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Innumeracy: Mathematical Illiteracy and Its Consequences

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Dozens of examples in innumeracy show us how it affects not only personal economics and travel plans, but explains mis-chosen mates, inappropriate drug-testing, and the allure of pseudo-science.

208 pages, Paperback

First published January 1, 1988

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John Allen Paulos

23 books150 followers

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Displaying 1 - 30 of 385 reviews
Profile Image for Jerzy.
467 reviews104 followers
March 29, 2008
Most of the book is a collection of examples commonly seen in other pop math books: how a particular gambling game or con trick lets the house win most of the time; tricky things about Bayes' Theorem and Simpson's Paradox; how raising the price by 40% and then lowering the new price by 40% does not give you back the original price; the difference between statistical correlation and causation; etc.

I hoped the book would be an in-depth look at where innumeracy stems from and how to prevent it. There is a chapter about this, but it's not the meat of the book. He mentions standard things like poor math education, psychological blocks like "math anxiety", and popular misconceptions that math is just cold spiritless arithmetic. He does propose a few solutions here and there, like getting more non-mathematicians writing about math and highlighting the warmth and passion of the subject to get rid of negative stereotypes... but this is definitely not an overarching policy to improve the standing of math in this country like I'd been hoping.

But I do really like his idea of placing more emphasis on estimation in schools, and especially that people should build personal mental libraries of collections of things for every power of 10 up to at least a trillion. (In other words, you should be able to visualize how many is a thousand of something vs a million of something vs a trillion of something. For example, the stadium in our town seats 1,000 people; a wall nearby has 10,000 bricks; etc.) It would be handy for people to be able to judge for themselves whether or not a number cited in the newspaper is realistic.
Another cool idea is his (logarithmic) risk scale or safety scale. For example, if 1 out of every 5,300 Americans dies in a car crash each year, then driving a car has a low safety index of log(5300) = 3.7. If 1 out of 800 die due to smoking annually, then smoking has an even lower safety index of log(800) = 2.9. If only 1 in 5 million US kids is kidnapped each year, the safety index is a much higher 6.7, and so on. If newspapers and TV started to use this kind of scale, it would be an easier way for people to compare the relative risk of various activities.

I also liked his discussion of coincidences - for example, hearing in the morning that vivid details of your previous night's dream match what you hear on the news. Assuming that there's only a one-in-ten-thousand or one-in-a-million chance of this happening on a given night, over the course of a year in a big country like the USA you'd still get plenty of people to whom this happens simply due to plain chance - not any sort of ESP or anything. So the fact that this has occasionally happened to you or someone you know should not be surprising in the least.
The author goes on to bash more pseudoscience in detail; I agree with him but doubt that anybody who believes that stuff in the first place is going to be convinced otherwise by something as simple as facts and math. (Anyway, reasonable people often believe total crap too. It cracks me up that, at one point, phrenological exams were commonly a precondition of employment in big corporations!)

There's also an interesting comment about "winners" and "losers". A given coin toss has a 50% chance of landing on heads and 50% of tails, and in the long run if you toss a coin many many many times, the ratio (number of heads) / (total number of tosses) will approach 1/2. HOWEVER! That only applies to the ratio - the absolute difference between (number of heads) and (number of tails) is NOT guaranteed to approach zero. If an initial large absolute difference arises due to chance, it's not likely to go away. So if Harry is betting heads and Tom is betting tails, and after the first 100 tosses Harry just happens to be ahead 60 to 40, Harry is likely to stay ahead for a long time. The next 100 tosses are likely to split about 50-50, so he'd end up ahead 160-140, and so on; at 1000 tosses Harry's still most likely to be ahead 510-490. The ratio keeps getting closer to 1/2 (60/100 = .6, but 510/1000 = .51), but not the absolute difference. This doesn't mean that one side or the other is necessarily likely to get that far ahead - but if someone DOES, by pure chance, then they're likely to stay ahead. Perhaps in real life some people end up treated like "winners" or "losers" in general because they've ended up on the wrong side of the difference in wins; Harry here always seems to be ahead of Tom, even though Tom and Harry are each successful at only about half the things they attempt.

Another good section is about reward and punishment. Say that each of us tends to perform at some mean level on a particular task (for example, if I throw darts, assume I'll tend to hit near the bullseye 10 times out of 50). I may do particularly well or particularly poorly (40/50 or 0/50) in one session, but the next time I'm still most likely to be back around my mean score of 10/50. So if I do poorly today I'm likely to do better tomorrow; and if I do well today I'm likely to do worse tomorrow. This is called regression to the mean. Now, if we reward good performance and punish poor performance, and regression to the mean occurs, we are likely to assume that punishment causes improvement while praise causes a lapse - even if the punishment or reward had no effect on the next day's performance.

Finally, he also says mathematicians tend to have a particular sense of humor - they take things literally when they're not meant to be, or they take a premise to extremes with comical result. And indeed it makes sense that this kind of play is exactly what you do when solving math problems or coming up with proofs. See, Katie? My puns and bad jokes aren't pathological - I'm just studying!
Profile Image for Andrew.
564 reviews116 followers
December 23, 2020
Let's say there was a <1% chance that I would buy an unknown book after stumbling randomly upon it on a bargain shelf (something I haven't done in almost a decade after perusing dozens of such shelves in that time), and then a 30% chance that I would then like that book (giving myself some credit for taste while taking into account the vast quantities of extant crap). Those are two dependent scenarios, meaning I'd have to multiply them to get the likelihood that I ever might have liked this book, which comes out to .3%. As Mr. Paulos probably would have warned me (>95% chance), I should have trusted the percentage and skipped the book.

