The Power of Mathematical Thinking
Life gets busy. Has How Not to Be Wrong been on your reading list for a while? Learn the key insights now.
We’re scratching the surface in this How Not to Be Wrong summary. If you don’t already have Jordan Ellenberg’s popular book on science and mathematical thinking, order it here or get the audiobook for free on Amazon to learn the juicy details.
Wouldn’t you love to never be wrong? You’re not alone, and it’s hardly a surprise. In our work, relationships, and even our hobbies, we learn from a young age that being wrong is a mistake. Constant correct thinking is a concept that seems almost impossible, right? Wrong (ironically!).
How Not to Be Wrong by Jordan Ellenberg discusses ways we can make life simpler by thinking mathematically.
The book dives into the world of mathematics and explores its applications in everyday life. By looking at simple and complex decisions, Ellenberg reveals our mistaken beliefs that lead to common errors in our thinking.
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About Jordan Ellenberg
Jordan Ellenberg is a mathematician and author. He is a professor at the University of Wisconsin-Madison and has written several books on math and its application in various fields. He received his Ph.D. in math from Harvard University in 1998 and has written for publications like The New York Times, The Washington Post, and Wired.
How Not to Be Wrong is a popular book that explores the ways mathematical thinking can help us understand and solve everyday problems.
StoryShot #1: Think in a Nonlinear Way
Nonlinear thinking means thinking logically about what you can and cannot control.
Consider the following statement: “Where you should go depends on where you are.” This nonlinear way of thinking helps you develop the skill of critical thinking and be better equipped to avoid mistakes. Imagine yourself in a car at a crossroads. When the light turns green, you drive across the road directly in front of you, not diagonally to the opposite side. This is linear thinking.
Thinking in a nonlinear way gives us freedom to make choices and move forward with our lives. It also prompts more questions, which leads to more answers. This allows us to acknowledge changes in our lives, even if we have no control over these changes.
Linear regression is a statistical technique that looks for a linear relationship between two or more variables. For example, there is a statistic that shows that for every extra $10,000 someone earns, they are 3% more likely to vote Republican. Linear regression can help you understand how different factors influence an outcome and make predictions based on new data.
However, to avoid reaching false conclusions, we must be aware that linear regression can’t be used for every set of data and, if misused, produces misleading results.
StoryShot #2: Understand That Math Is Part of Everything You Do
Your teachers weren’t lying when they said you would use math every day. And you probably aren’t even aware of when and how you use it. Calculating the length of your work commute or the budget for a night out, and even the timing of your French press coffee, requires basic math.
With this in mind, being right constantly is possible. At least theoretically. If math and its fixed rules are at the heart of everything we do, then following the rules of math should always lead to the correct outcome. This suggests that you can avoid being wrong if you follow these rules step by step. This concept is not naive and provides a sense of comfort in its simplicity.
The problem is that people tend to guess and estimate rather than look at the cold, hard facts. This is how mistakes happen, and it’s why people are sometimes wrong.
If we simplify a problem, it’s easier to find an answer. If you take a big problem in your life and break it down, you may find a solution for the simple version that can lead you to an answer for the bigger one.
Math is a powerful tool at the heart of almost everything we do. As math is at the forefront of our lives, it is vital to improve our critical reasoning skills.
This gives you the chance to be right more often than you’re wrong. Through a basic application of simple math, you will arrive at more accurate conclusions. And you can increase your chances of being consistently correct in everything you do.
StoryShot #3: Math Can Help You Win the Lottery
An “expected value” is the average of the values that a random variable has over many trials. For example, it describes the probability of winning versus losing money in the long run. If you were to visit a casino and play roulette, you could calculate your expected value to inform your decision-making.
Consider the probability of winning the lottery and the expected value of lottery tickets. Ellenberg recounts the story of how MIT students managed to “win” the lottery every time in their town and revisits the law of large numbers (LLN). The students began analyzing well-known lottery games such as Powerball and MegaMillions, but soon became intrigued by Cash WinFall. The game was designed so that when the jackpot reached $2 million without a winner, it would “roll down” and be distributed among players who matched fewer numbers. The MIT students used computer software to analyze historical data and predict when roll downs would occur. They exploited a flaw in the game’s design by purchasing thousands of tickets during roll downs. Eventually, their scheme was exposed, and the game was suspended.
Risk can be quantified, but uncertainty cannot. For example, if an urn contains 90 balls, 30 of which are red and the rest yellow and black, the risk of not pulling out a red ball is 2/3. However, it is impossible to quantify the chance of pulling out a black ball.
StoryShot #4: Math Can Help Us Make Better Decisions
One of the key concepts explored in the book is utility. This is a measure of the satisfaction or happiness that we derive from a particular action or decision.
Have you ever considered measuring your utility in standard units, known as utils? Imagine if you valued an hour of your time at home as one util. In that case, arriving two hours before your flight would cost you two utils, while arriving just one hour before would cost only one. It’s easy to see that missing your flight is far worse than simply wasting an hour of your time. By valuing your time in utils, you’ll have a better understanding of the true cost of your decisions.
