Як ніколи не помилятися. Сила математичного мислення

by Jordan Ellenberg, Андрій Іщенко (Translator)

Бестселер The New York Times, який рекомендує прочитати співзасновник компанії Microsoft Білл Ґейтс.

ТЕМАТИКА
Математика, логіка, мислення, психологія, популярна наука

ПРО КНИЖКУ
Математика — це не просто послідовність обчислень, які потрібно зубрити, поки у вас не забракне сил, як вам могло здатися в школі. Професор Джордан Елленберґ доводить, що увесь наш світ пронизаний математикою. Вона демонструє його приховану красу та логіку й має прямий стосунок до нашого щоденного життя. Автор показує, як проблеми в політиці, медицині, бізнесі, богослов’ї вирішуються за допомогою математики. А ще математично пояснює, за скільки часу потрібно приїжджати в аеропорт, чим є насправді «громадська думка», чому у високих батьків низькі діти, як виграти в лотерею та безліч інших речей.

ДЛЯ КОГО КНИЖКА
Для усіх, хто цікавиться тим, як влаштовано наш світ.

ЧОМУ ЦЯ КНИЖКА
Джордан Елленберґ переконаний, що, озброївшись математичними інструментами, кожен зможе краще й глибше зрозуміти світ.

ПРО АВТОРА
Джордан Елленберґ — професор математики в університеті Вісконсін-Медісон і письменник. Отримав докторський ступінь в Гарварді. Досліджує теорію чисел і алгебраїчну геометрію. Пише статті на теми, пов’язані з математикою, до видань New York Times, Washington Post, Wall Street Journal та інших.

Genres: Nonfiction, Science, Mathematics, Psychology, Self Help, Business, Philosophy

408 pages, Hardcover

First published May 29, 2014

About the author

Jordan Ellenberg is the John D. MacArthur Professor of Mathematics at the University of Wisconsin-Madison. His writing has appeared in Slate, the Wall Street Journal, the New York Times, the Washington Post, the Boston Globe, and the Believer.

Reviews

[Stuart]

Here's the deal. If you're a social scientist or a physical scientist (me) who works outside the world of controlled laboratory data, you have to make sense of the world with imperfect experiments. You often have limited data, you can't repeat your experiments, and the differences between your subject and control are sometimes very fuzzy. Yet you have to try to make some inferences even though imperfect data are all you have. How do you do that in an honest and careful way? That's what How Not To Be Wrong is about.

How Not To Be Wrong is, in terms of quality of prose, the best written book on applied math and statistics I've ever read. The author has an MFA as well as a math Ph.D., so maybe that's not surprising. The title isn't quite right, though. The book is really about "how to try to be right even though you know you're going to make mistakes now and then." That's not as catchy a title, I know. There's a lot of useful and thoughtful material here mixed in with the elegant writing. It's not really a layman's book although it's being sold that way. That's OK. Those kinds of books, like Freakonomics, are usually awful and filled with junk analyses. This one, on the other hand, is filled with good stuff. It isn't a perfect book. There are occasional glitches. That's OK, too. Math geeks, especially math geeks who love a good sentence, will love it. I hope that social scientists and scientists who use (and often abuse) math and statistics read it as well.

[Will Once · Rating 3 out of 5]

I so wanted to like this book.

It's a topic I enjoy. I flicked through the book and the author was saying things that I agree with. Jordan clearly knows what he is talking about. All the signs were good.

So why the 3 stars? Because the book is unfortunately quite dull. There are long sections where Jordan spends ages proving some mathematical point or other, but then he doesn't draw any conclusions from it.

He starts with a story about school kids not liking mathematics because they can't see the relevance to their lives ... and then he gives us a book which largely proves that the kids were right. Some of his explanations are linked to something useful and real world, but most are not.

And when he does make a point, sometimes he wants to hammer that point in with explanation after explanation. It's as if he is trying to batter us into submission with repeated hammer blows. Yeah, yeah, I got it.

The writing varies from quite readable to fairly turgid. At times it feels like we are wading through a text book.

The title is a complete misnomer. If you are looking for a practical guide on the use of mathematics, then look elsewhere. This does not tell you how not to be wrong. Please, please, either write a book to fit the blurb or change the blurb. This does not do what is says on the tin.

Disappointing. There is a good book in here, but it needs a much stronger edit to make it readable. And it needs to be linked more to real life.

So it's a three star for me. The content ought to make it a five star book. The writing and limited conclusions drag it back to three.

[David Rubenstein · Rating 5 out of 5]

This is a wonderful book about mathematics and its application to everyday life. Jordan Ellenberg shows that the certainty that people associate with math is often misplaced; some areas of math are devoted to uncertainty, and that's where things get very interesting.

Ellenberg starts the book with a beautiful example of application of mathematics, logic, and thinking out of the box. During World War II, a group of mathematicians working for the Statistical Research Group were given a problem by some Air Force officers. Fighter planes returning from missions were analyzed for bullet holes. The number of bullet holes per square foot were counted. For example, there were 1.11 bullet holes per square foot in the vicinity of the engine, 1.73 in the fuselage, 1.55 in the fuel system, and 1.8 in the rest of the plane. The officers wanted to add some armor to the planes; the question was where? The planes could only support so much weight, and where would additional armor be most advantageous? The officers thought that since the fuselage had the greatest density of bullets, that would be the logical location for more armor. A mathematician named Abraham Wald said exactly the opposite; more armor is needed where the bullet holes aren't, namely, around the engines. Planes with lots of bullet holes in the engine did not return at all!

The book discusses the issue of statistical significance. Scientific experiment often use a 95% confidence threshold as an indicator of statistical significance. This means that if a truly random outcome were expected, a positive correlation would be seen only 5% of the time. Ellenberg includes an xkcd cartoon that shows how easy it would be to perform a set of experiments that could come up with statistically significant results like "Green jelly beans linked to acne! at the 95% confidence level.

Some of the section and chapter titles are hilarious. For example, in the chapter titled "Are you there, God? It's me, Bayesian Inference", Ellenberg brings up a scary example of the use of "big data". Based on a teen-age girl's purchases of unscented lotion, mineral supplements, and cotton balls, the retail store "Target" began sending her coupons for baby gear, because of the (correct) inference that she was pregnant. Another great section title is "One more thing about God, then I promise we're done."

