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Terence Tao explains 6 essential mathematical concepts [video]

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Re: Terence Tao explains 6 essential mathematical concepts [video]

#61

Post by moffkalast »

prolly97 wrote: post #54
_fizz_buzz_ wrote: post #52

Earlier quoted context omitted.

> 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.

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.

Re: Terence Tao explains 6 essential mathematical concepts [video]

#62

Post by andsoitis »

alansaber wrote: post #32
binyu wrote: post #25

Earlier quoted context omitted.

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.

Isn't every summarization a reduction?

> Isn't every summarization a reduction?

Broadly yes, but summarization can simultaneously be compression of data and revelation of structure, hence increasing understanding.

Re: Terence Tao explains 6 essential mathematical concepts [video]

#63

Post by andsoitis »

orphereus wrote: post #51
binyu wrote: post #23

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 disagre…

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.

> Obviously, it would take a long time, but it would happen at some point.

Has someone done the calculation whether it would happen before the end of the universe?

Re: Terence Tao explains 6 essential mathematical concepts [video]

#64

Post by neuroticnews25 »

esalman wrote: post #50
vismit2000 wrote: post #40

Earlier quoted context omitted.

I also loved this talk (went through printed version: https://news.ycombinator.com/item?id=49362728 ) 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 i…

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.

Do you think all the providers on openrouter are burning money to provide a commodity?

Re: Terence Tao explains 6 essential mathematical concepts [video]

#66

Post by igravious »

brokencipher wrote: post #42
stillpointlab wrote: post #22

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…

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

Exactly; couldn't have put it better myself.

Re: Terence Tao explains 6 essential mathematical concepts [video]

#67

Post by kgeist »

UpsideDownRide wrote: post #58
AdamN wrote: post #55

Earlier quoted context omitted.

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.

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.

Re: Terence Tao explains 6 essential mathematical concepts [video]

#68

Post by eps »

stillpointlab wrote: post #22

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…

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

It holds up very well in a lot of situations.

Re: Terence Tao explains 6 essential mathematical concepts [video]

#69

Post by theteapot »

stillpointlab wrote: post #22

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…

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

Doesn't that fit under abstract algebra?

Re: Terence Tao explains 6 essential mathematical concepts [video]

#70

Post by tgv »

moffkalast wrote: post #61
prolly97 wrote: post #54

Earlier quoted context omitted.

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…

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.
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