I recently read Jordan Ellenberg's book titled How Not to Be Wrong: The Power of Mathematical Thinking. Ellenberg packs this book with several read world examples of mathematical concepts that bring to light when we actually use mathematics in daily life. Ellenberg defines math as the extension of common sense by other means. To illustrate this idea he says, "Dividing one number by another number is mere computation; figuring out what you should divide by what is mathematics." So throughout the book he attempts to illustrate how we use mathematical concepts and thought in daily life as an extension of common sense. |

The text is split up into five sections that each tackle a different type of mathematics. The five topics are linearity, inference, expectation, regression, and existence. Each topic is crammed with several real life examples (some more complex than others) to show how math plays a part in everyday life including the lottery, politics, the height of fathers compared to the height of their sons, the relationship between smoking and lung cancer, as well as many more.

Part 1:Linearity was my favorite section of this book. Linearity seems so simple, and we as humans tend to think linearly because it comes so natural to us. The idea of change at a constant rate is just comfortable and easy to understand. However, Ellenburg points out that not everything is linear - a very important concept to remember when making generalizations or predictions using data (like estimating what percent of the population will be overweight in 2048). My big take away from part 1 was exactly this concept. You can't assume everything changes at a constant rate. Linear models are great to use with some things, but not with all things. Something everyone, not just mathematicians, should keep in mind. Some other interesting topics discussed in this section are the Law of Large Numbers, and the fact that percentages tend to act funny when negative numbers are thrown into the mix.

Part 2 was all about making inferences. This is where statistics and p-value starts to be used quite frequently in the book, and if you are like me and frequently get list in statistics jargon this can be pretty confusing. However, when the text gets heavy in the mathematical computations the author takes you through, skimming is enough and he will explain in more detail at the end so you can still understand. The big reoccurring theme in this section, and my big takeaway, is that improbable things happen a lot, so much so that "one might argue that what is improbable is probable." The author stresses that impossible and improbable are not the same thing since impossible things can never happen. Other big concepts were statistical significance, the idea of the Boston Stockbroker, and Bayesian Inference.

Part 3 discussed expectation. The hot topic here is expected value. Ellenburg's big example in this section is the lottery. This was another one of my favorite parts of the book. The book tells the story of some MIT students who figured out how to pretty much always win the lottery and figuring out if it is worth playing the lottery based on how the expected value of the ticket compares with the purchase price. Essentially if the expected value is less than the cost of the ticket-do not play the lottery. However, "in order to play the lottery without risk, it's not enough to play hundreds of thousands of tickets, you have to play the right hundreds of thousands of tickets." This is exactly what those MIT students figured out how to do. My big take away from this section was that the expected value is not actually what you expect for that specific thing, rather the "average value" if we repeated something several times.

Part 4 deals with regression to the mean. This idea is that overtime both things above and below the average with move toward the average. We can see this phenomena clearly in the shape of scatter plots formed by two correlated variables - there are more points near the mean. Correlation is discussed at length in this section, and it is important to know and understand that correlation is not the same as causation. Two things can be related without one variable causing the changes in the other. My big takeaway from this section were that correlation is not transitive. The idea of transitivity seems to apply in so many areas of mathematics it took me a while to wrap my head around the fact that correlation is not transitive.

The last part, Part 5, discusses the idea of existence. This is the part of the book that lost me, although I did find many of the topics to still be very interesting. My takeaway from this section is that majority rules doesn't always work. Ellenburg uses examples from US politics to explain this concept, but the explanation kind of lost me a few times. To illustrate this, let's say that in a poll 40% vote that their favorite color is blue, 35% of people vote that their favorite color is red, and 25% vote that their favorite color is orange. None of the options received 50% or more of the vote so there was no clear majority. However, 75% voted that their favorite color was NOT orange, 60% voted that their favorite color was NOT blue, and 65% voted that their favorite color was NOT red. Now we have 3 different majorities - who rules? This concept was fascinating to me because I had never seen majority rules in this light before. But, honestly, after this topic I was almost completely lost and had a hard time keeping up with the rest of the book.

Overall, I really enjoyed

And, possibly one of my favorite part of reading any book, quotes that really spoke to me:

"An important rule of mathematical hygiene: when you field testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something is wrong with your method" -pg. 64

"Just because we detect it doesn't always mean it matters." pg. 121

"Impossible and improbable are not the same-not even close. Impossible things never happen. But improbable things happen a lot." -pg. 137

"If you're not giving advice until you're sure it's right, you're not giving enough advice." -pg. 357

"If you try to improve your boyfriend's complexion by treating him to act mean, you've fallen for Berkson's fallacy." -pg. 362

"Ever tried. Ever failed. No matter. Try again. Fail again. Fail better."

