoI've heard it said that true understanding is demonstrated when someone can explain difficult concepts well. Tao manages to convey complex ideas without making me feel like he is condescending to me. His depth of understanding is unmistakable. My changes to his list would be s/Geometry/Topology/ and I might have found a place for logic and type theory. I am especially glad he brought to mind Dynamics since that is a field I know I need to pay more attention to. Great video, we're lucky to have this kind of content so easily and widely available.
- eps 16d agoThe expression is "To know is to be able to explain". It holds up very well in a lot of situations.
- kaffekaka 15d agoNot least in the sense that to know something well _and also_ be able to explain it well is worth so much more than to just be an expert
- brokencipher 16d ago« Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisément » Boileau
- igravious 16d agoExactly; couldn't have put it better myself.
- bbeonx 16d agoagreed! it's funny, i'm returning to some of the maths i studied in undergrad with a bit more "worldly" knowledge (if you can call comp sci academia knowledge "worldly"), and i'm seeing that so so much of the stuff that was confusing was actually just trying to frame really intuitive properties. of course, the language to do so rigorously can be very dense and terse and difficult to get through, but at the end of the day, mathematicians are often trying to do some very simple things. for example, intuitively, i and j are _basically_ the same "shape" as one another, and f and c and s and v are _basically_ the same "shape" as each other, but the two sets of shapes are definitely _not_ the same as one another...to quantify this and actually capture it in math you gotta do topology, and once you get past the point set stuff it gets real abstract real fast. but they really just wanna say "hey, my donut kinda looks like my coffee mug".
- binyu 16d agoVery good resource. However, I fail to understand how a monkey writing the Hamlet is akin to a brute-force problem, in Terence's own words. The thesis seems to be here that given enough time, a monkey will be able to reason as a human. Edit: Probably refers to the evolutionary aspect of the problem. My criticism is that he compares the time for a monkey needed to learn the hamlet to merely "quadratic time". I disagree that such degree of non-linearity applies here. I think it is grossly simplified/underestimated.
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- hmokiguess 16d agoCould you be conflating writing hamlet with understanding how to write hamlet? as in, what it takes to end up with the same raw output though devoid of intent and meaning
- orphereus 16d agoTo be fair, I didn't watch the video to the part where he mentions monkeys writing Hamlet, but I thought the point of the monkeys is that they will stumble upon a Hamlet by accident at some point in time, by randomly pressing buttons on a keyboard. Obviously, it would take a long time, but it would happen at some point.
- andsoitis 16d ago> Obviously, it would take a long time, but it would happen at some point. Has someone done the calculation whether it would happen before the end of the universe?
- eyeundersand 15d agoIt would involve multiple, multiple universes (but not infinite) coming and going. Even with infinite monkeys.
- mc32 16d agoMaybe he was joking -not sure but around the 5:45 mark he says "irrational" in irrational numbers comes from the Latin for insane or unreasonable. But just before that he defines the numbers as not being able to be expressed as a ratio (that's what we all learn). Just seems odd he'd juxta that. Or it's dry wit.
- thaumasiotes 16d agoWell... https://www.etymonline.com/word/irrational https://www.etymonline.com/word/irrational > The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid. https://www.etymonline.com/word/ratio https://www.etymonline.com/word/ratio > The mathematical sense of "relation between two similar magnitudes in respect to quantity," measured by the number of times one contains the other, is attested in English from 1650s (it also was a sense in Greek logos) We can cross-check dictionary entries. The standard dictionary of Ancient Greek fully backs this up: > λόγος > II. 2 Math., ratio, proportion The standard dictionary of Latin doesn't mention this particular sense. (A negative is harder to cite, but you can check it here: https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dratio https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext... ) The senses that the Greek word and the Latin word have in common are those of reasoning in general and numeric computation in specific. You might guess that "irrational numbers" are named for their inability to be computed. (Or, if you're only looking at Lewis and Short, you might guess that they are named by reference to an inability to think methodically; this sense exists in Latin and indeed still persists in the English word "irrational". Insanity would usually be represented by another word, presumably something more like dementialis than irrationalis.) However, ratio is the conventional translation of the Greek logos, and since we're told that "rational" (of numbers) comes from a translation of Greek, it seems fair to attribute an existing Greek sense to the translated word too. So the best analysis does appear to be that "irrational numbers" are named, as you might expect, for the fact that they cannot be represented as integer ratios.
- thedreammachine 16d agoRelated and a great read: "The Joy of X: A Guided Tour of Math, from One to Infinity" by Steven Strogatz: https://www.goodreads.com/en/book/show/13356649-the-joy-of-x https://www.goodreads.com/en/book/show/13356649-the-joy-of-x
- bobthe3 16d agoThe Joy of X and I think now renamed to “The Joy of y” is a great podcast for general knowledge building!
- tzury 16d agoYou can pre order his book Six Math Essentials at https://a.co/d/0e89Jcjf https://a.co/d/0e89Jcjf
- marcy_74 16d agoHis lectures are gold, always manage to unlock a new perspective. Wish he taught my undergrad courses!
- nexacoreai 16d agowow, that's really nice
- Tbarlow 16d agoSaw him explain Fourier transforms once; it finally clicked. Probably another masterclass in clarity.
