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.
> ... 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.
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 !
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.
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.
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.
"The title is the same as that of a very well-known book by Professor
L. E. Dickson (with which ours has little in common). We proposed
at one time to change it to 'An introduction to arithmetic', a more novel
and in some ways a more appropriate title; but it was pointed out that
this might lead to misunderstandings about the content of the book."
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."
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'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 :(
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.
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.
True. Math is drier than physics. But Terrence Tao is well known, and there are a lot of nerds on the planet. The last popular math book I read wasn't good (Humble Pi; a Comedy of Math Errors), but I did enjoy e.g. How to Lie with Statistics, and, to a lesser extent, Innumeracy. I would buy Tao's book, probably.
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.
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/
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:
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.
If one wants to study Mathematics (or any Science for that matter) sincerely, one has to put in conscious self-effort/time/be-persistent and have the attitude of "knowledge for the sake of knowledge". The goal should be personal development/understanding and not competition/ego-boosting/fame/money/etc. (these are only relevant for the job market) which will not sustain motivation over the long-term. The emphasis should be on one's self-effort and not on the merits/demerits of the Professor/Teacher/Books i.e. "The Master can only show the way, but it is the Student who must walk the path".
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.
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.
counterpoint to
> easily four or five years of study just to play around a bit
it depends significantly on the branch of maths you choose! I've been told by a professor of fluid mechanics that he has difficulty posing and approving subjects of undergrad dissertations because the knowledge threshold for contributing meaningful ideas reliably is so high, but in my primary interest (combinatorics) this is very much not the case.
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
> Genuinely, what is it that you get from studying mathematics?
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.
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.
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.
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.
> but I do spend a lot of time listening to music. I've spent a lot of money on audio equipment.
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".
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
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.
> ... 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.
> 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.
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...
Interview with the author about that book
https://www.econtalk.org/a-mind-blowing-way-of-looking-at-ma...
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
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
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.
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-...
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.
Sorry for the stupid question but is Elliptic Tales your favorite or is it Summing it up?
Elliptic Tales, but maybe it is because I read it first
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
"The title is the same as that of a very well-known book by Professor L. E. Dickson (with which ours has little in common). We proposed at one time to change it to 'An introduction to arithmetic', a more novel and in some ways a more appropriate title; but it was pointed out that this might lead to misunderstandings about the content of the book."
"The word `basic’ in the title is closer in meaning to `foundational’ rather than `elementary’ [...]" (quoted from that same wikipedia page).
The one math majors joke about is Serra’s A Course in Arithmetic, which is definitely not for young children.
I remember a joke along the lines of "elementary" meaning that someone somewhere has solved it before.
"Any question in maths is either unsolved or obvious."
> 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."
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".
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...)
Looks like an inspiration from Richard Feynman's "Six Easy Pieces"
Hopefully we shall get a Feynman type math book from a true Master.
Six Math Essentials as a title reminds me of Six Easy Pieces. I wonder if that's intentional.
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 :(
https://web.archive.org/web/20260216165954/https://terrytao....
Jeremy Kun's A Programmer's Introduction to Mathematics is also a good one.
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.
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.
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.
True. Math is drier than physics. But Terrence Tao is well known, and there are a lot of nerds on the planet. The last popular math book I read wasn't good (Humble Pi; a Comedy of Math Errors), but I did enjoy e.g. How to Lie with Statistics, and, to a lesser extent, Innumeracy. I would buy Tao's book, probably.
The Pelican UK version[1] looks a lot nicer
[1] https://www.penguin.co.uk/books/482167/six-maths-essentials-...
Don't judge a book by its cover ?
I kinda like the cover, but maybe I'm just a boring person myself.
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.
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/
Again, thank you for the recommendation.
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.
If one wants to study Mathematics (or any Science for that matter) sincerely, one has to put in conscious self-effort/time/be-persistent and have the attitude of "knowledge for the sake of knowledge". The goal should be personal development/understanding and not competition/ego-boosting/fame/money/etc. (these are only relevant for the job market) which will not sustain motivation over the long-term. The emphasis should be on one's self-effort and not on the merits/demerits of the Professor/Teacher/Books i.e. "The Master can only show the way, but it is the Student who must walk the path".
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.
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.
counterpoint to > easily four or five years of study just to play around a bit it depends significantly on the branch of maths you choose! I've been told by a professor of fluid mechanics that he has difficulty posing and approving subjects of undergrad dissertations because the knowledge threshold for contributing meaningful ideas reliably is so high, but in my primary interest (combinatorics) this is very much not the case.
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
> Genuinely, what is it that you get from studying mathematics?
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.
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. 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.
> but I do spend a lot of time listening to music. I've spent a lot of money on audio equipment.
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
Which ebook provider should I use to get an actual epub file?
Dynamics? What branch of mathematics does it refer to? Is he talking about differential equations?
probably dynamical systems, ergodic theory, etc.
Is this Terry's 17th book?
the only math essential is Hessians
Why?
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