Here’s one difference between me and Paul Krugman: He enthusiastically supports President Obama’s new immigration policy, which he calls a matter of human decency. I grudgingly support President Obama’s new immigration policy, which I call a bit less indecent than the policy it replaces.

Here’s another difference between me and Paul Krugman: I believe it’s the job of an economics journalist to call attention to unpleasant tradeoffs and offer frameworks for resolving those tradeoffs. Krugman apparently believes it’s the job of an economics journalist to sweep all tradeoffs under the rug in the name of advancing your policy agenda — appealing, if you will, to the stupidity of the American op-ed reader.

Krugman, for example, tells us that he opposes deportations because they’re cruel, but also opposes open borders because they’d make it both economically and politically impossible to maintain the modern American welfare state.

In furtherance of which, he offers this kind of claptrap:

Second, there are large numbers of children who were born here … but whose parents came illegally, and are legally subject to being deported.

What should we do about these people and their families? There are some forces in our political life who want us to … deport the undocumented parents of American children and force those children either to go into exile or to fend for themselves.

But that isn’t going to happen, partly because, as a nation, we aren’t really that cruel

Dammit, I hate this stuff. Krugman says (and I agree with him) that it’s cruel to deport people. He ignores the fact that it’s also cruel to keep other people out. Krugman says (and I agree with him) that letting more people in would put pressure on the welfare system. He ignores the fact that allowing people to stay also puts pressure on the welfare system. Why should we prioritize kindness to those who are already here over kindness to those who are clamoring to get here?

There might be a really good answer to that question, but you’d never know it from reading Krugman. In fact, the takeaway from Krugman’s column is that the cruelty of deportations is unacceptable **only** because Krugman says so, and the cruelty of closed borders is a necessary evil **only** because Krugman says that too. So the next time you want to know whether some other policy is unacceptably cruel or not, the only way to find out is to ask Paul Krugman.

And then there’s more:

The truth is that sheer self-interest says that we should do the humane thing. Today’s immigrant children are tomorrow’s workers, taxpayers and neighbors.

Ummm…Paul? They are tomorrow’s workers, taxpayers and neighbors only if we let them stay. Do you know who **else** are potentially among tomorrow’s workers, taxpayers and neighbors? **The ones we’re not letting in**.

Once again, there might be some reason why we benefit more from those who are already here than we’d benefit from those who have not yet arrived — but if Krugman knows that reason, he’s keeping it a secret. He makes absolutely no attempt to **quantify** his cost-benefit analysis, or even, for that matter, to be explicit about what he’s counting as a cost or a benefit. His arguments — both his moral arguments and his arguments from self-interest — **apply equally well** to current residents and to current non-residents. They are arguments either for mass deportations or for open borders, but not for the Obama policy.

If you want to make an honest case against open borders, you’ve got to start with this acknowledgement: **Even if we grant for the sake of argument that the modern American welfare state is a good thing**, and **even if we grant for the sake of argument that open borders would fully undermine it**, it does not follow that the enormous benefits of open borders would fail to offset that enormous cost. That requires an argument. Here’s what an argument would consist of:

1. Either a) some estimate of the benefits of open borders, a separate estimate of the costs, and a comparison between the two or b) some clever way of proving, without any actual measurement, that the costs must exceed the benefits, say by showing that each individual benefit comes packaged with a larger cost.

2. A clear statement of how much weight you’ve given to costs and benefits felt by Americans as opposed to the costs and benefits felt by Mexicans, preferably along with some justification for your weighting and a fair accounting of how your conclusions might change if you’d chosen different weights. This would, for example, lead to some arithmetic along the lines of what you see in Chapter 19 of *The Big Questions*. That arithmetic is surely not the last word on the matter, but that **kind** of arithmetic is precisely what economics can contribute to this debate.

Not only does Krugman offer no answers; he **pretends the questions don’t exist**. His agenda, for whatever reason, is to stop deportations without loosening up the borders. Rather than defend that agenda, he pretends that

a) It needs no defense.

b) And if you think otherwise, you’re a bad person. Sneer, sneer.

Look: **The essence of Krugman’s position is that current non-residents should be treated more cruelly than current residents.** That position is probably defensible. Economics teaches us that life is full of uncomfortable trade-offs, and that sometimes you’ve got to be cruel in one way to avoid being even crueler in another. But economics also teaches us that it’s important to **face those trade-offs honestly**, even to call attention to them, so that we don’t make our choices with blinders on.

That’s where Krugman becomes the anti-economist. As is his right, he supports the Obama policy. But he has far too much contempt for his readers to fashion an argument that might actually illuminate that policy. Instead he throws out a bunch of rhetoric that, when analyzed with an even slightly critical eye, **offers exactly zero support for his position**.

