Is Maths Real? How Simple Questions Lead Us to Mathemat… (2024)

As soon as I saw this book, I knew I had to read it, if only because I wrote a book called Are Numbers Real?. The maths content of Eugenia Cheng's book is brilliant: where I was covering the history of mathematics, she focuses on what real, pure mathematicians do. (Funnily, I had called my book Is Maths Real?, but it was changed to the less accurate Are Numbers Real? so we wouldn't need different covers for the UK and US editions.)

The mathematical journey that Cheng takes us through is mesmerising. She starts by showing the power of abstraction - how by thinking about the nature of, say, something basic like addition or multiplication it is possible to extend the concept into something other than numbers. We also discover that, in some ways, the answer '2' is the least interesting response to 'What is 1 + 1?' - real maths isn't about the answer per se, but about digging into the processes, mechanisms and definitions to get a deeper understanding of the underlying logic.

From these simple beginnings, we are then helped to get over what has proved a stumbling block for many: the abstraction of what we'd call variables in computing. Using an x (say) instead of a specific number. I loved algebra at school, because it felt like code breaking, but I absolutely understand why this step is one that defeats many young people in their exposure to mathematics. We then get on to formulae and their relationship to other mathematical structures like geometric ones, and different visual representations. And this was all a delight.

I wouldn't entirely agree with Cheng on why so many people aren't interested in, or openly dislike, maths. I suspect a lot of it is about the very thing that makes it attractive to her - that abstraction. If we look at the rest of STEM, engineering and medicine give us practical things. Physics and chemistry tells us how the world works. Biology tells us about interesting animals and feeds into medicine. But most of maths is neither useful in ordinary life nor telling us anything about the world, because it exists within its own abstracted universe. This book makes clear how beautiful that can be - but it's a hard sell to teenagers.

Originally, I was going to give this book five stars, because the maths coverage is brilliant, but most of Cheng's books I've read have the flaw of containing (for me, at least) too much about her and her opinions. I want to read about maths, not Cheng's interest in food, her politics or her cultural inclinations. I do want to know about why mathematicians find maths interesting - but that’s a very different thing. This sort of personal content does appeal to some readers, just as, say, reading a magazine about celebrities is fascinating for some, but it puts me off.

Overall, then, do read it if you want to know more about the nature of pure mathematics and about being a mathematician - the mathematical content is great - but you may need to occasionally grit your teeth over the rest.

Yes and no. Quite a lot to recommend in this book. I found parts of it quite fascinating. I'd certainly recommend this if you are a teacher, as much of it is about the pedogalogical aspect of mathematics.

I particularly like the "from this point of view this, but from this point of view this" kind of style. Especially the argument about how pi=4 from a certain perspective. That tickled me nicely just how I like it.

There are a few sections that simply start explaining certain sections of maths. This is a book that purports to be exploring the basis of maths, but there's a long section on trigonometry, that really is just a primer on trigonometry, and seems to serve no purpose to the overall argument. Could be useful if you wanted to learn trigonometry I guess.

Then there are the diversions into "politics". It really feels like these are being crowbared into book, and often feel a bit contrived and add very little to the argument.

"Not thinking that 'i' can be a number is like thinking that women can't be mathematicians."

"If you don't think that x can represent a number you're like someone who thinks that "they" can't represent a non binary individual.

"If you think that 3 is greater than 4 you're like someone who thinks that white people are greater than black people, because white people are not greater than black people and 3 is not greater than 4"

OK I made up one of those examples for comic affect. But the other two are real.

It's all very touching, but so contrived as to be distracting.

Then there's the section on decolonising maths. Where the idea that revelation from a Hindu godess is as valid as a formal proof is floated. This had me scratching my head a little.

There's actually quite a lot to recommend, all told. Some fascinating mathematics. And I love the emphasis on embracing uncertainty, and making maths more accessible. So kudos there. Certainly recommend for some, I think maybe I'm just not the target audience?

