Welcome to the earth. Congratulations on being born in the 21st century, where the odds are excellent that you’ll live a richer, more prosperous and more fulfilling life than almost any of the 100 billion or so who preceded you — and paved the way for your prosperity with their investments and their inventions. Would that there had been more of them.

As you go through life, you will almost assuredly contribute to the world’s stock of ideas, diversity and love in ways your parents never contemplated — which is why the rest of us are a little sad that you might be their last child.

There’s certainly such a thing as a population that’s too large. Nobody disputes that. The interesting question is: Given the incentives faced by parents, it the population size we actually get too large or too small? And there are good reasons to think it’s too small.

In fact, population growth is a lot like pollution in reverse. Polluters don’t care about the damage they impose on strangers, so they pollute too much. Parents and potential parents don’t care about they joy and prosperity their chidren bring to strangers, so they reproduce too little.

When you’re older, maybe you’ll want to understand these arguments a little better. If so, you might want to read Chapter 3 of my book More Sex is Safer Sex. Or, if reading has gone out of fashion by then, you might want to watch the first video on this page. You’ll have to sit through eleven minutes of talk about sex before you get to the part about population, but I’m guessing sex will still be an interesting topic in twenty years.

Or maybe you’ll have no interest at all in this issue. But I’m guessing you’ll have some interest in something and that both friends and strangers will have reasons to be glad you did. when Baby Eight Billion comes along, the world is likely to be an even richer place, thanks partly to you.

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(If this were a less serious post, I might suggest that this famous letter was the first example of a Galois Correspondence.)

Now, two centuries later, every first year graduate student in mathematics spends a semester studying Galois Theory, and many devote their subsequent careers to its extensions and applications. Many of the greatest achievements of modern mathematics (for example, the solution to Fermat’s Last Theorem) are, at their core, elucidations of Galois’s 200-year-old insight.

As every high school student knows (or should know), a quadratic equation (like, say, x² – 4x – 1 = 0) can be solved by applying the quadratic formula (which, in this case, gives x = 2 ± √5). The quadratic formula uses only addition, subtraction, multiplication, division, and the extraction of square roots.

What about a cubic equation, like, say, x³ + 2 x² – 5 x – 3 = 0 ? The less well-known cubic formula finds the solutions, using only addition, subtraction, multiplication, division, and the extraction of square and cube roots.

And, yes, there’s a quartic formula, for equations of degree 4. But it stops there. Galois’s contemporary Niels Abel (who, unlike Galois, survived to the ripe old age of 26) showed that no formula can consistently solve equations of degree 5 using only addition, subtraction, multiplication, division, and the extraction of roots.

On the other hand, some equations of degree 5 and higher can be solved by such formulas. Call those equations solvable. Galois figured out how to identify the solvable equations. It all comes down to understanding symmetry. Galois was the first to see clearly that the solutions to any equation satisfy certain symmetries. (For example, the solutions 2-√5 and 2+√5 are symmetric under the interchange of the plus and minus signs on the square root.) The nature of those symmetries differs from equation to equation, and dictates whether the equation is solvable. This in turn leads to a much deeper appreciation of the importance of symmetries throughout the theory of equations and throughout algebra more generally.

Today, algebraists take it for granted that understanding an equation, or a system of equations, entails understanding its symmetries. The development of that instinct was a key advance in the history of thought. After almost two centuries, we still use it to discover new insights and to solve old problems every single day.

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If you’re interested in mathematics at the research level, MathOverflow is a place to learn something new and fascinating every single day. (If you are not doing mathematics at a research level, feel free to read but please don’t feel free to join the fray; questions at anything below about a second-year graduate level should be directed to MathStackExchange, another massively collaborative site aimed, roughly, at the college level — which reminds me that it’s not just mathematical research, but also mathematical education, that is being revolutionized before our eyes.)

It has been absolutely fascinating to watch MathOverflow and its sister sites develop. The mathematical content is awesome, but so is the sociological phenomenon — problems solved in hours instead of months via virtual brainstorming among some of the smartest and most knowledgable people in the world.

