Innumeracy Summary and Review

by John Allen Paulos
Has Innumeracy by John Allen Paulos been sitting on your reading list? Pick up the key ideas in the book with this quick summary. For most people, mathematics is a difficult and tedious subject and even a feared one. We’ve all struggled with logarithms, sine functions, and probability distributions in school, and we’re still not comfortable with the concepts. Mathematics is generally considered a challenging discipline which is why so many of us shy away from it. In our society, it is quite common for people to boast that they were always bad at maths. Not being a “numbers person” is extremely common and so is not being able to grasp even the most basic concepts of probability and numbers. Consequently, it is safe to say that innumeracy has become a major issue. From our summary, you will learn about the negative effects of innumeracy and why it has become a major risk for our society. You will also see how people who are not familiar with maths struggle with simple day-to-day situations and how understanding basic mathematical concepts can be extremely helpful. Throughout this summary of John Allen Paulos’ Innumeracy, we’ll discuss:
  • how by owning just five shirts and three pairs of trousers you have many outfit choices;
  • why seeing a woman with blonde hair and a ponytail sitting in a yellow car next to a black man from Los Angeles is expectable; and
  • why astrology is considered a pseudoscience.

INNUMERACY CHAPTER #1: People who are innumerate might fail to react appropriately due to their inability to grasp basic mathematical principles.  

People don’t often admit to illiteracy, but they don’t have a problem acknowledging that they are not a numbers person or that they were very bad at maths in school. But lacking the basic notions of mathematics, or in other words, being innumerate shouldn’t be something to boast about. There are many negative consequences that being innumerate has, including not being able to make correct judgments and not having appropriate reactions in circumstances that involve probability and numbers. Because they are incapable to determine whether the figures that are given in different contexts are small or big, innumerate people will often personalize. Thus, their own experiences will often prejudice their numerical intuition. For example, the odds of being eaten by a predator are quite low, although alligators do sometimes eat people. The most common reaction that an innumerate person will have when reading about such an event in the news will be to develop a crippling fear of alligators. This fear stems from the fact that they will ignore the statistics that indicate how rare such alligator attacks are. Another disadvantage of being innumerate consists of not being able to grasp simple maths principles and their implications. Let’s take a moment to think of the multiplication principle. According to this principle, we have m ways to make a choice and n distinctive ways to make a subsequent choice. So, we have m x n combinations of choices. Let’s try to apply this simple mathematics concept to a very common situation. If a woman has three pairs of trousers and five shirts, she has fifteen (5 x 3) different outfit combinations. If we go even further and calculate the number of outfit choices that this woman would have for one week, then she’d have fifteen outfits to choose from each day of the week or a total of 15⁷ choices. This means that she actually has a whopping 170,859,375 different options! People who are innumerate are extremely likely to reject this conclusion, as they won’t be able to grasp how only a few items can result in such a huge number of choices.

INNUMERACY CHAPTER #2: People who are innumerate have a difficult time understanding coincidences, which although improbable, are quite common.

Christopher Columbus discovered the New World in 1492, and Enrico Fermi discovered the atom in 1942. Is this a coincidence? If you were to ask Sigmund Freud, he would probably say “I think not!” Freud, one of the most famous psychoanalysts of all times claimed that coincidences do not exist. But in order to discuss this concept, we need to define it first. A coincidence is an event that is extremely improbable but that happens very often. Coincidences are so common that people even expect them to happen. Coincidences have become such a well-established concept that they were even used in court. For example, in 1964, a woman with blond hair and ponytail stole a purse in Los Angeles. She then sped off with a bearded black man in a yellow car. Two suspects who fit this description perfectly were quickly apprehended and brought to the Supreme Court of California. Interestingly enough by using math, the court was able to make the argument that in a very large city like Los Angeles, there were many similar couples and all except one were innocent. Lucki the court used math to argue that in a city as large as LA, it was probable that there were many such couples, most of whom were innocent. So, the suspects were allowed to walk free. The likelihood of coincidences can sometimes be explained by probability and statistics. But, because innumerate people will have a hard time understanding these concepts, they will confuse simple nuances and ignore the statistical evidence. For example, they are often shocked by the fact that while some coincidences are extremely likely to happen, to them, it seems extremely unlikely that a specific coincidence will happen. If 23 people are attending a party, the chances that at least two of them have the same birthday is a good fifty percent. However, the chances that two people will share a birthday that has been specifically selected are slim.  In order for two people attending the party to be born on September 21st, the party should have at least 253 guests for the chances to reach a level of fifty percent.

INNUMERACY CHAPTER #3: People who are innumerate tend to fall in the traps of pseudosciences.