A rare DNF that I’ll briefly comment on.
For me, it started well. An experienced and recognised academic in Mathematics tries to explain her subject from about the most fundamental starting points she can find. In particular, with her interest in STEM education, which interests me too, she often seemed to be orientating her explanations to those with no mathematical grounding at all.
For example, it might seem bizarre to discuss 1+1=2, its validity and occasions where you could expect a different result (a trivial example would be one pile of sand added to another pile still gives one pile of sand, albeit bigger). It was a tool for her to explain the basic need to define the basis of your maths if you expect to get something out of it; it’s as much a language of human invention as any other and definitions are critical.

However, I personally found the pace slow. When I abandoned at about 60% we’d just got onto algebra after some number theory. The author is also fairly verbose and brings in all sorts of analogies, and sometimes goes off at a tangent, often related to her interests in cooking and music. Although I’d like to think I’m on roughly the same page as her politically and her social concerns I rarely felt that their involvement to be of use to the mathematical core of the book.
So I just got rather bored..!

But I’d recommend it perhaps to students with an arts background, who may find maths rather confusing, as perhaps one entry route into understanding it without involving the traditional proofs and problem solving aspects you find at school. You could maybe see the point of it without starting to become a practitioner.

I am sad about it, but I have to move this book to my DNF shelf. This is a case of bad timing, and "it's me, not you." May 2024 was just not the month for me to take on books that required a lot of brainpower and focus to get through.

To be fair, in the 17% (89 pages) I read, Cheng was doing great at making math concepts approachable. And while it required some thinking on my part, I was enjoying it.

So while I'd like to think I'd come back and finish this, without a book club to push me through, I have to be real and say I probably will not. Too much else out there to read that qualifies as pure fun that I'll prioritize. But I hope this book finds appreciative readers, which it deserves.

In this book the author wants to let people know that if you hate math it's more likely because of the way you were taught. In fact, if there are things that don't make sense to you and you've asked "dumb questions" then you probably think more like professional mathematicians.

I found this book through to Ali Ward's Ologies podcast

Dr. Cheng talks about many mathematical concepts with an explanation of why and how mathematicians come up with concepts. I was blown away in the section "why is 1 + 1 = 2" because it turns the question on its head. We should think about "When does 1 + 1 not equal 2" for example, when you are painting, if you have one color, adding another color gives you a new combined color, not two colors, in this case 1 + 1 = 1. So then a good definition of *when* does 1 + 1 = 2 is in order.

I like that Dr. Cheng hates it when people try to use math to make themselves sound smarter as if knowing these things makes you a better, superior person somehow (it does not, it's your character)

I liked the little commentary on current events sprinkled here and there, I suppose that some people will balk at that, that's fine by me.

I didn't like that the material for me was introductory, I didn't walk about with new mathematical theory (well, at the end she goes into her category theory research) but it's an introductory book aimed at people who don't do math.

Overall, this is an excellent book. Highly recommend it. Math if for you and for all of us.

Eugenia Cheng is a very talented science/math communicator. I've discovered her through "x + y" a book I absolutely loved. "Is Math Real" didn't capture me quite as much as "x + y", but it still made me think about the world through different, rapidly changing lenses. The beginning and ending of the book are very strong in my opinion, but the middle sags. This is where we discuss calculus and graphing, where I found myself skimming because the explanations felt too simple for me, and I didn't discover anything I didn't already know or consider. The middle was also prone to changes of topic that weren't well channeled and prepared. Cheng's work touches on a lot of pressing social issues, but sometimes the way she brings those up feels a bit "out of the blue." And I'm saying this despite being in complete agreement with her statements. Take the example of discussing casting thin actors in movies in the chapter about graphing--it really had nothing to do with the chapter, and even the author admitted it wasn't fitting in by a simple "I digress." I also thought that the middle was a bit too focused on saying that people who take mathematical 'rules' (the ones taught in school) for granted might not be good at math after all. This felt like an oversimplification; I know in my case I felt the rules 'made sense' and I've always had an easy time performing certain mathematical abstractions and operations in my mind. This doesn't make me a mathematician, but doesn't make me a rule follower either. Her point was that mathematicians will dig deeper than that; agree, but some may still share that 'intuitive' feel with numbers. Since Cheng likes parallels with real life I'll use this example: my dad is super talented at painting and drawing; it's eerie how he intuitively knows how to draw. He was so good that in 4th grade he got in trouble with the art teacher who accused him of having an adult doing his homework--he painted a super realistic war scene. At age 10. His classmates had seen him paint incredibly accurate pictures before and they rose to defend him. At that point my dad did not yet know/understand perspective and colors the way an expert artist does, he learned some of that later in school. And he never persuaded an art degree (primarily because he wanted to have a salary). But does that mean that his intuitive painting makes him not an artist? How deep did he need to interrogate before we considered him an artist? Anyway, like the author, I'm digressing a bit. Toward the end the book was more moderate in its proclamations of what makes a mathematician, and the awe returned to the pages.
" As an abstract concept, it is real. This is the level at which math is real. We can’t touch it, but there are plenty of other real things that we can’t touch either. Sometimes it’s for logistical reasons, like the center of the earth or the inside of our own brains. But there are some real things we can’t touch because they’re abstract, like love, hunger, population density, greed, grief, kindness, joy."
Overall, an enjoyable thought-provoking read. For those who were 'good at math' in school, some parts may feel over-explained, but I think it's worth your time.