When MathOverflow first came on line, I thought it would be a superfluous addition to the many electronic resources already available. I couldn’t have been more wrong. I now suspect that like all other toddlers, it will be awesome at age five in ways that are only dimly imaginable at age two. If you like mathematics, these are very good times to live in.

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It was Emmy (I use her first name in order to distinguish her from her mathematician father Max) who first fully recognized the power of abstraction, which became the driving force of 20th century mathematics. She demonstrated time and again that it can be easier to solve a general problem than a specific one, and therefore the best way to attack a specific problem is often to generalize. Do you want to prove a fact about polynomial functions? First notice that polynomial functions can be added together, and they can be multiplied, and they obey certain laws along the way (like associativity and commutativity). Now prove a theorem that applies to anything that can be added and multiplied subject to those laws. Do it right, and you’ll replace intricate calculations with simple logical deductions. What was hard becomes easy. You get your result for free, and a whole lot of other results as a bonus.

Or, if you that doesn’t quite work, figure out what additional properties you’re using about polynomials, beyond associativity and commutativity, and prove a theorem about everything that has those properties.

To get a sense of how revolutionary this was, consider the Hilbert Basis Theorem, one of the foundational results of modern algebra. Have a look at Hilbert’s original proof — though you might not want to work through every detail in the 62 pages of equations and formulas. By contrast, Noether’s proof of a more general, more powerful and more useful version occupies all of one paragraph on Wikipedia.

I daresay most of the great mathematical triumphs of the past 100 years came about because people took Noether’s dictums and ran with them. The greatest of all those triumphs, the oeuvre of Alexandre Grothendieck, takes all its inspiration and all its power from the philosophy and the work of Emmy Noether.

As a minor avocation, Noether dabbled in mathematical physics, where her main contribution was the insight that conservation laws (like conservation of mass) and symmetries (like “the laws of physics don’t change from one location to another”) are two different ways of looking at the same thing. Uncover a new symmetry and you can automatically deduce a new conservation law. Much of what we know about particle physics was discovered using Noether’s principle, which, according to the Nobel-prize-winning physicist Leon Lederman, is “certainly one of the most important mathematical theorems ever proved in guiding the development of modern physics, possibly on a par with the Pythagorean theorem”. And physics was her sideline!

Emmy Noether’s portrait hangs in a place of honor on my home office wall. Every day I look on it with awe.

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For clarity: When Stigler refers to an academic “field”, he is referring to a sub-discipline. Economics is a discipline; industrial organization and public finance are fields. Physics is a discipline; particle physics and solid state physics are fields.

We cannot build universities that are uniformly excellent … I shall seek to establish this conclusion directly on the basis of two empirical propositions.

The first proposition is that there are at most fourteen really first-class men in any field, and more commonly there are about six. Where, you ask, did I get these numbers? I consider your question irrelevant, but I shall pause to notice the related question: Is the proposition true? And here I ask you to do your homework: gather with your colleagues and make up a numbered list of the twenty-five best men in one of your fields — and remember that these fields are specialized. Would your department be first-class if it began its staffing in each field with the twenty-fifth, or even the fifteenth, name? You have in fact done this work on appointment committees. I remember no cases of an embarrassment of riches, and I remember many where finding five names involved a shift to “promising young men”, not all of whom keep their promises. I leave it to the professors of moral philosophy and genetics to tell us whether the paucity of first-class men is a sort of scientific myopia, a love of invidious ranking, or a harsh outcome of imprudent marriages. But the proposition is true.

My second proposition is that no one school has much in the way of financial resources … No school, not even the richest, has a wages-fund sufficient to hire one of the six best men in each field within the traditional arts and sciences. Fifty or a hundred institutions seriously seek such men, and even the fiftieth in wealth — which is about one-fourth as rich as the first in wealth — can bid enough for one or two such leaders to make them prohibitively expensive to others. The richest museums cannot acquire all the Rembrandts, and the richest school cannot hire all the leaders.

…

Universities will make their peace with the forces of specialization by making a choice that falls somewhere between two poles: a universal mediocrity, at one end; a select and none too lengthy list of truly distinguished departments, at the other. I diffidently interpret the tradition of Chicago to be that which I, too, desire: the preservation of pre-eminence in a dozen of the most durable and basic disciplines, with at least respectable competence in the remainder of the basic disciplines — and nothing more.