Another interesting title is "The Cat in the Hat, the Cleanest man in school, and the creation of the universe", in which Ellenberg reviews some of the probabilistic arguments for and against the existence of god. And I love the famous quote by Richard Feynman:

You know, the most amazing thing happened to me tonight. I was coming here, on the way to the lecture, and I came in through the parking lot. And you won't believe what happened. I saw a car with the license plate ARW375. Can you imagine? Of all the millions of license plates in the state, what was the chance that I would see that particular one tonight? Amazing!

I also love the chapter title, "If Gambling is exciting, you're doing it wrong". Ellenberg describes how several groups capitalized on several state lotteries. Due to some strange lottery rules, it is (was?) possible to reliably make a profit, given enough investment of resources. No illegal shenanigans--the states make money no matter what you do. You could make a profit by taking advantage of the rules, and of the people who buy lottery tickets without a coherent strategy. And, I did not realize that Voltaire made his fortune by taking advantage of state lotteries!

Ellenberg brings up the phenomenon of Nate Silver predicting the outcome of the Obama-vs.-Romney election. Silver predicted the probability of both candidates winning state by state, along with the margin of error. By adding up the probable errors, he estimated that he would be wrong by 2.83 states. Critics seemed to have ignored the fact that he was not wrong by this many states--in fact he correctly predicted the outcome in all 50 states!

I highly recommend this book to all people who are even vaguely interested in math, probability, logic, and the application to everyday life. This is an excellent book!

[Kara Babcock · Rating 5 out of 5]

I math for a living. I mathed, both amateurly and professionally, at school. I math quite a bit. And as a math teacher, I like reading "pop math" books that try to do for math what many science writers have done for science. So picking up How Not to Be Wrong was a no-brainer when I saw it on that bookstore shelf. I’ve read and enjoyed some of Jordan Ellenberg’s columns on Slate and elsewhere (some of them appear or are adapted as chapters of this book). And he doesn’t disappoint.

I should make one thing clear: I mainlined this book like it was the finest heroin. Partly that’s because I just love reading about math, but in this case I was also days away from moving back to Canada from the UK when I started this, and luggage space was at a premium, so I was on a deadline to finish this book. I injected chapters at a time into my veins, revelling in that rush as Ellenberg charismatically and entertainingly explores the math behind a lot of everyday concepts and ideas. Unlike similar attempts, however, Ellenberg doesn’t pull the punches. He’s more than willing to go into the higher-concept ideas behind the math, and when it starts getting too esoteric or academic even for this venue, he’s always ready with a book recommendation for those interested in some further reading.

Early in my reading, I tweeted I had already decided to give this book five stars because Ellenberg alludes to Mean Girls in a footnote. (Specifically, he says, “As Lindsay Lohan would put it, ’the limit does not exist!’”) That’s really all you need to know about Ellenberg’s writing style and sense of humour. Actually, I’m not all that enamoured with the footnotes in general; they interrupted the flow of my reading and the symbols used to mark them were slightly too small, so I kept missing them in the text—but that’s a design issue. The content of the footnotes themselves is often informative or, as in the case above, humorous. Ellenberg might be a university math professor, but he also has a sense of humour and an awareness of pop culture that helps to make his writing accessible.

I’m impressed by the way Ellenberg effortlessly straddles pure and applied mathematics. The child of two statisticians, he clearly has a good grasp and appreciation of the way applied math drives so many areas of society. From economics to gambling, he makes passionate appeals for informed perspectives over simplistic analogies or fallacies. His first chapter criticizes analogies that promote linear thinking about taxation when the very same economists writing these analogies know that taxation probably isn’t linear. He doesn’t argue for or against an increase in taxes, but rather he points out that it’s wrong to oversimplify the concept when trying to sell it to the public. Is a curve really all that much harder to understand than a line?

There’s also some great chapters on odds and the lottery, in which Ellenberg recounts how a group of MIT students set up a legitimate operation to bulk buy lottery tickets from a certain game that actually gave them good odds of winning. They made a profit, because they used math to turn a game of chance into a predictable investment strategy (which is more than we can say for the stock market). So, you know, stay in school kids.

But actually, the parts about the lottery that impressed me were more towards the purer end of the math spectrum. Ellenberg started discussing, for example, how best to pick the numbers on one’s tickets so that one could maximize the chance of winning at each tier of prizes. It turns out that it’s possible to represent the way of picking these numbers geometrically (yes, as in pictures) and that it’s related to the way we create error-correcting codes (which allow us to send instructions to spacecraft, and compress data in JPEGs, MP3s, and on discs). He goes into quite a bit of detail about the more advanced concepts behind these ideas. Later, he points out how correlation on scatter plots corresponds to an ellipse—and we know how to deal with ellipses algebraically, which gives us a good toolset for talking about correlation algebraically too.

So, How Not to Be Wrong makes an effort time and again to belie the impression that we often get in school that math consists of a series of discrete topics: arithmetic, geometry, statistics, and the dreaded algebra. We teach it that way because it’s easier to lay out as a curriculum and focus on the essential skills of each discipline. And also because we are boring. If you’re lucky, like me, then as a student you’ll start to see the connections yourself. Circles and pi start showing up everywhere, to the point where suddenly you feel like you’re being stalked, and no amount of infinite series or integration is going to save you. But really, good teachers start showing these connections as soon as possible. We fail students and leave them behind because, in our rush to equip them with the skills we’ve been told they need, we rob them of the idea that math is a creative process, instead fostering this false impression that math is a sterile, difficult, procedural slog. If it is, then you might be a computer.

