What does David get out of this? Well first, of course, he wants you to read his book. But second, he’s about to start preparing a third edition and welcomes reader feedback. If you post your comments here, I’ll make sure he sees them.

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I have never tried to count the times I have read a newspaper article explaining that students are bored with math that has no visible practical application, and follows with an example of a teacher, or club, that rectifies the situation in some novel and engaging way.

In the present case a class has built a sculpture that resembles a graph of a modulated wave motion. Of all the practical, real-world
applications of mathematics! It is as practical as a snowman.

Why doesn’t anyone ask for real-world applications of table tennis? What a bore any game must be, that has no real-world application! Why do kids stand for it? Ping-pong again? Ugh.

But I can think of something: Let’s all make a model of a ping-pong ball in the school yard, seventy feet high, blocking all the entrances and thus preventing all us students from entering the (ugh) school. Then we can take our fishing poles and torn straw hats out from under our beds and, with the hats on our heads and fishing poles over our shoulders, all traipse together down the dusty road to Norman Rockwell’s house.

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Like everyone else I know, I am of course a longtime fan of the webcomic XKCD. But somehow it took me until last week to become aware of the frequently brilliant competitor Luke Surl, of which the above is a delectable example. What else out there am I missing?

Hat tip to Harry Brighouse of Crooked Timber.

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Today I read it. Wow, was I wrong. This is Caitlin Flanagan at her blistering best. I’ll offer you a few choice quotes, but my real recommendation is to leave now and go read the entire piece.

With the Edible Schoolyard..the idea of a school as a venue in which to advance a social agenda has reached rock bottom. This kind of misuse of instructional time…has been employed to cheat kids out of thousands of crucial learning hours over the years, so that they might be indoctrinated in whatever the fashionable idea of the moment or the school district might be. One year it’s hygiene and the another it’s anti-Communism; in one city it’s safe-sex “outercourse” and in another it’s abstinence-only education.

Does the immigrant farm worker dream that his child will learn to enjoy manual labor, or that his child will be freed from it?…If this patronizing agenda were promulgated in the Jim Crow South by a white man who was espousing a sharecropping curriculum for African American students, we would see it for what it is: A way of bestowing field work and low expectations on a giant population of students who might become troublesome if they actually got an education.

Until our kids have a decent chance at mastering the essential skills and knowledge that they will need to graduate from high school, we should devote every resource and every moment of their academic day to helping them realize that life-changing goal. Otherwise we become complicit—through our best intentions—in an act of theft that will not only contribute to the creation of a permanent, undereducated underclass, but will rob that group of the very force necessary to change its state.

There’s much more where that came from. Why are you still here?

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I’ve just finished reading an excellent book by Mario Livio which is entirely about the invention/discovery question, though he’s chosen the (somewhat unfortunate) title Is God a Mathematician? Much of the book is a lively romp through mathematical history, with a well chosen mix of biography and exposition. Although he parts company with them in the last chapter, Livio gives a more than fair hearing to the many great mathematicians who have insisted that they are discoverers, from Pythagoras through Galileo, G.H. Hardy, Kurt Godel, and the contemporary Fields Medalist Alain Connes (among others). Here, for example is Connes:

Take prime numbers, for example, which as far as I’m concerned, constitute a more stable reality than the physical reality that surrounds us. The working mathematician can be likened to an explorer who sets out to discover the world…We run up against a reality every bit as uncontestable as physical reality.

Readers of The Big Questions will know that I am entirely in Connes’s camp on this issue, for reason I’ll blog about later in the week. And as I’ve said, it seems that most mathematicians sit in this camp. But there are notable dissenters, including the great Sir Michael Atiyah, another Fields Medalist who I might well have included in my gallery of heroes. Despite my great admiration for Atiyah, I believe he’s wrong on this issue. But more fundamentally, I believe his primary argument proves exactly the opposite of what he thinks it does. Here is that argument (slightly condensed):

Any mathematician must sympathize with Connes. We all feel that the integers really exist in some abstract sense and the Platonic view is extremely seductive. But can we really defend it? It might seem that counting is really a primordial notion. But let us imagine that intelligence had resided, not in mankind, but in some vast solitary and isolated jellyfish, buried deep in the Pacific Ocean. It would have no experience of individual objects, only of the surrounding water. Motion, temperature and pressure would provide its basic sensory data. In such a pure continuum the discrete would not arise and there would be nothing to count.

Here Atiyah has envisioned a world where the natural numbers get neither invented nor discovered. I’m not sure why that’s supposed to prove they don’t exist. On the contrary, it seems to me that quite unbeknownst to Atiyah’s Jellyfish, the earth would still have exactly one moon and exactly two magnetic poles, and two would still be twice one. Two would still be a prime number, and 1729 would still be the smallest number that is the sum of two cubes in two different ways.