I listened to Eugenia Chang on Ologies and was inspired by her description of mathematics to pick up this book. I think her explanations translated better over an audio medium for me, and ultimately her description of math concepts didn't quite grab me. On the whole, I do wish her math pedagogy had been around when I was still in coursework. Had someone approached math from her gentle, curious, and inclusive perspective, I think my school experiences would have been drastically different.

When does 1 + 1 not equal 2? Bend your mind around this as the answers are real, surprising and rather fun. Basic math gives the impression there is only one 'right' answer to a problem. That's true for practical everyday calculations (if it wasn't, our checkbooks would be unmanageable). But that often isn't true if the problem is looked at in another way.
A strong, basic understanding of math is essential for trying to follow the questions and answers in this book. The reason for calculus and what it does, different kinds of triangles, even genetics come into play on these pages. Some of the asides add to understanding the math questions. Others are only opinions of the author.
This book is not an easy read. Even with a good math background, some of it is not understandable. It does explain, sort of, what mathematicians do and how they approach their work.
The purpose of the book is to get past the rigid way math is normally viewed and this is done well. Some of the different ways of approaching a math question leave the impression it is all playing around with the question with no practical ends. Yet, some of this, as with calculus, ends up having major implications in the 'real' world most of us live in.

Starts well then depends into esoteric drivel

There are some interesting views expressed by the author in the first half of the book, if you can get past her obsession with “Eurocentric white male mathematicians’ influence on modern maths”. She goes on a bit about it. Unnecessary and could just go in and tell us about the ancients and their maths, whether Greek, Egyptian or Neanderthal and not go on about how they may have discovered it first. I got to 80% of the book then gave up (very unusual for me, because I insist on finishing a book no matter how painful) because the author was using left and right handed hair braids to demonstrate a principle, which eluded me for the remaining 20%. Don’t waste your money.

tries to take mathematical frustration towards curiosity and understanding, though i found i could not finish it. it doesn't pass muster for my non-fiction aesthetics, nor is its subject interesting enough for me to push past the obstacles it continually throws (cloying political correctness).

I had hoped this book would turn me into a math person. I’m still not a math person, although it was an interesting read and I learned new things.

Ratings, being opinions, relate as much toward the person who rates as the book itself. This could be the book for you. It's not the book for me.

So, Eugenia Cheng has written a book that I think is somewhat aimed toward people who (1) hated math classes and, as a result, did badly in them and (2) others who just think it would be fun to see possible answers to questions like, "Why does 1+1=2?"

I'm not in the first group and Ms Cheng's mind and mine don't travel on the same highway so I was disappointed and occasionally confused by the way she went about demonstrating things.

Delightfully retracing basic math concepts to show mathematicians motivations and enthusiasm, emphasizing an openness to not assuming things are "obvious", an educator's deep interest in honest innocent questions that do not one correct answer, and the relevance of math's interest in contextual "why and when is such true?" in seeing similarities and differences in analogy to current political and cultural rifts.