Ellenberg never demands a knowledge of integral calculus, of set theory, or of transfinite numbers. What he does demand is an open mind, a willingness to be convinced that not only does math have a useful place in life (it’s pretty obvious to most people that someone needs to know how to math; they just don’t see why it should be them) but that a deeper understanding of the roles and uses of math can enrich anyone’s life. One can be a believer in the power of mathematics without necessarily worshipping at its altar, and it’s this quest for adherents rather than acolytes that makes this popular math book successful. It helps that Ellenberg’s style is witty. It helps that he is passionate without sounding too evangelical. He weaves in enough history, anecdotes, and allusions to demonstrate that mathematicians’ journeys and the development of mathematics as a discipline has been just like everything else in life: alternately dramatic and dull, intense, occasionally acrimonious. We don’t like to admit it, but we mathematicians are people too. And occasionally we’re wrong, very wrong (like those nineteenth-century French eugenicists…). The title here is tongue-in-cheek, and How Not to Be Wrong can’t guarantee your future correctness with great certitude. All it can do is help you think more critically, more logically, but more creatively about the problems and questions that you’ll face in the future. Because mathematics is a tool for helping us to do amazing things. You can be a novice, or you can be a proficient user of this tool, but either way you’ll need to pick it up at some point to do a little handiwork. Don’t fear it: embrace it.

Oh, and read this book.

[Ian · Rating 5 out of 5]

This book was an excellent guide to the many ways in which our intuitions and poorly understood statistical training can lead us astray. One of the areas that it covers is regression to the mean, a concept which pretty much everyone needs to be aware of, since a better awareness of its ubiquity would prevent a lot of errors. Among other things, this concept explains why a successful pilot study is likely to give worse results when rolled out, why a good performance is often followed by a worse performance (and vice versa), why sequels are less successful and so on. The book also explains why a lot of medical research is effectively inconclusive despite statistically significant results and how p-values are generally misinterpreted, so we should take medical research with a large grain of salt. Some of the other areas the author discusses are counter-intuitive. One topical example is that if there are three or more options then unless one option has an absolute majority in its favour then ALL of the options will have a majority opposing them, which explains why politicians can never please everyone. For example, if the government offers three options for reducing the deficit: (a) increase taxes (b) cut health funding (c) cut welfare funding and equal numbers of people favor each option then 66% of people will be opposed to each option, so nothing the government does will please a majority.

This book is full of gold for anyone who hasn't encountered some of these concepts before, such as the story of Abraham Wald's insight regarding how the missing bullet holes determined where bombers should be armoured during WWII or how the Laffer curve governs a lot of phenomena rather than linearity or how a zero correlation doesn't necessarily mean that there is no relationship between two variables ( correlation co-efficient does not detect non-linear relationships) or the story of why scientists took so long to be certain of a link between smoking and lung cancer (not because of obstruction by the tobacco companies).

Although I am trained both as a mathematician and statistician, you don't need a strong mathematical background to understand and benefit from this book. Most examples require basic arithmetic and the author has a talent for producing crude but enlightening graphics that help to guide the reader's intuitions.

In addition to this book, I would recommend pretty much any recent book by Gerd Gigerenzer, whose work shows how scarily ignorant doctors are of how to properly interpret results of medical tests and why mass screening programs do significant harm while not significantly reducing mortality.

[Maryanne · Rating 4 out of 5]

Where language and math meet is where my head explodes.

That's this book.

Fortunately, the author has a funny, down-to-earth style that keeps me going even when my eyes glaze over and start to roll back into my head. That has nothing to do with him; it's all me. He and I have a fundamental difference in wiring: he loves numbers and the things they can do. For him they sing. For me, they are instruments of torment and deceit.

Let me give you an example. Here's one from page 44 et seq., where he demonstrates that the sum of an infinite string of ones (1+1-1+1-1...) equals zero, except that it might also equal 1. Or maybe it's actually 1/2. You heard me, the sum of an infinite string of whole numbers is a fraction. And they say that numbers are immutable and true and solid, unlike MY stock in trade, words, with their shades of meaning and the ease with which they can be manipulated. HA!

But you cannot frighten me away so easily, sir!

This is a more challenging book than, say, Nate Silver's, because it gives you the method -- the math -- behind the theories. That should not scare you. It should dare you. Along the way, you'll be confused and befuddled, but you'll also laugh and be intrigued and remind yourself that the ways in which you think you know -- really KNOW -- the world are hopelessly flawed, and this guy can prove it. If nothing else, this book is endlessly valuable for that.

[Amit Mishra · Rating 4 out of 5]

Mathematics is a piece of music the deeper you allow yourself to understand its lyrics you will understand the practicality of it in real life. It can be a dull and unimaginative concept that only deals with some of the already established formulas. It paved a way for people to live their life hassle-free.
It brings the practicality and scientific conclusion on any topic whether it's about calculation for about judging a person. With probability and numbers, it makes us our life comfortable.
The author has done a tremendous job to bring out the real-life benefits of Maths.

[WarpDrive · Rating 4 out of 5]

Enjoyable, entry-level book, particularly recommended to any lover of applied maths who did not get prior significant exposure to the main concepts of statistics and probability calculus.
The author writes in a very engaging and conversational manner, and his enthusiasm for maths is quite contagious; I like how he manages to compellingly convey the message that math is a creative process, not a sterile, procedural slog.
While the book is designed to be understood by a wide audience, so it is necessarily kept at a pretty popular level (which disappointed me a little bit, to be honest, as I was expecting something more meaty from a purely mathematical perspective), I must nevertheless admit that there are some subjects of the book that are brilliantly explained with lucid clarity: the author's treatment of the application of statistical techniques to number theory is nothing short of fascinating, for example; and his explanations of the basic concept of Bayesian inference, of projective geometry, and of the Buffon needle problem, are masterful. Chapters 18 (when he deals with the concept of axiomatic postulating, deduction and self-contradiction) is a real gem (where, by the way, his neo-Platonist view of mathematics comes to fore, view to which I full subscribe). The final chapter is also great, and it represents a passionate defence of mathematics and rationality as fundamental tools for a sceptical, commonsensical, balanced, realistic view of reality, as opposed to the false ideologically motivated "certainties" that permeate some circles (and at which the author pokes some great fun).

[Atila Iamarino · Rating 4 out of 5]

Um ótimo livro sobre como pensamento matemático é importante e onde ele muitas vezes falha. Um "O Homem que calculava", mas com exemplos do mundo real (e muita explicação didática também). Tem alguns exemplos comuns de outros livros, como a relação entre cigarro e câncer (para discutir causa e efeito), retorno à média e afins. Que ficam um pouco batidos para quem já leu outros livros do tema, mas são uma ótima introdução ao assunto.