It's a breezy enough introduction to the problem of innumeracy, but ultimately it has less to do with its subtitle -- Mathematical Illiteracy and its Consequences -- and more to do with Mr. Paulos flaunting his intellectual superiority via a litany of schizoid statistical and probabilistic scenarios. It feels like a precursor to Malcolm Gladwell, what with his "Did you know. . ." and "You may think X, but really Y. . ."

I can't help but lament that Gladwell must have been just a tyke when this was written because he could have offered some much needed focus and social relevancy to the author. Even had Paulos focused more on the last word of that subtitle, the consequences and implications, this might have felt more worthwhile. As is it felt like little more than a talent show.

Paulos actually admits that mathematicians have a deserved reputation for arrogance, and also that he was attracted to math mainly because it gave him a way of feeling superior to others (p. 99). Both traits are obvious here, and in that sense I have to think that he is one of the least effective ambassadors that Math could have wound up with. His arrogance is particularly off-putting when casually insulting educators (a population to which I belong) and also when dismissing dreams; though I accept his point about their predictability, I respect the human mind enough to acknowledge we probably don't understand exactly how they work yet.

He does offer some important reminders and warnings about the misuse of statistics, probability and averages -- really interesting were two of the last points he makes, about the difference between statistical significance and practical significance, and then introducing his unique and highly useful safety logarithm. But each of these topics could have benefited from a much deeper treatment, with both more examples and a more structured argument of how the problem affects us and what we can do to prevent it. Similarly, the first two chapters could have been condensed into an introduction.

Basically, this book was not well organized at all, especially puzzling coming from a "coldly rational" mathematician. To put it in terms Paulos might appreciate, it was about 35% useful and 65% gloating crap.

Not Bad Reviews

Profile Image for Jorge Zuluaga.
261 reviews275 followers
October 18, 2022
Dentro de todos los libros que un estudiante universitario lee por solicitud de sus profesores o de su plan de estudios, hay piezas importantes de la literatura de su carrera y otros que solo sirven para saber dónde está la información cuando se necesita (libros índice como los llamo).

Sin embargo, poco se habla de aquellos textos que deberían ser de obligatoria lectura para todos, sin distinción de carrera o nivel. Que aportan mucho a cualquiera que quiera convertirse en un buen profesional, por no decir tan solo en un buen ciudadano del mundo. No me cabe la menor duda que este librito de John Allen Paulos es uno de ellos.

No es un libro para leer en las noches como reemplazo de algún somnífero. Tampoco es uno para leer con descuido en un metro o en los ratos libres en una cafetería. La importancia del tema, así como la profundidad de algunas de las cosas que discute merecen una buena dosis de atención.

No se trata tampoco de un libro aburrido o de interés solo académico. El librito tiene una prosa fluida, está lleno de anécdotas y acotaciones chistosas. Sin embargo, a pesar de la increíble capacidad de su autor, el matemático John Allen Paulos, para divulgar las matemáticas, hay apartes qué hay que leer con cuidado e incluso con un papel en la mano. Pero como dice el autor, a diferencia de los textos académicos, el que se sienta agobiado con un ejemplo o una explicación que supera la capacidad de concentración del lector promedio, puede “saltarse” el respectivo aparte. No hay examen al final del libro pero yo les recomiendo leerlo como si lo hubiera.

Sobre el contenido sólo quiero mencionar una situación que ilustra bastante bien alguna de las cosas increíbles que enseña el libro.

Está usted reunido con un grupo de 30 personas y les dice “entre ustedes estoy casi completamente seguro qué hay al menos dos que cumplen años exactamente el mismo día”. Este es un ejemplo del tipo de afirmaciones genéricas que hacen los astrólogos y los pseudocientíficos (sabiendo o no sabiendo lo que hacen). Tal como lo demuestra el libro hay una probabilidad el 50% de que en un grupo de 23 personas hayan dos con el mismo día de cumpleaños. Así que afirmar entre 30 que existe esa coincidencia no tiene gracia.

Lo que esperaríamos todos los escépticos es que esos “profetas” modernos en lugar de decir algo tan vago como “hay al menos dos con la misma fecha”, dijera que está seguro que en el grupo dos personas cumplen años el 4 de julio. La probabilidad de acertar en ese caso es muy baja y en caso de que lo hiciera sistemáticamente demostraría realmente lo que quiere probar.

¿Que le falta al libro? Me hubiera encantado que discutiera más a fondo el hecho de que los humanos no somos “naturalmente numéricos”. Se adivina en las reflexiones del autor el supuesto de que el “numerismo” se puede conseguir con educación u otras influencias sociales.
Profile Image for Lisa.
320 reviews16 followers
July 10, 2012
On page 94, Paulos bemoans the fact that people attribute combination to causation: "...when people reason that if X cures Y, then lack of X must cause Y."

But just a few pages later, on 108, he states: "In short, there is an obvious connection between innumeracy and the poor mathematical education received by so many people. [...] Still, it's not the whole story, since there are many quite numerate people who have had little formal schooling."

For those who are only as bright (or dim) as Paulos, he's skipping a step here - he's comparing students with a bad education to students with no education, which doesn't follow even his own logical rules he's set forth. He is trying to segue into other reasons for innumeracy, but fumbles, because to make the strongest logical statement, he should have said, "Still, it's not the whole story, since there are many quite numerate people who have had such poor mathematical education."

I'm a linguist, not a mathematician, but there is a need for good logic in both fields. I believe I'm not being nit-picky to say that his poor control over logic is a deterrent to finishing his book about the poor state of logic in America. Or maybe, it just proves his point even more strongly...

So, I'm on page 108 of 180, and I refuse to read any further. Additionally, I will never pick up any book by Paulos again, nor will I read anything he recommends.
Profile Image for Mark.
132 reviews5 followers
December 1, 2020
First off let me say that I endorse any book that advances the general knowledge of math in society. This book is in that vein.

