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boerseth
> ... six of the fundamental concepts in mathematics ... and how they connect with our real-world intuition
While the connections are interesting, I would be as interested in the disconnects, as there's a bunch of cases where our human intuitions can fail us in subtle ways. This is actually one of the lessons I treasure from mathematics: it has helped me grow a healthy set of alarm bells for those unintuitive cases. Especially for probability and statistics.
rramadass
> Especially for probability and statistics.
For the general reader, two books by David Spiegelhalter (https://en.wikipedia.org/wiki/David_Spiegelhalter) are relevant here;
1) The Art of Uncertainty: How to Navigate Chance, Ignorance, Risk and Luck.
2) The Art of Statistics: Learning from Data.
And of course, all the books by Nassim Taleb.
leethargo
There's also the argument that human intuition can and should be trained, rather than just dismissing it, in particular when it comes to mathematics.
See, for example, the book "Mathematica" by David Bessis, or this blog post: https://davidbessis.substack.com/p/thinking-fast-slow-and-su...
socalgal2
Interview with the author about that book
https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...
rramadass
Nice.
There is also Intuition in Science and Mathematics: An Educational Approach by Efraim Fischbein - https://link.springer.com/book/10.1007/0-306-47237-6
rramadass
Seems similar to John Stillwell's classic Elements of Mathematics: From Euclid to Gödel - https://press.princeton.edu/books/hardcover/9780691171685/el...
It has one chapter each for Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.
He positioned it as a sort of a modern update to Felix Klein's Elementary Mathematics from an Advanced Standpoint series of books.
From the preface;
This book grew from an article I wrote in 2008 for the centenary of Felix Klein’s Elementary Mathematics from an Advanced Standpoint. The article reflected on Klein’s view of elementary mathematics, which I found to be surprisingly modern, and made some comments on how his view might change in the light of today’s mathematics. With further reflection I realized that a discussion of elementary mathematics today should include not only some topics that are elementary from the twenty-first-century viewpoint, but also a more precise explanation of the term “elementary” than was possible in Klein’s day.
So, the first goal of the book is to give a bird’s eye view of elementary mathematics and its treasures. This view will sometimes be “from an advanced standpoint,” but nevertheless as elementary as possible. Readers with a good high school training in mathematics should be able to understand most of the book, though no doubt everyone will experience some difficulties, due to the wide range of topics...
The second goal of the book is to explain what “elementary” means, or at least to explain why certain pieces of mathematics seem to be “more elementary” than others. It might be thought that the concept of “elementary” changes continually as mathematics advances. Indeed, some topics now considered part of elementary mathematics are there because some great advance made them elementary...
Note: "Elementary" here does not mean Easy.
ColinWright
I find Stillwell's writings to be exceptionally clear and accessible, and I recommend them.
It will be interesting to see if Tao's writings are as clear, though possibly he is targetting a different audience.
rramadass
From Book Details;
a brief tour of six core ideas—numbers, algebra, geometry, probability, analysis, and dynamics—that capture the beauty and power of mathematical thinking for everyone.
In Six Math Essentials, the renowned mathematician and Fields Medalist Terence Tao introduces readers to six central concepts that have guided mathematicians from antiquity to the frontiers of what we know today and now help us make sense of our complex world. This slim, elegant volume explores
numbers as the gateway to quantitative thinking;
algebra as the gateway to abstraction;
geometry as a way to calculate beyond what we can see;
probability as a tool to navigate uncertainty with rigorous thinking;
analysis as a means to tame the very large or the very small; and
dynamics as the mathematics of change.
Six Math Essentials—Tao’s first popular math book
Terence Tao's comment :- This book is for a general audience, without necessarily having a college-level math education. It is aimed more at adults than at children, but some children with an interest in mathematics may be able to get something out of it.
It is just 160 pages so must be information dense with no fluff. I am sold !
