I learned this from Macaulay’s History of England, which I just pulled up on my Kindle, which of course gives me instant — and searchable! — access to pretty much everything that’s ever been published. But the Kindle, and its brother e-readers, are more revolutionary than that. Not only do they give us easy access to existing literature; they call forth entirely new literary genres, such as the Kindle e-book, which brings to market extended essays that are too long to be magazine articles but too short to be traditional books — and are priced to sell.

All of which brings me to Alex Tabarrok’s Launching the Innovation Renaissance: A New Path to Bring Smart Ideas to Market Fast, which is both a product and a celebration of the innovation revolution, along with a recipe, or rather a set of recipes, for nurturing that revolution.

This is a great book. It’s fast-paced, fun to read, informative as hell, and it gets everything right. At first I wished I’d written it— until I realized I could never have written it half so well.

Those of us who write about economics know that the search for perfect real-world examples can be excruciating. Readers crave them, but most of the time a given example ends up illustrating a point that’s close to, but not exactly, what the writer is trying to get at — and so ends up muddying the waters. Tabarrok’s got an amazing knack for finding exactly the right examples, so that we’re simultaneously enthralled by his storytelling and clued in to exactly what he wants us to understand.

His recommended policies — patent reform, prize funds, better education through better teachers, trade schools instead of colleges for many students, liberalized immigration policies for skilled workers, globalization — are all backed by formidable intellectual consensus. Among people who know what they’re talking about, almost none of this stuff is controversial. For the public at large, this book will explain why.

In fact Tabarrok is extremely careful to stick to those subjects where he’s surely right. For example, he mentions in passing that liberalized immigration for unskilled workers would also be a good thing — but that he’s chosen not to dwell on that because, compared to his other proposals, it’s not quite so widely endorsed by the cognoscenti.

My only quibble is that he left out one of my favorite hobbyhorses, namely tax reform and the urgency of reducing taxes on capital — another pro-innovation policy with a substantial but too-little-publicized intellectual consensus behind it.

I wish everyone in the world would read this book. It only takes a couple of hours, and it is by far the best introduction I know of to the topic that towers above all others in its importance for the happiness of human beings everywhere, now and in the future, namely how to foster and accelerate the kinds of innovation that lead to economic growth. It will, I hope and expect, make you an enlightened advocate for enlightened policies. And it will arm you with a bundle of fun facts and anecdotes to share with your friends. This book might turn you into a proselytizer, but it will surely not turn you into a bore.

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As I Lay Dying went down pretty easily, but I remember many nights struggling my way through The Sound and the Fury, Cliff notes at my side. It felt like scaling Everest, and the vistas at the top were worth the climb.

A couple of weeks ago, as part of my ongoing project to read great novels, I decided to revisit The Sound and the Fury, and I’m more than glad I did; I finally have an answer to give the next time I’m asked what one novel I’d bring to a desert island. But what I’m flabbergasted by is this: How did this book ever get assigned to high school students in the first place? I ask for at least two reasons:

• The story is saturated with sex and racism (societal racism, that is, not Faulkner’s). It’s all “whore” this and “nigger” that, to the point where I’d have thought the PC police — you know, the guys who banned Huckleberry Finn — would have intervened long ago. (Come to think of it, high school was long ago. Maybe they have intervened.)
• What high school student has the patience to figure out what’s happening in a book like this? The first quarter is narrated by an idiot with no sense of time, so that he jumps back and forth between periods of his life mid-page, mid-paragraph, and sometimes mid-sentence, as he starts describing one event and finishes describing another similar event that took place twenty years earlier. Nobody (except maybe the Cliff Notes) ever warns you about the ever-shifting time frame. (I have no idea how the Cliff Notes people figured it out.) The narrator of the second quarter is no idiot, but seriously disturbed, and obsesses on events that are never described, but which we have to infer from the obscure references in his internal monologue. There are multiple characters with the same name, and single characters with multiple names — and not a shred of of warning about all this.