As a math nerd I have had my eye on this book for awhile. After getting through it, though, I am less impressed.

Where it succeeds is touching on a very broad set of topics. Where it fails is that it never gets into enough detail on many of those topics to come away with an increased appreciation. Also, some of the topics don't fit in with the overall theme of Innumeracy such as detours into game theory. While game theory may fit under the broader category of mathematics, here it seemed out of place and did not sufficiently acknowledge the role that personal preferences play in many decisions.

The first part of the book dealt with numbers in general and the examples given were loaded with calculations and estimating using primarily arithmetic that were difficult to follow even as someone who uses all types of math every day.

The author also brought in many public policy considerations related to risks and there was a strong undercurrent of the author's personal opinions creeping into the discussion. While addressing this topic the most egregious omission in my opinion, was discussing societal and personal risks without mentioning the economic consideration of marginal costs.

There was mention of both Daniel Kahneman and Darrel Huff here and I would recommend books by both these authors for increasing your mathematical awareness.

Overall this book certainly helps math literacy but may add confusion on certain topics rather than clarity.

Profile Image for TheF7Pawn.
84 reviews6 followers
September 2, 2014
Never judge a book by its cover or, in this case, by its title. The author purports to explain numerical illiteracy ("innumeracy") and the consequences of it. However, he skates from there to explaining formal logic, probability theory, estimations, critiques of psuedo-science, and then to the reasons why so many people just don't like math. Although his points are valid, and at times slyly humorous, the tone is at times condescending and and self-pitying. I wanted to like this book but the opening pages nearly made me quit reading. The author engaged in dry parlor tricks like estimating the number of grains of sand on earth. Clever, but impractical. If you, like me, were looking for a book describing how experts manipulate numbers to advance specious public policy, this book isn't for you. It barely scratches the surface. If that makes me "innumerate," so be it.
Profile Image for Jonathan.
184 reviews14 followers
October 4, 2014
An easy little read about mathematical illiteracy.

The author, it is eventually revealed, was a mathematical prodigy as a child, and still takes immense pleasure in doing things like deftly computing the volume of all the blood in the world in terms of how deep it would fill Central Park, or how fast human hair grows in miles per hour. He seems genuinely surprised that there might be people for whom these questions are not interesting.

He also has some ideas for improving the state of mathematical education, some of which seem plausible and some of which seem downright wacky. For instance, I think we can all agree that math education has been biased towards the rote memorization of formulas and terms at the expense of fluency and "playful" exploration of numerical and geometrical concepts. On the other hand, his suggested solution is hiring mathematicians as full-time staff in every school to oversee the teaching of math and to rotate in as teachers, which seems quite heavy-handed.

The best parts of this book are the bits on probability and patterns. A little discrete math is all it takes to have a toolset for evaluating all kinds of claims, and the author's analysis of the popular pseudoscientific fads of his day are applicable to all sorts of poppycock that happens now.

In the end though, it's hard to say just who this book is for. It feels less like a friendly exhortation directed to the mathematically illiterate (or innumerate in the author's parlance) than a jeremiad more likely to be read by the literate, explaining why the hoi polloi are so easily duped by cheap parlor tricks. There's a occasional air of smug superiority, and the author suggests outright that the innumerate students of today are just too lazy to learn about math.

Profile Image for Aaron.
Author 4 books13 followers
April 25, 2021
This book is useless. I hoped it would be a serious discussion of causes of and possible solutions to mathematical illiteracy, but it’s not. All it is is a collection of random facts, most of which I already knew, and it lacks a clear thesis or a scheme of organization.
Profile Image for Lyndall Rae.
76 reviews26 followers
February 19, 2020
Not my usual read but my maths/science/politics loving brother gave me Innumeracy to read and I felt obliged 😋

I skimmed majority because I just found it boring and there were too many mathematical equations for me.
Profile Image for Michelle.
577 reviews23 followers
November 14, 2014
Innumeracy is a great book for the era of Ebola panic (even if it is quite dated). Paulos expounds on mathematical concepts as they relate to everyday life - the true nature of particular risks, gambling chances, and understanding extremely large and small numbers. There are a lot of mathematical puzzles (always fun) and real-world examples of the (mis)application of seemingly abstract concepts. There's also some overlap with Thinking, Fast and Slow regarding cognitive blocks to thinking mathematically.

But Paulos comes off as a bit disparaging of "innumerates" - this is definitely not a book for them, but more for those in the know to tsk-tsk at the state of mathematical literacy. It is certainly sad, though, how little progress the American public has made in this domain since this book was originally published. Evidenced by today's popular obsessions and myth-making, educating on these concepts remains a difficult task indeed.
Profile Image for DiscoSpacePanther.
329 reviews10 followers
July 12, 2019
Sitting in an independent coffe shop, sipping a latte and listening to the soundtrack to The Phantom Menace or Taylor Swift’s 1989 is a great way to devour John Allen Paulos’s Innumeracy: Mathematical Illiteracy and Its Consequences.

I picked up this short volume after hearing it repeatedly praised by Matt Dillahunty—a speaker whose ability to cut through bullshit is legendary—so I had some pretty high expectations.

Innumeracy is a quick read, and Paulos takes aim at poor education, poor media presentation of facts, lazy thinking, as well as pseudoscience (which sits at the confluence of these three factors). Nothing is dealt with in great depth—the writing mostly acts as a primer for the reader to not accept face value assertions—and leaves it to the reader to heed the warning and educate themselves.

Apart from his rather frustrating and parochial aversion to the metric system, John Allen Paulos writes rather engagingly. Some of the examples chosen have become much more familiar to the general public in the years since this book was first published, but that should be taken more as an indicator of this work’s influence than its lack of originality.