PS: Another book in the same (but easier) vein would be Ian Stewart's classic Concepts of Modern Mathematics - https://store.doverpublications.com/products/9780486284248
gtani
Stillwell's books are very good, as are Courant/Robbins What is Math and Ian Stewart's several books(one with David Tall as collaborator). My dad gifted me What is Math in grade school and i return to it every couple of years.
nhatcher
I'm sure it's a great book :).
I find good popular books on higher mathematics difficult to come by. A nice exception is the trilogy written by Avner Ash and Robert Groß:
Elliptic Tales, Fearless Symmetry and Summing it up (in my order of preference)
the-mitr
Of possible interest
https://mirtitles.org/2024/05/11/little-mathematics-library-...
Guestmodinfo
Mir titles is a feeling. It's nostalgia. It's childhood for many including me. The books are excellent written by some of the finest of their times talking to the lowest highschool level student or even a child and making him or her understand fully what all is going on. Indians love Mir.
My favorite author is Landsberg. He is in Mir titles. He got defeated by our main man C V Raman by 2 weeks to publish the same research (independently) which got C V Raman the only Physics Nobel Prize for India.
digital55
Terence Tao: Just a brief announcement that I have been working with Quanta Books to publish a short book in popular mathematics entitled “Six Math Essentials“, which will cover six of the fundamental concepts in mathematics.
alok-g
From the brief description, this sounds to be quite basic. Looking forward to hearing if Terence has treated the explanations differently. :-)
ngcc_hk
Basic … that kind of word give me nightmare in my mind when you talked about maths … still remember a book called “elementary set theory” …
esafak
My favorite example is https://en.wikipedia.org/wiki/Basic_Number_Theory
raegis
The one math majors joke about is Serra’s A Course in Arithmetic, which is definitely not for young children.
ecshafer
I remember a joke along the lines of "elementary" meaning that someone somewhere has solved it before.
hirvi74
> this sounds to be quite basic.
It should be according to Tao's own comment at the bottom of the blog:
"This book is for a general audience, without necessarily having a college-level math education. It is aimed more at adults than at children, but some children with an interest in mathematics may be able to get something of it."
travisjungroth
I just really liked that question and response.
peter_d_sherman
This is a highly interesting comment from user "thoughtfullyd4c9a86b93" on the above site:
>"My two cents worth — Logic is fundamental. Most of mathematics does not treat infinities nor singularities as first class citizens. Yet, there are a lot of problem classes in which you can actually reason with a set that includes those limits. My preference is a strict axiomatic hierarchy where you can not blend “levels”. Each level is a gatekeeper for the next tier.
The idea that mathematics is a language of its own does not work until you completely disambiguate mathematics in your language of choice — and logic is a language that facilitates complete understanding.
* ⟨T⟩0: ZFC (The Material). The box that contains the idea of a box. If you aren’t starting here, you aren’t even playing the game.
* ⟨T⟩1: Topology (The Stage). This defines “nearness” before you own a ruler. It’s the rubber floor where a donut is a coffee cup—and where singularities (like the zero-point or the absolute) are perfectly admissible inhabitants.
* ⟨T⟩2: Geometry (The Ruler). Adding distance and angles. It’s just Topology after it’s been forced to commit to a specific measurement.
* ⟨T⟩3: Algebra (The Syntax). The ledger for people who trust the ruler more than the stage. It tracks the symmetries the geometry allows.
* ⟨T⟩4: Analysis (The Measure). Measuring the vibration of a string to prove the violin is real. Great for change, but “blind” to the structural admissibility of the stage itself."
rramadass
Reminds me of this quote by Oscar Wilde;
"I am so clever that sometimes I don't understand a single word of what I am saying".
peter_d_sherman
Here is what Grok says about the above (I asked it to explain it better):
Grok:
"⟨T⟩0: ZFC (The Material) — Zermelo–Fraenkel set theory with the axiom of choice (the standard foundation for most modern mathematics). Called "The Material" and metaphorically "the box that contains the idea of a box," highlighting how ZFC provides the basic "stuff" (sets) out of which everything else is built. Without this, "you aren’t even playing the game."
