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.
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."
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."
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...)
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.
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...
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 !
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.
Genuinely, what is it that you get from studying mathematics?
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.
For me, math was a way to study structure. I find this sort of thing tremendously beautiful on its own, but as it happens "finding the structure in things" turns out to be quite lucrative in the professional world as well, and I often use various ideas and strategies I chanced upon as a student of mathematics.
I do, yes. I won't call it a hobby because I don’t create anything, I'm just a mindless rabid stupid cunt of a consoomer who doesn't know how to differentiate his ass from a hole on the fucking ground, but I do spend a lot of time listening to music. I've spent a lot of money on audio equipment.
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.
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.
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.
Absolutely! Any person willing to study and think can understand the above books. As i mentioned, they cover a broad swath of mathematics and are meant for the general reader. You can checkout reviews on Amazon and elsewhere on the web.
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/
Six Math Essentials as a title reminds me of Six Easy Pieces. I wonder if that's intentional.
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)
Of possible interest
https://mirtitles.org/2024/05/11/little-mathematics-library-...
Sorry for the stupid question but is Elliptic Tales your favorite or is it Summing it up?
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.
From the brief description, this sounds to be quite basic. Looking forward to hearing if Terence has treated the explanations differently. :-)
Basic … that kind of word give me nightmare in my mind when you talked about maths … still remember a book called “elementary set theory” …
My favorite example is https://en.wikipedia.org/wiki/Basic_Number_Theory
I remember a joke along the lines of "elementary" meaning that someone somewhere has solved it before.
> 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."
I just really liked that question and response.
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."
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...)
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.
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.
Which ebook provider should I use to get an actual epub file?
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.
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.
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
Is this Terry's 17th book?
How exciting!
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.
Genuinely, what is it that you get from studying mathematics?
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.
For me, math was a way to study structure. I find this sort of thing tremendously beautiful on its own, but as it happens "finding the structure in things" turns out to be quite lucrative in the professional world as well, and I often use various ideas and strategies I chanced upon as a student of mathematics.
I see. I suppose it makes sense if you're in a career position that allows you to freely explore the world.
Do you listen to music ?
I do, yes. I won't call it a hobby because I don’t create anything, I'm just a mindless rabid stupid cunt of a consoomer who doesn't know how to differentiate his ass from a hole on the fucking ground, but I do spend a lot of time listening to music. I've spent a lot of money on audio equipment.
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.
Sounds a damn good hobby to me.
I don't think hobby requires building anything. Spending time actively engaged is enough.
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.
You might find the couple of books that i mention in my other comments here useful;
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.
Could a clever idiot understand such books? If so, I might be willing to check them out. Thank you for the recommendations either way.
Absolutely! Any person willing to study and think can understand the above books. As i mentioned, they cover a broad swath of mathematics and are meant for the general reader. You can checkout reviews on Amazon and elsewhere on the web.
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/