* Which, among pros, is now called Wiles’s Theorem, since Andrew Wiles proved it (with a critical assist from Richard Taylor) and Fermat did not. But the traditional name will probably never be dislodged.
* To be honest, I did spend some part of my early twenties thinking I might want to be a Serious Literary Novelist. I even finished a Serious Literary Novel, called The Grasshopper King, and got it published. But in the process I discovered that every day I devoted to Serious Literary Novel-writing was a day half spent moping around wishing I were working on math problems.
* Here “Swedishness” refers to “quantity of social services and taxation,” not to other features of Sweden such as “ready availability of herring in dozens of different sauces,” a condition to which all nations should obviously aspire.
* Or a line segment, if you must. I won’t make a big deal of this distinction.
* Laffer disputes the napkin portion of the story, recalling that the restaurant had classy cloth napkins that he would never have vandalized with an economic doodle.
* Somewhere between a half million and a million dollars a year in today’s income.
* Like I’m one to talk.
* Whether the increased tax receipts are because the rich started working harder once less encumbered by income tax, as supply-side theory predicts, is more difficult to say for certain.
* Or, more likely still, it might not be a single curve at all, as Martin Gardner illustrated by means of the snarly “Neo-Laffer curve” in his acid assessment of supply-side theory, “The Laffer Curve.”
* By the way, we don’t know who first proved the Pythagorean Theorem, but scholars are almost certain it was not Pythagoras himself. In fact, beyond the bare fact, attested by contemporaries, that a learned man by that name lived and gained fame in the sixth century BCE, we know almost nothing about Pythagoras. The main accounts of his life and work date from almost eight hundred years after his death. By that time, Pythagoras the real person had been completely replaced by Pythagoras the myth, a kind of summing up in an individual of the philosophy of the scholars who called themselves Pythagoreans.
* It can’t, in fact, but nobody figured out how to prove this until the eighteenth century.
* Actually, silos weren’t round until the early twentieth century, when a University of Wisconsin professor, H.W. King, invented the now-ubiquitous cylindrical design in order to solve the problem of spoilage in the corners.
* Rather, each of the four pieces can be obtained from the original isosceles right triangle by sliding and rotating it around the plane; we take as given that these manipulations don’t change a figure’s area.
* At least if, like me, you live in the midwestern United States.
* Apart from the effects of gravity, air resistance, etc., etc. But on a short timescale, the linear approximation is good enough.
* Admittedly, these particular people were teenagers at a summer math camp.
* So as not to leave you hanging: there is a context, that of 2-adic numbers, in which this crazy-looking argument is completely correct. More on this in the endnotes, for number theory enthusiasts.
* The surreal numbers, developed by John Conway, are especially charming and weird examples, as their name suggests; they are strange hybrids between numbers and games of strategy and their depths have not yet been fully explored. The book Winning Ways, by Berlekamp, Conway, and Guy, is a good place to learn about these exotic numbers, and lots more about the rich mathematics of game playing besides.
* Like all mathematical breakthroughs, Cauchy’s theory of limits had precursors—for instance, Cauchy’s definition was very much in the spirit of d’Alembert’s bounds for the error terms of binomial series. But there’s no question that Cauchy was the turning point; after him, analysis is modern.
* Ironic, considering Grandi’s original theological application of his divergent series!
* In the famous words of Lindsay Lohan, “The limit does not exist!”
* If you’ve ever taken a math course that uses epsilons and deltas, you’ve seen the descendants of Cauchy’s formal definitions.
* More details on these studies can be found in the Journal of Stuff I Totally Made Up in Order to Illustrate My Point.
* “Closest,” in this context, is measured as follows: if you replace the actual tuition at each school by the estimate the line suggests, and then you compute the difference between the estimated and actual tuition for each school, and then you square each of these numbers, and you add all those squares up, you get some kind of total measure of the extent to which the line misses the points, and you choose the line that makes this measure as small as possible. This business of summing up squares smells like Pythagoras, and indeed the underlying geometry of linear regression is no more than Pythagoras’s theorem transposed and upgraded to a much-higher-dimensional setting; but that story requires more algebra than I want to deploy in this space. See the discussion of correlation and trigonometry in chapter 15 for a little more in this vein, though.