⟨T⟩1: Topology (The Stage) — Introduces the primitive notion of "nearness" or continuity without any rigid measurement (no distances or angles yet). Famously, topology is "rubber-sheet geometry," where continuous deformations are allowed, so a donut and a coffee mug are equivalent (both have one hole/handle). Singularities/infinities (e.g., zero-point in physics or the point at infinity in projective geometry) can exist naturally here without causing foundational issues.
⟨T⟩2: Geometry (The Ruler) — Builds on topology by adding concrete measurements (distances, angles, metrics). It's topology "forced to commit" to specifics.
⟨T⟩3: Algebra (The Syntax) — Focuses on symmetries and structures (groups, rings, fields, etc.) that geometry permits. It's more abstract and rule-based ("the ledger" tracking allowed operations).
⟨T⟩4: Analysis (The Measure) — Deals with limits, continuity, change, integration/differentiation, etc. ("measuring the vibration of a string"). It's powerful for dynamics but "blind" to deeper structural issues in the underlying topology or sets.
(Or, phrased another way, it's one set of possibilities for a "Math/Mathematics Stack" (AKA "Abstraction Hierarchy", "Math Abstraction Hierarchy") built level by level, on top of the foundation of Logic...)
max_
Looks like an inspiration from Richard Feynman's "Six Easy Pieces"
Hopefully we shall get a Feynman type math book from a true Master.
Agentlien
Six Math Essentials as a title reminds me of Six Easy Pieces. I wonder if that's intentional.
tosti
I'm not a math expert, but if I want to pre-order the book I can save money and dead trees on the eBook. It's half the price but comes with DRM. I'm not selling my soul and decrypting a book is quite a math problem.
So I'll be downloading this one from Anna and save even more money. I'm poor :(
jawns
I greatly admire Tao's work.
But for a book intended for a popular audience, it sure does have a bore-you-to-death cover.
plaguuuuuu
I don't think a popular audience is buying a book on mathematics.
But, the world is huge. Even if this is kind of niche (people who didn't really get into maths in school or college, but now have a strange impulse to pick it up for shits and giggles) the audience is still thousands of people. Or just, people who want to see how Tao connects everything up, because the way he sees and explains stuff is amazing.
There are levels to what's worth publishing or working on in general. Hardly anyone is going to be the next Steven Hawking but this obsession with the most popular or successful celebrity creators ultimately leads to this highly homogenised global media landscape. The most exciting thing about the internet for me was always accessing the long tail of truly unusual shit that you wouldn't find in book/record stores, tv, etc.
tgv
Thousands? You might be surprised. The Order of Time by Rovelli sold 1 million copies. Hawking sold 10 million. I think 100k for Tao is feasible.
ekjhgkejhgk
I have a PhD in physics and I read Hawking's book as a child.
You just got me to realize that while I've read many physics popular books that have been "simplified enough that the common person can get something out of them, but not so much that they become meaningless", maths books that achieve the same are much rarer, I think.
wmwragg
The Pelican UK version[1] looks a lot nicer
[1] https://www.penguin.co.uk/books/482167/six-maths-essentials-...
rramadass
Don't judge a book by its cover ?
kleiba
I kinda like the cover, but maybe I'm just a boring person myself.
gessha
Jeremy Kun's A Programmer's Introduction to Mathematics is also a good one.
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This is an interesting profile of Terence Tao as an ~8 year old: https://gwern.net/doc/iq/high/smpy/1984-clements.pdf, written by somebody who seems to have been well-versed in working with mathematically precocious children. It's interesting less for how good at math Tao already is but a peek into how he went about learning and doing math at a time when the subject matter was still accessible to, say, most users here. Among other things, what might be called his "openness to math experience" and independence are both remarkable.