As I said, my high-school-self relied on Cliff Notes for guidance. I am not ashamed to tell you that my adult self required Cliff Notes, Barron’s Notes, Spark Notes, and the full power of the Internet. My strategy was to read a few pages, then seek help to find out what just happened, then reread. When I got to the halfway mark, I went back to the beginning and read the book straight through (the second half is mostly downhill). I am eager to return to the beginning and read it straight through one more time.

Make no mistake; this is perhaps the best novel I’ve ever read. The rewards are surely commensurate with the effort, but I can’t help believing that few high school students would invest enough effort to earn the rewards.

And yet—something magical did happen to me back in high school (belated thanks, Mrs. Schreiber!), something that left me with a decades-long intention to reread this book someday, and now I have, and I’m about to read it again. Maybe I should have also dipped into the Hemingway.

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The Adventures of Augie March (Saul Bellow): Not an easy read, even with the iPhone’s automatic links to dictionaries and Wikipedia. But well worth the effort, I think. Rewarding in the small and in the large — while I was reading, I kept caring what would happen next, and after I’d finished, I felt like I’d had a good workout and I was glad I’d gone to the gym. This made me want to read more Bellow (I think I will tackle Herzog next, if they ever bring out an electronic edition) but with long breaks between novels.

The Age of Innocence (Edith Wharton): Extremely good. Hard-to-put-down good. And a real insight into not just a vanished way of life, but a vanished way of seeing the world. All about the blinders that people wore in that time and place (the book is set in the 1870s, but was written in 1920, after those particular blinders had come off) and a real inspiration to think about what new blinders have taken their place.

The Ambassadors (Henry James): Quit after a couple of pages; it was failing to grab me. I should come back and try again.

American Pastoral (Philip Roth): I am currently 1/3 of the way into this, but I am already sure it is a great book. I will now read everything Philip Roth ever wrote.

Anthem (Ayn Rand): The first Ayn Rand I’ve ever read. Less tendentious and a better read than I’d expected.

Barchester Towers (Anthony Trollope): Terrific fun! To paraphrase one of the Amazon reviewers—I’d never have believed you if you’d tried to tell me that I’d be utterly gripped by a story about who was going to get which of the various positions in the heirarchy of the Church of England that I have never heard of. This is the second in a six-part series. After I’d read it, I read the earlier and much shorter The Warden, which is sort of like going back to read The Hobbit after you’ve already read Lord of the Rings. The Warden was okay, but if I’d read it first, I’d probably have stopped there. I’m glad I started with this one, after Trollope had hit his stride. I remain most eager to get to the remaining four in the series, and then everything else he ever wrote, which should take me a while, since he wrote at least 50 books.

The Brothers Karamazov (Fyodor Dostoevsky) (for the fourth time!). Arguably the greatest novel ever. The issue with the Brothers K is always the choice of a translation. The last time around (maybe ten years ago or so), I went with Pevear and Volokhonsky, and pronounced it by far the best. This time I went back to the classic Constance Garnett translation that I last read at age 16 — and reminded myself that this one is also great.

Man and Superman (George Bernard Shaw): Another re-read; more tedious more often than I’d remembered. I once saw the third act performed by Agnes Moorehead and the great Ricardo Montalban; even with Montalban, it seemed clear that this was a play that worked better on the page than in the theater. But even on the page, it still kind of drags.

The Man Who Changed His Skin (Harry Stephen Keeler): It was definitely weird. I’ll give it that.

Nightmare Abbey (Thomas Love Peacock): The only Peacock I’d never read before, and probably the best of them. More of a long short story than a novel, very funny in places, I was glad to be reading it, but also glad it was short. (That is, it was fun, but the kind of fun that’s best in small doses — which, since it comes in a small dose, is just fine.)

Pride and Prejudice (Jane Austen): Surprisingly tedious. I was glad when it ended. The only Jane Austen I’ve ever read is Northanger Abbey, which seems considerably spritelier.

Snow Crash (Neal Stephenson): It depresses me that there’s a market for a book this bad. I slogged through the whole thing, but oh my God it was painful. What kept me going was the sprinkling of brilliant passages, and the occasional brilliant chapter — and when they were brilliant, they were very very brilliant. (Also, the vision of future technology, though dated, was clever and insightful.) But there was precious little of that, compared to the endless passages that made me feel like I was trapped in a theater, watching a really really bad movie directed by a no-talent hack who thought he could make me care about the heroine trapped in the basement by dragging the scene out for twenty minutes.