Most science books follow a bit of a progression: they start with simple explanations, which I know already most likely, they move into more difficult concepts, which I follow with ease, they ramp up the complexity, which I follow with difficulty, and then at about the 80% mark they lose me and I no longer have any idea what they're talking about.

In this case I lasted until about 86% of the way through, but to be fair I also didn't understand the trigonometry chapter. Not her fault: despite being "good at math," in school, I never had, never will, understand or like trigonometry. I understand some of the beauty of math is abstract and you needn't always have practical applications to appreciate it, but the beauty of trigonometry escapes me.

As does category theory, which I ought to like, because it's the abstract qualities of math that appeal to me the most. I like principles (e.g. (a + b = b + a). I like talking about sets, and qualities, and I eagerly read her book on infinity. But category theory always starts like this: "you can write 30 as 5 x 6, and you can write 6 as 2 x 3, and of course it's all divisible by 1, and you can draw arrows between them, and you can arrange it all so it looks like a cube," and I think "yes, you can, but why on earth would you want to do that?" And I can never understand if you (a) have to do that to make something else work, or if (b) it's illuminating you can do that because it means something about 30 that's not true for other numbers, or (c) it's just one of many ways to illustrate 30-ness and it seems an easy one to start with (it's not), or (d) something else I don't understand. And then it moves on to abstraction very quickly, and instead of 30 -> 6 it's A -> B and I think I can draw a diagram of George Clooney -> velociraptor but I don't know that that makes it mean anything ...

However: for the first 80%ish, she's great. I especially appreciated understanding (for the first time, and it helps illuminate my difficulty with category theory) that math has increasingly difficult areas of abstraction, and most people will indeed be comfortable only up to a certain point. For some it's imaginary numbers, or irrational numbers, or negative numbers, but even simple numbers like 1, 2, 3 etc are abstractions, and I'd never thought about math like that before. It is, indeed, all abstract ways of representing things, even the humble and much used 1.

Like most readers, despite being largely liberal (I like being fiscally conservative, but that's about it) I wouldn't mind if she dialed back the frequent asides (or entire chapters) on subjects often dismissed as mere 'political correctness.' I agree with her, but they're raised too often in a book purportedly about maths. It's a bit like reading an Italian cookbook where the author keeps raising the political situation in the Gaza strip, it's out of place.

(Note: I'm a writer, so I suffer when I offer fewer than five stars. But these aren't ratings of quality, they're a subjective account of how much I liked the book: 5* = an unalloyed pleasure from start to finish, 4* = really enjoyed it, 3* = readable but not thrilling, 2* = disappointing, and 1* = hated it.)

The author presents a set of excellent and accessible descriptions and explanations of the ideas and meanings of mathematical argument. She takes the next step and shows how many mathematical approaches can be applied to areas in which you would not normally expect to find application. Demonstrates that maths is not only real but insightful.

In that way it is a very good work.

I stopped reading the half way through because I became annoyed to the point of distraction by the political and social agenda interlaced in the book.

By way of illustration, she spends several pages building a solid argument for rigour in mathematical statements. Her explanation is excellent and brings the reader along quite effortlessly to some fundamental insights with broad utility. She is an exceptional communicator. However, what follows is a few paragraphs of unsupported statements and ambiguous labeling of types of people. In this latter part, she is at best inconsistent in language and arbitrary in the labels she uses. This is in stark contrast to the earlier part of the chapter.

In another place she complains about students challenging her and makes a blanket statement about how student challenges are more common for her due to her gender and race. Others have experienced what she describes who did not have the same labels. The experience could be as due to being a young lecturer with unenthusiastic, or perhaps overly confident, audiences as it is about the particular lecturer.

In still another place she uses an an example set the types of privilege one racial group has. The internal logic of the arbitrary types is correct. The choice of the category types and the language used is from a specific social perspective and detracts from her good mathematical point.

If she wants to write a book about social issues she should do so. Given her ability to communicate about how to think well it might prove an insightful work. If she is going to insert a political argument into a book on maths my strong suggestion is to treat all her arguments with the same rigour.

