How Not To Be Wrong: The Power Of Mathematical Thinking (2014) - Plot & Excerpts
A better than average "Math is cool and is about helping you be a better thinker" type of book. Some notables:A … teacher’s least favorite thing to hear from a student is “ I get the concept but I couldn’t do the problems.” Thought the student doesn’t know it, this is shorthand for “I don’t get the concept.”Fields Medalist David Mumford has suggested that we replace plane geometry with a first course in programming…Dangers of linear regression: "by 2050, we'll all be fat..."Describing things in terms of percentages "90 percent of americans are against X"The confusion of divergent series: instead of the sums being described as “what is”, they should be “how shall we define” (Cauchy)“When am I going to use this?” (students complaining about math class). Doing drills at soccer – why do it if you are not going to be a professional? Healthier, happier, and more appreciative of the professionals…Same for math.Doing arithmetic in your head: “You can’t write a sonnet if you have to look up the spelling of each word as you go.”“If your government isn’t wasteful, it’s probably spending too much money on eliminating waste!”Not all money is the same: utility curves need to be used to offset the variable impact of $1000 for a poor man vs for a rich man. I am, by training, a mathematician. However, for the most part, although still fascinated by the beauty and power of mathematics, I don’t read mathematical books for pleasure. But every once in a while, a new mathematical book is published and I read it, and often enjoy it immensely. The last such book I truly liked and had trouble putting down was Here Comes Euclid by Alex Bellos. Jordan Ellenberg’s How Not To Be Wrong: The Power of Mathematical Thinking falls into that category. I found this to be a very interesting and quite enjoyable read.What Ellenberg, a professor of mathematics at the University of Wisconsin-Madison, has done is by using stories and anecdotes both historical and from current events, written a book on how mathematics (particularly statistics) is used to analyze, solve and make correct decisions. To do this, Ellenberg has divided the book into five parts: Linearity, Inference, Expectation, Regression, and ExistenceIn Part I: Linearity, Ellenberg’s main focus is that how many mathematical models are assumed to be linear and in doing so, major mistakes and decisions will be made. His own summary of the section is, “Includes: the Laffer curve, calculus explained in one page, the Law of Large Numbers, assorted terrorism analogies, ‘Everyone in America will be overweight by 2048,’ why South Dakota has more brain cancer than North Dakota, the ghosts of departed quantities, the habit of definition.” (pg. 19)Ellenberg begins his first chapter with a quote from the debates over the Affordable Health Care Act in which a member of the libertarian Cato Institute wonders why the US was trying to be more like Sweden when Sweden was trying to be more like the US. Sweden known to have universal health care, in attempting to become more like the US might argue that Sweden wants to back away from that position. However, to draw this conclusion, one would need to assume the models comparing prosperity and “Swedishness” are linear in that as the amount of “Swedishness” increases, the overall prosperity decreases. In rough terms, the model would be linear with a negative slope (going down as we move from left to right). However, this is not accurate. The actual model is more like a parabola (think of the Gateway Arch as a rough approximation). In this case, the US lies to the left of the top of the Arch and so as it moves right (more “Swedishness”), prosperity increases. Sweden, on the other hand lies to the right of the top of the Arch. In this case, as it moves left (more “USishness”), prosperity also increase. Only if one or the other is sitting at the top does a move left (“USishness”) or right (“Swedishness”) cause prosperity to decrease in both directions. This curve is essentially the Laffer curve, originally created by Arthur Laffer in 1974 to explain the relation between government tax rates and revenue. Depending on where the US is on the curve, one should either increase the tax rates to raise revenue (if we’re on the left side of the peak) or decrease the tax rate to increase revenue (if we’re on the right side of the peak). So where are we? This is a question of considerable importance. Unfortunately, there is no actual exact answer due to so many factors and the changes in tax rates. This now has become ensnarled in partisan politics.The next chapter takes on the issue of how if we are looking at a curve (not a line) but we zoom in closely, the curve resembles a line. This idea is behind the invention of calculus. It is here that Ellenberg explains calculus in one page. The idea is that if we zoom in close enough, our curve will resemble a line. Using what we know about lines and basic geometry, we can make conjectures and then use these to make statements about the curve. For example, suppose you wanted to find the area between a horizontal line and a curve above it. We could zoom in until the curve looks like a piece of a line segment. Then using geometry (rectangles, trapezoids, triangles, etc.), we can find the area and then “add” all these up to get the total area. This idea is also used to add up an infinite collection of numbers. This was done in our early school days when we converted decimals to fractions. It’s also used to resolve some famous problems in Greek antiquity called Zeno’s Paradoxes.His next chapter tackles the issue that all Americans will be obese by 2048. This claim is from a paper published in Obesity, “Will