Although absolute truths are the foundation of mathematics, its applications are not always one hundred percent correct. In fact, there are entire industries that misuse mathematics in order to build pseudosciences. If you’ve ever used phrases like “We are not compatible because he was a Leo” or “I am feeling under the weather because Mercury is in retrograde”, then you might have fallen into the trap of astrology, which scientists consider to be a pseudoscience. According to some of the basic concepts of astrology, each one of us is affected by the ways in which gravity pulled the planets when we were born. Astrologists claim that our personalities, our everyday lives, our moods, and pretty much everything that happens to us can be explained through this pseudoscience. Interestingly enough, the field of astrology is based, to some extent, on physical and mathematical concepts, which state that a body or an object’s mass is proportional to its gravity pull. In other words, as the square distance between two bodies increases, the gravitational pull between them decreases. It’s easy to prove that the field of astrology is a complete sham since astrologists misuse and bastardize the laws of physics and mathematics. The doctors and the nurses who helped your mother bring you into the world have a much greater gravitational pull than some planets that are millions of miles away. A 986 Gallup poll revealed that more than fifty percent of young Americans believe in astrology, which indicates that this topic is extremely alluring. But it’s not just young folk who don’t have a good understanding of mathematics and physics that fall in the trap of pseudoscience. Even Freud himself, who was a great psychoanalyst, fell into such a trap when his close friend Wilhelm Fliess managed to convince him of the special properties that the numbers 23 and 28 had. Fliess argued that these numbers were special because if you add and subtract their multiplies, you can obtain any number. If you want to get the number 6, you can make the following calculations: 6 = (23 x 10) + (28 x -8). The calculations that Fliess made were correct, but not because those numbers have anything special. You can take any two numbers that don’t have any common factors and get the same results. So you can replace 23 and 28 with 11 and 24 or with 2 and 3. But if even a man as brilliant as Freud was eventually convinced that those numbers were special, the fact that pseudosciences have such success shouldn’t come as a surprise. Unfortunately, because math gives the impression of universal truth, it can be very easy for tricksters to use it in order to manipulate those who are not familiar with maths.

INNUMERACY CHAPTER #4: Poor education, misconceptions about maths, and psychological blocks are the leading causes of innumeracy.

So what are the leading causes of innumeracy? An important factor that leads to innumeracy is the way in which the subject of mathematics is thought in schools. Although students become familiar with the basic principles of maths such as substracting and adding, they do not learn how these concepts can be applied in real life. Let’s say the students are given the following equation: (1-¼) x ⅕ =? For most students, this problem might feel completely irrelevant. But if they were asked to solve the following problem: Calculate the percentage of Indian global population if of the total world population, one quarter is Chinese and one-fifth of the remaining population is Indian. Much more students will get the answer to this question right than the answer to the equation. It would be much easier for students to understand the importance of maths if maths teachers would use more examples like the one above. Understanding how maths can be applied is much more effective than mechanically learning abstract concepts. A second issue that can lead to innumeracy consists of the psychological blocks that can be paralyzing for many people, especially for those who already think that they are bad at maths. This fear of mathematics is also known as math anxiety and it can be triggered when emotional scarring and intimidation is associated with the subject. Whether it’s an aggressive teacher who would raise their voice when the students had difficulties, or maybe it’s a parent who always told their children that they are bad at maths, the result is the same - an adult with math anxiety. Like Freud, a lot of capable and highly intelligent people can be afraid of maths, but just because they have difficulties understanding certain concepts doesn’t necessarily mean that they are incapable. If a person has math anxiety, the first step towards becoming friends with maths is becoming more confident. Even if it seems silly, becoming friends with mathematics can be quite easy. Once your confidence has been boosted, the only thing that needs to be done is solving many simple problems. Slow but steady progress will surely follow. Innumeracy can also result from the misconceptions that are perpetrated by people who have a natural aversion towards it. A common misconception about mathematics is that it’s a mechanical, cold subject that numbs people from fully embracing and appreciating humanistic studies. But this couldn’t be further from the truth. Saying that maths prevents us from being good at humanistic studies is like saying that eating salty foods prevents us from liking sweets.

INNUMERACY CHAPTER #5: If we understand maths, we can make better decisions in our daily lives.

Mathematics isn’t restricted to numbers and abstract ideas. It is actually closely related to many different activities that people do in their daily lives. Take the concept of trade-offs for example. When faced with making a decision, we often need to make a trade-off between two or more different concerns. Whether it’s determining which charity we should donate to, or what friends to invite to our party, these trade-offs can be made more efficiently and wisely by people who understand maths. There are two types of statistical errors that need to be taken into account when we make a decision: type-1 and type-2 errors. The errors that happen when we reject a true hypothesis are type-1 errors. A good example of a type-1 error is not believing that smoking causes cancer. When we accept a false statement as being true, we are dealing with a type-2 error. Believing that the earth is flat is a great example of a type-2 error. When people need to answer a question or to make a decision, they will have a unique understanding of what the error actually is. For instance, when it comes to capital punishment, people who have a more liberal approach would focus on avoiding type-2 errors because they want to avoid any unfair suffering. However, a conservative would probably gravitate towards avoiding the type-1 error because they want to make sure that the criminal gets what he deserves. Not only can math provide fascinating insights about our society, but it can also be used to make more informed decisions. For instance, just a basic understanding of statistics and probability can prevent people from wasting their money on fake discounts. If you find a dress with a forty percent discount, and the store makes and additional forty percent discount, you might jump at the occasion to buy it, thinking you just scored an eighty percent discount.  In truth, you only got a sixty-four percent discount from the original price, as the item was already reduced by forty percent once, and an additional forty percent is equal to just twenty-four off the original price.


What is the key message of John Allen Paulos’ book Innumeracy? The key message of the book is that having a good understanding of mathematics can improve our everyday lives. Whether it’s having a better perspective over the news that we hear or about knowing how to detect a pseudoscience or a fake discount, there’s no denying that maths is an essential part of our existence and that innumeracy is a real problem.