Fiquei especialmente marcado pela discussão sobre as chances da loteria e a demonstração que ele dá de algumas que tinham calculado mal as chances de acerto e teve quem realmente fez dinheiro apostando usando estatística.

[Maria Espadinha · Rating 4 out of 5]

Uma Deusa por Defeito e Feitio

Embora invisível, a Matemática encontra-se em quase tudo. É a melhor aproximação (por defeito) de Deus que conheço! 😉👍

[Paul · Rating 2 out of 5]

Almost everything that we do these days has some sort of mathematical element to it, from analysis by companies that are looking for patterns, voting, the stock market and ways of winning the lottery.

Ellenberg does make some reasonable arguments; I particularly liked the explanations on the three way voting where the favoured guy can end up being eliminated purely because of the first past the post method, and the way that groups were able to exploit a badly designed lottery.

And most of the time he does a reasonable job of getting his points across using mathematical explanations and details revealing the hidden maths of every day life. But the book suffers from a lack of direction at times it and it regularly jumps into very complex explanations, which some will find difficult. In this sort of book, you also need to stick to one subject at a time, and it sadly flits back and forth as you go through the book.

There are other books out there that are much better at explaining the way that maths affects us.

[kartik narayanan · Rating 4 out of 5]

I am one of those fortunate individuals who cherishes and loves Mathematics, in all its forms. But, I know, a lot of people for whom the Maths is a dreaded specter.

Why is that so? Inevitably, this is a problem that arises from the way the subject has been taught. And this is what the book tries to dispel. This book takes us behind the numbers, equations, theories and abstruse concepts to show the practical applications of whatever we have been taught. Along the way, the history of these various ideas are explained as are various anecdotes, which are informative and amusing.

This book is written along similar lines to Metamagical Themas and GEB, while not at the same level. Think of this book as a stepping stone to the fore mentioned books.

The book deals with concepts that we have been taught in our 11th/12th & Graduation. The author doesn’t really dumb down the concepts – this means that while they have been explained well, the reader really has to concentrate while reading the book.

Another facet of this book which made me love it were the lovely quotes from history. A couple of examples are below.

When talking about the romantic notion of how mathematicians are portrayed as genius, loner types, the author quotes Mark Twain – “It takes a thousand men to invent a telegraph, or a steam engine, or a phonograph, or a telephone or any other important thing—and the last man gets the credit and we forget the others.”

When talking about the need to focus on practical applications, the author quotes Theodore Roosevelt –

“It is not the critic who counts; not the man who points out how the strong man stumbles, or where the doer of deeds could have done them better. The credit belongs to the man who is actually in the arena, whose face is marred by dust and sweat and blood; who strives valiantly; who errs, who comes short again and again, because there is no effort without error and shortcoming; but who does actually strive to do the deeds; who knows great enthusiasms, the great devotions; who spends himself in a worthy cause; who at the best knows in the end the triumph of high achievement, and who at the worst, if he fails, at least fails while daring greatly, so that his place shall never be with those cold and timid souls who neither know victory nor defeat.”

The author, Jordan Ellenberg, is a mathematical prodigy. He has deftly weaved the concepts with good writing to bring out the inherent joy in maths.

People interested in Big Data have to definitely read this book. For those who hated mathematics when growing up and now want to figure out what the fuss is all about and those who just want to enjoy a good read, this book is brilliant.

[สฤณี อาชวานันทกุล · Rating 5 out of 5]

หนังสือเกี่ยวกับคณิตศาสตร์และหลักสถิติเบื้องต้นที่ดีที่สุดเล่มหนึ่งที่เคยอ่าน อธิบายด้วยภาษาที่เข้าใจง่าย ตัวอย่างน่าสนใจ แถมคนเขียนยังมีลีลาแพรวพราว เชื่อมโยงต่อจุดเรื่องราวต่างๆ ที่ตอนแรกดูไม่รู้ว่าเกี่ยวอะไรกัน เข้าด้วยกันอย่างสนุกสนาน (ตอนอ่านทำให้นึกถึงสารคดีโทรทัศน์ชุด Connections ของ James Burke ที่เคยดูตอนเด็กๆ)

ชอบครึ่งแรกของหนังสือมากเป็นพิเศษ โดยเฉพาะบทที่อธิบายแนวคิด expected value ผ่านการเล่าเรื่องกลุ่มคนฉลาดที่รวยจากล็อตเตอรี่ได้เพราะคำนวณ expected value เป็น, บทที่อธิบายความแตกต่างระหว่าง “สหสัมพันธ์” (correlation) กับ “ความเป็นเหตุเป็นผล” (causation), คำอธิบายเข้าใจง่ายเรื่องข้อจำกัดของการทดสอบสมมุติฐาน (hypothesis testing) ผ่านการเล่าทฤษฎีเกี่ยวกับงานเขียนของเชคสเปียร์ ฯลฯ ฯลฯ

น่าสนใจว่าหนังสือเกี่ยวกับวิชาที่คนทั่วไปมองว่ามีตรรกะแน่นหนักปานภูผาอย่างคณิตศาสตร์จะให้ความสำคัญกับประเด็นความน่าจะเป็น ความไม่แน่นอน และขีดจำกัดของวิชานี้ มากกว่าเรื่องกฎหรือทฤษฎี แต่เรื่องนี้ก็สะท้อนความตั้งใจของคนเขียนที่กล่าวไว้ตั้งแต่บทนำว่า อยากให้คนอ่านเข้าใจว่าคณิตศาสตร์และนักคณิตศาสตร์จริงๆ ทำงานอย่างไร

บางตัวอย่างในเล่มไม่ค่อยเวิร์ค อย่าง Zeno’s Paradox อ่านแล้วยังไม่ค่อยเข้าใจ ส่วนประเด็น “ความเป็นสวีเดน” ของนโยบายรัฐ (ระดับการเก็บภาษีไปใช้จ่ายในสวัสดิการสาธารณะ) ก็ดูจะเป็นตัวอย่างที่ยกมาเพื่อหลอกด่า (หรือด่าตรงๆ) เศรษฐศาสตร์สำนักเสรีนิยมใหม่ มากกว่าจะเป็นตัวอย่างที่ดีของ nonlinearity อย่างที่ผู้เขียนอ้าง (ส่วนตัวไม่ได้ชอบเสรีนิยมใหม่ แต่มีหนังสือเศรษฐศาสตร์ที่ด่าอย่างมีเหตุมีผลและตรงประเด็นกว่าเล่มนี้มาก) แต่โดยรวมแล้วก็ต้องนับถือผู้เขียนที่สามารถย่อยเรื่องยากให้คนทั่วไปเข้าใจได้อย่างแจ่มแจ้ง และฝากเกร็ดเล็กเกร็ดน้อยทางประวัติศาสตร์ที่น่าสนใจเป็นของแถมคนอ่านมาหลายเรื่อง

[D · Rating 5 out of 5]

The book's description gives a fairly accurate account of its contents.

