Terence Tao explains 6 essential mathematical concepts [video]
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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.
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".
I think this not generally. I worked together with this amazing engineer, but he really struggled to sometimes explain what he was trying to do. He came up with great solutions, but often took us some time to figure out what he was trying to get at.
You can do things extremely well without having the foggiest about the actual underlying principles, just from observations and intuition. Doubly so if the process can be machine automated, which by this point encompasses nearly everything to some extent. Sufficiently advanced overfitting is indistinguishable from generalization.
Binary is very simple, but scaled up: look what we've created with software.
When it comes to explanation: pulling from rote memory, requires someone to attempt to hold all the short-term details in mind.
There are biological limitations to how well we can do this, but we can also exercise our brains to improve this ability.
But when something is deeply learned, in long-term memory, the effort of recall is much less than rote memory of short-term details. Our context window is limited, fills up, and we must recover. When you're remembering long-term details, context seems easier to swap in and out (sorry to sound like an LLM, but they do simulate thinking).
Whether or not someone is a master of any given domain of knowledge comes from demonstration. Maybe that is teaching the essence of a subject in a way that demonstrates you can visualize and move around the subject with ease. Or maybe you can create something very useful, or tasteful.
We accept that you have spent time in this area and probably can revral truth to us. You are credible.
If you can't demonstrate mastery through teaching, exchanging ideas to bring me closer to your level: them other forms of credentials are sought: like how well they code, or how useful their products become.
But life isn't about usefulness and will just lead to unhappiness. Just be the best version of yourself you can be. Life is too much to understand all at once.
Up to the limits of the Goedels incompletness theorem.
Dependenting on the used notation the formal proof can be very long. For example, Principia Mathematica took about 300 pages to prove that 1 + 1 = 2.
https://commonplacefacts.com/2022/07/27/principia-mathematic...
Yeah, there's this thing called the curse of knowledge. If an engineer has a deep understanding of something, it's not a given that they can explain it well. For them, the topic feels so simple, and they've done it so many times that they may have forgotten other people aren't as knowledgeable. They will throw terms around without explaining them, etc.
An example from something I've had to iterate on: When explaining an event loop multiprocessing runtime sort of thing, I eventually found I had to hand-wave "and your CPU hates that" to establish an appropriate premise to the problem and solution (referring to item-by-item dynamic dispatch with a large number of task types as the specific demon which needed to be slain while discussing that subset of the design). People in the know didn't need more understanding. People not in the know were happy to brush their lack of microarchitectural understanding under the rug. With that premise, both crowds were able to understand what followed.
That wasn't my first attempt. I have a bad habit of trying to explain those missing details as well, especially when it's clear the listener doesn't know them yet -- trying to get them into a position where they could've built the thing themselves -- but that only lands well with like 1-5% of people I've met.
Critically, agreeing with you, that's a communication failure, not an engineering failure. I understood the problem just as well in both cases; I just didn't understand the full extent of the people problem.
It holds up very well in a lot of situations.
Doesn't that fit under abstract algebra?
I'm sure that an admittedly great mathematician who is sponsored by the "AI for Math" fund and math.inc (which literally wants to corporatize mathematics!) is very appealing to LLM startups.
Roger Federer would say he never knew what kind of grip he used on his shots(which is one of the first things one learns as a beginner), and I think Roger might not be an elite coach, because so much of his greatness may have come from a very intuitive understanding of tennis. (Would I still take him as my coach, heck yeah).
I think some people have really intuitive understanding of their subjects and can express that understanding in amazing applications, but they lack the communication skills, patience, or language to properly pass on the knowledge to others.
I remember watching Tao's video on the IOI (or math olympiad, I don't recall), and I couldn't understand anything he said :D
But maybe I'm just not his target audience :shrug:
Not a single sentence conveys any knowledge to me. I have a theoretical physics degree so I am not afraid of math but still.
Sometimes even Fields Medalists are not sure if a complex theorem about a complex mathematical object is 100% correct.
From Peter Scholze:
"— I spent much of 2019 obsessed with the proof of this theorem, almost getting crazy over it. In the end, we were able to get an argument pinned down on paper, but I think nobody else has dared to look at the details of this, and so I still have some small lingering doubts."
https://xenaproject.wordpress.com/2020/12/05/liquid-tensor-e...
Vladimir Voevodsky started research program centered on formalizing Homotopy theory, because he feared possible bugs in his proofs.
"This story got me scared. Starting from 1993, multiple groups of mathematicians studied my paper at seminars and used it in their work and none of them noticed the mistake. And it clearly was not an accident. A technical argument by a trusted author, which is hard to check and looks similar to arguments known to be correct, is hardly ever checked in detail.
But this is not the only problem that allows mistakes in mathematical texts to persist. In October 1998, Carlos Simpson submitted to the arXiv preprint server a paper called “Homotopy Types of Strict 3-groupoids.” It claimed to provide an argument that implied that the main result of the “∞-groupoids” paper, which Kapranov and I had published in 1989, cannot be true. However, Kapranov and I had considered a similar critique ourselves and had convinced each other that it did not apply. I was sure that we were right until the fall of 2013 (!!)."
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.
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?
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
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.
Broadly yes, but summarization can simultaneously be compression of data and revelation of structure, hence increasing understanding.
i'm sure you could come up with another breakdown, but these also have the benefit of tracking roughly with history. numbers and geometry (euclid), then eventually algebra. probability was a fundamentally new way of looking at the world. analysis comes via newton/leibniz and then dynamics tries to tackle complex systems (kinda where newton left off, e.g., 3-body problem type stuff).
also: dimension reduction is not a bad thing. this is like a decomposition: we can decompose so much of what research mathematics is doing into 6 different basis elements. that's pretty darn neat.
(edit: liebnitz -> leibniz)
From a previous HN discussion of Terence Tao's Six Math essentials book (to be published) my comments pointing to a similar book by John Stillwell (covers Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic) - https://news.ycombinator.com/item?id=47116399
If you are interested in Terence Tao's personal mathematical inner world, he touches on that during his interview with Lex Friedman, which I think you might find interesting. Interviews with Kevin Buzzard sometimes touch on these themes too.
Disclaimer: I built this tool a couple of weeks back.
https://www.goodreads.com/en/book/show/13356649-the-joy-of-x
[1] 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
[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.
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.
Has someone done the calculation whether it would happen before the end of the universe?
The monkeys don’t understand hamlet. They just bash enough keys that it appears by chance (eventually). Hence “‘It was the best of times, it was the blurst of times…’ Stupid monkey. “
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
> 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... )
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.
Terence Tao.
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-id...