Expected utility is a measure of the average utility that an action or decision will produce if it is repeated many times. Utility and expected value are both ways of evaluating choices in the event of uncertainty.
Ellenberg uses the Laffer curve as an example of how mathematical thinking can help us avoid being wrong about complex issues. He argues that we need to use mathematics to examine real-world issues with more rigor and nuance.
Laffer’s curve is a graphical representation of the relationship between tax rates and government revenue. It has played a prominent role in Republican economic theory for almost 40 years. The Laffer curve represents the idea that an increase in taxes does not necessarily result in an increase in government revenue. The curve suggests that when the tax rate is close to zero, raising taxes increases government revenue, but when the rate is close to 100%, raising taxes decreases it.
These days, most reputable economists believe that the level of taxation is currently on the left-hand side of the Laffer curve, which suggests that an increase in taxes can still result in an increase in government revenue. This opinion stands in contrast to the Reagan era, where the top tax rate was merely 35%, an amount that would have seemed absurdly low for most of the twentieth century.
StoryShot #5: Consider the Triumph in Mediocrity
We should question our desire to be right, even perfect, all the time. A perfect outcome cannot always be guaranteed.
Despite appearances, this point of view is not self-defeating. Instead, it is an insight into how we can, and should, praise mediocrity. Normality leads to many of the theories and math questions Ellenberg discusses. Some math is born out of a simple need to solve a problem. The resulting theorems have shaped our world for decades.
And what is normal? Do any of us experience a normal life? You might not be rich or famous, but why would these concepts make one life more or less ordinary than another? Math is as normal and central to everything as life itself.
Mediocrity is rarely encouraged in the modern world, especially for the younger generation. But what might be considered a “normal” life almost always has an element of the extraordinary.
Even the rarity of life itself means it has some innate value.
This approach helps us see mediocrity as a superpower that can often lead to the most extraordinary creativity.
StoryShot #6: Public Opinion Doesn’t Exist and Doesn’t Matter
The power of mathematics can teach us that public opinion does not exist, and, therefore, it doesn’t matter.
To illustrate this point, consider forums in which public opinion seems to matter most, for example, elections.
Election statistics can demonstrate the idea that “there are no answers” in a controversial way.
Everyone is different, so everyone has their own opinion. Therefore, public opinion cannot exist. Sure, there might be popular opinions amongst groups of people, but there are always those with adverse opinions.
What’s more, we can make mistakes when analyzing statistics. Public misunderstanding can also affect particular outcomes. This gives us further insight into how public opinion shouldn’t matter, as it can stem from inaccurate information.
Of course, we all know that’s not the way of the world. Public opinion is likely to always impact politics and other areas.
StoryShot #7: It’s Okay Not to Know Everything
It’s more than okay; it’s impossible. Humans do not know everything there is to know about our world and others. Knowing everything is a goal that can never be met.
And that is okay. After all, if we were born knowing everything, wouldn’t life be incredibly dull? To know nothing is power, because it gives us the chance to ask questions. And questions lead to one of two things: direct and conclusive answers, or the need to find out the answer through experimentation.
It is the latter that has resulted in human advances in science, technology, and even art. Without questions, there could be no answers. Answers lead to further questions, and the cycle inevitably continues. And it is important for humans as a species that it does. Knowing everything would mean no discoveries.
Instead of relying on public opinion, nurture your internal hunger for knowledge. True geniuses listen more than they speak, and active listening is a consistent way to learn. Based on this new knowledge, you can form your own conclusions, using the power of math to ensure these conclusions are based on logic. This will give you an increased chance of being right more often.
StoryShot #8: Anything Can Be Proven from a Contradiction
If you state it is October, you can logically conclude that next month is November. However, you can’t reasonably believe that next month will be January. Logic dictates that if it is currently October, November must follow, and to argue anything else is foolish.
There is, of course, a degree of math behind this concept, which Ellenberg goes into in some detail. But the takeaway is that being wrong has value because it has the power to lead us to a new conclusion.
Science and math rely on contradictions to research and prove (or disprove) new theories. The contradictions we experience every day can teach us valuable life lessons.
Being right means facing things you don’t know to understand them better, which can be daunting. But, through math and the power of contradiction, facing these uncertainties can be freeing and informative.
Elements of math we don’t understand can land us in trouble, especially when pursuing being right all the time. Studying life through contradictions can help us to develop other ways of thinking and will also increase our critical analysis skills. All with the simple power of math.
StoryShot #9: You Will Learn from Failure
Learning almost always comes from making mistakes. When you think about it, this has been true since your first lesson.
Being right about everything all the time is a pleasant idea. But it makes for a far more valuable life to fail occasionally.
If you had only ever passed exams and had never studied for them, you would have no experience of the value of study. When the time came for a test you weren’t ready for, failure would come quickly, knocking you out of your upward trend with devastating results.
When we start learning math, we’re guaranteed to fail at first. They say talent is born from pursuing an interest, and the same is true for math. But the key word here is “pursued.”
Ellenberg’s honesty about failure as a mathematician teaches us that clever people fail often. However, they push toward success and come up with new and creative ways to solve their initial problem. This is a mark of their intelligence.