All Americans Become Overweight or Obese? Estimating the Progression and Cost of the US Obesity Epidemic” by Yousfa Wang, et. al., in 16, no. 10 (October, 2008) pgs. 2323-30. The gist is that is the data is plotted and a straight line fitted to it (a process known as linear regression), the extrapolated line would lead to 100% of Americans being obese by 2048. Herein lies the problem, although the data may look like the line is a nice fit, in reality, the data is better fitted with a curve.Ellenberg next tackles the use of statistics in one area (deaths in Isreal) to predict the values in another (deaths in the US). This proves to be very problematic in that in order to project these numbers, circumstances and basic assumptions would need to match. This is not the case. This leads to his discussion of brain cancer rates in South and North Dakota. The resolution of the discrepancy in cancer rates is tied into the actual populations of these states. The smaller the overall population, the higher the rates will appear. This leads into a discussion of the Law of Large Numbers. This basically says the more times an experiment is done, the more likely it will be closer to the theoretical expected value. The fewer number of times something is done, the more likely the experimenter will see extreme outcomes. The example Ellenberg uses is flipping coins. If you flip a coin 10 times, you may get only 1 head or 10 heads, with some frequency. However, if you flip the coin 1000 times, you will most likely get something very close to 500 heads. This law applies to many things. He uses it to analyze performance in school districts. Schools with few students can show remarkable success while larger schools with vastly more resources show only moderate success. A quick look at the discussion in David and Goliath, Malcolm Gladwell’s recent book, regarding class size, is an interesting addition to this argument.I have gone into considerable details for the first section to give a reader an in-depth sense of what is in this book. For the remaining sections, I can only say they are very similar in nature and make for fascinating reading. I will quote Ellenberg’s summaries of each section so you can get some sense of what is discussed.In Part II: Inference, Ellenberg looks at the basic tools of statistical inference (hypothesis testing, etc). He says the Part “Includes: hidden messages in the Torah, the dangers of wiggle room, null hypothesis significance testing, B. F. Skinner vs. William Shakespeare, ‘Turbo Sexophonic Delight,’ the clumpiness of prime numbers, torturing the data until it confesses, the right way to teach Creationism in public schools” pg. 87Part III is Expectation. This Part “Includes: MIT kids game the Massachusetts State Lottery, how Voltaire got rich, the geometry of Florentine painting, transmissions that correct themselves [information and coding theory], the difference between Greg Mankiw and Fran Lebowitz, ‘I’m sorry, was that bofoc or bofog?,’ parlor games of eighteenth-century France, where parallel lines meet, the other reason Daniel Ellsberg is famous, why you should be missing more planes” pg. 193Part IV is Regression. This “Includes: Hereditary genius, the curse of the Home Run Derby, arranging elephants in rows and columns, Bertillonage, the invention of the scatterplot, Galton’s ellipse, rich states vote for Democrats by rich people vote for Republicans, ‘Is it possible, then, that lung cancer is one of the causes of smoking cigarettes?.’ why handsome men are such jerks.” pg. 293Part V is Existence. This “Includes: Derek Jeter’s moral status, how to decide three-way elections, the Hilbert program, using the whole cow, why Americans are not stupid, ‘every two kumquats are joined by a frog,’ cruel and unusual punishment, ‘just as the work was completed the foundation gave way,’ the Marquis de Condorcet, the second incompleteness theorem, the wisdom of slime molds” pg. 363Ellenberg concludes his book with a chapter, “How To Be Right.” In this he offers a way to think about and use mathematics to get to the correct decision. As an illustration he discusses the methods used Nate Silver to accurately predict the results of the 2012 Presidential Election.As I wrote at the beginning of this now lengthy review, I thoroughly enjoyed the book. I did find it had to put down. His explanations and illustrations are for the most part clear and quite readable. Overall, I have very little to criticize in it. However, I did find a couple of mistakes: one I am sure is a typographic error and the other was an incorrect scatterplot. The first was on pg. 37. In his calculation of the area of a circumscribed octagon in a circle, he has the area is 8sqrt(2-1). This was clearly wrong for if it were correct, the answer would just be 8. Working through the mathematics (and thus, reinforcing my nerd cred), the correct answer is 8((sqrt(2)-1). The incorrect scatterplot is at the top of pg. 332. I did my own scatterplot for the data and have inserted it into the book. But, for a book of this magnitude and span, these are certainly forgivable and should not detract from the value of the book. I highly recommend it to anyone who wants to see how mathematics is critical in today’s world and would like an answer to the question of when am I going to use this?
What do You think about How Not To Be Wrong: The Power Of Mathematical Thinking (2014)?
Accessible, well written book, but definitely a slow read for me, not being a math person.
—Aroun
Would like to buy a copy of this and have it to reread parts. Very informative.
—Jenni
The rare book better read in chunks, separated by another book. Finishes strong.
—spence
This book is a hilarious, interesting read. I highly recommend it.
—naja
