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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]

#51

Post by orphereus »

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.

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

#52

Post by _fizz_buzz_ »

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'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.

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

#53

Post by seanhunter »

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…

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. “

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

#54

Post by prolly97 »

_fizz_buzz_ wrote: post #52
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'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.

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

#55

Post by AdamN »

_fizz_buzz_ wrote: post #52
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'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.

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.

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

#56

Post by red_admiral »

not2b wrote: post #49
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…

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 w…

I guess there's an exception for startups where you have to have something before you reach the end of the runway, and as you say, _mature code_.

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

#57

Post by rramadass »

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?

Related:

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

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

#58

Post by UpsideDownRide »

AdamN wrote: post #55
_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.

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.

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

#59

Post by RossBencina »

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 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.

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

#60

Post by andsoitis »

MrbroJangles wrote: post #28
Isamu wrote: post #18

Earlier quoted context omitted.

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 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…

> Numbers are just syntax

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.

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