“Is Math Real?” makes the case for the study of abstract math and demystifies it (as much as it can, anyway). It critiques the math education status quo and explicitly argues against the importance of real world applications as part of that process. The author argues for using math as a way to sharpen our intellectual “core muscles”; the point of learning trigonometry or algebra is not just to calculate angles and solve equations, but to build logical frameworks and work inside them. She laments a focus on producing correct answers quickly, at the expense of the open questioning process that is critical for mathematical research. She reinforces this by poking at declared truths, demonstrating situations where 1+1 doesn’t equal 2, or walking through the logic of why 1 is not a prime number. She points out the paradox of how an “obvious” truth often means something that is “so clearly true that I can’t explain it”. She discusses the value of rigorously proving something over merely observing a likely pattern.

The author’s style relies heavily on digressions, making analogies to cooking, social behavior, and cultural forces. She explicitly calls out the gatekeeping of traditional white European male control of academic math, and the way that diminishes other cultures' mathematical traditions. These asides can be jarring; one page you are stepping through a logical derivation and the next you are considering the extent to which medieval African mathematicians were erased from history. I don’t have one single opinion on these. Sometimes I found them interesting, other times distracting. They varied in depth and sometimes felt incomplete.They serve to reinforce the idea that math is not monolithic, but I don’t think they were always effective. They do make the book feel very human.

Math is not concrete, and “Is Math Real?” celebrates this as its great strength. I don’t know that this book will convince many math skeptics, but it may present a new perspective for them.

3.5 stars

Cheng thinks that how we teach math in schools is off-putting, and doesn't really get at what the subject really is. We teach it as a series of formulas to memorize and equations to brute force into our skulls. We spend so much time on WHAT that we ignore the WHY and she thinks that addressing the WHY questions make math easier to understand, less intimidating, and can help people who think of themselves as "not math people" or "bad at math." understand it better.

OK, sounds good - but there's just one problem. I had more trouble understanding this than I did understanding traditional math teaching methods.

I did like the point about how sine is vertical and cosine is horizontal.

Beyond that, the book rambles into various bits of commentary on all sorts of other matters. And I get it - she wants to make points on these other matters. But it continually comes off disjointed, awkward, and not really flowing with anything else.

But my main issue is I just plain didn't get most of what she was talking about.

Mix two parts illuminating math, one part overly-judgmental worldview.

Very interesting book that leads the reader to reassess basic mathematics (e.g., 1 + 1 does not always equal 2, it depends on the context). There are a number of illuminating examples provided across addition/subtraction, multiplication/division, letters, formulas, etc. I found the discussion of prime numbers to be very interesting and useful, along with short biographies of various lesser known mathematicians, including why they are lesser known.

Where the book struggles is pointing out, several times per chapter, the author's worldview on gender, religions, politics, race, sexuality (etc.). While I likely agree with the author's point of view on these topics, the presentation was overly judgmental, to the point of being off-putting (e.g., the author points out that she could never make true challah bread, being non-Jewish, while also pointing out that certain Jewish traditions are "sexist and idiotic"). If these asides were toned down, the book would be much more effective

The maths content in this book is interesting and if I am being honest some of it went over my head. This book is written well and a clear passion for mathematics shines through. However I found the fairly regular intersections about the authors ideas of race and politics very off putting. This probably reads as someone on the different side of the political spectrum as the author but I would imagine if I lived in the same country as them I would probably vote for the same party but I didn't buy a book on maths to hear twitter comment level explanations of white privilege or to hear why black people voting for the republican party is illogical.

I skimmed some of this because I want to get it back to the library on time, and because I can already tell I need to buy my own copy. Any time I find myself pulling out my phone to take a picture of a page, I should just buy the book!
I adore Cheng's brain, and her writing style here. Math has always come easily to me, and she writes about it the same ways I think about it (ways I didn't even realize were how I think about it) and always wish I could better share with people.
I would give this book to anyone, child or adult, who is frustrated by having to do things a certain way for math without understanding why. Understanding, questioning, should be the whole point <3

For anyone who has gone throught the 'drill and kill' process of 'learning' math(s), this book is a godsend of what is possible. Dr. Chang looks at several questions that children (and often others) ask about math, and then lucidly and entertainly explains how they fit into the math process. She is thoughtful, rigorous, funny, thorough, and challenging. I wish she could do a text book beginning with 1 + 1 = 2 on to abstract mathematics. This book is an indication of what a wonderful offering that would be to the world of math(s). Her insights and connections are invaluable and often downright fun.