…

But the goal of selective eminence cannot be pursued effectively if one ignores its selectivity. The goal cannot be achieved if we fail to be ruthless with proposals to increase our comprehensiveness: it is a fact of life that a vote for a school of journalism or an institute of automation is a vote to get rid of one or two first-class men in physics or anthropology or law. The goal cannot be achieved if we insist that every department be almost pre-eminent: a vote to hire two expensive number-twenty men is a vote to be rid of a number-one man. These are different ways of saying that we must steer the difficult course between easy achievement and romantic impossibility. Some women are not fastidious, and others insist upon marrying only perfect men. I know Chicago will not become a harlot; I do not want it to become a spinster.

I would add a word concerning a very troublesome lot who insist upon intruding into the discussions of their betters — I refer to the students. The student cannot achieve the best possible instruction in every specialized field at any one institution; this I shall now treat as a corollary. Though a student does not study every specialized field even within one department, he would often profit by dividing his time between institutions whose strengths complement one another. There would be much merit in the development, at the graduate level, of spending a half year or a year at a second institution. This practice, you will recall, was prevalent during the fourteenth century; and, on balance, transportation has improved since then (aside from parking). The student would also gain perspective by living in a different intellectual atmosphere, and the professors — for whom things must be good if they are to be good for the country — would also gain by the diversity of students.

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One year ago today, he celebrated with a 100th birthday party, though he was only 99. I’m not sure what festivities are planned for today but I hope it’s a very good day for him.

Also one year ago today, I published a 99th birthday tribute here on this blog. I’m re-running it today.

In the theory of externalities—that is, costs imposed involuntarily on others—there have been exactly two great ideas. The first, forever associated with the name of Arthur Cecil Pigou (writing about 1920) is that things tend to go badly when people can escape the costs of their own behavior. Factories pollute too much because someone other than the factory owner has to breathe the polluted air. Nineteenth century trains threw off sparks that tended to ignite the crops on neighboring farms, and the railroads ran too many of those trains because the crops belonged to someone else. Farmers keep too many unfenced rabbits when they don’t care about the lettuce farmer next door.

Pigou’s solution—and it’s often a good one—is to make sure that people do feel the costs of their actions, via taxes, fines, or liability rules that allow the victims to sue for damages. Do a dollar’s worth of damage, and you’re charged a dollar.

Pigou endorsed this policy not because it seems fair, though it does seem fair to many, but because it yields, under what he believed to be very general conditions, the optimal amounts of damage. We don’t want too much pollution, but we don’t want too little, either, given that pollution is a necessary by-product of a lot of stuff we enjoy. Pigou offered a proof—now standard fare in all the textbooks—that his policies lead to the perfect compromises, in a sense that can be made precise.

The second great idea about externalities sprang full-blown from the mind of a law professor and subsequent Nobel prize winner named Ronald Coase, who stunned the profession in 1960 by pointing out that Pigou’s argument runs both ways. If you breathe the pollution from my factory, I’m imposing a cost on you—but at the same time, you’re imposing a cost on me. After all, if you lived somewhere else, you wouldn’t be complaining about the smoke and I wouldn’t be getting punished for it.

This insight—so simple once stated, but thoroughly astonishing to the economists of 1960 (I’ve heard tales of this astonishment from several of the participants in Coase’s historic seminar)—means that in a case of externalities, pure theory can never tell you who should bear the costs; you’ve got to look at the specifics of the case. Take those spark-throwing railroad trains. Pigou says: There are too many fires because the railroads don’t care; let’s make them reimburse the farmers for all the crop destruction, and then they’ll care. Coase says: Wait a minute. Often, farmers can prevent fires at very low cost by not planting quite so close to the tracks. True, the railroads don’t currently care about the crop damage. But if you reimburse the farmers, then the farmers won’t care, and you’ll get too many crops planted too close to the tracks. The best way to prevent fires might (or might not) be to grant the railroads complete legal immunity.