However, there is another distracting stylistic element that distracts from the universality of the message: the author’s writing is permeated with a difficult-to-ignore Amerocentrism—from bringing Reagan into every chapter (betraying the book’s era of origin as well as the author’s—perfectly justifiable—distaste for that US president) to using baseball or basketball as the source of clichéd statistical examples. The author doesn’t waste any opportunity to establish his credentials as an AMERICAN, and the reader can feel quite bludgeoned by it.

Though the Amerocentrism isn’t sufficent to damage the quality of the writing, I had to deduct one star for the ridiculous adherence to American Customary Measures—no-one who wants to be taken seriously globally as a thinker should be avoiding the metric system.

This is a great book that works as a good companion to Sagan’s The Demon Haunted World, which delves more into the why of people believing weird things rather than Paulos’s focus on how lack of mathematical competence leads to errors or exploitation.
436 reviews
December 30, 2017
I already believe that numbers are beautiful and just make sense but it's always nice to read a book that agrees with you. The author does a great job of putting perhaps non-intuitive concepts in perspective. (For example, when describing magnitude he says, a million seconds is about eleven and a half days, but a billion seconds is almost thirty-two years! I had never thought of it quite like that.) He also does a great job pointing out the negative impact of innumeracy on society in general - how misunderstandings about things like coincidence and cause and effect can lead to harmful beliefs and behaviors about things like the pseudosciences, gambling, and healthcare. Overall a very readable book that could have a big impact if only the right people would read it.
Profile Image for Mary Slowik.
Author 1 book20 followers
September 24, 2015
Quick, essential reading.

While it hasn't completely killed my interest in coincidences, it tried valiantly to do so. The author's anger at the popularity of pseudosciences (astrology, mediums, fortune-telling, etc.) comes across pretty clearly, and it's hard not to agree with him. In fact... it's impossible to disagree with him in certain sections, since he's using cold, hard mathematics. (He'd hate that I used the adjective 'cold,' there...)

I don't have much more to say beyond the fact that this would make worthwhile reading for just about everyone. There are a lot of innumerates out there. It's difficult to follow him in certain passages if you're not taking notes or have a calculator handy, but for the most part he uses tight, compressed, powerful writing, boiling everything down to the essence.

Many of the examples used are adorably dated, late-80's and early-90's.
Profile Image for Roman .
55 reviews7 followers
July 22, 2016
Моя 2 книжка на англійській яку прочитав повністю.
Вердикт - непогано. Деякі глави були дуже цікавими і змусили здивувати і згадати те, що знав. Інші - не такі цікаві, бо я факти з них знаю, проте написані відмінно. Читається легко.
Рекомендується до прочитання всім, особливо тим, хто все життя жаліється на математику.
Profile Image for Doug.
389 reviews3 followers
June 21, 2019
I read this when first published in the late 80s and was then struck by how important the ideas in were to everyone. I'd not thought of the book again until reading Factfulness by Rosling. Although sometimes a bit of a slog and with a few dated references, Innumeracy remains a must-read for anyone who values rational thinking about the world around them.
Profile Image for Gabriel.
312 reviews18 followers
January 31, 2019
As a math teacher, I think about the subject of societal constructs against math a lot. I think about how often phrases like: "I gave up on math a long time ago," "I was never good at math," "I don't need any math for what I do," or "What's so important about insert precalculus topic here anyway" can be uttered with less backlash than saying Shakespeare is hard or various other difficult reads (those tend to get arguments of, "but it's worth it in the end" or similar). So I picked up this book cheap a long while back thinking that this would talk about the various branches of math and how knowing math is important, especially in today's data driven age.

I wanted a personal essay that dove into the general ideas of math and illuminated for the innumerate the immense possibilities of logical thinking and got a treatise on the importance of estimation, big numbers, and statistics. This book is full of specific examples of how poor number sense and poor statistical sense can lead to misinterpretations ... but it never felt like enough. I kept wanting more from the book - more passion, more generalized ideas, more generalized concepts, more examples of everyday math folks ignore when saying they "can't" do math - and didn't get it.

Which means, I think, I came into the book with expectations beyond what the author could offer. What's present is well written - albeit the examples are short and explained so quickly that it took me, a high school math teacher who just taught basic probability, a couple of rereads per section to fully follow the logic - and entertaining. It's also slightly dismissive of the innumerate; tending towards the "lukewarm water" (a la The Bible's "Revelations" letter) of anger. Either get boiling hot and talk about how innumeracy can translate into stupidity OR move the dial to the other side and focus on the dangers of misunderstanding the concept (or listening to purposeful misinterpretations) without mentioning a willful ignorance of math. Trying to walk the line between the two extremes just fell flat for me.

Maybe there's another book out there that will better suit my wants. Something that will go beyond the statistics and bring up grids/graph theory/person hours/tax brackets/etc. else we tempt the pedagogical gods to saying "math must be a path to stats" (right now it's the equally false "math must be a path too calculus"). Why don't we dive into the possible sources of our society's math phobia/reverence? Why do we allow the myth that only smart people can "do math" (a phrase as silly as "do science")? Where did that myth come from? What is the fallout from that myth? Why do we allow the myth that calculus is an end-goal for math to exist? Gatekeeping is not healthy ... where did this come from and what cycles does that perpetuate?

