How Not to Be Wrong: The Power of Mathematical Thinking by Jordan Ellenberg
In How Not to Be Wrong: The Power of Mathematical Thinking, Jordan Ellenberg demonstrates how mathematical reasoning can be used in real-world settings. Ellenberg attempts to use layman terms to describe what are some deep mathematical ideas. In many ways he succeeds, often using simplified examples to demonstrate the principles. But at times his explanations require if not understanding of technical mathematics, then at least the ability to follow mathematically-logical paths of reasoning. Many of his topics relate to data, statistics and probability. He discusses, among other things, linearity, inference, expected value, correlation, regression to the mean and, in general, the uncertainty of mathematics in relationship to everyday phenomena. This article presents a small sample of the concepts and examples that Ellenberg discusses.
Data and statistics are often used to make predictions. Data are collected and analyzed for trends, and the trends are used to predict what the future will hold. The problem, Ellenberg says, is that we often use “close-up” data that show a linear relationship to make predictions about real-world phenomena that in the long run are not linear. One of the examples he uses to illustrate this point is a study on overweight Americans in the journal Obesity in which the authors used a few data points to make predictions. The data showed a near-linear progression over the relatively short (35 years) timespan:
Using this model, the authors concluded that all Americans will be overweight by 2048. Ellenberg points out, however, that the trend cannot continue this way – that the trend line will eventually curve – because if it did not, it would mean that by 2060, 109% of Americans would be overweight. It is one of the instances of common sense overcoming application of mathematics that Ellenberg discusses.
Ellenberg tells the parable of the Baltimore stockbroker to illustrate that making inferences can lead to unwarranted conclusions. In this parable, you, a consumer, receive ten tips – one for each of ten weeks – from a stockbroker (from Baltimore) that predicts a particular stock will rise or fall, and each week the stockbroker is correct, and the stockbroker solicits your business. On the eleventh week, thinking that the stockbroker can predict the future, you invest in a stock he predicts will rise and pay a commission to the stockbroker – and then the stock drops and you lose a great deal of money while the stockbroker still makes money. What happened?
What happened is that the stockbroker set you – and many, many other people – up for failure and set himself up to make a great deal of money by sending out the tips to a very large number of people. However, the tips are different: Half the people receive a “stock-will-rise” tip and the other half receive a “stock-will-fall” tip. The stockbroker then sends the next tip only to those people who received the correct tip. He continues this process, sending new tips to only those who receive a correct tip. Since the probability of being correct on any one prediction is , then to be correct ten times in a row, there is a chance. This means that for every 1,024 people the stockbroker sends tips, one will have received all 10 correct tips. As the recipient of the 11th tip, you, of course, do not know you are only 1 in 1,024 people to receive all correct tips – and so you infer, incorrectly, that the stockbroker is always correct.
The concept of expected value is used in a variety of ways, with application to a lottery being one to which most readers can relate. Ellenberg tries to make it clear that expected value, as concerns the lottery, is the average value you would expect to win each time you play over the long run of playing the lottery; it is not the average value you could expect to win on any particular lottery draw. Expected value is rather simple to calculate: multiply the probability of each particular outcome by the value of the outcome and add those products. Here is the calculation for the Powerball example Ellenberg gives (p. 202):
This means that, on average, you would expect to win about 94 cents for each time you play. But, if it costs $2 to play, that means you would expect to LOSE $1.06 for each time you play. The conclusion is that while the lottery may make a few people rich, for most, playing the lottery is a bad bet.
Writes Ellenberg in the final chapter of the book: “Mathematics is also a means by which we can reason about the uncertain, taming if not altogether domesticating it” (p. 425). Many people think of mathematics as an exact science: Put specific numbers into a formula and get specific numbers out. But that is not really what mathematics is about. It is about trying to use mathematical processes and mathematical thinking to make sense of the world around us. So much of our world is uncertain that mathematics can only assist us in making sense of it. It cannot provide certainty when certainty does not exist.
Ellenberg uses the example of predicting winners of elections to demonstrate this idea. Elections are just the collection of data (votes) in a few categories. Basically, he says, there is no perfect voting system. The 2000 presidential election is one of the foremost examples of this, and in the end, no math could settle the argument, and it was left in the hands of the learned men and women of the United States Supreme Court. In the election, the recipient of the electoral votes of Florida were in dispute. George Bush had received 48.85% of the popular vote, Al Gore had received 48.84%, and Ralph Nader had received 1.6%. One of the mathematical arguments was: If Nader was not in the election – only Bush and Gore were – then who would those 1.6% of Florida voters chosen? Ellenberg describes various scenarios and voting schemes that would apply to such situations (where there are really only two viable choices, but a third, unlikely choice, is included). In the end, we see that there is no one way that would solve the problem – and thus shows the uncertainty that mathematics leaves on the table no matter what analysis and computation are carried out and the reason human judgment is not always trumped by mathematics.
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Cited in How Not to Be Wrong:
Wang, Y., et al. (2008). Will all Americans become overweight or obese? Estimating the progression and cost of the U.S. obesity epidemic. Obesity, 16(10), 2323-2330.