Except that calling it "the freakonomics of math" does not do it justice. I find this book much more interesting.

A joy to read. No formulas in sight while conveying the essence of the many topics. I loved how the author starts a story about a concept, then gets distracted into another also very interesting story, to finally return to the first thread. Fascinating and never boring. I hope that his other book is just as good.

[Ana · Rating 4 out of 5]

Having come back to math in my late twenties, this book was comforting and gave me hope that learning the equations and complicated language would not be for nothing. It's also a lot of fun to read.

[Andrey S · Rating 1 out of 5]

While the math parts of this book were OK and sometimes even pretty good, it has many issues.
We spend too much time on the lottery example and mostly on parts that have nothing to do with math (why didn't the state stop the lottery? If it has nothing to do with differential equations or anything like that, why should we care?).
But the most striking problem is the one you can find in many bad popular science books written in the USA. Basically it goes like this: "Some math stuff or physics or whatever and then hey! Democrats are great and Republicans are poopyheads". Did you know that some Republican senator from some state made a stupid decision? Did you know that some stupid Republican said that Obama was wrong, but Obama was right (because he is cool)? And most importantly, Al Gore should have won Florida, yes, you will read about that at least three times in every chapter. I'm afraid to think what would have happened if this book had been written after Trump won the election.
And if you're not an American (like me) and don't care about this stuff, don't worry. North Korea is also terrible (maybe Kim Jong Un is a Republican?). And while we don't go to the USSR in this book, which is great, there is a mention of Stalin killing 50 billion people, because of course he's evil (I would suggest to the author to shit on his own country's history in the future, being good at math doesn't make you an expert in history or political science for that matter).
And not only do these examples make you feel like someone is trying to shove their political views down your throat, but they are not very good examples either.
Let's take one about Syria. There's a guy, an author friend from university, who used to sell T-shirts and he had piles of these T-shirts in his room and he decides to wear them because he's lazy and he doesn't want to bother with laundry. Everyone thinks that he is the dirtiest guy because he wears the same T-shirt every day, but actually he is the cleanest because it is a new T-shirt every day. The conclusion is that you shouldn't make assumptions unless you know all the possibilities. Makes sense.
So there were elections in Syria, and Bashar Assad won (and Assad is a dictator, evil guy, probably killed many billions of people, maybe he is even a republican). And the voting numbers looked strange, not random enough for the author's taste. Then we are told that while this is not hard evidence, it looks like the elections were rigged. But isn't that the same situation as a few chapters before? There could be other reasons why the numbers look that way, but because of the author's political stance, he tells us that Bashar Assad wears dirty t-shirts every day. He could have taken any other election that was proven to be rigged and show us numbers, but no, this book is about how bad Assad is, the math is just for background.
The author's political views have more influence on him than the math. This book is not about mathematical thinking, this book is about a political position. And that was not what the notes said it was. It has some math, and the math parts are good, but there are many books that are much better. For example, "Thinking, Fast and Slow" covers most of this stuff (but with less math in it, unfortunately), "Algorithms to Live By" was also good. Read those and skip this one.

I've noticed that if a book about science mentions Stalin, it's a bad book, like this one. Stalin is not an astronomical object, you can't make Stalin in a particle accelerator, you can't find him in your DNA, so you won't find him mentioned in books by Neil deGrasse Tyson or Stephen Hawking, in Smashing Physics or The Selfish Gene, because those are good books about science and Stalin is a historical and political figure and his place is in books about history and politics. So if you open up a popular science book and the author is trying to convince you that Stalin is evil or that Al "He actually won in Florida" Gore actually won in freaking Florida, then that book is probably a shitty book.

[Lisa · Rating 2 out of 5]

Have you ever heard the joke, "I'm an English Major. You do the math." or "There are three kinds of people in this world. Those who are good at math, and those who aren't."?

Both of those apply to me. Anyone who knows me knows that I hate math, that my mind draws blanks when it comes to anything relating to it. So why did I read this book? It was a book club selection that I wouldn't have picked up otherwise.

I respect what Eilenberg is trying to do, which is to make math more accessible. He succeeds to some extent. But he's also trying to appease hard-core mathematicians and math nerds. He can't do both. I wish he had written two books: One for people like me, and one for people like him.

I liked his humor and found some stories interesting (the Torah, the plane, and lottery), but I had to skim much of the book because there were figures. As in, actual mathematical equations. I felt like I was in school and reading this became too much of a chore.

[Ettore1207 · Rating 5 out of 5]

"Ogni matematico crea nuove cose, alcune grandi, alcune piccole. Tutta la scrittura matematica è scrittura creativa. E le entità che possiamo creare per via matematica non sono soggette ad alcun limite fisico; possono essere finite oppure infinite, possono essere realizzabili nel nostro universo osservabile oppure no. A volte questo porta chi non è del mestiere a immaginare che i matematici siano esploratori in un regno psichedelico di pericoloso fervore mentale, in cui tengono lo sguardo fisso su visioni che condurrebbero degli esseri inferiori alla follia e che, in effetti, talvolta conducono anche loro alla follia.
Non è così, come abbiamo visto. Noi matematici non siamo pazzi, e non siamo alieni, e non siamo mistici.
Quel che è vero è che la sensazione data dalla comprensione matematica – dal fatto di sapere d’un colpo come stanno le cose, di saperlo con una certezza assoluta, fino in fondo – è qualcosa di speciale, che si può realizzare in pochi altri ambiti della vita, se mai si può. Hai come l’impressione di aver raggiunto il cuore dell’universo e di aver messo la mano sul filo dell’alta tensione. È una sensazione difficile da descrivere a chi non l’ha provata."

