Six Math Essentials
terrytao.wordpress.com
terrytao.wordpress.com
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.
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.
See, for example, the book "Mathematica" by David Bessis, or this blog post: https://davidbessis.substack.com/p/thinking-fast-slow-and-su...
https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...
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
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.
It will be interesting to see if Tao's writings are as clear, though possibly he is targetting a different audience.
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
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)
https://mirtitles.org/2024/05/11/little-mathematics-library-...
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.
G.H. Hardy and E. M. Wright "An Introduction to the Theory of Numbers"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."
>"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."
"I am so clever that sometimes I don't understand a single word of what I am saying".
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...)
Hopefully we shall get a Feynman type math book from a true Master.
So I'll be downloading this one from Anna and save even more money. I'm poor :(
But for a book intended for a popular audience, it sure does have a bore-you-to-death cover.
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.
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.
One book that I can't recommend more highly is William Dunham's "Journey Through Genius". He picks ten or so the greatest proofs in math over the centuries, then proceeds to give you all the historical background about why they were created, who created them, etc, and then proceeds to give the full details of the proofs themselves. Including showing places the proof is considered wrong by modern standards.
It's my favorite "semi-pop-math" book, I highly recommend it.
[1] https://www.penguin.co.uk/books/482167/six-maths-essentials-...
I am atrocious at mathematics and held much contempt for the field until I was in college and 'saw the light,' if you will. Since college, I have absolutely fallen in love with mathematics. I learned it was not mathematics I always hated, but the U.S. public education system's method of teaching mathematics.
While I am still quite weak in the matter, I do believe that I will be preordering a copy of this book. Thank you for sharing this.
Concepts of Modern Mathematics by Ian Stewart - https://store.doverpublications.com/products/9780486284248
Elements of Mathematics: From Euclid to Gödel by John Stillwell - https://press.princeton.edu/books/hardcover/9780691171685/el...
Both of them give a nice tour of various domains within modern mathematics and their inter-relationships which is what i believe is most important to understand for a general reader.
Mathematics can be approached in two ways; 1) For understanding 2) For techniques of usage.
The above books help with (1). Textbooks focus on (2). A very good succinct (< 150 pages!) introductory text for (2) is George Simmons' Precalculus Mathematics in a Nutshell: Geometry, Algebra, Trigonometry. It is available at https://github.com/enilsen16/The-Math-Group
One word of advice; most people's phobia of mathematics arises from not knowing/understanding the notation. It is just a shorthand language which you need to get familiar with. When you come across a formula, just expand and read it out aloud in your own version of easy English. You will understand better and lose your fear of mathematics. A book like Mathematical Notation: A Guide for Engineers and Scientists by Edward Scheinerman is of great help here. There are of course lots of free resources for this on the web starting with https://en.wikipedia.org/wiki/Glossary_of_mathematical_symbo... and https://mathvault.ca/hub/higher-math/math-symbols/
I think I want a mix of both options you listed. Though, I suppose techniques of usage could perhaps be derived if one's understanding is sufficient.
> most people's phobia of mathematics arises from not knowing/understanding the notation.
That does describe me at times. I have found Wikipedia to be a poor resource on math topics because it tends to get quite complex rather quickly. For example, take a look at this section on Wikipedia:
https://en.wikipedia.org/wiki/Bloom_filter#Probability_of_fa...
I can understand everything until the series notion starts. From that point on, I have no idea what any of that means.
I am going to pick of those books though. My interest is piqued.
Also, do you happen to have any recommendations for 'Proofs for Idiots?' I have realized that I like starting with proof's to some degree and working up the stack. I got in this habit when I tried to work through the "Algorthim Design Manual" by Skiena. The book began with trying to prove various algorithms, and I found that to be a great way to approach the topic. It was like the missing part of my Data Structures and Algorithms course in College.
You have to understand that a lot of my math education was taught by people that probably did not understand the topics well themselves. I grew up with a lot of questions I had in math being answered with, "Well, that's just the way things are" or "because the formula states <insert whatever>." College changed my opinion because I had a professors who started to get me to ask the questions of "Why?" For me, that was always the missing piece of enjoyment.
Also Mathematics should be approached from multiple perspectives including (but not limited to) Imagination, Conceptual, Graphical, Symbolic, Applications, Modeling, Sets/Relations/Definition/Theorem/Proof.