And as for that rabbit farmer—the one who lives next to the lettuce farmer and lets his rabbits run wild—Pigou would have insisted that the rabbit farmer cover the damages. Coase is more evenhanded. There are a lot of ways for the rabbit farmer to solve this problem: Put the rabbits in cages, or file their teeth down, or raise a different breed of rabbit, or move away, or switch to keeping geckos. There are also a lot of ways for the lettuce farmer to solve this problem: Fence the lettuce, or spray it with rabbit repellent, or move away, or switch to growing barley. If the rabbit farmer is immune from lawsuits, he’ll have no incentive to implement his solutions. But if the lettuce farmer is routinely reimbursed for lost lettuce, then he’ll have no incentive to implement his solutions. Which outcome is worse? That depends on whose solutions are better. Pure theory can’t answer that question.

How did Pigou—and every other economist in the world—manage to miss this point until Coase came along? According to Coase, it’s because they were obsessed with the faulty notion of “fault”—the idea that if there’s a problem, it must be someone’s fault, and we should begin by identifying that someone. But in the rabbit/lettuce example, who’s really at fault? It’s true that if there were no rabbits, the lettuce wouldn’t get eaten. But it’s equally true that if there were no lettuce the lettuce wouldn’t get eaten. The problem is that rabbit farmers should not be next to lettuce farmers, and when you put it that way, you’re forced to recognize the fundamental symmetry of the situtation.

What, then, should courts and legislators do? Coase had a lot to say about this also, beginning with the observation that it’s sometimes a really really good idea to encourage antagonists to talk to each other. From this beginning sprang the entire intellectual framework usually called Law and Economics.

Coase’s Nobel Prize winning paper is surely one of the landmark papers of 20th century economics. It’s also entirely non-technical (which is fine), and (in my opinion) ridiculously verbose (which is annoying). It’s littered with numerical examples intended to illustrate several different but related points, but the points and the examples are so jumbled together that it’s often difficult to tell what point is being illustrated. I frequently assign my students the task of distilling all of the main ideas into two or three pages, and they frequently succeed.

But pioneering work is rarely presented cleanly, and Coase is a true pioneer. Today is his 99th birthday, and a day to celebrate.

[Two other related posts are here and here.]

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If Dylan Thomas hadn’t drunk himself to death in 1953, he might be celebrating his ninety-sixth birthday today, perhaps with a successor to the grand and glorious poem he wrote to celebrate his thirtieth.

He left us with a small number of poems so heart-wrenching that I cannot read them, even for the two hundredth time, without all of the symptoms of an emotional crisis. Take In Country Sleep, where a father reassures his daughter that she has nothing to fear from fairy tale villains—but only from the Thief who comes in multiple guises to take her faith and ultimately to leave her “naked and forsaken to grieve he will not come”. In Country Sleep was a standard bedtime poem in our house, and my daughter soon learned to anticipate “the part where Daddy cries”.

Then there’s the prose. Nobody is better at nostalgia and grief for time’s relentlessness:

The lane was always the place to tell your secrets; if you did not have any, you invented them. Occassionally now I dream that I am turning out of school into the lane of confidences when I say to the boys of my class ‘At last, I have a real secret!’

“What is it? What is it?”

“I can fly!”

And when they do not believe me, I flap my arms and slowly leave the ground, only a few inches at first, then gaining air until I fly waving my cap, level with the upper windows of the school, peering in until the mistress at the piano screams, and the metronome falls to the ground and stops, and there is no more time.

And finally there’s the voice, the great booming melliflous irresistible voice lovingly preserved by Caedmon on about a dozen CDs that you will thank yourself for buying. The Caedmon collection includes a performance of the haunting “play for voices” Under Milk Wood narrated by Thomas himself (free e-book here); for an even greater treat, get the BBC Radio version with Richard Burton (Warning: Do not rent the highly regrettable movie version with Burton and Elizabeth Taylor.)

For a brief 39 years, as Time held him green and dying, Dylan Thomas spun words and images of surpassing beauty that will live as long as the English language. He sang in his chains like the sea.

(First written and posted six years ago today; this is a slightly updated version.)

