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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.
> ... 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...
I've read through Tao's Analysis I and II textbooks, and his writing is actually quite different from Stillwell's approach. Tao tends to frontload more rigorous epsilon-delta formalism early on, which makes it harder going at first but pays off when you hit the measure theory chapters. Stillwell keeps things more conversational throughout.
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've had good luck with Pugh's "Real Mathematical Analysis" and Spivak's "Calculus" for bridging that gap between pop-math and graduate texts. They're rigorous but written with actual humans in mind. The Ash books are solid too—Fearless Symmetry especially clicked for me when I was trying to wrap my head around Galois theory without drowning in abstraction.
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.
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 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.
Those books won't help when prod breaks and you need to estimate cache hit ratios at 2am. Math that matters is the stuff you actually use under pressure.
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.
But which part of the teaching method actually broke for you? Was it the pacing, the lack of motivation for why we care, or something about how definitions were introduced? Because "the system fails at math" is something everyone agrees on, but I rarely see someone pin down what specifically would've worked better in their own case.
I'm curious how Tao frames "abstract algebra" for a general audience—in my experience, the gap between intuitive examples (symmetries, permutations) and the formal definitions is where most people get lost. The challenge isn't the algebra itself but making the motivation for abstraction feel necessary rather than arbitrary.
