I was really expecting to enjoy this book. Unlike most high school kids, I enjoyed learning calculus. It just made sense to me. After all, derivatives are essentially subtraction taken to an extreme, and integrals are addition taken to an extreme. Things get complicated when you actually have to derive equations using limits or when you get into proving theorems in calculus, but that's true of math in general. The basic ideas of calculus are fairly straightforward.
Fernandez did a decent job of bringing that across, but something about the book didn't sit well with me. I thought his treatment of concepts was too superficial, and he never went beyond the trivial exploration of applications of calculus to everyday life. The general format of the book was to walk through a day in the life of Fernandez from waking up to going to bed and examine things like the sleep cycle, coffee, and television from the perspective of calculus. The concept was okay, but I just couldn't get into it. I wanted to see some more substantial analysis than what was there, and I thought that should have been possible while still keeping the book accessible to the wider audience that it was intended for.
As an example, in chapter 3 he shows how the derivative can be used when calculating the efficiency of driving a car with different rates of acceleration, but he assumed that the efficiency of the car itself was constant when calculating fuel consumption for a given trip. It would have been more interesting—and more realistic—to model the efficiency of the car dependent on acceleration and integrate over the resulting curve to figure out a more accurate fuel consumption for the trip. Then he could have showed how much of an impact the acceleration rate had on fuel economy. Granted, he didn't cover integration until later in the book, but still, he could have revisited it. I would have loved to see a more in depth analysis of that.
Some parts of the book did pique my interest. I thought the equation for sustainability analysis was pretty cool, and I enjoyed contemplating the questions that came to mind when Fernandez was describing how integration was developed. It took 2000 years to fully develop the concepts and structure of calculus, which makes it seem like it was very difficult to do. Indeed, the use of limits and infinitesimals was rejected as a valid method of calculation for a long time, and Isaac Newton had to overcome a lot of resistance to his ideas. Now calculus is routinely taught to millions of students around the world, and it's a basic requirement for many fields of study.
It's amazing to me that calculus has gone from a field that took two millennia to develop to the point that even a few people could understand and use it, to something that a significant portion of the population is expected to know. Are we getting smarter as a species, or is it more a matter of us collectively standing on the shoulders of giants? Is it inherently easier to learn something once it is already known? That's likely, considering that once an idea is discovered or developed, it can be shaped and refined until it is much more easily accessible to more people. It's also interesting to think that we're getting smarter, not that we're all smarter than Newton, but on average we might be smarter than the average person a thousand years ago. It would be hard to determine if such a trend had to do with anything more than better hygiene, nutrition, and education. It's still fun to think about, though.
In any case, Everyday Calculus had a few interesting nuggets, but overall I didn't really enjoy it. However, it was a quick read, and it did make me want to get out my old calculus and numerical methods textbooks, so it wasn't a total loss.
How Not To Be Wrong
Despite its pretentious title, this book was full of awesome. With chapter titles like "Everyone Is Obese" and "Dead Fish Don't Read Minds," I knew I was in for an entertaining read, and I was not disappointed. Ellenberg does a great job covering a number of real-world issues from a mathematical perspective, and clearly explains the logic and reasoning behind many of the mathematical methods that are used in analysis.
The book had a definite focus on probability and statistics, which makes a lot of sense when talking about how to make money off of a poorly designed lottery or how the link between smoking and lung cancer became undeniable as the evidence mounted against cigarettes. Ellenberg periodically brought other fields of mathematics into the discussion, such as linear algebra and non-Euclidean geometry, but he always circled back to statistics. It's fitting since statistics is probably the most applicable field of mathematics to our everyday life, and it's something everyone would benefit from knowing more about. So many of our personal experiences that guide our intuition can lead us to the wrong conclusions when we try to extrapolate them into broader contexts, and statistics provides the tools to correct our thinking.