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Of all the essays ever written, the one I most wish every voter could read and understand is Bastiat’s That Which is Seen and That Which is Not Seen. A boy breaks a window. Someone in the crowd observes that it’s all for the best—if windows weren’t occasionally broken, then glaziers would starve. This can’t be right, says Bastiat. If it were, we’d have no reason to diapprove of a glazier who pays boys to break windows. But why is it wrong? It’s wrong because it focuses on what is seen—six francs in the glazier’s pocket—and ignores what is unseen, namely the shoemaker who is deprived of a sale because those six francs come from what would have been the homeowner’s shoe budget.

Bastiat’s great insight in this essay is that exactly the same fallacy, in only slightly subtler form, underlies many of the public policy positions that were taken seriously in the 19th century—and, we might add, in the 21st.

The disbanding of troops (in Bastiat’s time) or a reduction in military procurement (in ours) is said to create great hardship for those who who sell bread to the troops, or parts and labor to the military contractors. It’s said that if we dismiss 100,000 unnecessary soldiers (or a 100,000 unnecessary workers), we’ll drive them into other industries where wages must fall. That’s what you see.

But what you do not see is this. You do not see that to dismiss a hundred thousand soldiers is not to do away with a hundred millions of money, but to return it to the tax-payers. You do not see that to throw a hundred thousand workers on the market, is to throw into it, at the same moment, the hundred millions of money needed to pay for their labour; that consequently, the same act which increases the supply of hands, increases also the demand; from which it follows, that your fear of a reduction of wages is unfounded. You do not see that, before the disbanding as well as after it, there are in the country a hundred millions of money corresponding with the hundred thousand men. That the whole difference consists in this: before the disbanding, the country gave the hundred millions to the hundred thousand men for doing nothing; and that after it, it pays them the same sum for working. You do not see, in short, that when a tax-payer gives his money either to a soldier in exchange for nothing, or to a worker in exchange for something, all the ultimate consequences of the circulation of this money are the same in the two cases; only, in the second case, the tax-payer receives something, in the former he receives nothing. The result is—a dead loss to the nation.

And on he goes, relentlessly applying the same observation to taxation, public support of the arts, trade restrictions, credit markets and more. Nobody has ever said it better or more accurately.

Edited to Add: Commenter Cloudesley Shovell observes that I appear to have gotten the date of Bastiat’s birthday wrong, but suggests that Bastiat is worth celebrating for the entire month of June. I regret the error and gratefully accept both the correction and the exit strategy.

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What does it mean for some infinities to be larger than others? Well, for starters, some infinite sets can be listed, while others are too big to list. The natural numbers, for example, are already packaged as a list:

The integers, by contrast (that is, the natural numbers plus their negatives) aren’t automatically listed because a list, by definition, has a starting point, whereas the integers stretch infinitely far in both directions. But we can fix that by rearranging them:

So the integers can also be listed.

The positive rational numbers (that is, numbers expressible as fractions) appear even harder to list than the integers, because they have no immediate successors. What is the next rational number after 1/2, for example? Answer: there is no next number. Between any two rationals lie infinitely many more.

We can still list them, though—after a suitable rearrangement of course. There are many ways to do this; here’s probably the simplest: First list all the fractions whose numerator and denominator add to 2, then all the fractions whose numerator and denominator add to 3, then all the fractions whose numerator and denominator add to 4, then 5, and so on, like so:

and finally string them all together:

There’s some repetition here (for example, 2/2 is the same number as 1/1), but just cross out the repeats and you’ve got your list.

What if you wanted to list all the rational numbers, both positive and negative? Easy! Just combine the two tricks we’ve already used. Start with the list just above, and stick in the negatives:

Now the only missing rational is zero, which you can throw in anyplace you like—say at the very beginning.

At this point you should begin to suspect that any infinite set can be listed, given enough cleverness. Not so, though. Let’s try to list all the real numbers—that is, all numbers expressible as (possibly infinite) decimals—between 0 and 1.

Now offhand, I can’t think of any way to do this, but that doesn’t prove anything about the real numbers; it might just prove I’m not as clever as I ought to be. But Cantor, with one incredibly simple argument, demonstrated that no attempt to list those real numbers can be successful.

