[Thanks for Aaron Adair for this. The original post on his excellent blog can be found here]
Connecting to my previous post on the need to be well-read-up in order to do critical thinking, I am going to be doing a series of book reviews.
Here I want to discuss a few books published recently and which I have read in the last few months that are focused on math. They are not books on how to do math (i.e., textbooks), but instead they discuss mathematical concepts and their relations to ways of thinking about the world. Sometimes they touch on theological issues, sometimes a lot. But all three are good reads.
Infinity is a really, really weird concept. It takes any intuitions we have and makes us say apparently silly things. But there are rigorous ways of dealing with infinity, but there are also limitations, even for the most brilliant mathematicians.
One of the points is that you never really reach infinity. No matter where you start on a number line or how long you count forward, you never even get closer to infinity. This means that it is not possible to use something finite to create an infinite set. That is, you cannot construct infinity from finite sets and operations. Hence we get lazy when writing a set that is supposed to go on forever with … (hence the title of the book). And yet we can talk about infinite sets. In fact, we can talk about different sized infinities. If that didn’t make sense to you, then you are getting the point about how weird infinity is.
In this book, mathematician James Lindsay shows many important points about how infinity is used and understood by mathematicians and how the terminology is poorly used in other contexts, especially when applied to God. In many ways the book is focused on problems with the infinite god concept, but what I found as one of the more interesting threads running through the book is the problem with mathematical Platonism. What Lindsay shows very well is how much math is a human project. We chose the various axioms and definitions, and those different choices can lead to all sorts of amazing conclusions. But showing how much math is a human invention, it shows that there isn’t really a “true form” of the set of all rational numbers and the like. We chose the rules. Historically, there have been arguments about whether negative numbers are really numbers, or if i is a number or not. Or even if zero is a number! Why do most people consider these objects numbers in the end? Because of what we can do with them. They are practical, even imaginary numbers (I couldn’t do the physics I learned in grad school without them).
Seeing the human side of math (rather than the human side of certain mathematicians) was excellent, especially when it comes to the sorts of concepts that bugger human comprehension. I value the volume for doing more than just showing what makes an infinite God incoherent, but it shows how much math is truly a human adventure and not simply that boring stuff forces on you in school.
Infinity blows our intuitions away because of how far it is beyond our experience, but we live in a world where apparently random things happen. Nonetheless, we get confused and confounded by probabilities all the time. Here physicist Leonard Mlodinow shows many aspects of the historical threads that touch on how we developed and understand probability. There is plenty of talk about gambling with games of chance, but that’s largely because a lot of probability theory was first developed to understand just that. Sometimes we joke that lotteries are a tax on those that are bad at math, but really understanding math (especially statistics) belies that argument (a really good anecdote for that is talked about in the next book below).
When it comes to thinking about things in the political arena, this book is great because of the focus it provides on judicial cases. There are plenty of ways that probabilities can be abused, where meager evidence is made to look extraordinary by making sins of probability calculation. In particular, not all probabilities are independent. Consider the following numbers that I have made up: 1/3 of men have a mustache, 1/3 of men have a beard. What is the probability of a man having a mustache and a beard? If you multiple those two fractions together, you get 1/9. However, these are hardly independent probabilities; if you have a mustache, you are more likely than average to have a beard as well. Heck, with current fashion trends as I have noticed it may be more rare to have a mustache without a beard–perhaps it makes people think porn-stache. Why would this sort of consideration be important? Well, suppose you have an eyewitness say a suspect has a mustache and beard. If you nab a person fitting that description he won’t be nearly as likely a suspect as you think; it is not 1 in 9. Moreover, considering that in a given city there could be a million people, then the chances of nabbing the right suspect by using these criteria are really bad. And yet a case like this (with some additional details) was used at first to convict a couple of a significant crime.
It’s number games like this that make it in fact difficult to know what are the real numbers if you are on a jury. You can be told that a DNA sample matching a random person is less than one in a million, and so you think a positive match in a court case means that person’s blood being at the scene of the crime is really high (more than 99.9999%). But a more relevant probability is the lab making a mistake, like mixing up which sample to test. That is much more probable. Still not likely (less than 1 in a 100), but not as super-solid as you think.