As i mentioned, for studying Mathematics you need both 1) Overview/General books which help in building interdisciplinary intuition/insight and 2) Textbooks which teach methods/tools and put them to use in solving real-world problems. You will find plenty of recommendations for textbooks both on HN and elsewhere on the web (especially college/university websites).
One excellent must-have book that straddles both is; Mathematics: Its Content, Methods and Meaning by Aleksandrov/Kolmogorov/Lavrent'ev. It covers almost all domains of mathematics until the early 20th century in an introductory succinct form up to undergraduate level. You can then look for individual textbooks for each of the domains given there. See the ToC at - https://store.doverpublications.com/products/9780486157870
W.r.t. books on Proofs, my first suggestion would be to not focus too much on the formal mechanics of it but try to understand the reasoning/logic behind it in a informal way i.e. the "proof idea" problem-solving process.
The first book to read here is George Polya's classic; How To Solve It. It gives a problem-solving process with heuristics and thumbrules which is the prerequisite to mathematical formalization - https://en.wikipedia.org/wiki/How_to_Solve_It
With the problem-solving process in hand, you can now get a gentle introduction to Logic/Discrete Maths leading to Proofs. One very accessible book with broad coverage and a bent towards CS is Nimal Nissanke's Introductory Logic and Sets for Computer Scientists. The author wrote it as the needed background mathematics for formal methods and hence contains everything (including doing Proofs) within one pair of covers - https://www.amazon.com/Introductory-Computer-Scientists-Inte...
I highly recommend getting all of the above before looking for more. Given your background (as i understood it) i think this would be the best and easiest path.
I get that it's a hobby, but what do you even do with the knowledge you acquire?
I don't exactly fear math (even though I'm complete shit at it) but the time investment required is absolutely massive for something with questionable utility, even just for playing around with. You need a super strong base to even attempt bashing basic problems, so that's easily four or five years of study just to play around a bit.
the OEIS is replete with old sequences that no-one has considered in much detail in a decade or two, and have a lot of 'low-hanging fruit' for one willing to toy with them.
https://oeis.org/A185105 is a good example of such a sequence; "sample the elements of a random permutation of [n] in a random order and record each one's cycle (under repeated iteration), then T(n,k)/n! is the expected of the kth distinct cycle recorded," which seems like it would have been of some interest to someone in the last ≈13 years (since ie. it's well-known that the first cycle's length is uniform in [1..n]), but didn't receive any formulas until I happened upon it recently with my own toolbelt (which is quite modest and certainly could be learned in less than 4 years).
the OEIS is an excellent resource for both readinh and sharpening one's amateur teeth on novel (ie. unexplored, or at least undocumented) problems and very rewarding, if that's your goal with learninh maths
GP here, I would say that I gain understanding. I know that might seem vague, but that is the truth. For example, while not technically traditional math, I have been trying to brush up on stats a bit. I like to read research journals about health, psychology, etc.. I want to be able to make my own inferences about the journals I read with an informed opinion.
Even so, if you wanted to bring up time signatures, microtonality or something like math rock… I'm aware of those, but I still think the only thing that matters is that they're tools meant to allow you to express a certain message in the most appropriate ways, not so much an end in themselves.
I don't think hobby requires building anything. Spending time actively engaged is enough. One can enjoy mathematics the way one enjoys listening to music.
On the other hand if you do want to make something, and you happen to know related math then suddenly you can use it.
For example, https://news.ycombinator.com/item?id=47112418
Building these are neither my hobby, not did I learn the relevant math for the exclusive purpose of making it. But once you acquire a few math razors you start seeing inviting fluffy yaks that were invisible before.
This is a great domain to motivate oneself to delve deeper into Mathematics. For example;
1) What parameters do you look at in audio equipment before you buy?
2) Somebody is trying to sell you "Hi-Res" music and equipment; Are they worth the money? Why? Why Not?
All of the above need mathematics to comprehend at even a basic level. There are both complicated objective (physics/mathematics) and subjective (our auditory system) parameters to understand eg. logarithms, harmonic series, frequency modulation, tuning, impedance, human hearing frequency range and sensitivity etc.
Having some mathematical idea of the above not only saves you money but also helps you enjoy music "optimally".
References:
Sound: A Very Short Introduction by Mike Goldsmith (also see his other related book on Waves) - https://global.oup.com/academic/product/sound-9780198708445?...
The Science of Musical Sound by John R. Pierce. An old classic (also checkout his other books on Waves, Signals and Information Theory). They are all written in a semi-technical and clear manner for the general audience. - https://en.wikipedia.org/wiki/John_R._Pierce