Here’s the argument. Suppose you believe you have managed to list all those real numbers. Maybe your list looks something like this:

(For convenience, I’ve displayed this list vertically instead of horizontally, so the first item on the list is .8410729…., the second is .1415926…, and so on.) Now I’m going to prove you wrong—that is, I’m going to prove your list is incomplete—by writing down a number that’s definitely missing. First I write down a decimal point. Then I write down any first digit other than 8 (say 6). This insures that the number I’m writing down is not the same as .8410729….. Then I write down any second digit other than 4 (say 5). This insures that the number I’m writing down is not the same as .1415926….. Then I’ll write down any third digit other than 3 (say 7). This insures that the the number I’m writing down is not the same as .3333333….. Continuing in this way, I get a number

that is definitely nowhere on your list.

(If it bothers you to imagine that I could make infinitely many choices, just imagine that I make all the choices at once by applying some fixed rule. For example: Whenever I need a digit other than 1, I’ll pick 7; whenever I need a digit other than 2, I’ll pick 5…)

If you say “oops, I forgot that number; I’ll stick it in my list somewhere”, I’ll just pull the same trick again and find another number that’s not on your list. So no matter how clever you are, you can never list all the real numbers—or even just the real numbers between 0 and 1.

But we saw that there is a way to list all the rational numbers. So in what turns out to be a profound and fundamental sense, the infinity of real numbers is bigger than the infinity of rational numbers.

With that discovery, Cantor taught the world how think about infinity, rocked the foundations of mathematics, and, with a lag of a hundred and some years, changed my life.

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To my mild surprise, the face that generated the most controvery—in both comments and email—was that of Abraham Lincoln, who was born 201 years ago on this day. Readers pulled no punches. ScottN wrote: “Lincoln is on a different list I have: People Who Caused the Most Unnecessary Deaths.” Peter wrote: “[Lincoln] was a tyrant and a racist to boot.” And the consistently provocative and thoughtful Bob Murphy wrote:

I would love to hear your reasons for including Lincoln. I have the same misgivings as the other commenter above, though I was going to introduce them with levity. (E.g. “I know you like math, Steve, so is that why you included the guy who maximized the wartime deaths of Americans?”)

I replied to Bob (and others) by email, with some sketchy thoughts and a promise to blog about Lincoln sometime on or before his birthday. With the deadline looming, I realize that I have little to add to those sketchy thoughts. So here, with only some minor editing, is the email I sent to Bob Murphy:

Dear Bob:

First, I do believe that a perfectly reasonable person could come to the judgment that Lincoln was a bloodthirsty madman.

Second, however, I am not one of those reasonable people. My reading of history is that Lincoln was motivated by a passion to end slavery, above all else. I agree that alternative readings are reasonable (see above). I also agree that there are a lot of other people who are more well-versed in this history than I am.

Third, even if we grant that freeing the slaves was a noble cause and Lincoln’s sole motivation, we might well take the position that the victory was not worth the bloodshed.

Fourth—on the other hand, 3/4 of a million people died to save 4 million slaves. That’s over five slaves freed per war death, which does not seem to me to be an unreasonable ratio. And that doesn’t even count all the future generations who would otherwise have been enslaved. (Of course it also does not account for the possibility that slavery could have been brought to a more peaceable end.)

Fifth—moreover, I think it’s clear that neither Lincoln nor any other reasonable person had any way of anticipating that there would be casualties on that order, so the war looks like a much better bet ex ante than ex post; and I think ex ante is the right standard for judgment.

Sixth—On the other hand, Lincoln had a lot of pretty ugly secondary motivations. (Protectionism, something like a national industrial policy, etc.) Well, so did Reagan—but you’ll never take Reagan’s portrait off my wall.

Seventh: Bottom line is that I think Lincoln was out, above all, to end slavery (though a reasonable person might disagree) and that the price he paid, while enormous, was a reasonable one (though a reasonable person might disagree), and that the clarity of his vision was a form of greatness and a force for good (though a reasonable person might disagree). I am also acutely aware that I might change my mind about these things if I were better educated.

**************************

So ends the email. Those readers who care to contribute to my better education are invited to fire away.

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