Terence Tao explains 6 essential mathematical concepts [video]

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 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.

[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.

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.

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

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?

You can pre order his book Six Math Essentials at https://a.co/d/0e89Jcjf

this is great foundational content! I also enjoyed his earlier appearance on 3Blue1Brown

https://www.youtube.com/watch?v=YdOXS_9_P4U

Saw him explain Fourier transforms once; it finally clicked. Probably another masterclass in clarity.

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.

His lectures are gold, always manage to unlock a new perspective. Wish he taught my undergrad courses!

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.

Apologies for the typo. I’ve asked mods to fix it.

wow, that's really nice

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

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At least spell his name correctly, my goodness...

Terence Tao.