The Sunlight Dialogues (John Gardner): In my 20s, John Gardner was my favorite author and the Sunlight Dialogues was one of my favorite books. I decided to read it again, and gave up a quarter of the way through. Part of what drove me crazy was my inability to distinguish among the various characters named “Hodge”; there would be long passages about what “Hodge” was up to, and I could never figure out whether this was Hodge the father, Hodge the son, Hodge the brother, or (for all I know) Hodge the Holy Ghost. I do not remember having this problem back in my 20s, so I suspect the fault lies not in the book but in the aging brain of the reader.

Tom Jones (Henry Fielding): Loads of fun.

Young Lonigan and The Young Manhood of Studs Lonigan (James T. Farrell): The first two books in the Studs Lonigan trilogy. The tone is as unlike Edith Wharton as you can get, but there’s the same sense of looking back on a time when people wore different blinders than they do today (in Wharton’s case, the upper crust WASPs in 1870’s New York; in this case the working class Irish in early 20th century Chicago). Farrell made me want desperately to jump into the action and guide these people toward a better way to be. Good enough to make me want to read the third part of the trilogy, but not good enough to make me want to read it immediately. The (clearly deliberate) choppiness of the writing kept taking me aback; sometimes it worked, sometimes it didn’t.

Still on my iPhone waiting to be read (I am too lazy to insert links for these):
Absalom, Absalom! (William Faulkner)
An American Tragedy (Theodore Dreiser)
A Bend in the River (VS Naipaul)
The Bonfire of the Vanities (Tom Wolfe)
Brideshead Revisited (Evelyn Waugh)
Cannery Row (John Steinbeck)
Dr Jekyll and Mr Hyde (Robert Louis Stevenson)
The Finkler Question (Howard Jacobson)
The Ginger Man (JP Donleavy)
Jonathan Strange and Mr Norell (Susanna Clark)
Lonesome Dove (Larry McMurtry)
The Magnificent Ambersons (Booth Tarkington)
Motherless Brooklyn (Jonathan Lethem)
Pale Fire (Vladimir Nabokov)
The Postman Always Rings Twice (James Cain)
Sons and Lovers (DH Lawrence)
Spooner (Pete Dexter)
Stone’s Fall (Iain Pears)
To the Lighthouse (Virginia Woolf)
Uncle Tom’s Cabin (Harriet Beecher Stowe)
Under the Net (Iris Murdoch)
The Way of All Flesh (Samuel Butler)
Winesburg Ohio (Sherwood Anderson)

— and, recently added, a whole bunch of Edith Wharton, Anthony Trollope and Philip Roth.

It might take me a while to get through these.

And finally: There are a lot of very old books on this list, partly because old books are cheap (or often free) in electronic formats. In the not-so-distant past before I got my first Kindle, I leaned more toward the contemporary. The last two dead-tree books I read were The Thirteenth Tale (Diane Setterfield) (reading hint for this one: keep a list of all the little unexplained mysteries, and cross them off as they get explained) and Water for Elephants (Sara Gruen), both of which were so good and so gripping and so memorable that I couldn’t resist mentioning them even though they’re slightly off topic for this post.

What else should I read?

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But there’s a better explanation. Glassman’s story never made sense in the first place, for reasons Paul Krugman explained when the book first came out.

Glassman has a substantial history of confusion about how financial markets work. Ten years before he wrote Dow 36,000, he was explaining in The New Republic that stocks are better investments than real estate:

if you bought a $200,000 home in Foggy Bottom [a neighborhood in Washington, D.C.] in 1979, it would have been worth $316,000 [ten years later]. But if you’d bought $200,000 worth of stock in 1979, it would be worth $556,000 [ten years later]—and you’d have another $68,000 in dividend income.

Edit: I’d left a few key words out of this quote; it’s fixed now.