I want a book heavier on the social science view of mathematics in culture. This book is simply quite a few examples of logical fallacies that are easy to fall into without a clear understanding of the conceptual underpinnings of statistics + a chapter of big numbers and making sense of them. It's good, but not what I want.
Profile Image for Anastasia Tuple.
143 reviews
April 21, 2022
JAP, a Maths genius since his childhood (as he writes), has compiled this book to enlighten all of us, poor non-Maths geniuses, and show us "how it's done"! Even though JAP insists from the very beginning of his book that he doesn't want to sound condescending, he does. I often felt he rushes through the Maths & statistics calculations, without explaining as much as I would like (and needed) to, so I was left with feeling frustrated and struggling to keep up. Well, to be specific, he does explain sometimes, but mostly in the beginning chapters; then he just goes on and on calculating various (mostly irrelevant to me) stuff, this is why I was unable to follow his train of thoughts with ease. The reason of my inability is not that I am "extremely intellectually lethargic", as JAP claims (p. 89), but that in my everyday life I never need (or care) to calculate probabilities or use multiple fractions in order to calculate the most efficient dice rolls (I stopped playing board games with dice since High School!) or the most favourable gambling outcomes (I don't gamble, and I never have!). Nonetheless, this book has a few interesting concepts (e.g. the prisoner's dilemma), which you can enjoy reading about only if you manage to avoid getting distracted and exhausted by the detailed calculations.

Anyways, if you 'd like to read (and enjoy) similar content about human psychology, probabilities and Mathematics that are based on real life, I definitely recommend that you read the works by the very reader-friendly author, Daniel Kahneman.
Profile Image for Chik67.
188 reviews
November 25, 2017
Quali sono i costi sociali dell'analfabetismo matematico? Alti certamente se la maggior parte delle persone non è in grado di tradurre informazioni importanti per la vita di tutti in criteri utili e comprensibili per effettuare delle scelte. Dati statistici sono la base su cui dovremmo compiere molte scelte relative alla nostra salute, l'analisi del rischio ci dovrebbe guidare in piccole decisioni quotidiane e in grandi scelte politiche, la capacità di valutare a occhio gli ordini di grandezza di alcune misure ci permetterebbe di capire al volo perchè alcune idee che sembrano ragionevoli sono gran bischerate.
Il cervello umano, se non allenato, fatica a comprendere tutti i fenomeni che riguardano "grandi numeri", cresciuto com'è per inferire informazioni dai pochi casi che può afferrare, e proprio questo, frequentemente, ci porta a ragionamenti pseudologici fallaci. Se non ci ragioniamo su, se non cresciamo con una attitudine ad analizzare più profondamente certe false credenze siamo facili vittime di chi usa i numeri per manipolare la nostra percezione della realtà.
Paulus svela questi meccanismi con ricchezza d'esempi e semplicità del discorso, ricordandoci che l'analfabetismo matematico, al pari di quello linguistico, limita pesantemente la nostra capacità ci comprendere il mondo che ci circonda.
Profile Image for Gina.
Author 5 books22 followers
April 16, 2017
There are some good ideas and points in here. If part of the book's purpose is to raise the comfort level of the reader with certain concepts, then there are probably too many places where it throws in a formula too quickly, causing less numerate minds to glance away.

I do appreciate the author's clear love of math, and there are several good examples. There are also things he misses. For example, one of the sections focuses on normal fluctuations, like how one shooter in a basketball game may have a good streak and then a bad streak without either having a lot of significance. That's true to a point, but there are many factors that can affect that besides normal ups and downs - amount of practice, sore muscles, more aggressive guarding. Part of making math relevant should be getting away from pure math, but then when it is applied you can't just ignore the factors that are not easily broken down into math. He does acknowledge that to some extent, but the book still has various frustrations.
Profile Image for Remo.
2,271 reviews126 followers
November 18, 2021
Interesantísimo ensayo sobre varios temas bastante relacionados. Por un lado el autor critica constantemente la definición de persona culta, que se atribuye a gente con dominio de la Literatura, la Historia, el Arte, pero que puede admitir carencias del tamaño de un portacontenedores en la zona Ciencia y Matemáticas. Por otro, nos pone mil ejemplos, algunos muy escogidos y otros de la vida diaria, en los que un mínimo conocimiento de matemáticas, unido a un sano sentido escéptico y a algo de sentido común, pueden ayudarnos a descartar directamente un montón de afirmaciones a las que nos vamos a ver expuestos.
Paulos siempre fue de buenas ideas, aunque tiene ejecuciones a veces sobresalientes y a veces no tanto. Este libro me ha encantado.
Profile Image for Paige Cash.
14 reviews2 followers
June 16, 2022
I loved this book. I am by no means a mathematician so the Perspective that Paulos offers is eye-opening. In this day and age, we need rational thought and an ability to judge the probability of single events becoming widespread events just because some politician or some pundit has focused on them.
9 reviews
February 26, 2021
Summary: the lack of footnotes renders the book incomplete

In the preface to the 2001 edition, the author states his reason for not
including an index. Fair enough. But the lack of citations supporting
facts and figures throughout can not be excused when the entire thesis
of the book is mathematical (and logic) imprecision needs to be eradicated.

Example: no mention whatsoever of significant digits that carry the
meaningful contribution to a measurement resolution. No mention
of rounding error. The author states as fact (p. 11) approximately
5x10^11 cigarettes are smoked each year in the US. Problems: only one
significant digit. How much is this figure rounded? Truly smoked or
merely manufactured (and exported)? Results from a survey? In which
case, the measured data are the number of cigarettes respondents *say*
they smoke, not observation. But most glaringly, where did this 5x10^11
figure come from?

Example: The author points out the fallacy of "broad base" (p. 168)
then goes on to use this exact fallacy himself on p. 179, "a military
that spends more than one quarter of a trillion dollars each year on
ever smarter weapons." See, he wants the reader to experience, "Wow,
0.25 trillion dollars! Such a big number I must therefore be aghast."
He does not use the more meaningful, relative framing of percent of
budget or percent of GDP. But most glaringly, where did this "quarter
of a trillion" figure come from?

