I love the fact he mentioned the Reimann rearrangement theorem 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.
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.
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.
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
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.
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
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.
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.
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]
I love the fact he mentioned the Reimann rearrangement theorem 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.
https://en.wikipedia.org/wiki/Riemann_series_theorem
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.
« Ce qui se conçoit bien s'énonce clairement, et les mots pour le dire arrivent aisément » Boileau
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.
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 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.
Related and a great read: "The Joy of X: A Guided Tour of Math, from One to Infinity" by Steven Strogatz:
https://www.goodreads.com/en/book/show/13356649-the-joy-of-x
The Joy of X and I think now renamed to “The Joy of y” is a great podcast for general knowledge building!
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?
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 $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
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?
Saw him explain Fourier transforms once; it finally clicked. Probably another masterclass in clarity.
You can pre order his book Six Math Essentials at https://a.co/d/0e89Jcjf
His lectures are gold, always manage to unlock a new perspective. Wish he taught my undergrad courses!
this is great foundational content! I also enjoyed his earlier appearance on 3Blue1Brown
https://www.youtube.com/watch?v=YdOXS_9_P4U
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.
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
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.
wow, that's really nice
Apologies for the typo. I’ve asked mods to fix it.
Looks like it’s been fixed. Thanks.
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.
i blinked and read terrence howard. i was like good god
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]
[1] https://www.youtube.com/watch?v=1uLi1I3G2N4
At least spell his name correctly, my goodness...
Terence Tao.
Can't believe we missed that. Fixed now. Thanks!