* It’s a little reminiscent of Orson Scott Card’s short story “Unaccompanied Sonata,” which is about a musical prodigy who is carefully kept alone and ignorant of all other music in the world so his originality won’t be compromised, but then a guy sneaks in and plays him some Bach, and of course the music police can tell what happened, and the prodigy ends up getting banished from music, and later I think his hands get cut off and he’s blinded or something, because Orson Scott Card has this weird ingrown thing about punishment and mortification of the flesh, but anyway, the point is, don’t try to keep young musicians from hearing Bach, because Bach is great.
* In the research literature, “overweight” means “BMI at least 25 but less than 30” and “obese” means “BMI 30 or above,” but I’ll refer to both groups together as “overweight” to avoid having to type “overweight or obese” umpteen times.
* I’m not going to do these computations on the page, but if you want to check my work, the key term is “binomial theorem.”
* And yes, shooting percentage is as much a function of which shots you choose to take as your intrinsic skill at hitting the basket; the big man whose shots are mostly layups and dunks starts with a big advantage. But that’s orthogonal to the point we’re making here.
* Experts will note that I am carefully avoiding the phrase “standard deviation.” Non-experts who wish to go deeper should look the term up.
* To be precise, it’s a little less, more like 95.37%, since 31 is not quite the square root of 1,000 but a little smaller.
* Actually, closer to 51.5% boys and 48.5% girls, but who’s counting?
* Safety warning: never divide by zero unless a licensed mathematician is present.
* Actual cutest Beatle.
* Math pedantry: in order to claim some phenomenon is “ever-more-startling,” you have to do more than show that it is startling; you have to show that its startlingness is increasing. This issue is not addressed in the body of the op-ed.
* The analysis here is indebted to that of Glenn Kessler, who wrote about the Romney ad in the April 10, 2012, edition of the Washington Post.
* Which is only a tiny fraction of the possible permutations of thirty-two dates, of which there are 263,130,836,933,693,530,167,218,012,160,000,000.
* Which was supposed to happen in 2006, so, whew, I guess?
* There’s a useful principle, the product rule, hiding in this computation. If the chance of foo happening is p, and the chance of bar happening is q, and if foo and bar are independent—that is, foo happening doesn’t make bar any more or less likely—then the chance of both foo and bar happening is p × q.
* This story certainly dates back to the days when this process would have involved reproducing and stapling ten thousand physical documents, but is even more realistic now that this kind of mass mailing can be carried out electronically at essentially zero expense.
* There is a long-standing and profoundly unimportant controversy abo
ut whether the term “natural number” ought to be defined to include 0 or not. Feel free to pretend I didn’t say “0,” if you are a die-hard antizeroist.
* You might object here that Fisher’s methods are statistics, not mathematics. I am the child of two statisticians and I know that the disciplinary boundary between the two is real. But for our purposes, I’m going to treat statistical thinking as a species of mathematical thinking, and make the case for both.
* Arbuthnot saw the propensity for the slight excess of boy children as itself an argument in favor of Providence: someone, or Someone, had to have set the knob just right to make extra infant boys to cancel out the extra adult men killed in wars and accidents.
* We’ll assess this argument in more detail in chapter 9.
* And the actual mathematical definition of “group” has still more to it than that—but, sadly, this is another beautiful story we’ll have to leave half told.
* The paper doesn’t address the interesting question of what the corresponding rates are for children in the care of their own parents.
* Not everyone has the language we have, of course. Chinese statisticians use (xianzhu) for significance in the statistical sense, which is closer to “notable”—but my Chinese-speaking friends tell me that the word carries a connotation of importance, as the English “significance” does. In Russian, the statistical term for significance is , but the more typical way to express the English-language sense of “significant” would be .
* It’s said that David Byrne wrote the lyrics to “Burning Down the House” in a very similar way, barking nonsense syllables in rhythm with the instrumental track, then going back and writing down the words that the nonsense reminded him of.
* Some people will insist on the distinction that the argument is only a reductio if the consequence of the hypothesis is self-contradictory, while if the consequence is merely false the argument is a modus tollens.
* As a good rule of thumb, you can figure that each of the fifty subjects contributes a 1/20,000 chance of finding an albino in the sample, yielding 1/400; this isn’t exactly right, but is usually close enough in cases like this one, where the result is very close to 0.
* Indeed, it’s a general principle of rhetoric that when someone says “X is essentially Y,” they generally mean “X is not Y, but it would be simpler for me if X were Y, so it’d be great if you could just go ahead and pretend X is Y, sound good?”
* Disclosure: I used to read Numb3rs scripts in advance to check their mathematical accuracy and provide comments. Only one line I suggested ever made it on the air: “trying to find a projection of affine three-space onto the sphere subject to some open constraints.”