Example: (p. 120) "Rousseau's disparagement of the English as 'a nation
of shopkeepers' persists..." The Feb 2021 wikipedia page for this
quote is rife with ambiguity as to the true attribution of this quote,
mentioning Napoleon yet with no mention of Rousseau. In fact, a full web
search and gutenberg.org search using Rousseau+"nation of shopkeepers"
results in no hits whatsoever. So quite glaringly, where did this
attribution to Rousseau come from?

Example: (p. 130) "one American in 96,000 dies each year in a bicycle
crash." Now consulting a wikipedia page, which itself cites a March 2018
"Traffic Safety Facts" analysis published the NHTSA, we see that actually
there were 911 bicycle fatalities in the US in 1988. Referencing the
population of the US in 1988 (244.50 million), one can compute with simple
arithmetic, this is 1 out of 263,390 (to five significant digits).
The author has committed a 64.2% error (to 3 significant digits).
Perhaps the author really means to write, "one American in 96,000 who
engages in cycling dies..." But that is not what he wrote. Result:
imprecision that results in deceit. Again, and finally, where did this
"one in 96,000" figure come from?

The entire point of footnotes is that only those inquisitive readers
who wish to pursue the topic need to follow the citation. The wee
superscript does not hinder the casual reader who chooses to ignore notes.

The author proceeds through many terms of the Drake Equation on p. 80,
but doesn't name it. He cites the quintessential examples of the
Simpson Paradox on p. 55, but again doesn't name it.

I do laud the writing style, with just the right seasoning of vocabulary
stretching word choice (eg. "jeremiad" and "gematria"). And I found
the final sentences breathtaking, like true literature:

I think there's something of the divine in these feelings of our
absurdity, and they should be cherished, not avoided. They provide
perspective on our puny yet exalted position in the world, and are
what distinguish us from rats. Anything which permanently dulls
us to them is to be opposed, innumeracy included. The desire to
arouse a sense of numerical proportion and an appreciation for the
irreducibly probabilistic nature of life -- this, rather than anger,
was the primary motivation for the book.

Review of book Innumeracy 2nd Ed. (2001) by John Allen Paulos
Read and reviewed in Feb 2021.
Profile Image for Boosh.
35 reviews
June 19, 2017
I think perhaps the best part of this book is that it exists - the very concept of innumeracy, just hearing the word, is one of those things that blows a hole wide open in your mind. The author does a great job of explaining how and why math is important and how we can suffer without a good understanding of it. The writing style is casual and engaging, and the examples are relevant and interesting.

I have a few relatively minor complaints, which keep it from being 5 stars, but don't keep it from being an amazing, must read book. For one, the author is a bit soap-boxy, and despite his expressed hope in the beginning that his tone is not scolding, he does come off as a bit condescending, much like he is a contemporary of Dawkins. But, that doesn't mean he isn't right, and for those who enjoy the acerbic comedy of George Carlin, it wouldn't be a problem at all. My other complaint is that some of his arguments are poorly structured, but I am willing to forgive it because he is not really making any arguments that any reasonable person who would be reading the book would need to be air tight. They are just examples to carry fairly obvious, unarguable points, so their relevance is more important than their quality. Finally, a few careless and condescending comments about women in science with merely the "required" mention of the underlying sexism behind the reduced ratio of women in STEM left a sour taste in my mouth and makes me hesitant to google the author for fear of another Hitchin's style character take down. Anyway, those issues are minimal compared to the enormous value this book can bring by painting a broad picture of the various social consequences of a lack of mathematical fervor in our culture. It certainly made me want to learn calculus...
Profile Image for Dan Richter.
Author 14 books40 followers
June 23, 2016
Paulos betont, wie wichtig es ist, ein Gefühl für Wahrscheinlichkeiten zu entwickeln, was natürlich nur funktioniert, wenn man sich mit dem Instrumentarium einigermaßen auskennt. Das Buch von 1990 scheint mir immer noch sehr aktuell, zumindest für Deutschland. Die meisten Schüler begegnen der Wahrscheinlichkeitsrechnung und dem Umgang mit sehr großen und sehr kleinen Zahlen erst sehr spät – in der 11. oder 12. Klasse. Wenn man davon ausgeht, dass nicht alle Fächer beliebig ausgedehnt werden können und man die Unterrichtszeit für Mathematik beibehielte, könnte man nach meinem Gefühl zum Beispiel bei der Trigonometrie abspecken, die im Grunde nur noch für Designer und Ingenieure relevant ist. Der Umgang mit Wahrscheinlichkeiten hingegen müsste so früh wie möglich in den Unterricht eingebaut werden, da er dermaßen alltagsrelevant ist und letztlich auch persönlichkeitsprägend. Mathematische Analphabeten, so zeigt Paulos, neigen dazu, Ereignisse persönlich zu nehmen, können Risiken schwerer abschätzen, fallen leichter auf statistische Tricks, geschwätzigen Humbug und Pseudowissenschaften herein. Sie vermögen nicht zwischen Korrelation und Kausalität zu unterscheiden. Sie sind diejenigen, die ihre dunklen Theorien mit dem Satz Halbsatz „Es kann doch kein Zufall sein, dass…“, beginnen. Und während man Legasthenikern oft mit Herablassung begegnet, gilt es oft sogar als schick, mit mathematischer Unkenntnis zu kokettieren.
Paulos gibt sich redlich Mühe, auch für jene verständlich zu bleiben, die sich der Mathematik nur mit Vorsicht nähern. Für den mathematisch und statistisch halbwegs geschulten Geist bietet Paulos trotzdem unterhaltsame Überraschungen.
Profile Image for Manoj Arora.
Author 8 books143 followers
September 10, 2020

24 Innumeracy Examples from the Book

1/ TV weathercaster announced that there was a 50 percent chance of rain for Saturday and a 50 percent chance for Sunday, and concluded that there was therefore a 100 percent chance of rain that weekend.

