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stillpointlab 12 hours ago [-]
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.
_fizz_buzz_ 5 hours ago [-]
> 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.
prolly97 4 hours ago [-]
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.
moffkalast 3 hours ago [-]
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.
tgv 1 hours ago [-]
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.
AdamN 4 hours ago [-]
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.
UpsideDownRide 4 hours ago [-]
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 2 hours ago [-]
>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.
eps 2 hours ago [-]
The expression is "To know is to be able to explain".
It holds up very well in a lot of situations.
theteapot 2 hours ago [-]
> I might have found a place for logic and type theory.
Doesn't that fit under abstract algebra?
brokencipher 7 hours ago [-]
« Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisément » Boileau
igravious 2 hours ago [-]
Exactly; couldn't have put it better myself.
bbeonx 6 hours ago [-]
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".
esalman 13 hours ago [-]
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.
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.
not2b 5 hours ago [-]
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.
red_admiral 4 hours ago [-]
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_.
esalman 5 hours ago [-]
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.
neuroticnews25 3 hours ago [-]
Do you think all the providers on openrouter are burning money to provide a commodity?
SMEbooop 13 hours ago [-]
[flagged]
iTokio 14 hours ago [-]
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?
RossBencina 4 hours ago [-]
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.
iTokio 1 hours ago [-]
Thank you, I will watch these interviews!
kian 6 hours ago [-]
Read Polya's "How to Solve It" :)
Isamu 13 hours ago [-]
So… numbers, algebra, geometry, probability, analysis, dynamics are not the primitives or essentials ?
8 hours ago [-]
MrbroJangles 11 hours ago [-]
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 $anything; 1.5 can be half a pill and a whole pill or T or A; numbers are euphemism.
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
Geometry is addition or subtraction of shape
It's all built on 4 operations we see in daily life all the time
andsoitis 3 hours ago [-]
> 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.
binyu 11 hours ago [-]
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.
iTokio 6 hours ago [-]
It’s like principal component analysis, the main axis, I don’t mean reduction in a negative way.
alansaber 9 hours ago [-]
Isn't every summarization a reduction?
andsoitis 3 hours ago [-]
> Isn't every summarization a reduction?
Broadly yes, but summarization can simultaneously be compression of data and revelation of structure, hence increasing understanding.
rramadass 4 hours ago [-]
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
bbeonx 6 hours ago [-]
these might not be 100% complete, but i think this does a reaaalllly good job at capturing the vast majority of mathematics.
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).
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.
(edit: liebnitz -> leibniz)
SMEbooop 11 hours ago [-]
[dead]
thedreammachine 10 hours ago [-]
Related and a great read: "The Joy of X: A Guided Tour of Math, from One to Infinity" by Steven Strogatz:
The Joy of X and I think now renamed to “The Joy of y” is a great podcast for general knowledge building!
seanhunter 6 hours ago [-]
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.
[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 https://en.wikipedia.org/wiki/Riemann_series_theorem
[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.
tim-kt 52 minutes ago [-]
You can also think of conditional convergence as convergence under the condition of a specific order. It then turns out that unconditional 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).
jonplackett 3 hours ago [-]
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.
laybak 14 hours ago [-]
this is great foundational content! I also enjoyed his earlier appearance on 3Blue1Brown
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 disagree that such degree of non-linearity applies here. I think it is grossly simplified/underestimated.
hmokiguess 10 hours ago [-]
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
seanhunter 4 hours ago [-]
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. “
orphereus 5 hours ago [-]
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.
andsoitis 3 hours ago [-]
> 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?
11 hours ago [-]
11 hours ago [-]
bobajeff 13 hours ago [-]
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.
Tbarlow 7 hours ago [-]
Saw him explain Fourier transforms once; it finally clicked. Probably another masterclass in clarity.
marcy_74 8 hours ago [-]
His lectures are gold, always manage to unlock a new perspective. Wish he taught my undergrad courses!
matthewsinclair 15 hours ago [-]
Apologies for the typo. I’ve asked mods to fix it.
matthewsinclair 15 hours ago [-]
Looks like it’s been fixed. Thanks.
mc32 10 hours ago [-]
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.
> The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid.
> 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 logos)
We can cross-check dictionary entries. The standard dictionary of Ancient Greek fully backs this up:
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 only 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 dementialis than irrationalis.)
However, ratio is the conventional translation of the Greek logos, 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.
Agentus 14 hours ago [-]
i blinked and read terrence howard. i was like good god
stuxnet79 13 hours ago [-]
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]
It was expected that people without established credibility would be looked down on for using AI, but it's been weird to see people with all the credibility in the world lose it for embracing AI.
15 hours ago [-]
orf 15 hours ago [-]
What credibility has he lost?
femiagbabiaka 14 hours ago [-]
To sensible people? None. But the amount of criticism of his work that boils down to "he uses AI" is undeniable.
orf 14 hours ago [-]
Can you share some of the criticism? I’ve not seen any from anyone real.
pfdietz 14 hours ago [-]
It's weird to believe he's lost credibility.
nutjob2 14 hours ago [-]
Tao gives or takes credibility from AI, not the other way around.
oliculipolicula 9 hours ago [-]
Is Tao the rising angel? Or AI the falling ape?
--Another Terry, Perhaps
AI-avatar-of-Tao wielded by an internet rando lies to or gets fooled by Tao, not the other way around
--Me, attempting to take your Untouchable* joke all the way to a (sociopathic, one-party) joke
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.
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.
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.
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.
It holds up very well in a lot of situations.
Doesn't that fit under abstract algebra?
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".
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.
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?
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.
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 $anything; 1.5 can be half a pill and a whole pill or T or A; numbers are euphemism.
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
Geometry is addition or subtraction of shape
It's all built on 4 operations we see in daily life all the time
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.
Broadly yes, but summarization can simultaneously be compression of data and revelation of structure, hence increasing understanding.
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
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).
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.
(edit: liebnitz -> leibniz)
https://www.goodreads.com/en/book/show/13356649-the-joy-of-x
[1] https://youtu.be/OOMx2BHHWtE?is=M1lqZI2gxNqWqg6G&t=18m35s
[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 https://en.wikipedia.org/wiki/Riemann_series_theorem
[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.
https://www.youtube.com/watch?v=YdOXS_9_P4U
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.
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. “
Has someone done the calculation whether it would happen before the end of the universe?
https://www.etymonline.com/word/irrational
> The mathematical sense "inexpressible in ordinary numbers" is from late 14c. in English, from use of the Latin word as a translation of Greek alogon in Euclid.
https://www.etymonline.com/word/ratio
> 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 logos)
We can cross-check dictionary entries. The standard dictionary of Ancient Greek fully backs this up:
> λόγος
> II. 2 Math., ratio, proportion
The standard dictionary of Latin doesn't mention this particular sense. (A negative is harder to cite, but you can check it here: https://www.perseus.tufts.edu/hopper/text?doc=Perseus%3Atext... )
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 only 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 dementialis than irrationalis.)
However, ratio is the conventional translation of the Greek logos, 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.
[1] https://www.youtube.com/watch?v=1uLi1I3G2N4
Terence Tao.
--Another Terry, Perhaps
AI-avatar-of-Tao wielded by an internet rando lies to or gets fooled by Tao, not the other way around
--Me, attempting to take your Untouchable* joke all the way to a (sociopathic, one-party) joke
*https://archive.ph/2023.11.28-152809/https://www.astralcodex...