* Down here at the bottom of the page I can safely reveal the real definition of log N; it is that number x such that ex = N. Here e is Euler’s number, whose value is about 2.71828 . . . I say “e” and not “10” because the logarithm we mean to talk about is the natural logarithm, not the common or base-10 logarithm. The natural logarithm is the one you always use if you’re a mathematician or if you have e fingers.
* Fermat wrote a note in a book claiming he had a proof, but that it was too long to fit in the margin; no one nowadays believes this.
* This condition may seem a bit out of the blue, but it turns out there’s a cheap way to generate lots of “uninteresting” solutions if you allow common factors among A, B, and C.
* Most notably, Pierre Deligne’s results relating averages of number-theoretic functions with the geometry of high-dimensional spaces.
* Following a path laid out by Goldston, Pintz, and Yıldırım, the last people to make any progress on prime gaps.
* I was disappointed to find that this study has not yet spawned any conspiracy videos claiming that Obama’s support of birth control coverage was aimed at suppressing women’s biological drive to vote GOP during ovulation. Get on the stick, conspiracy video producers!
* Chabris is perhaps most famous for his immensely popular YouTube video demonstrating the cognitive principle of selective attention: viewers are asked to watch a group of students passing a basketball back and forth, and usually fail to notice an actor in a gorilla suit wandering in and out of the shot.
* All these examples are drawn from the immense collection at the blog of health psychologist Matthew Hankins, a connoisseur of nonsignificant results.
* All the numbers in this example are made up, partially because the actual computation of confidence intervals is more complicated than I’m revealing in this small space.
* Oversimplification watch: Fisher, Neyman, and Pearson all lived and wrote for a long time, and their ideas and stances shifted over the decades; the rough sketch I draw of the philosophical gap between them ignores many important strands in each person’s thinking. In particular, the view that the primary concern of statistics is making decisions is more closely associated with Neyman than with Pearson.
* The basic method here is called logistic regression, if you’re looking for further reading.
* In this context, the confusion between question 1 and question 2 is usually called the prosecutor’s fallacy. The book Math on Trial, by Coralie Colmez and Leila Schneps, treats several real-life cases of this kind in detail.
* Complicating factors: Beber and Scacco found that numbers ending in 0 were slightly rarer than would be expected by chance, but not nearly as rare as in human-produced digits; what’s more, in another data set of apparently fraudulent election data from Nigeria, there were lots of extra numbers ending in 0. Like most forms of detective work, this is far from an exact science.
* Admittedly, this is not a very compelling theory about conventional roulette wheels, where the slots alternate in color. But for a roulette wheel you can’t see, you might theorize that it actually has more red slots than black.
* Of course, if we were doing this for real, we’d have to consider more than three theories. We’d also want to include the theory that the wheel is weighted to come up 55% red, or 65%, or 100%, or 93.756%, and so on and so on. There are infinitely many potential theories, not just three, and when scientists carry out Bayesian computations in real life, they need to grapple with infinities and infinitesimals, to compute integrals instead of simple sums, and so on. But these complications are merely technical; in essence the process is no deeper than the one we carried out.
* More precisely, it tends to kill T + not-U.
* No, seriously, this was actually fashionable.
* Paley himself was surely aware of this issue; note how careful he is to say “artificer or artificers.”
* People who, of course, might themselves actually be simulations engineered by a yet higher order of people!
* A more refined analysis of “the right price” would also take into account my feelings about risk; we’ll return to this issue in the next chapter.
* This job still exists! But it is now a largely honorary position, since the annual salary of one hundred pounds sterling has remained unchanged since Charles II established the post in 1675.
* Other states, as far back as third-century Rome, had understood that the proper price of an annuity needed to be greater when the purchaser is younger.
* Or so it seems to me. I wasn’t able to get official statistics for ticket sales, but you can get pretty good estimates of the number of players from the data Powerball releases about the number of winners of the lower-tier prizes.
* For readers who want to go even deeper into the decision-theoretic details of the lottery, “Finding Good Bets in the Lottery, and Why You Shouldn’t Take Them,” by Aaron Abrams and Skip Garibaldi (The American Mathematical Monthly, vol. 117, no. 1, January 2010, pp. 3–26) is a great resource. The title of the article serves as an executive summary of their conclusions.
How Not to Be Wrong : The Power of Mathematical Thinking (9780698163843) Page 50