[☘Misericordia☘ ⚡ϟ⚡⛈⚡☁ ❇️❤❣ · Rating 5 out of 5]

Is math really twice removed from our lives? Nope.
The very incredibly incredible math story from a math child prodigy (in his day), now a professor (a sensible one! a rara avis!). Fun and readable and readily comprehensible tale making math closer and WAY cooler!

Q:
“Mathematics is pretty much the same. You may not be aiming for a mathematically oriented career. That’s fine—most people aren’t. But you can still do math. You probably already are doing math, even if you don’t call it that. Math is woven into the way we reason. And math makes you better at things. Knowing mathematics is like wearing a pair of X-ray specs that reveal hidden structures underneath the messy and chaotic surface of the world. Math is a science of not being wrong about things, its techniques and habits hammered out by centuries of hard work and argument. With the tools of mathematics in hand, you can understand the world in a deeper, sounder, and more meaningful way. All you need is a coach, or even just a book, to teach you the rules and some basic tactics. I will be your coach. I will show you how.” (c)

[Kylie Rae · Rating 3 out of 5]

I'm definitely someone very interested in math -- I have a BA in math and I'm pursuing a Master's in Analytics. However, this book got a bit too abstract even for me. I was hoping for more examples of applied mathematics, and while this book definitely had that, there were often theoretical/abstract/historical asides that seemed distracting. I also think there was too much information about mathematicians that wasn't needed; it just detracted from the examples and things got muddled. I would have loved to hear about female mathematicians as well; I understand that they are in the minority, especially in the past. But women mathematicians have contributed a lot to our understanding of math today and I didn't get to hear about any of them.

However, I did appreciate Ellenberg's passion for math, especially when applied in everyday contexts. We all use math everyday, even if we don't realize it. He also did a fantastic job of breaking down frequently misunderstood concepts (statistical significance in research, probability and chance, expectation, regression, etc)

[Farhana · Rating 5 out of 5]

This is a very practical, useful, and beautifully-written book on mathematics, particularly about mathematical thinking. I learnt about this book back in 2016 from the review of Bill Gates [https://www.gatesnotes.com/Books/How-...]. Since then it had been sitting idly in my to-read list of 2017. However, I finally picked it up at the end of 17. But I wasn't paying enough "attention"! I believed in its worth but I felt I wasn't being committed enough to the contents of this book! Cause to enjoy a book fully you need not only read it at the RIGHT time but also with RIGHT kind of approach! So, I restarted this book again and again (3times actually) and finally I completed it this time! And the book has finally worked its wonder!


The introduction of the book phrases a very common question of ours: "When will I use this math?". I realized a part of this when I needed to use matrix multiplications and vectors for completing the beautiful graphics assignment of Ray Tracing! Or on other occasions, when I got messed up with some statistical tool for the purpose of my thesis. Since then I have believed that all these maths have their use somewhere out there and stopped questioning about their existence!

Professor Ellenberg's book is not absolutely about practical implementations of maths! But it's more about the elegance and beauty of mathematical thinking and concepts that underlie many common events and observations of modern life. Ellenberg hardly uses any mathematical jargons or calculations. He explains the stuffs so naturally and qualitatively that the book is really a comfortable read! And obviously he never oversimplified or understated any concept!

The book starts with the story of mathematician Abraham Wald & World War II when Wald was presented with the problem of how & where to optimally shield the fighting jets so that they could return unharmed - more or less! Wald's mathematical insight beyond the presented data was so simple to think of - yet elegant. Then follows the story of Laffer curve & tax-cuts during the period of President Ronald Reagan. Earlier this year I read nonfiction comics "Economicx" where President Reagan's tax-cut policy was heavily criticized. Now presented with the concept of laffer curve and the mathematical stand of economists who support tax-cuts - it was a different picture. In fact, laffer curve is a good example how mathematics either get used or abused in politics. As economist Hal Varian put it, "You can explain it to a Congressman in six minutes and he can talk about it for six months." Or as Ellenberg puts it,

"It could be the case that lowering taxes will increase government revenue;
I want it to be the case that lowering taxes will increase government revenue;
Therefore, it is the case that lowering taxes will increase government revenue."

Then Ellenberg plays with the concept of law of large numbers, probability, statistical significance tests, and their real life implications. Then and there comes one of the most interesting parts of the book: Expectation and the story of Cash Winfall. How some lottery cartels (some MIT kids & other math enthusaists) exploited the loophole in the game to win major prizes at the expense of other players. And learning about how the concept of expectation works like a magical tool in lottery mathematics was really amazing! The most mindblowing part was how the concept of projective gemotery from graphics aligns with the concept of picking "right" combinations of ticekts to maximize expected return by minimizing risk factors! This was really a new thing to learn - geometry, combination & probaility - how these seemingly different concepts go together! Then there was again the geometry of error-correcting codes like hamming codes from theory of digital signal processing! And this thing always amazes me - how concepts from one field is redefined to solve problems in a seemingly different field! And in the last part the story of Galton, the father of Eugenetics & Darwin's cousin appears again. How he used the concept of correlations and their elliptical nature to introduce the concept of modern fingerprinting (from the idea of principal component analysis) used in criminal investiagtions. And last of all, follows the fallacies and contradictions associated with "public opinion" or polls.

Overall, this book is an amazing package for both popular and ardent math-lovers or at least for math-enthusiasts! His narratives are so informal, delightful, yet serious! This obviously reflects the command of the author in his field!

[E · Rating 3 out of 5]

The press for this book seems a little overblown. It is decidedly not the "freakonomics of mathematics." Rather than hitting a plethora of topics, like Innumeracy and other popular books have done, Ellenberg homes in on just a few: linearity (consider: most trend lines are Laffer curves, not straight lines); inference (consider: an FBI algorithm determines that you are probably a terrorist; what are the odds that you are a terrorist? very very low; false positives almost always vastly outnumber true positives when rare events are in view; take heart next time your doctor tells you that you tested positive for cancer); expectation (takeaway: if you never miss a flight, you spend way too much time in the airport; great discussion of "expected value"); regression; and existence (or, why Bush won Florida in 2000--because when you have more than one candidate, the majority rarely rules, even when it agrees most with itself--go figure that one out).