- kelvo_ran 16d ago[dead]
- seanhunter 16d agoI love the fact he mentioned the Riemann rearrangement theorem [1] briefly in his examples about analysis. That is (in my opinion) one of the coolest and least intuitive consequences of infinities. Requires some intro to different types of convergence to fully appreciate. More about the theorem here if you’re interested. [2] Weird as it seems it’s definitely true and one of the things you would prove in a typical undergrad sequence on analysis. [1] https://youtu.be/OOMx2BHHWtE?is=M1lqZI2gxNqWqg6G&t=18m35s https://youtu.be/OOMx2BHHWtE?is=M1lqZI2gxNqWqg6G&t=18m35s [2] Formally, I think the normal way to state the theorem is if you have an infinite series of real numbers which is “conditionally convergent”[3], then the terms can be rearranged so that the sum converges to any arbitrary real number, or diverges https://en.wikipedia.org/wiki/Riemann_series_theorem https://en.wikipedia.org/wiki/Riemann_series_theorem [3] Meaning it converges but does not converge absolutely. a_n = 1 - 1/2 + 1/3 - 1/4 + … is an example of such a series. It converges but if you take sum of the absolute values of each term you get the harmonic series which does not coverge.
- tim-kt 16d agoYou can also think of conditional convergence as convergence under the condition of a specific order. It then turns out that unconditional convergence (that is, convergence where it doesn't matter what order you choose) is equivalent to absolute convergence (that is, the sum over the absolute values converges).
- jonplackett 16d agoBeen really enjoying this channel in general. The fall of civilisations one was good and I think they had one with Brian Cox too. Just so nice listening to an hour of an intelligent person.
- irchans 15d agoHere is another list of ideas in math made by a lesser math person. (top 100 most useful) counting zero integer decimal positional notation 100, 1000, … the four arithmetic operations + – * / fractions decimal notation 0.1, 0.01, … basic propositional logic (Modus ponens, contrapositive, If-then, and, or, nand, …) negative numbers equivalence classes equality & substitution basic algebra – idea of variables, equations, … the idea of probability commutative and associative properties distributive property powers (squared, cubed,…), – compound interest (miracle of) scientific notation 1.3e6 = 1,300,000 polynomials first order predicate logic infinity irrational numbers De Morgan’s laws statistical independence the notion of a function square root (cube root, …) inequalities (list of inequalities) power laws (i.e. abac=ab+c ) Cartesian coordinate plane basic set theory random variable probability distribution histogram the mean, expected value & strong law of large numbers the graph of a function standard deviation Pythagorean theorem vectors and vector spaces limits real numbers as limits of fractions, the least upper bound continuity Rn, Euclidean Space, and Hilbert spaces (inner or dot product) derivative correlation central limit theorem, Gaussian Distribution, Properties of Guassains. integrals chain rule modular arithmetic sine cosine tangent π, circumference, area, and volume formulas for circles, rectangles, parallelograms, triangles, spheres, cones,… linear regression Taylor’s theorem the number e and the exponential function Rolle’s theorem, Karush–Kuhn–Tucker conditions, derivative is zero at the maximum the notion of linearity Big O notation injective (one-to-one) / surjective (onto) functions imaginary numbers symmetry Euler’s Formula eiπ+1=0 Fourier transform, convolution in time domain is the product in the frequency domain (& vice versa), the FFT fundamental theorem of calculus logarithms matrices conic sections Boolean algebra Cauchy–Schwarz inequality binomial theorem – Pascal’s triangle the determinant ordinary differential equation (ODE) mode (maximum likelihood estimator) cosine law prime numbers linear independence Jacobian fundamental theorem of arithmetic duality – (polyhedron faces & points, geometry lines and points, Dual Linear Program, dual space, …) intermediate value theorem eigenvalues median entropy KL distance binomial distribution Bayes’ theorem 210≈1000 compactness, Heine – Borel theorem metric space, Triangle Inequality Projections, Best Approximation 1/(1−X)=1+X+X2+… partial differential equations quadratic formula Reisz representation theorem Fubini’s theorem the ideas of groups, semigroups, monoids, rings, … Singular Value Decomposition numeric integration – trapezoidal rule, Simpson’s rule, … mutual information Plancherel’s theorem matrix condition number integration by parts Euler’s method for numerical integration of ODEs (and improved Euler & Runge–Kutta) pigeon hole principle mathematical used less often: Baire category theorem, Banach Spaces, Brouwer Fixed Point Theorem, Carathéodory’s Theorem, Category Theory, Cauchy integral formula, calculus of variations, closed graph theorem, Chinese remainder theorem, Clifford algebra (quaternions), Context Free Grammars, countable vs uncountable infinity, Cramer’s Rule, cohomology, Euclidean algorithm, fundamental group, Gauss’ Law, Grassmannian algebra , Graph Theory, Hahn-Banach Theorem, homology, Hairy Ball Theorem, Hölder’s inequality, inclusion-exclusion, Jordan Decomposition, Kalman Filters, Markov Chains (Hidden Markov Models), modules, non-associative algebras, Picard’s Great Theorem, Platonic/Euclidean solids, Principle of Induction, Probabilistic Graphical Models (Bayesian Networks, Markov Random Fields), Pontryagin duality, Quaternions, Spectral Theorem, Sylow p subgroup, repeating decimals equal a fraction, ring ideals, sine law, tensors, tessellation, transcendental numbers, Uniform Boundedness Theorem, Weierstrass approximation theorem. From http://artent.net/2012/11/27/100-most-useful-theorems-and-ideas-in-mathematics/ http://artent.net/2012/11/27/100-most-useful-theorems-and-id...