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.

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_fizz_buzz_ 18 hours ago | parent | next [-]

> 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.

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hansvm 9 hours ago | parent | next [-]

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.

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prolly97 18 hours ago | parent | prev | next [-]

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.

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Terr_ 5 hours ago | parent | next [-]

To reuse a useful phrase, understanding a topic is "necessary but not sufficient" for explaining.

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moffkalast 17 hours ago | parent | prev [-]

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.

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tgv 15 hours ago | parent | next [-]

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.

	▲	itsalwaysgood 12 hours ago \| parent \| next [-]
		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.
	▲	leonidasrup 4 hours ago \| parent \| prev [-]
		Expert mathematicians may not be able to explain how they find a proof of a difficult mathematical statement, but once a correct proof is found, it can be, with sufficient work, be formalized (most mathematicians don't do this part). This formal proof can be mechanicaly checked, without any creativity, step-by-step according to the axioms and inference rules of a mathematical logic. 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...

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ambicapter 7 hours ago | parent | prev [-]

It doesn't matter if you can explain it in a novel way if your novel way doesn't correspond to your listener's ability. So it is possible they can explain it in a different way but still don't know how to explain it in a way that their listener can gain understanding from it.

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AdamN 18 hours ago | parent | prev [-]

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.

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UpsideDownRide 17 hours ago | parent [-]

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.

	▲	kgeist 16 hours ago \| parent [-]
		>but also of other people. 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.

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brokencipher 20 hours ago | parent | prev | next [-]

« Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisément » Boileau

	▲	igravious 16 hours ago \| parent [-]
		Exactly; couldn't have put it better myself.

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mym1990 11 hours ago | parent | prev | next [-]

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.

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theteapot 16 hours ago | parent | prev | next [-]

> I might have found a place for logic and type theory.

Doesn't that fit under abstract algebra?

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kokon 8 hours ago | parent | prev | next [-]

I don't know about that. Some concepts are just genuinely hard to understand.

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:

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kaffekaka 7 hours ago | parent [-]

In math (and other subjects too of course) many things simply cannot be understood without deep prior knowledge. I came across the Wikipedia page for the modularity theorem a while ago and it reads truly satirical to me: https://en.wikipedia.org/wiki/Modularity_theorem

Not a single sentence conveys any knowledge to me. I have a theoretical physics degree so I am not afraid of math but still.

	▲	leonidasrup 4 hours ago \| parent [-]
		It took many years for Andrew Wiles and Richard Taylor to prove the modularity theorem for semistable elliptic curves. 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 (!!)." https://www.ias.edu/ideas/2014/voevodsky-origins

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eps 16 hours ago | parent | prev | next [-]

The expression is "To know is to be able to explain".

It holds up very well in a lot of situations.

	▲	kaffekaka 7 hours ago \| parent [-]
		Not 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

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Zsfe510asG 13 hours ago | parent | prev | next [-]

stillpointlab.com: "Get more from your LLM"

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.

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bbeonx 19 hours ago | parent | prev [-]

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".