For example, Ellenberg goes into a discussion on Big Data and how when you're filtering for some particular type of person that's a very small percentage of the population (his example is terrorists, but it could be any small group), it doesn't matter how accurate your filter is. Even if it's 99% accurate in filtering out people not in the group you want to detect, you're going to end up with a lot of false positives because 1% of hundreds of millions of people is still millions of people. Then he lays out the argument for the right to privacy:
You might well think that Facebook would never cook up a list of potential terrorists (or tax cheats, or pedophiles) or make the list public if they did. Why would they? Where's the money in it? Maybe that's right. But the NSA collects data on people in America, too, whether they're on Facebook or not. Unless you think they're recording the metadata of all our phone calls just so they can give cell phone companies good advice about where to build more signal towers, there's something like the red list going on. Big Data isn't magic, and it doesn't tell the feds who's a terrorist and who's not. But it doesn't have to be magic to generate long lists of people who are in some ways red-flagged, elevated-risk, "people of interest." Most of the people on those lists will have nothing to do with terrorism. How confident are you that you're not one of them?It doesn't matter if innocent people should have nothing to hide, which is one of the arguments the people make that are trying to create these lists and provide Security For All. Innocent people should not live in fear that their government will wrongly implicate them in criminal activity, especially without their knowledge via these secret lists. It's not really improved security when a large number of citizens are in danger of wrongful incrimination by their government.
Ellenberg also talks eloquently about education in mathematics, both from his perspective as a professor teaching students the importance of practicing calculation and learning how to think mathematically, and his experience as a student learning about hard work:
The cult of the genius also tends to undervalue hard work. When I was starting out, I thought "hardworking" was a kind of veiled insult—something to say about a student when you can't honestly say they're smart. But the ability to work hard—to keep one's whole attention and energy focused on a problem, systematically turning it over and over and pushing at everything that looks like a crack, despite the lack of outward signs of progress—is not a skill everybody has. Psychologists nowadays call it "grit," and it's impossible to do math without it. It's easy to lose sight of the importance of work, because mathematical inspiration, when it finally does come, can feel effortless and instant. I remember the first theorem I ever proved; I was in college, working on my senior thesis, and I was completely stuck. One night I was at an editorial meeting of the campus literary magazine, drinking red wine and participating fitfully in the discussion of a somewhat boring short story, when all at once something turned over in my mind and I understood how to get past the block. No details, but it didn't matter; there was no doubt in my mind that the thing was done.I can totally relate to this type of problem solving. I experience it all the time when I have a hard design problem I'm working on, and it's constantly turning over in the back of my mind. I call it bedtime debugging.
This passage also made me think about another aspect of studying that I pretty much avoided in high school and college. As I get older, I become more interested in the history of math and science, the lives of the great thinkers that discovered and developed the ideas we use today, and the difficulties involved in the process of discovery. This kind of knowledge gives context to the theorems and laws and ideas of math and science. Understanding where these ideas came from can give you a greater appreciation for the humanity involved in our development of knowledge, that it didn't all come about in the perfect form that it's presented in classrooms and textbooks. There was intense struggle, debate, and uncertainty behind it all, so you can take comfort in the fact that your own struggles are normal. The pursuit of knowledge has always been a battle.
Clearly, this book has made me think about some deep topics, and there were many more instances like these of discussions in the book sending me off thinking on wild tangents. I really enjoy books that do that, and Ellenberg was especially good at it. He's also an exceptional writer. He has a way of starting off a topic, developing it for a while, and then summarily dropping it on the floor to describe some other thing. At first you wonder what just happened, but you keep on reading because this new topic is also well written and interesting. Eventually, he'll pick the first topic up off the floor and tie it in with his current explanation, and everything becomes clear. Every once in a while he'll also drop little quips about old topics into the current discussion to keep you on your toes. It was completely engaging, and I burst out laughing a number of times while reading particularly witty parts. My wife probably thought I was crazy, laughing at a book about math like that. It truly was a treat to read.
Surrounded By Math
We are surrounded by math in our daily lives. It permeates everything we experience, and it can explain a lot about why things happen, if we're only willing to pay attention and understand. I went looking for a book that would capture the pervasiveness of mathematics in the real world, and I definitely found what I was looking for. While Everyday Calculus fell flat, and I couldn't get into it, How Not To Be Wrong delivered the goods. It has everything I want in a good popular math book. It's well written, engaging, and above all, it made me think hard on a number of topics. That always makes for a satisfying read. If you're in the mood to see how mathematics, and statistics in particular, shape our world and our understanding of it, go read How Not To Be Wrong. I can't recommend it enough.