Indeed, all these things were true back in the days when the world was populated by creatures who were unaware of them. It doesn’t matter for the argument whether those creatures are highly intelligent in other ways. In that sense, Atiyah’s Jellyfish is more like a Red Herring.

Or to put this another way: Atiyah says that an intelligent creature might be unaware that one plus one makes two. Sure—so might any creature. To me, this indicates that 1+1=2 is not an invention; it’s simply a truth. Can anyone explain why Atiyah thinks his story proves otherwise?

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When my sister asked me this question over Thanksgiving dinner, I answered, with the utmost confidence, that it causes (quite instantaneously) an enormous flood. (Note the exact wording. This will be important later.) My sister nodded sagely and said “That’s what I thought, too.” My sister and I had the same mother, you see.

And then she asked, quite innocently, “So. How exactly does that work?”. And I was stunned—absolutely stunned—to realize not only that I had no answer to this question, but that there could not plausibly be an answer. Which somehow had never occurred to me in the half century or so that I’d been harboring this ridiculous notion.

My sister had seen the light just a few weeks earlier when her husband was filling the bathtub. The water level had gotten perilously close to the metal plate, and my sister had frantically cried out that he was about to cause an “enormous flood”. (My mother’s contention that she never taught us any such thing is severely undermined by the fact that my sister and I shared not only the same bizarre misconception, but the same phrase to describe it). Her husband, puzzled, had responded appropriately with something like: “Huh?”. And she’d realized she had no answer.

Readers of The Big Questions will be aware that in my opinion, much religious belief is very like my belief in the mystical power of bathtub hardware. That is, it survives only because it is unexamined. It does not, in other words, run deep. And as a consequence, it doesn’t affect the way most people live their lives—because as soon as it starts interfering with your life (or with your husband’s bath) you set it aside.

Such “beliefs”, it seems to me, do not deserve to be called beliefs at all. In the words of Samuel Taylor Coleridge, “You do not believe; you only believe that you believe”.

I was recently pointed to a wonderful column by the philosopher Jamie Whyte, making the same point so vividly that I rushed to order two of his books from Amazon. As it turns out, they’re both the same book. Crimes Against Logic is the Americanized version of the original and very British Bad Thoughts. I randomly chose to read the British version, and it’s a blast. Whyte lists a dozen logical fallacies so blatant you’d think nobody could ever fall for them, and then gives you multiple examples of people who have fallen for them. Much snarky commentary ensues.

As Whyte documents, people speak a lot of nonsense. (I am one of those people. So are you.) You might be tempted to conclude that people are dumb, but I’m more inclined to conclude that people are busy. We’re all working so hard to be good carpenters, or good taxi drivers, or good teachers, or good parents, that we don’t have the time and energy to think hard about bathtub hardware or God or the consequences of a protective tariff. And most of the time, we don’t suffer for getting this stuff wrong. But every now and then it pays off—in some combination of enlightenment and entertainment—to pull out one of your cherished “beliefs” and ask: “So. How exactly does that work?”

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First, with a hat tip to my sister, three from Amazon.com.

• For your ex’s divorce lawyer: A laptop desk to attach to your steering wheel! Proceed as follows (you’ll thank me, really): First cursor over the customer images on the left side of the page. Then read the customer reviews.
• For the political activist on your list: Uranium ore!. Again, read the customer reviews. Again, you’ll thank me.
• For your oddball cousin: Wolf urine!. Not a common taste, but for those who indulge, there simply is no substitute. And of course: Read the reviews.

And speaking of Amazon customer reviews, I was more than pleased to stumble on this quote in a review of The Big Questions:

Also, if you are a parent and are blessed with a math/science inclined child, please, please, please buy them a copy!

It’s not too late!

Finally, as a Christmas gift to my readers—or at least to that vocal subset of my readers who have been clamoring for answers to the honors questions I posted a couple of weeks ago: Your wish is my command.

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The American Economics Association has chosen, among others, Karl Marx, John Maynard Keynes, Milton Friedman, George Stigler, Friedrich von Hayek, Joseph Schumpeter, Adam Smith and Joan Robinson. (Can you identify the rest?) I haven’t yet held the Economists Calendar in my hands, but the reliable Mark Skousen has, and his verdict (via private email) is “Bravo!”

The calendar includes, according to Skousen, “an amazingly complete listing of top economists through the ages”, along with the featured 18. There’s still time to order before Christmas.

[Click here to comment.]

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