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.
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.
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.
It holds up very well in a lot of situations.
Doesn't that fit under abstract algebra?
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".
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?
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.
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. “
Has someone done the calculation whether it would happen before the end of the universe?
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...
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.
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.
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
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
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)
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.
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.
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.