This is a wondrous example because it goes so far beyond being simply wrong all the way to the exact opposite of the truth. Here’s how I exploited that example in my book The Armchair Economist:

Well, yes, but if you’d bought the house you would have had a place to live for those ten years, whereas if you’d bought the stock you’d have been making rental payments to a landlord. This renders Glassman’s comparison meaningless. All he shows is that if you compare some of the benefits of owning stock to some of the benefits of owning real estate, then the stock comes out ahead. Big deal.

Glassman’s … conclusion is exactly the opposite of the truth. He explains that “stocks appreciate faster than real estate; they always have and they always will. The reason is that a share of stock is a piece of a company in which minds are producing value. Real estate just sits there.” The truth is that stocks appreciate faster than houses precisely because a house does not just sit there; it provides shelter, warmth, and closet space every single day that you own it. Stocks need to appreciate faster to compensate for the fact that they don’t provide any comparable stream of services. If stocks and real estate appreciated at the same rate (counting the dividends as part of the appreciation, as Glassman does), nobody would own stocks.

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To understand the universe at the deepest level, we need to know not only how the universe behaves, but why.

• Why is there something rather than nothing?
• Why do we exist?
• Why this particular set of laws and not some other?

So say Stephen Hawking and Leonard Mlodinow in their book The Grand Design, and so say I.

The Big Big Question is the first one: Why is there something rather than nothing? Hawking’s answer: The laws of physics — and especially the form of the law of gravity — allow for the spontaneous creation of universes out of nothing at all. We live in one of those spontaneously created universes. But this, of course, only serves to raise a new Big Big Question, namely: Why are the laws of physics as they are? Hawking’s answer: The laws of physics must be consistent and must predict finite results for the quantities we can measure. It turns out that those criteria pretty much dictate the form of the laws of physics.

So unless I’ve misunderstood him, here is Hawking’s position: In order for us to be able to measure the things that we measure, the laws of physics must have a certain form, and in order for them to have that form, universes must be able to arise from nothing. Therefore our universe was able to arise from nothing. But this does not seem to answer the question of why things couldn’t have been very different. Why couldn’t there have been no us, no measurements, no laws of physics and no anything?

I know of only one satsfying (to me) answer to this question, and Hawking comes tantalizingly close to it without ever quite going there. He spends a lot of pages reviewing current physical theories but never mentions the one glaring feature they all share: Every modern physical theory, taken literally, predicts that our universe is a mathematical object. For example, the simplest version of special relativity posits that we live in a four-dimensional geometric object called “spacetime”. More sophisticated theories posit that spacetime is part of some larger geometric object whose properties we perceive as “forces” or “particles”. According to modern physics, everything is made of math.

Now you might say that physical theories aren’t meant to be taken that literally; that instead they describe mathematical objects with properties that are analogous to the properties of the physical universe. But it seems to me that if, like Hawking, you trust in theories to explain the mystery of creation itself, then you ought, at least provisionally, to take those theories literally. Otherwise, what you’ve got is not a theory. It’s a theory plus a bunch of ad hoc and arbitrary choices about which parts of that theory you choose to believe.

Once you believe the universe is a mathematical object, its existence ceases to be a mystery—at least if you believe, along with most mathematicians, that mathematical objects can’t help but exist. Hawking embraces M-theory, which tells us that the universe is a particular 11-dimensional object (with a whole bunch of additional geometric curlicues that appear to our senses as everything from stars to bacteria. M-theory also says there are a whole bunch of other 11-dimensional universes, all of which were spontaneously created, and we just happen to live in this one.

What I’m suggesting is that the universes of M-theory are only a tiny fraction of the universes out there, because anything that exists mathematically is a universe, though most of them (like most of the universes of M-theory) are far too simple to contain anything like sentience. This is essentially the view of cosmologists like Max Tegmark of MIT.