2/ I once had a conversation with a doctor who, within approximately twenty minutes, stated that a certain procedure he was contemplating (a) had a one-chance-in-a-million risk associated with it; (b) was 99 percent safe; and (c) usually went quite well.

3/ It takes only about eleven and a half days for a million seconds to tick away, whereas almost thirty-two years are required for a billion seconds to pass, gives one a better grasp of the relative magnitudes of these two common numbers. What about trillions? Modern Homo sapiens is probably less than 10 trillion seconds old.

4/ Two strangers from opposite sides of the United States sit next to each other on a business trip to Milwaukee and discover that the wife of one of them was in the tennis camp run by an acquaintance of the other. This sort of coincidence is surprisingly common. If we assume each of the approximately 200 million adults in the United States knows about 1,500 people, and that these 1,500 people are reasonably spread out around the country, then the probability is about one in a hundred that they will have an acquaintance in common, and more than ninety-nine in a hundred that they will be linked by a chain of two intermediates.

5/ There’s always enough random success to justify almost anything to someone who wants to believe.

6/ Some would-be adviser puts a logo on some fancy stationery and sends out 32,000 letters to potential investors in a stock index. The letters tell of his company’s elaborate computer model, his financial expertise, and inside contacts. In 16,000 of these letters, he predicts the index will rise, and in the other 16,000, he predicts a decline. No matter whether the index rises or falls, a follow-up letter is sent, but only to the 16,000 people who initially received a correct ‘prediction.’ To 8,000 of them, a rise is predicted for the next week; to the other 8,000, a decline. Whatever happens now, 8,000 people will have received two correct predictions. Again, to these 8,000 people only, letters are sent concerning the index’s performance the following week: 4,000 predicting a rise; 4,000, a decline. Whatever the outcome, 4,000 people have now received three straight correct predictions. This is iterated a few more times until 500 people have received six straight correct ‘predictions.’ These 500 people are now reminded of this and told that in order to continue to receive this valuable information for the seventh week they must each contribute $500. If they all pay, that’s $250,000 for our adviser.

7/ There is a strong general tendency to filter out the bad and the failed and to focus on the good and the successful. Casinos encourage this tendency by making sure that every quarter that’s won in a slot machine causes lights to blink and makes its own little tinkle in the metal tray. Seeing all the lights and hearing all the tinkles, it’s not hard to get the impression that everyone’s winning. Losses or failures are silent.

8/ Because people usually focus upon winners and extremes whether they be in sports, the arts, or the sciences, there’s always a tendency to denigrate today’s sports figures, artists, and scientists by comparing them with extraordinary cases. A related consequence is that international news is usually worse than national news, which in turn is usually worse than state news, which is worse than local news, which is worse than the news in your particular neighborhood.

9/ Coincidences or extreme values catch the eye, but average or ‘expected’ values are generally more informative

10/ Astrology maintains that the gravitational attraction of the planets at the time of one’s birth somehow has an effect on one’s personality. This seems very difficult to swallow, for two reasons: (a) no physical or neurophysiological mechanism through which this gravitational (or another sort of) attraction is supposed to act is ever even hinted at, much less explained; and (b) the gravitational pull of the delivering obstetrician far outweighs that of the planet or planets involved. Remember that the gravitational force an object exerts on a body – say, a newborn baby – is proportional to the object’s mass but inversely proportional to the square of the distance of the object from the body – in this case, the baby. Does this mean that fat obstetricians deliver babies that have one set of personality characteristics, and skinny ones deliver babies that have quite different characteristics? These deficiencies of an astrological theory are less visible to the innumerate, who are not likely to concern themselves with mechanisms, and who are seldom interested in comparing magnitudes.

11/ Most diseases or conditions (a) improve by themselves; (b) are self-limiting; or (c) even if fatal, seldom follow a strictly downward spiral. In each case, intervention, no matter how worthless, can appear to be quite efficacious. This becomes clearer if you assume the point of view of a knowing practitioner of fraudulent medicine. To take advantage of the natural ups and downs of any disease (as well as of any placebo effect), it’s best to begin your worthless treatment when the patient is getting worse. In this way, anything that happens can more easily be attributed to your wonderful and probably expensive intervention.

If the patient improves, you take credit; if he remains stable, your treatment stopped his downward course. On the other hand, if the patient worsens, the dosage or intensity of the treatment was not great enough; if he dies, he delayed too long in coming to you. In any case, the few instances in which your intervention is successful will likely be remembered (not so few, if the disease in question is self-limiting), while the vast majority of failures will be forgotten and buried. Chance provides more than enough variation to account for the sprinkling of successes that will occur with almost any treatment; indeed, it would be a miracle if there weren’t any ‘miracle cures.’

12/ What’s wrong with the following not quite impeccable logic? We know that 36 inches = 1 yard. Therefore, 9 inches = ¼ of a yard. Since the square root of 9 is 3 and the square root of ¼ is ½, we conclude that 3 inches = ½ yard!

13/ Not being able to conclusively refute the claims does not constitute evidence for them.

14/ If you walk down the main street of a resort town any summer night, for example, and see happy people holding hands, eating ice-cream cones, laughing, etc., it’s easy to begin to think that other people are happier, more loving, more productive than you are, and so become unnecessarily despondent. Yet it is precisely on such occasions that people display their good attributes, whereas they tend to hide and become ‘invisible’ when they are depressed. We should all remember that our impressions of others are usually filtered in this way and that our sampling of people and their moods are not random.