Biggest takeaways: 1) the Parsons code. Look it up on wikipedia. You can identify nearly every song on earth merely by notating how the notes change in pitch, either up, down, or stationary. Cruise on over to musipedia and check it out.

2) correlation is not transitive. Just because A is correlated to B, and B to C, it does NOT follow necessarily that A is correlated to C. I really should have known this, and common sense dictates it, but I'm not sure I'd ever thought of it that way before. For instance, taking niacin is correlated to higher HDL levels; higher HDL levels are correlated with fewer heart attacks. So should taking niacin lower your risk of heart attack? No; it does not. Correlation is not transitive.

[Ryan Bergen · Rating 1 out of 5]

In “How Not to be Wrong” Jordan Ellenberg commits the cardinal sin of not knowing his audience. I really don’t know who he wrote this book for. For those with little math experience, the concepts in this book will be relatively complex and the explanations poor. For those who have a background in math and are passionate about the subject, the lack of proper notation and convoluted explanations for things you likely have already learned will most likely drive you as crazy as it drove me. He’s clearly quite brilliant and the writing itself is above average for the genre but that doesn’t make up for the lack of focus and poor structure of the book

2/10

[Devika · Rating 5 out of 5]

The most amazingly insightful yet simply written book on the importance of math in daily life, simply because math is present even in the most unassuming of places!

There are many things that I absolutely loved about the book. First, the discussion on how a Jewish mathematician Abraham Wald helped refine the strategy of placing armour on WW2 Planes with his counterintuitive yet eureka-esque approach. Second, analyses differ because of the way the math is involved i.e. linear vs. curve graph approaches to the same problem yield completely different conclusions (e.g. application of Laffer Curve on learning how to be more like Sweden). Third, and most intriguingly, Pythagoreans believed that beans were the entrapment of dark souls. Fourth, Law of Large Numbers works by diluting outcomes and not balancing/cancelling them out.

Another highlight was the Baltimore Stock Broker problem i.e. how people are duped into believing a complete stranger's advice regarding stock investment purely because they end up being the victims of receiving the certain portfolios that have his most accurate predictions. Yet, all along, this Baltimore Stock Broker is just hedging his predictions across stocks by sending different people a different set of predictions. Thus, he's bound to be right for some people at the very least!

I've always been a Math geek, but I don't think I truly appreciated the nuances of the subject till now. Even better is how Ellenberg discusses that genius is not the product of an individual's labour but a communal enterprise. The work of different mathematicians spanning generations helps someone build on to the ideas and offer that stroke of insight/genius. Unfortunately, it's the last guy that gets the credit even though he built off of the ideas of others.

People always ask - "when am I going to use this?" The real and harder question is - "when am I not going to use this?"

There's something in this book for everyone because it offers great perspective on mathematical reasoning in daily life.

[ScienceOfSuccess · Rating 2 out of 5]

The ultimate test for being total math-nerd.

I found out that I'm not one.

[Cheryl · Rating 2 out of 5]

(Done - those of you who have already liked this may want to reread now!)

I like math. I want to be reminded of how cool it can be, and how relevant. But all the books like this, including this, that I've attempted to read have too much explication of the maths and not enough of what it actually means. For example, a chapter will start by explaining that c + a = a +c and just a paragraph later will expect us to know what a quadratic equation is, what it means, and how to solve it. What I'm saying is, these books need to be vetted by regular ppl... like Ellenberg's reluctant students, perhaps?

I'm gonna read it all, even if by 'read' I mean 'skim' for significant portions, because there are interesting tidbits. But gosh. So far so disappointing.
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Ok done. So, the thing is, by 'skimmed' I mean 'read.' And by 'read' I mean 'studied.' And by done, I mean analyzed enough to record all the book-darts you can see below. Oy.

What does that footnote about the 'real divide' between statistics and mathematics mean? I always assumed that statistics were a kind of maths, like logic and probability and topology... are those mathematics or am I just so wrong that I need to go back to Jr. High instead of reading books like this?

And besides, the title is wrong not only because it misleads about the content, but because the idiom only makes logical sense if it's stated as How to Not Be Wrong. (My son, who is at uni. to be a math teacher at the high school or college level, agrees to the point where it's been not easy to ask him to help me get through the sticky points of this book.)

Some bookdarts, mostly from footnotes:

InvestigateF.H. King, the inventor, one hundred years ago, of the cylindrical silo.
Investigate Winning Ways for Your Mathematical Plays.
Investigate Martin Gardner's "The Laffer Curve" from The Night Is Large.
My son did explain the difference between standard deviation and normal distribution to me, but I need to Investigate for reinforcement->mastery.
Investigate Charles Minard's chart of Napoleon's retreat from Russia and Florence Nightingale's 'coxcomb graph' that revealed the high rate of fatal infections re' the Crimean War for early examples of Data Visualization.
Investigate Musipedia.org - is it easy enough for me to use? Not likely, but maybe my son could.
Investigate Tom Lehrer's "Lobachevsky."

Two points about the necessity of replication: "The significance test is the detective, not the judge" which means that if you get a Significant Result, it means more research is needed, and yet if "your study measures a four-year-old's ability to delay gratification and then relates these measurements with life outcomes thirty years later, you can't just pop out a replication."

ManyLabs, addressing the problem of insufficiently performed and/or published replication trials, did find, in Nov. 2013, that of the first 13 studies addressed, 10 were successfully replicated. I need to see if I can find out which 3 were *not* happy-making!

One thing Ellenberg explained to me that I'd never mastered before: You know how it is that many school classrooms have two kids with the same birthday, even though it's a couple of dozen kids and 365 possible days? Well, the thing is, the relevant number of *pairs* of kids is what matters... when there are 30 people, the possible number of pairs is 435, and each pair has a 1 in 365 likelihood of a match, which is 70% odds.