Hawking is 90% of the way there. The many universes of M-theory are mathematical objects, and all are pieces of a bigger mathematical object called the multiverse. “Spontaneous creation” means that the multiverse is structured in such a way that it must contain these universes. But why is there a multiverse and why is it structured in that way? That’s the part Hawking seems not to address. Proposed answer: The multiverse itself is only one of many multiverses. They all exist for the same reason the natural numbers exist: The laws of mathematics require it. And unlike the laws of physics, which differ from multiverse to multiverse, the laws of mathematics, which live outside any universe, could not have been otherwise.

(For more on this subject, read Chapter 1 of The Big Questions !)

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]]> http://www.thebigquestions.com/2010/09/27/the-grand-design/feed/ 41 http://www.thebigquestions.com/2010/09/03/the-harvard-classics/ http://www.thebigquestions.com/2010/09/03/the-harvard-classics/#comments Fri, 03 Sep 2010 06:01:16 +0000 Steve Landsburg http://www.thebigquestions.com/?p=4601 If you happen to be attending Harvard this semester, one of your course options is Greg Mankiw’s Freshman Seminar 43j, “The Economist’s View of the World”:

This seminar probes how economic thinkers from the right and left view human behavior and the proper role of government in society. Each week, seminar participants read and discuss a brief, nontechnical, policy-oriented book by a prominent economist. Regular writing assignments are also required. Students should have some background in economics, such as an AP economics course in high school or simultaneous enrollment in Social Analysis 10.

The ten books on tap for this semester are:

• The Worldly Philosophers, by Robert Heilbroner
• Reinventing the Bazaar: A Natural History of Markets, by John McMillan
• Thinking Strategically, by Avinash Dixit and Barry Nalebuff
• Capitalism and Freedom, by Milton Friedman
• Equality and Efficiency: The Big Tradeoff, by Arthur Okun
• Nudge, by Richard Thaler and Cass Sunstein
• How the Economy Works, by Roger E.A. Farmer
• The Return of Depression Economics, by Paul Krugman
• The Road to Serfdom, Friedrich Hayek
• The Myth of the Rational Voter, by Bryan Caplan
• The Big Questions, by Steven Landsburg

What would your list have been?

Click here to comment or read others’ comments.

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]]> http://www.thebigquestions.com/2010/09/03/the-harvard-classics/feed/ 22 http://www.thebigquestions.com/2010/08/26/living-in-the-future/ http://www.thebigquestions.com/2010/08/26/living-in-the-future/#comments Thu, 26 Aug 2010 06:01:07 +0000 Steve Landsburg http://www.thebigquestions.com/?p=4522
My treasured copy of the humor classic Science Made Stupid, copyright 1985, contains a Wonderful Future Invention Checklist. Who in 1985 would have thought that just 25 years later, I could check off a third or so of the entries?
• Househould Robot. Does my Roomba count?
• Magnetic Train. Check.
• Flat-Screen TV. Check.
• Flat-Screen 3-D TV. Check.
• Two-Way Wrist Radio. We are so far past this.
• Two-Way Wrist TV. Ditto.
• Intelligent Computer. My computer’s a lot smarter than it looks, honest. It just acts dumb when it has to run Microsoft products.
• Instant Access to All Human Knowledge. Check!
• Human Clones. Getting there.
• First Woman President. Does Secretary of State count?
• First Black President. Check!!
• Universal Language. That would be English.
• X-Ray Specs. My infrared camera sees through clothes.
• World War III. By some accounts, we’re about 9 years into it.
• Access to Other Dimensions. Talk to the string theorists.
• Immortality. Check?
• Spelling Reform. OMG! I cn chk ths 1 off 2.

Some of the other entries, like “Home Holographs”, “Personal Rocket”, and “My Trip to Other Galaxy” might be a bit farther off. But things sure change in a hurry.

What else about your current life do you think would most surprise a time traveler from 25 years ago?

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]]> http://www.thebigquestions.com/2010/08/26/living-in-the-future/feed/ 29 http://www.thebigquestions.com/2010/08/06/65-years-later/ http://www.thebigquestions.com/2010/08/06/65-years-later/#comments Fri, 06 Aug 2010 06:01:48 +0000 Steve Landsburg http://www.thebigquestions.com/?p=2344 65 years ago today, the world changed. In his magnificent World War II memoir Quartered Safe Out Here, George McDonald Fraser looks back on what might have been:

I led Nine Section for a time; leading or not, I was part of it. They were my mates, and to them I was bound by ties of duty, loyalty and honor… Could I say, yes, Grandarse or Nick or Forster were expendable, and should have died rather than the victims of Hiroshima? No, never. And the same goes for every Indian, American, Australian, African, Chinese and other soldier whose life was on the line in August, 1945. So [I'd have said]: drop the bomb.

