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.<p>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.<p>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.<p>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'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.<p>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.
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.
>but also of other people.<p>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.
stillpointlab.com: "Get more from your LLM"<p>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.
The expression is "To know is to be able to explain".<p>It holds up very well in a lot of situations.
> I might have found a place for logic and type theory.<p>Doesn't that fit under abstract algebra?
« Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisément » Boileau
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.<p>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.
I also loved this talk (went through printed version: <a href="https://news.ycombinator.com/item?id=49362728">https://news.ycombinator.com/item?id=49362728</a>)<p>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.
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Numbers<p>Algebra<p>Geometry<p>Probability<p>Analysis<p>Dynamics<p>I loved this talk, but these concepts are like an attempt at dimensional reduction of math research, science, the academics knowledge.<p>I would have loved to have his thoughts on the mathematical mind, the process, how to reason, infer vs deduct, abstract, prove..<p>I don’t really know, what are the primitives, essential concepts of math reasoning?
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."<p>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.
Read Polya's "How to Solve It" :)
So… numbers, algebra, geometry, probability, analysis, dynamics are not the primitives or essentials ?
Imo it's not numbers at all but linked to our awareness of physical relationships<p>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<p>Which is why I never liked numbers before algebra which then chucks numbers in the bin more or less.<p>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.<p>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<p>Geometry is addition or subtraction of shape<p>It's all built on 4 operations we see in daily life all the time
> Numbers are just syntax<p>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.<p>> It’s all built on four operations<p>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.
Why do you feel this is a reduction? If anything, it is an attempt to summarize the various areas of math and how they relate to each other.
It’s like principal component analysis, the main axis, I don’t mean reduction in a negative way.
Isn't every summarization a reduction?
Related:<p>From a previous HN discussion of Terence Tao's <i>Six Math essentials</i> book (to be published) my comments pointing to a similar book by John Stillwell (covers Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic) - <a href="https://news.ycombinator.com/item?id=47116399">https://news.ycombinator.com/item?id=47116399</a>
these might not be 100% complete, but i think this does a reaaalllly good job at capturing the vast majority of mathematics.<p>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).<p>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.<p>(edit: liebnitz -> leibniz)
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Related and a great read: "The Joy of X: A Guided Tour of Math, from One to Infinity" by Steven Strogatz:<p><a href="https://www.goodreads.com/en/book/show/13356649-the-joy-of-x" rel="nofollow">https://www.goodreads.com/en/book/show/13356649-the-joy-of-x</a>
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.<p>[1] <a href="https://youtu.be/OOMx2BHHWtE?is=M1lqZI2gxNqWqg6G&t=18m35s" rel="nofollow">https://youtu.be/OOMx2BHHWtE?is=M1lqZI2gxNqWqg6G&t=18m35s</a><p>[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 <a href="https://en.wikipedia.org/wiki/Riemann_series_theorem" rel="nofollow">https://en.wikipedia.org/wiki/Riemann_series_theorem</a><p>[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 <i>under the condition</i> of a specific order. It then turns out that <i>unconditional</i> 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).
this is great foundational content! I also enjoyed his earlier appearance on 3Blue1Brown<p><a href="https://www.youtube.com/watch?v=YdOXS_9_P4U" rel="nofollow">https://www.youtube.com/watch?v=YdOXS_9_P4U</a>
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.
You can pre order his book Six Math Essentials at <a href="https://a.co/d/0e89Jcjf" rel="nofollow">https://a.co/d/0e89Jcjf</a>
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.<p>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.<p>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. “
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.
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.
Saw him explain Fourier transforms once; it finally clicked. Probably another masterclass in clarity.
His lectures are gold, always manage to unlock a new perspective. Wish he taught my undergrad courses!
Apologies for the typo. I’ve asked mods to fix it.
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.
Well...<p><a href="https://www.etymonline.com/word/irrational" rel="nofollow">https://www.etymonline.com/word/irrational</a><p>> The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek <i>alogon</i> in Euclid.<p><a href="https://www.etymonline.com/word/ratio" rel="nofollow">https://www.etymonline.com/word/ratio</a><p>> 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 <i>logos</i>)<p>We can cross-check dictionary entries. The standard dictionary of Ancient Greek fully backs this up:<p>> λόγος<p>> II. 2 <i>Math.</i>, ratio, proportion<p>The standard dictionary of Latin doesn't mention this particular sense. (A negative is harder to cite, but you can check it here: <a href="https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext%3A1999.04.0059%3Aentry%3Dratio" rel="nofollow">https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext...</a> )<p>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 <i>only</i> 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 <i>dementialis</i> than <i>irrationalis</i>.)<p>However, <i>ratio</i> is the conventional translation of the Greek <i>logos</i>, 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.
i blinked and read terrence howard. i was like good god
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]<p>[1] <a href="https://www.youtube.com/watch?v=1uLi1I3G2N4" rel="nofollow">https://www.youtube.com/watch?v=1uLi1I3G2N4</a>
At least spell his name correctly, my goodness...<p>Terence Tao.
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wow, that's really nice