If you embrace failure, you will feel more satisfaction from success than if you passed every test in your life, whether real or metaphorical.
StoryShot #10: Mathematics Is Just Common Sense
People sometimes feel intimidated by mathematics, but that’s because they can’t see how common it is. Math is, at its core, the study of following rules to lead to an accurate conclusion. If you follow these rules to the letter, you will find the correct answer.
The rules of math are common sense, so anyone can learn them. The difficulty seems to come because math is rarely viewed objectively. The reason for this is that whenever we are using math, it is for something pressing in our lives that requires an immediate answer.
But take a step back. It makes sense that two plus two is four, doesn’t it? You have two of something. When you get two more, you will always have the same number of items in the end. And the same rules are seen in almost all areas of math.
This is why you learn addition, subtraction, multiplication, and division at such an early age; you will use these concepts as you grow up. What five-year-old wouldn’t want to work out how many years it will be until they’re ten? When you understand math as a common-sense concept, it becomes less intimidating. You can then employ it in your life to always keep you on the right track.
It gets more complex when dealing with high-end physics and complex numbers, as these can be difficult concepts to grasp. But in everyday life, the math you find yourself using makes sense, even if you can’t always see it.
Be careful when dealing with proportions and negative numbers, as they can lead to incorrect or misleading information. Negative numbers do not represent quantities like positive numbers do, so operations such as percentages can lead to incorrect or misleading information.
StoryShot #11: Don’t Use Probability Alone to Assess Risk
We often think of risk and probability as interchangeable. However, they are more different than you might expect. Using probability to calculate an outcome is helpful, but using it alone to assess the risk of any particular decision is foolish.
The issue arises because risk doesn’t rely on probability alone. Physical circumstances, random luck, and even the kinds of people involved all contribute to the outcome.
Probability can be a vital tool for various problems, both mathematically and in the wider world. But using it to assess risk without considering other factors leaves you open to getting things wrong. And to danger.
Use probability with the other factors involved in a particular situation to assess the outcome as precisely as possible.
StoryShot #12: Question Everything
Question everything, not only in areas of math, but in life. Ask yourself two questions:
- What assumptions are being made?
- Are these assumptions justified?
This is particularly true of scientific and statistical-based conclusions. Many people hear a statistic on the news and believe it without question. But experts make errors more often than we’d like to think, which is why asking difficult questions is so important.
Human error is also a significant issue. It’s a fact of life that people mess up. However, this is no reason to treat everyone like they will mess up. When something seems wrong, question it. That way, you will eventually find the correct answer.
Data might be changed and edited to suit the person or organization using it. Language can also prove deceptive, as words can insinuate one idea while neglecting another.
Question everything you know and do your research to draw conclusions that make the most sense. You’ll be amazed at what you can teach yourself simply by asking, “Could this be wrong?” This allows you to draw conclusions that lead you on the right track more often than not.
You might never be right all the time, and you will almost certainly make mistakes. But there is value in being wrong. Mathematics can get us a little closer to the real answers we’re seeking.
StoryShot #13: Use Logical Reasoning to Arrive at Valid Conclusions
Understanding mathematics is not just about memorizing formulas. It is about using logic and reasoning to arrive at conclusions.
Ex falso quodlibet is a Latin phrase that means “from a falsehood, anything follows.” It refers to the idea that if you start with a false premise, you can prove anything. However, this is a fallacy. It is important to start with a true premise and use logical reasoning to arrive at a valid conclusion.
Theodore Roosevelt valued the importance of thinking critically and logically. He once said, “To educate a person in the mind but not in morals is to educate a menace to society.” This quote highlights the importance of not just knowing math but using it in a responsible and ethical way.
The book ends with an encouraging message. To truly love math is to use it for good, and to approach it with a spirit of curiosity and a commitment to reason.
Final Summary and Review
How Not to Be Wrong examines how mathematical thinking affects our lives. It is ideal for anyone interested in math, problem-solving, or critical thinking. By using non-linear thinking and the rules of mathematics, you can increase your chances of being right more often than wrong. You can do this by making problems simpler, forming conclusions based on logic, and applying basic math skills to everyday life.
Ellenberg presents complex mathematical concepts in a way that is easy to understand. He also uses a wide range of examples to illustrate his points. It is okay to not know everything. Nurture your internal hunger for knowledge by listening more and speaking less.
Let’s review the main takeaways from How Not to Be Wrong:
- Math is in everything we do. We can apply mathematical principles to everyday life so that we are wrong far less often.
- We shouldn’t be swayed by ‘public opinion’ as it doesn’t really exist.
- Failure is good for us as we learn from it.
- We can break down seemingly impossible problems and solve the smaller parts by applying logic and probability.
Would you like to be wrong less often? Tag us on social media and tell us if you have used mathematical concepts to increase your chances of being right!
We rate How Not to Be Wrong 4/5.
How would you rate Jordan Ellenberg’s book?
This unofficial summary and analysis was first published on 13/06/22. It was revised and updated on 24/03/23.
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