So if you are deciding a person’s life, an idea of what constitutes good evidence and what reasonable doubt would mean should be well-shaped. Even if you don’t have numbers, thinking about things in terms of less than, more than, equal, much less than, etc., is very important to making such decisions. And for a lot of other things. We live in a world that is governed by the unpredictable and probabilistic, and not just at the quantum level. Books likes this from Mlodinow help tune our probability intuitions. Also, Mlodinow is a great and humorous writer.
The title is ambitious, no? But there is truth in how powerful thinking mathematically can be. A refrain of this book is that math is the extension of common sense. Instead of thinking of math as a bunch of algorithms to memorize and steps to get to the “right” answer, what math is supposed to do is model ones thinking. And if you model your thinking clearly and consistently, then you can more confidently achieve accurate results.
More importantly, though, it to see if you project your reasoning does it lead to nonsense? In the first section of this book by mathematician Jordan Ellenberg there is a lot of emphasis on how not all curves are lines. By that, he means that not all trends are so straight-forward as have more, get more. For example, there is the concept of the Laffer Curve. This is about what should the tax rate be if the government wants to maximize its income. Obviously if the tax rate is zero, then the government will take in nothing. If you raise the tax rate, then you can bring in more money. But does that mean a higher rate rate always brings in more? If you made the tax rate 100%, then no one would have an incentive to work since all of their money is taken from them. Might as well not work at all, or find ways of hiding it. So again, the government will take in no income. Thus, the optimum tax rate for government income must be between 0 and 100%. Where that is is an empirical question (and if that is a worthy goal is another question), but clearly increases the tax rate is not going to necessarily mean more income. And of course, if you absurdly make the tax rate greater than 100% then no way will anyone work since you will owe more than you have no matter how much you work. In other arenas, it is clear that following a linear model of the relationship between things is not reasonable and even ridiculous.
Now, there is some overlap between this book at the previous two above, but Ellenberg’s volume covers a lot of areas in math and its relation to arguments and concepts in the real world, sometimes with surprising results. Would you guess that using a finite geometry you can figure out what are the best lottery tickets to pick are, given certain lottery rules for winning? A group buying up tickets in Massachusetts did, and they pulled in some serious dough.
One of the sections of the book I found most interesting is how our measure of public opinion can give contradictory results. Suppose on third of voters want to not cut spending, one third want to cut spending and cut it from defense, and one third want to cut spending and cut it from entitlements. If a politician doesn’t cut spending, 2/3rds will be unhappy; if a politician cuts spending in defense, 2/3rds will be unhappy; and if a polotician cuts spending on entitlements, 2/3rds will be happy. In other words, no course of action is the “right” one, as no matter what a majority will be against you. Now, you could instead rank priorities. For example, the first person may say their first choice is to no cut spending at all, but if spending had to be cut it should first come from defense before coming from entitlements. Another person can also rank their choices as they see fit. If you say the first choice is given 2 points, the second choice 1 point, and the third choice 0, then you can add up the choices of everyone and find where the ranked preferences lead. This method, called the Borda count, is similar to how your GPA is calculated (A = 4, E/F = 0), and it is used in some elections around the world. With it, you can get a different result than you would with the majority vote system. You could imagine how it would have affected the 2000 US Presidential elections; almost all of the votes for Ralph Nader would have gone to Gore and then Gore would have wold Florida (and thus the presidency), so a Borda count could have changes the history of the first decade of the 20th century (how much, I don’t know). But there exist other ways of potentially running a voting system, and they all have points to consider. If you consider all the options, you realize that it is possible to get different results with different polling/voting methods. So it’s almost like there isn’t really public consensus on issues, unless there is a significant majority.
I won’t get into everything Ellenberg goes into, and I want to leave his stories of the various mathematicians and statisticians as he tells them. But the key point is that you can see how you can build your own mathematical models of what you think is correct or fair and reasonable and see that it implies. Don’t consider math just stuff with calculators but a way of thinking. It won’t guarantee you are right, but it’s a far more useful and enlightening way to figure out the world. The real world, not just that of abstractions.