A concept I did find satisfying to read about here was about "The Triumph of Mediocrity," aka 'regression to the mean.' To simplify, any time chance or luck has any influence, over time or over subjects, given time or more subjects, less extreme outcomes will occur. A ballplayer who has a strong season has everything going for him, and he's not going to get stronger and stronger over more seasons because luck is involved. Past performance is no guarantee of future results. Larger sample sizes and longer trials will give more valuable data.

I really didn't like how the book got more and more about politics and sociology as it went along. For just one example, Ellenberg presents & uses a chart labelled "Average income within state." Any of us who have survived middle school know that 'averages' are pretty darn meaningless. And income? Compared to what? Cost of living? Lifestyle including percent urban and number of children/family compared to number of retirees? What? An immediate search with google gives an obviously better data set: https://www.infoplease.com/business-f...... an academic search of any effort would surely give something much more interesting.

Speaking of averages, my son, as a youngster, learned about "box and whiskers" aka "box plot." I wish I had. And I wish it was used more often in the media and popular culture as it seems interesting, easy, useful, unambiguous, and helpful. (But I do need to Investigate for mastery & confidence.)
-----------
The thing is, even though the final chapter is titled 'how to be right,' Ellenberg never really gives anything. He's stuck at what he told his editor he wanted to do, which is "... yell at people, at length, how great math is." A better thing to tell reluctant students is, imo, what Charles Darwin wrote. From Darwin's memoirs:

"I attempted mathematics, and even went during the summer of 1828 with a private tutor (a very dull man) to Barmouth, but I got on very slowly. The work was repugnant to me, chiefly from my not being able to see any meaning in the early steps in algebra. This impatience was very foolish, and in after years I have deeply regretted that I did not proceed far enough at least to understand something of the great leading principles of mathematics, for men thus endowed seem to have an extra sense."
----------
Ok, I admit it. Lots of bookdarts means lots of things to think about. I guess I won't discourage you from reading this. Just, don't try to understand every technicality or divergence, take your time with the bits you do find interesting, and don't have the high expectations that I did.

[Brian Clegg · Rating 4 out of 5]

In the preface to Jordan Ellenberg's chunky maths book (441 pages before the notes in the version I read) we are introduced to a hypothetical student moaning about having to work through a series of definite integrals and complaining 'When am I going to use this?' What Ellenberg sets out do is to show how we use mathematics all the time - and how important it is to understand it if we are not to get the wrong idea about the world. We'll see how well he does.

It was very interesting to read this book quite soon after Richard Nisbett's Mindware. Both cover how to interact with life better thanks to the support of mathematics. Nisbett drives from the psychology side and improving decision making, while this book drives from the maths. Perhaps surprisingly, How Not to be Wrong is the easier read of the two. Ellenberg has a delightful light touch and is often genuinely funny (it's important to read the footnotes, which Ellenberg, like Terry Pratchett, uses for a lot of his jokes).

Along the way he shows us the uses and risks of straight lines in forecasting and understanding data, the power (and danger) of using methods of inference, how to use expected value, the realities of regression to the mean and the interplay between correlation and causality, and some fascinating observations on why traditional statistics can be very misleading when it comes to public opinion. Here it is often not applied to either/or situations, and it's quite possible, for instance, for the public to both support the idea of cutting taxes while simultaneously supporting raising expenditure. Although there are a few cases where we lose the plot and the connection to the real world, mostly this all driven by real world examples - from lotteries where an appropriate strategy can result in big wins to the apparent prediction that everyone in America would be obese before the end of the century.

While I don't think is this as practical a book as Nisbett's, it is full of fascination for anyone who likes a bit of applied mathematics, but can't be bothered with the formulae - there is very little that is scary in that line here. What's more, if you have any exposure to scientists, this book contains by far the best explanation of p-values, what they really mean and where they are meaningless that I've ever seen.

So would the student from the preface feel after reading this book that there's no need to complain? Satisfyingly for a book that doesn't limit us to predictable mathematical answers, the response is both yes and no. Yes, because it becomes very clear that maths is hugely useful in understanding the world and responding to it. No, because the vast majority of maths you will have suffered at school and may have suffered at university, isn't required here. At least 90 per cent of the content depends on probability and statistics, topics that are rarely covered well enough in the curriculum, given how important they are in getting a grip on reality.

Although it felt a bit too long and used US sports rather too often as examples for my liking, this is a book for anyone with an interest in the way that mathematics can give us a better understanding of what's really happening in our complex world.

[Katia N · Rating 4 out of 5]

Interesting and elegant book about Maths (and data) and its applications in our daily life. I would not call it very original as there are quite a few recent books on the topic. But it was quite refreshing to read the book on the topic by the proper mathematician. He managed to explain Bayesian theory very clearly so I finally understood it while Nate Silver failed to do it for me in his book.

Overall if you read "Thinking fast and slow" by Daniel Kahneman and "The signal and the Noise" by Nate Silver you might not find a lot of new stuff in this book. But if you have not, it should be a pure delight.

Also this book shows to some extent how mathematicians reason which I found very interesting.



"Proving by day and disproving by night is not just for mathematics. I find it's good habit to put pressure on all your beliefs, social, political, scientific, and philosophical. Believe whatever you believe by day, but at night argue against the propositions you hold most dear. Don't cheat! To the greatest extent possible you have to think as though you believe what you don't believe. And if you can't talk yourself out of your existing beliefs you'll know a lot more why you believe what you believe. You'll be a little closer to a proof."

Such a simple and such a brilliant idea, but it is really hard to do because you have to be so brutally honest with yourself, that you might not like it:

[Jimmy Ele · Rating 5 out of 5]

I loved this book. Brilliant and funny as well as interesting, all mixed in with a touch of that feeling that you are actually learning something and furthering your pursuit of knowledge. 5 stars, great book recommendations throughout as well as a good aid for mathematical concept guidance. This is going in my foundation shelf and I highly recommend it for anyone who is interested in finding out many of the amazing actual world applications that math can be put to use for.

I might expound further on the merits of this book later, but this summary should suffice for now.

[Chip Huyen]

Most topics covered might be too familiar for people with CS/math backgrounds, but there are quite a few fun tidbits and the writing is easy to read.