…

And then I have another thought.

You see, I have a feeling that if—and I know it’s an impossible if—but if, on that sunny August morning, Nine Section had known all that we know now of Hiroshima and Nagasaki, and could have been shown the effect of that bombing, and if some voice from on high had said: “There — that can end the war for you, if you want. But it doesn’t have to happen, the alternative is that the war, as you’ve known it, goes on to a normal victorious conclusion, which may take some time, and if the past is anything to go by, some of you won’t reach the end of the road. Anyway, Malaya’s down that way … it’s up to you”, I think I know what would have happened. They would have cried “Aw, fook that!”, with one voice, and then they would have sat about, snarling, and lapsed into silence, and then someone would have said heavily, “Aye, weel” and got to his feet, and been asked “W’eer th’ ‘ell you gan, then?”, and given no reply, and at last, the rest would have got up, too, gathering their gear with moaning and foul language and ill-tempered harking back to the long dirty bloody miles from the Imphal boxes to the Sittang Bend and the iniquity of having to do it again, slinging their rifles and bickering about who was to go on point, and “Ah’s aboot ‘ed it, me!” and “You, ye bugger, ye’re knackered afower ye start, you!”, and “We’ll a’ git killed!”, and then they would have been moving south. Because that is the kind of men they were.

I’ll add no comments, though yours, of course, are welcome.

[A hat tip to my Mom and Dad, who told me to read this book. So should you.]

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]]> http://www.thebigquestions.com/2010/08/06/65-years-later/feed/ 17 http://www.thebigquestions.com/2010/07/16/the-girl-who-played-with-numbers/ http://www.thebigquestions.com/2010/07/16/the-girl-who-played-with-numbers/#comments Fri, 16 Jul 2010 06:01:43 +0000 Steve Landsburg http://www.thebigquestions.com/?p=3979 I’ve just finished reading The Girl Who Played with Fire, the second book in the series that begins with The Girl with the Dragon Tattoo. I’m not giving away any significant plot point when I tell you that there’s a character who works on Fermat’s Last Theorem as a hobby, or that the book was clearly written (or perhaps translated) by somebody with no clue how mathematics works or what Fermat’s Last Theorem is about. I particularly liked the reference to Andrew Wiles using the “world’s most complicated computer program” to solve the problem. It’s my understanding that Andrew barely even uses email. And certainly if you understood anything about the nature of the problem and/or the solution, you’d recognize the absurdity of trying to tackle it with a complicated computer program.

Be that as it may, I finished the novel with a few hours left to spare, so of course I was inspired to work on Fermat’s Last Theorem, or at least on the simplest cases. The problem, if you’ll recall, is to show that there are no integer solutions to any of the equations x³+y³=z³ , x⁴+y⁴=z⁴ and so on, except for the so-called trivial solutions in which one or more variables take the value zero.

This is relatively easy to prove in the n=4 case (that is, for the equation x⁴+y⁴=z⁴), and in fact I was able to reconstruct two separate proofs, one using elementary algebra and the other using a little geometry. (”Reconstruct” means that there was a time in my life when I knew these proofs well—and even taught them at a graduate level—but that was long long ago.) And I was able to reconstruct Lamé’s flawed proof, which, when supplemented with some more work, can be converted to a correct proof for a large class of exponents (beginning with n=5). The attempt to understand when Lame´’s argument can (or can’t) be patched up inspired a century of progress in algebraic number theory. Alas, that work reveals that there are plenty of exponents for which the proof is irreparable, beginning with n=37. The only known proof, associated in the popular imagination with the great Andrew Wiles, but more properly attributed to Frey, Serre and Ribet, is nothing like Lamé’s (and about one octillion times more difficult).

