I'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.
> I've heard it said that true understanding is demonstrated when someone can explain difficult concepts well.
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.
Being good at a thing and explaining it well are different skills. Part of the problem is that you have to empathize with your audience when explaining. It's not sufficient to break a problem down into its essence; its "essence" varies depending on who you're talking to.
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.
I've experienced something similar, I still think the quote works -
If you can explain difficult concepts well, it's a demonstration of true understanding. Not that poor communication demonstrates lack of it.
Oh but I think it can demonstrate the lack of it. There's a distinct difference between learning by memory and learning by understanding, and the only indicator is being able to explain it in a novel way. After all you can also memorize someone else's explanation. In practice, being able to explain it is the only actual difference, if you can't then it's the same as not understanding it and the burden of proof is on the one claiming otherwise. That's the approach upon which the world's education is generally built.
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.
For some tasks, this is absolutely true. We can identify e.g. a dog in picture in 100ms or so, and nobody is capable of explaining how. We can understand and speak languages, without the slightest idea of how it works. Maths is of course not comparable to these cognitive functions, but people with high levels of expertise do have a lot of their knowledge "automated", and not open to introspection.
You can also understand the very simple basic essence of something, but get lost in the complexity when scaling up.
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.
My sense (I could be wrong) is that once that amazing engineer really understood the class of problems (i.e. he handled it multiple times) and was really comfortable that they would in fact be able to communicate plainly about the decisions and architecture.
Being a good communicator is a skill in itself and requires understanding not only of the problem, not only of yourself, not only of the language but also of other people. And not everyone is capable in all of those at the same time.
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.
I think there is lots of nuance between theory and application. Some people are really good teachers, and good teachers can give us mental models are abstract enough to get the gist of things.
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'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.
agreed! 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".
I 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.
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.
Even 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.
Exactly my thoughts. There are people out there claiming code itself has become disposable, and maintenance and refactoring are cheap now. Which escapes the fact that a. all these are heavily subsidized now and unsustainable in the long run and b. when it comes to critical software and code, it's not really disposable and maintaining a vibe-coded codebase is going to be more costly.
The essential concepts of mathematical reasoning, if there are such things, are the concern of mathematical logicians, or maybe even psychologists, not, in general, of working mathematicians. One of my professors once told me something to the effect of "if you think you are going to learn any of that here, you are in the wrong place."
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.
"maybe even psychologists" -- unironically true, but maybe I am talking about something different from you. The basis of (higher-level) math seem to mostly be "am I psychologically (emotionally...?) comfortable with accepting annoying ideas?" At least from my experience.
Imo 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
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.
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
these might not be 100% complete, but i think this does a reaaalllly good job at capturing the vast majority of mathematics.
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.
I 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.
[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.
You 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).
Very 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.
Could 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
He doesn’t say quadratic time, he says infinite time. He is talking about the probability of any given string arising from a random sequence of letters. As t goes to infinity that probability approaches 1.
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. “
Here's something to ponder. Assuming the Riemann Hypothesis is true, would that monkey type out a proof of the RH before it got to Hamlet? I guess it depends on if the proof can be expressed in fewer words than the word count of Hamlet.
To 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.
Been 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.
I 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.
Maybe 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.
> 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.
> 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:
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.
For 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]
It 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.
Here 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-id...
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.
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