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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]

#81

Post by winwang »

RossBencina wrote: post #59
iTokio wrote: post #16

Numbers 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?

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 touche…

"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.

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

#82

Post by kokon »

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 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:

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

#83

Post by killerstorm »

iTokio wrote: post #16

Numbers 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?

This video actually explains math reasoning on a high level. Have you actually watched it?

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

#84

Post by ambicapter »

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…

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.

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

#85

Post by kaffekaka »

kokon wrote: post #82
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 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:

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.

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

#86

Post by kaffekaka »

eps wrote: post #68
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.

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

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

#87

Post by Terr_ »

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.

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

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

#88

Post by leonidasrup »

tgv wrote: post #70
moffkalast wrote: post #61

Earlier quoted context omitted.

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.

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...

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

#89

Post by leonidasrup »

kaffekaka wrote: post #85
kokon wrote: post #82

Earlier quoted context omitted.

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:

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.

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

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

#90

Post by leonidasrup »

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…

"The infinite monkey theorem states that a monkey hitting keys independently and at random on a typewriter keyboard for an infinite amount of time will almost surely type any given text, including the complete works of William Shakespeare."

https://en.wikipedia.org/wiki/Infinite_monkey_theorem

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