Just this once more, okay? Then I promise to stop. I'm sure I can. It's just that—well, how should I say it?—this is so damned weird! I'm scratching my head and need to get this off my chest. (Nice mixed metaphor there, if you'll excuse my patting myself on my own back. Ow!) I promise you will find this amazing.
Let's begin with a little math lesson. That's always a good way to gather a crowd. Come a little closer, kids, and Dr. Z will astonish you with his great wisdom! I take for my text a reading from the book of Tom the Apostol, Chapter 3, Section 2:
The equation limx→p f(x) = A is read: “The limit of f(x) as x approaches p is equal to A,” or “f(x) approaches A as x approaches p.” It is also written without the limit symbol, as follows:That was a perfectly straightforward description of the notion of a mathematical limit. Here's a concrete example to show you that you understand the basic idea:
f(x) → A as x → p.
This symbolism is intended to convey the idea that we can make f(x) as close to A as we please, provided we choose x sufficiently close to p.
Let f(x) = x2, which is often called the squaring function (the reason for this is left as an exercise for the student). Evaluating function notation is no big deal in a simple case like this. If you're told that x = 3, you just plug in 3 wherever you see x (the number you plug in is called the argument of the function), like so:
f(3) = 32 = 9.
Okay? If we revisit the limit idea with the squaring function in mind, replacing p with 3 and A with 9, we find ourselves looking at the statement that
x2 → 9 as x → 3.
Does that seem reasonable? As x gets closer and closer to 3, x2 gets closer and closer to 9. If you look at the accompanying graph of y = x2, you may be able to persuade yourself that this is true. You would be correct to think there are subtleties which we have avoided here, but feel free nevertheless to congratulate yourself on having grasped a key fact about the squaring function: when two numbers are close together, their squares are also close together. The closer the numbers, the closer their squares. It's not exactly rocket science at this point.
Once upon a time in Prague
I finally got around this year to reading David Berlinski's A Tour of the Calculus, the 1995 book that tried to popularize calculus and helped him make his mark as an author. Since I usually assign some outside reading to my calculus students in order to keep them from thinking that everything is in the pages of their textbook, I welcomed the appearance of the Berlinski book and its nontechnical approach. Before I found the time to read it on my own, however, some of my colleagues shared their disdain of Berlinski's prolix prose and metaphor mania. My ardor cooled and the book got shoved to the back of my stack of reading material, where it's languished the past decade.
In the intervening years, Berlinski has been busily pimping for the Discovery Institute, where he is a senior fellow with DI's Center for Science and Culture. Despite his yeoman service on behalf of the exponents of intelligent design, Berlinski claims a general agnosticism about the topic and does not profess to be a creationist. He prefers merely to lob his mathematically laden bombs at the theory of evolution. Such nonsense made me even less inclined to dig out his book on calculus, because he struck me more and more as a player of perverse intellectual games, pretending a kind of neutrality while focusing all of his attacks on the scientific side of the evolution/creation argument.
Berlinski has not escaped unscathed. I wrote about his wrongheadedness myself in David Berlinski vs. Goliath, but for a real nuts-and-bolts evisceration of his arguments, see what Mark Chu-Carroll has to say about Berlinski's Bad Math at Good Math, Bad Math:
In my never-ending quest for bad math to mock, I was taking a look at the Discovery Institute's website, where I found an essay, On the Origin of Life, by David Berlinksi. Bad math? Oh, yeah. Bad, sloppy, crappy math.This surfeit of criticisms of Berlinski, although I fully agreed with them, finally tweaked my conscience and reminded me of the unread calculus book. Perhaps Berlinski's seduction by the dark side of creationism had drawn him away from more constructive efforts. I had read enough of his prose and seen him on enough video to remain wary of his tendency to talk down to people, but A Tour of the Calculus was supposed to be an attempt to speak to the general public. I needed to read it to see for myself.
