I was drawn to How Not to Be Wrong: The Power of Mathematical Thinking, by Jordan Ellenberg, because I wanted to solidify my understanding of various mathematical topics. In addition, I enjoy reading books that are geared towards increasing your intuitive understanding of a technical subject.
A little background on me and my math career, I was a math major who was focused on theoretical math for my undergraduate degree. Once I reached my final year, I knew I needed more schooling but decided theoretical math was not for me. I realized my Master’s degree should be around applications of math, specifically in finance. Throughout my schooling, I took multiple higher level classes in Calculus, Probability, Statistics, and Algebra.
In addition to enjoying to think about the subject, I use statistical methods in my day job; How Not to Be Wrong:The Power of Mathematical Thinking could help me improve my performance.
Summary of How Note to Be Wrong
How Not to Be Wrong: The Power of Mathematical Thinking is an easy and enjoyable read. How Not to Be Wrong: The Power of Mathematical Thinking is split up into five sections: linearity, inference, expectation, regression, and existence. Ellenberg looks to explain each of these concepts using stories from the past and non-technical examples. To read this book, you do not need any formal mathematical training – there are no equations and many examples use illustrations to help the reader visualize the material.
One criticism I do have for the book is it could be organized better. There were some chapters where I thought Ellenberg got away from the point of the section.
I’ll give a brief description of each section below:
Many mathematical models are linear and are used incorrectly. The point of this section is to make the reader aware that non-linearity, i.e. curved lines, are much more common in the real world. Some topics Jordan Ellenberg looks at in this section include, but are not limited to, government fiscal policy (what tax rate maximizes tax revenue?), curves are straight locally but curved globally, and projecting obesity in 2048 (100% of Americans will be obese based on the current trend!).
Many mathematicians use hypothesis testing to make conclusions about statistical tests. This is commonly taught in entry level statistics courses. Ellenberg discusses how many people produced statistical “tests” showing that various messages were could be found in the Torah if you mix and match the letters in a mathematical way (Ellenberg says it is kind of bogus). Ellenberg also discusses the hypothesis testing, p-values, and how to appropriately draw conclusions (essentially proof by contradiction).
Expectation is the average value of a situation (put in layman’s terms). In this section, Ellenberg talks about how multiple groups won large sums of money playing a lottery game in Massachusetts, and looks at common misconceptions regarding the aggregation of the average value of various situations (the concept of additivity).
When mathematicians use regression and correlation analyses to to draw conclusions about two variables. A common mistake is drawing the conclusion that variables have a relationship based on the data where it could just be chance they are related. In addition, you have to be careful when assigning the direction of the relationship. In the 1950’s, there were multiple studies on smoking and lung cancer. A question a mathematician could ask is “Does lung disease cause smoking?” (this is probably not the smartest question to ask, but the data could suggest this relationship).
This section talks about a few different situations where unexpected results arise. One of the situations is in a three person political race and using different ways to tally the votes will result in different outcomes.
Remarkable Concepts from How Not to Be Wrong
Here are some significant passages, takeaways and concepts that were memorable to me:
- Oldest False Syllogism: It could be the case + I want it to be the case = It is the case.
- Basic Rule of Mathematical Life: If the universe hands you a hard problem, try to solve an easier one instead, and hope the simple version is close enough to the original problem that the universe doesn’t object.
- The good choices are the ones that settle unnecessary perplexities without creating new ones.
- Use caution when using linear regression.
- Smaller datasets have more variation than large datasets.
- Don’t talk about percentages of numbers when the numbers might be negative.
- Improbable things happen a lot – “What is improbable is probable.”
- Genius is a thing that happens, not a kind of person.
- It is very difficult to apply traditional methods to “unknown unknowns”.
- “That is the characteristic of great scientists, they have courage. They will go forward under incredible circumstances; they think and continue to think.” – Hamming
- Correlation is not transitive (no correlation does not imply no relationship)
I recommend this book for anyone who is analytical and enjoys applying math or statistics in their daily lives. Whether you are an analyst at a company working with numbers, someone who plays games of chance, or someone who wants to increase their intuition of the subject. How Not to Be Wrong: The Power of Mathematical Thinking is an easy and enjoyable read and will be influential to any reader.
Do you try to gain intuition about technical subjects? Are there any influential books you could suggest?
Check out our other book reviews!