But what really surprised me was that I didn’t have a clue how to solve the case n=3. And even now, I have no memory of ever having known how to solve the case n=3. I was aware that it took Euler to solve it in the first place, and that I am not as smart as Euler (by a factor of about one octillion), but I was also aware that I know a lot of fancy techniques that Euler didn’t have. So, like the character in the novel, I thought I’d give it a go.

My first idea was to use Fermat’s favorite technique: Pretend you’ve got a solution, and show that from that solution, you can construct a smaller solution. Keep repeating and your solutions get smaller forever, which is quite impossible with integers. (If your first solution involved x=100 and x gets smaller each time, you’re going to get stuck after 100 iterations—x can’t go below zero). This means you never had a solution in the first place. (Fermat called this the “Method of Infinite Descent”.)

So I pretended I had a solution—that is, a set of numbers x, y, z that satisfy x³+y³=z³—and used a little geometry to construct a new solution. I did this using what is, for a geometer, the obvious idea. Namely:

• Set X=x/z and Y=y/z, and observe that X³+Y³=1
• Observe that (0,1) and (X,Y) are both points on the curve defined by the equation x³+y³=1
• Draw the line connecting these two points. Because the curve is defined by a third degree equation, that line will hit the curve three times. We already know it hits at (0,1) and (X,Y). Compute the third point. Because everything else in sight is a rational number, that third point will have rational coordinates.
• Write the coordinates of that point as (a/c,b/c), where a, b and c are integers. (You can always make the two denominators equal by choosing a common denominator). Then because this point sits on the curve, it satisfies the equation (a/c)³+(b/c)³=1. This in turn implies that a³+b³=c³.

So starting with one solution (x,y,z), we get a new solution (a,b,c). If (a,b,c) is in any reasonable sense smaller than (x,y,z), we can keep repeating till we get a contradiction.

When I did this, I got a = x(1+y³), b = -y(1+x³) and c = x³-y³. (You can check by hand that if x,y,z solve the Fermat equation then so do a,b,c.) Sadly, this doesn’t help because the new solution is not smaller than the old solution in any reasonable sense that I can think of. (I’d expected as much, because if something this simple had any chance of working, it wouldn’t have taken Euler to solve the problem.)

So I futzed around with a few other ideas that didn’t work (e.g. instead of drawing the line that connects two points, you could draw the tangent line at the point (X,Y)) and finally looked up Euler’s proof, which I must say, rang absolutely no bells with me, meaning either that I must have been curiously uncurious about this when I was younger or that my memory is failing even more precipitously than I realized. On a side note, I also learned (for the first time, as far as I can recall) that Euler’s first published attempt was incorrect.

Well, at least I got a blog post out of this, and more importantly it was fun. Sometimes it pays to have a short memory. Every now and then (especially when I’m stuck in a boring meeting) I compute the sum of the infinite series 1 + 1/2ⁿ + 1/3ⁿ + 1/4ⁿ + … for various values of n, which is another problem that Euler got to before I did. The main idea stays with me, but the details are new every time.

Edited to add: For those who are playing along at home—I copied incorrectly from my notes. The a, b and c announced above come not from the line that connects (0,1) to (X,Y), but from the tangent line at (X,Y). If you use the line connecting (0,1) to (X,Y), you get a=-x, b= z, c=y, which is even less useful.

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]]> http://www.thebigquestions.com/2010/07/16/the-girl-who-played-with-numbers/feed/ 7 http://www.thebigquestions.com/2010/06/24/3816/ http://www.thebigquestions.com/2010/06/24/3816/#comments Thu, 24 Jun 2010 06:01:11 +0000 Steve Landsburg http://www.thebigquestions.com/?p=3816 David D. Friedman’s The Machinery of Freedom, a classic of libertarian thought, has long been out of print and hard to find. (Well, it’s easy to find, actually. But hard to find for less than about a hundred bucks.) It is therefore a very good thing that David’s gotten his publisher’s permission to post the entire book on the World Wide Web, for free.

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.

Click here to comment or read others’ comments.

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