15/ Any given individual, no matter how brilliant or rich or attractive he or she is is going to have serious shortcomings. Excessive concern with oneself makes it difficult to see this and thus can lead to depression as well as innumeracy

16/ Bad things happen periodically, and they’re going to happen to somebody. Why not

17/ Remember that rarity in itself leads to publicity, making rare events appear commonplace. Terrorist kidnappings and cyanide poisonings are given monumental coverage, with profiles of the distraught families, etc., yet the number of deaths due to smoking is roughly the equivalent of three fully loaded jumbo jets crashing each and every day of the year, more than 300,000 Americans annually. AIDS, as tragic as it is, pales in worldwide comparison to the more prosaic malaria, among other diseases. Alcohol abuse, which in this country is the direct cause of 80,000 to 100,000 deaths per year and a contributing factor in an additional 100,000 deaths, is by a variety of measures considerably more costly than drug abuse. It’s not hard to think of other examples (famines and even genocides scandalously underreported), but it’s necessary to remind ourselves of them periodically to keep our heads above the snow of media avalanches.

18/ Very intelligent people can be expected to have intelligent offspring, but in general, the offspring will not be as intelligent as the parents. A similar tendency toward the average or mean holds for the children of very short parents, who are likely to be short, but not as short as their parents. If I throw twenty darts at a target and manage to hit the bulls-eye eighteen times, the next time I throw twenty darts, I probably won’t do as well. This phenomenon leads to nonsense when people attribute the regression to the mean as due to some particular scientific law, rather than to the natural behavior of any random quantity. If a beginning pilot makes a very good landing, it’s likely that his next one will not be as impressive. Likewise, if his landing is very bumpy, then, by chance alone, his next one will likely be better. Regression to the mean is a widespread phenomenon, with instances just about everywhere you look.

19/ Judy is thirty-three, unmarried, and quite assertive. A magna cum laude graduate, she majored in political science in college and was deeply involved in campus social affairs, especially in anti-discrimination and anti-nuclear issues. Which statement is more probable? (a) Judy works as a bank teller. (b) Judy works as a bank teller and is active in the feminist movement. The answer, surprising to some, is that (a) is more probable than (b) since a single statement is always more probable than a conjunction of two statements. That I will get heads when I flip this coin is more probable than that I will get heads when I flip this coin and get a 6 when I roll that die.

20/ We find that detail and vividness vary inversely with likelihood; the more vivid details there are to a story, the less likely the story is to be true

21/ Choose between a sure $30,000 or an 80 percent chance of winning $40,000 and a 20 percent chance of winning nothing. Most people will take the $30,000 even though the average expected gain in the latter choice is $32,000 (40,000 × .8). What if the choices are either a sure loss of $30,000 or an 80 percent chance of losing $40,000 and a 20 percent chance of losing nothing? Here most people will take the chance of losing $40,000 in order to have a chance of avoiding any loss, even though the average expected loss in the latter choice is $32,000 (40,000 × .8). Tversky and Kahneman concluded that people tend to avoid risk when seeking gains, but choose risk to avoid losses.

22/ An item whose price has been increased by 50 percent and then reduced by 50 percent has had a net reduction in price of 25 percent. A dress whose price has been ‘slashed’ 40 percent and then another 40 percent has been reduced in price by 64 percent, not 80 percent. The new toothpaste which reduces cavities by 200 percent is presumably capable of removing all of one’s cavities twice over, maybe once by filling them and once again by placing little bumps on the teeth where they used to be. The 200 percent figure, if it means anything at all, might indicate that the new toothpaste reduces cavities by, say 30 percent, compared to some standard toothpaste’s reduction of cavities by 10 percent (the 30 percent reduction being a 200 percent increase of the 10 percent reduction). The latter phrasing, while less misleading, is also less impressive, which explains why it isn’t used. The simple expedient of always asking oneself: ‘Percentage of what?’ is a good one to adopt. If profits are 12 percent, for example, is this 12 percent of costs, of sales, of last year’s profits, or of what?

23/ According to government figures released in 1980, women earn 59 percent of what men do. Though it’s been quoted widely since then, the statistic isn’t strong enough to support the burden placed on it Without further detailed data not included in the study, it’s not clear what conclusions are warranted. Does the figure mean that for exactly the same jobs that men perform, women earn 59 percent of the men’s salaries? Does the figure take into account the increasing number of women in the workforce, and their age and experience? Does it take into account the relatively low-paying jobs many women have (clerical, teaching, nursing, etc.)? Does it take into account the fact that the husband’s job generally determines where a married couple will live? Does it take into account the higher percentage of women working toward a short-term goal? The answer to all these questions is no. The bald figure released merely stated that the median income of full-time women workers was 59 percent of that for men.

24/ If each of the ten items needed for the manufacture of something or other has risen 8 percent, the total price has risen just 8 percent, not 80 percent. As I mentioned, a misguided local weathercaster once reported that there was a 50 percent chance of rain on Saturday and a 50 percent chance on Sunday, and so, he concluded, ‘it looks like a 100 percent chance of rain this weekend.’ Another weathercaster announced that it was going to be twice as warm the next day since the temperature would rise from 25 to 50.
197 reviews5 followers
January 18, 2015
In "Innumeracy", John Allen Paulos argues that the level of mathematical illiteracy in the United States is shocking and unacceptable, that innumeracy has real and pernicious negative effects, and that it is promoted by poor teaching.

It's a pity: I share Paulos's love for mathematics, and I agree with the message of "Innumeracy", but I find his approach glib and pompous. At one point he says he finds it hard to write at length, preferring brevity and concision. That's all well and good, but it can lead to being condescending and dismissive, to grumpy ranting instead of a full and detailed argument. (In extreme cases it can even lead to intellectual bullying: "I know better than you, you're an idiot, convincing you is not worth my trouble, so I'm just going to bellow a bit". Thankfully Paulos doesn't go this far, but he does hint in that direction from time to time.)

Much of the same territory is covered better, and without the ranting, by Nate Silver in "The Signal and the Noise", and Leonard Mlodinow in "The Drunkards Walk".
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