-pg. 436

Part 1:Linearity was my favorite section of this book. Linearity seems so simple, and we as humans tend to think linearly because it comes so natural to us. The idea of change at a constant rate is just comfortable and easy to understand. However, Ellenburg points out that not everything is linear - a very important concept to remember when making generalizations or predictions using data (like estimating what percent of the population will be overweight in 2048). My big take away from part 1 was exactly this concept. You can't assume everything changes at a constant rate. Linear models are great to use with some things, but not with all things. Something everyone, not just mathematicians, should keep in mind. Some other interesting topics discussed in this section are the Law of Large Numbers, and the fact that percentages tend to act funny when negative numbers are thrown into the mix.

Part 2 was all about making inferences. This is where statistics and p-value starts to be used quite frequently in the book, and if you are like me and frequently get list in statistics jargon this can be pretty confusing. However, when the text gets heavy in the mathematical computations the author takes you through, skimming is enough and he will explain in more detail at the end so you can still understand. The big reoccurring theme in this section, and my big takeaway, is that improbable things happen a lot, so much so that "one might argue that what is improbable is probable." The author stresses that impossible and improbable are not the same thing since impossible things can never happen. Other big concepts were statistical significance, the idea of the Boston Stockbroker, and Bayesian Inference.

Part 3 discussed expectation. The hot topic here is expected value. Ellenburg's big example in this section is the lottery. This was another one of my favorite parts of the book. The book tells the story of some MIT students who figured out how to pretty much always win the lottery and figuring out if it is worth playing the lottery based on how the expected value of the ticket compares with the purchase price. Essentially if the expected value is less than the cost of the ticket-do not play the lottery. However, "in order to play the lottery without risk, it's not enough to play hundreds of thousands of tickets, you have to play the right hundreds of thousands of tickets." This is exactly what those MIT students figured out how to do. My big take away from this section was that the expected value is not actually what you expect for that specific thing, rather the "average value" if we repeated something several times.

Part 4 deals with regression to the mean. This idea is that overtime both things above and below the average with move toward the average. We can see this phenomena clearly in the shape of scatter plots formed by two correlated variables - there are more points near the mean. Correlation is discussed at length in this section, and it is important to know and understand that correlation is not the same as causation. Two things can be related without one variable causing the changes in the other. My big takeaway from this section were that correlation is not transitive. The idea of transitivity seems to apply in so many areas of mathematics it took me a while to wrap my head around the fact that correlation is not transitive.

The last part, Part 5, discusses the idea of existence. This is the part of the book that lost me, although I did find many of the topics to still be very interesting. My takeaway from this section is that majority rules doesn't always work. Ellenburg uses examples from US politics to explain this concept, but the explanation kind of lost me a few times. To illustrate this, let's say that in a poll 40% vote that their favorite color is blue, 35% of people vote that their favorite color is red, and 25% vote that their favorite color is orange. None of the options received 50% or more of the vote so there was no clear majority. However, 75% voted that their favorite color was NOT orange, 60% voted that their favorite color was NOT blue, and 65% voted that their favorite color was NOT red. Now we have 3 different majorities - who rules? This concept was fascinating to me because I had never seen majority rules in this light before. But, honestly, after this topic I was almost completely lost and had a hard time keeping up with the rest of the book.

Overall, I really enjoyed

*How Not to Be Wrong*. This book made me think about several mathematical concepts in ways I had never thought about them before, and I was able to take several things away from the text that I think will make me a better mathematician. I said before I got kind of lost toward the end of the book even with being a math major, so I think that some mathematical background would be helpful in understanding some of the more computational driven explanations. I would recommend this book to other math majors who would like to see things in a new light and learn a few things about the history of math without having to read something as content-dense as a traditional text book. This book will not teach you how to always be right. But it will teach you how to use mathematical thinking to give something your best guess. And, it will show you that it is okay to be wrong. as Ellenburg says, "we are trained to think of failure as bad, but it's not all bad. You can learn from failure."And, possibly one of my favorite part of reading any book, quotes that really spoke to me:

"An important rule of mathematical hygiene: when you field testing a mathematical method, try computing the same thing several different ways. If you get several different answers, something is wrong with your method" -pg. 64

"Just because we detect it doesn't always mean it matters." pg. 121

"Impossible and improbable are not the same-not even close. Impossible things never happen. But improbable things happen a lot." -pg. 137

"If you're not giving advice until you're sure it's right, you're not giving enough advice." -pg. 357

"If you try to improve your boyfriend's complexion by treating him to act mean, you've fallen for Berkson's fallacy." -pg. 362

"Ever tried. Ever failed. No matter. Try again. Fail again. Fail better."

-pg. 436