It didn't take too many pages for me to see that all of Berlinski's pomposity is at full play in A Tour of the Calculus. He lards the text with digressions and obscure allusions (perhaps I am not educated enough to appreciate him). The mathematical concepts are recast in odd ways that confuse rather than shed light. The writing is self-indulgent. These are disappointments rather than surprises. Nevertheless, I promised you a surprise, and it's time for me to deliver. I began with a short math lesson on the squaring function because it is at the center of a bizarre tale recounted by Berlinski in Chapter 15, A Prague Interlude.
Berlinski goes to Prague University to lecture on Tychnoff's theorem, a sophisticated result from topology. One of his hosts is Professor Swoboda, a mathematician. “Swoboda is extraordinarily intelligent,” Berlinski tells us. Please remember these things. Berlinski is giving a math talk attended by an extraordinarily intelligent mathematician. Here is an extended quote from the middle of his narration:
I am supposed to talk about Tychonoff's theorem, but to my surprise I find myself explaining the elementary calculus to a roomful of mathematicians, re-creating in my own mind the steps that Bolzano took in order to define continuity. For some reason I feel it absolutely crucial to explain how the concept of a limit is applied to functions. No one seems to mind or even notice.There's more. Much more. But enough. Can you explain this passage to me? Berlinski begins by admitting he is talking about elementary calculus, but then has his roomful of mathematicians rapt in awe as he reveals that values of the squaring function approach 9 as its arguments approach 3. College students would not be surprised by this result, let along a roomful of mathematicians.
“A function indicates a relationship in progress, arguments going to values. Given any real number, the function f(x) = x2 returns its square, tak? ... In go arguments 1, 2, 3, out come the values, 1, 4, 9.... As the arguments of f get larger and larger, its values get larger and larger in turn.* ... Now imagine,” I say, “arguments coming closer and closer to the number 3, tak?”
I walk back to the blackboard and show the men in my audience what I mean, writing, 2, 2.1, 2.2, 2.3, 2.4, 2.5, 2.6, ..., before the function.
“What then happens to the function? How does it behave?” I ask, realizing with a sense of wonder somewhat at odds with the hard-boiled pose I usually affect, that a function is among the things in this world that behaves—it has a life of its own and so in its own way participates in the drama of things that are animate.
“I mean,” I say, “what happens to the values of f as its arguments approach 3?”
I look out toward my audience. Swoboda and Schweik are looking at me intently, their faces serene, without irony. It is plain to me that they do not know the answer yet.
“They approach, those values, the number 9, so that the function is now seen as running up against a limit, a boundary beyond which it does not go.”
Swoboda leans back and sighs audibly, as if for the first time he had grasped a difficult principle. The room, with its wooden pews and narrow blackboard, is getting close.
I say, “The concept of a limit, as it is applied to functions, is forged in the fire of these remarks.”
I presume Berlinski intends this as an extended metaphor, because otherwise he is cruelly mocking his
I know how they feel.
Update: David Berlinski has complained of his rough treatment at the hands of the proprietor of Good Math, Bad Math. While trying to defend his mathematical arguments against Mark Chu-Carroll's criticisms, Berlinski mentions my comment about his visit to Prague University and objects:
If you check the index to my A TOUR OF THE CALCULUS under 'limits' you will find page references to a complete and precise definition. The section in Prague was, of course, intended to be dream-like. It is not the mathematicians who are diminishing in sophistication, but the narrator.Was that supposed to clear things up? If you have the stomach for more, Berlinski has returned to Good Math, Bad Math for Round 2. If, like me, you're ready for a rest cure, here's an opportunity to avert your gaze. (But don't forget to bookmark Good Math, Bad Math for future reference. Good stuff is added every week.)
*Berlinski is assuming that the x values are positive. If negative values are permitted, then larger x values do not always result in larger squares. For example, −1 is greater than −2, but (−1)2 is smaller than (−2)2.