The book's idea in itself is good, but throughout the book the author keeps giving examples with lgbtq, like if her society rules, and principles of her civilisation are the top, and everyone accept them.
So if at a point your civilisation accept cannibalism and son marrying his mother, you'll be OK with it, and giving them as examples as new ways of being as valid.
That's why I describe your way of comparisons as disgusting, and telling you that same sex marriage is still wrong even if you try to make it normal, and your civilisation is not the only one in this planet

I loved Cheng's 'Beyond Infinity' so much that when I saw her name on this new book, I snatched it up immediately -- even though I'm definitely not a "math person." (One of my most effective pieces for the college paper was an expose on how awful the school's math testing software was.)

I was the perfect audience for this book.

It felt like Cheng was leading me back through all the math I learned in grade school.. but actually *teaching* me instead of ordering me to memorize formulas. I easily learned more about math from this single volume than any of the textbooks I used.

This rating is absolutely bias, which is always true, but it feels more true here. I am not sure who this is written for. I enjoyed the maths part, but I also use calculus and trigonomitry regularly. The author says she writes for people, who don't like math. I'm not sure they would pick this up...
What I did not like, were the constant tangents, that had next to nothing to do with what was discussed. (Someone please count the "anyway"s in the text...) I think it would make it hard for people uncomfortable with math to follow. At least I found it a bit annoying.
I would recommend this, if you want an idea how a mathmatician's brain works.

As a person who love math, this book is explained math in the basic deep understanding. There are lots parts that author know what readers are thinking, and they are told immediately in just couple sentences. These are impressive for me that she knew what reader want. But some basic explaination is quite boring for me, but I understand it. This book is also answer the question for who didn’t in these field, but want to know “Is Maths Real?” Finally, I don’t like much how the book end, it should give a little deeper understanding.

En el libro la autora plantea simples preguntas como si 1+1=2 en todos los casos y a través de ellas empieza a desarrollar muchas ideas y a cuestionar como entendemos tanto las matemáticas como la vida en general.

El problema del libro es que si bien empieza fuerte se va diluyendo y en algunos momentos se pierde el hilo, en general me ha gustado pero si que es cierto que hay tramos del libro que cuestan un poco más.

Math whisperer Cheng tries to inspire a love of logic in all who struggle with long division. The Art Institute of Chicago resident scientist leaps from 1960s middle school New Math to Euclidean geometry, but the math underpinning of modern engineering makes at least her chapters on abstraction worth following. Cheng wanders further afield than in her past explainers, so your mileage may vary. If Covid didn't teach you about risk and exponential growth, Cheng won't be much help.

Learning math the "traditional" way resonated well for me so the way math is presented in this book was confusing to me but I'm not the audience for this book. Someone who has struggled with math or ridiculed for not being good at it will get more out of it.

I see that some people don't like the new culture references. It's the author's prerogative and I'm glad she wasn't afraid to express her opinions.

Misleading title. Should be: "Is Math Real? I Certainly Don't Know. But I 'Know' I'm Politically Correct!"

The math parts are very, very good. But the constant preaching is terrible, distracting, cringy.

Math: Don't focus on right answers.
Life: Look at me! I get all the (politically) correct answers!

I think she should ask herself: What's the difference between math & politics? Like, why isn't political philosophy just like research mathematics?

Heard Dr. Cheng talk on the Ologies podcast with my mom and she got me her book for Christmas (thanks mom!). Main takeaway will stick with me in how I think about questions (where and when something is true is more important that reducing it to if it is true or not). I love that I now understand why x^0 = 1! and I will confidently go forth knowing that my basic, whyy thoughh questions are endorsed by a Eugenia <3