His arguments, after all, come down to this: “Deportations are cruel and **for that reason alone** must be bad policy”, or “Open borders are costly and **for that reason alone** must be bad policy”. But if those were valid arguments, then (as Krugman knows perfectly well), one could just as easily switch deportations with open borders and reach exactly the opposite conclusions. But Krugman doesn’t care about logic, because he’s too busy bashing the morals of anyone who dissents from his apparently random value judgments.

According to Krugman, if you support the cruelty of deportations, you’re an evil person, but if you support the cruelty of closed borders, you’re a pragmatic adult. Why? Because Paul Krugman said so. Might there be a subject — like, oh, say, economics — that can help us think more clearly and systematically about such issues? If so, you’d never learn about it by reading Krugman. He wouldn’t want to risk teaching his readers to think.

]]>So let me get this straight. We drew this imaginary line in the desert. We’ll no longer use force to move people from this side to that side, but we will still use force to prevent movement from that side to this side.

This is good news for the people who are on this side at the moment, and I share their joy. But it does little for their less fortunate cousins who never made it here in the first place.

The president talks about “who we are as Americans”. I’d have hoped that we as Americans were not so basely hypocritical as to find it imperative that we stop bullying Group A (i.e. those who are here), but equally imperative that we continue to bully the even less fortunate Group B (i.e. those who aspire to be here).

This is a day to celebrate and a day to mourn.

Regarding Jonathan Gruber and the Cadillac tax, I think a little historical context will be useful:

1) Our tax system subsidizes employer-provided health insurance. That’s dumb. Pretty much all economists agree that it’s dumb.

2) On the other hand, it’s politically hard to eliminate a subsidy once people get used to it.

3) In 2008, we had an election. The candidates were named Barack Obama and John McCain. Exactly one of those candidates took the politically courageous step of proposing to eliminate the subsidies (and replace them with other subsidies, far more sensibly designed). The other candidate took the low road, leaping to the defense of subsidies he had to know were indefensible, playing to the crowd, and staking all on what could reasonably be called “the stupidity of the American voter” (though I myself would prefer to call it “the inattentiveness of the American voter”). That candidate won in a landslide.

4) Once elected, President Obama’s demagogy came back to haunt him. On the one hand, he knew that you cannot have sensible health care reform without curtailing those subsidies. On the other hand, he’d publicly committed himself to preserving them.

5) Therefore, Obama and his advisors resorted to sleight-of-hand, seeking to offset the subsidies with a new “Cadillac tax” that is the economic equivalent of curtailing the subsidies. This tax kicks in for policies that cost over about $23,000 for a family of four. It would be better if it kicked in for policies that cost anything over about ten cents, but it was still substantially better than nothing.

6) Obama and his advisors — at this point including Gruber — soon realized that it would be politically unpopular to impose the Cadillac tax directly on consumers, but politically palatable to impose it on insurance companies, who would undoubtedly pass it on to exactly the same consumers. There, once again, we see the inattentiveness (or, in Gruber’s unfortunate choice of words, stupidity) of the American voter.

7) So — here we have two different policies with exactly the same effects — tax the consumer or tax the insurance company. Gruber et.al. believed that the voters would oppose one and support the other. It does not seem to me, under those circumstances, that it’s especially dishonest to choose the package that you think will sell. If voters have opinions that are completely incoherent to begin with, then neither Gruber nor anyone else can be accused of confusing them. They were, after all, maximally confused in the first place.

8) I do think that those of us who are paid to teach economics have something of a moral obligation to, you know, teach economics. So it would have been better if Gruber, like so many of the rest of us, had made some effort to explain to the public that there is no difference between paying a tax and having a taxed passed on to you. On the other hand, the public is confused about enough different things that we can’t all be explaining all of those things all the time.

9) I want to say this again. If the voters favor a law that says all drivers must be licensed, but oppose a law that says nobody without a license is allowed to drive, then I don’t think it’s immoral to propose the first law instead of the second. That’s basically all Gruber did. I would prefer that he had tried to point out the inconsistency, but Gruber is under no obligation to live by my preferences.

10) Bottom line: The Cadillac tax is a good thing and Gruber found a way to make a good thing politically palatable, without telling any out-and-out lies (as far as I know, he never tried to claim that the tax would **not** be passed down). I see no sin in that.

11) The real sin was lying to the American people in the first place about the desirability of employer-based insurance subsidies, and doing it just to score political points against an opponent who had, in a burst of responsibility, dared to speak sense about them. That sin is entirely on Obama, not on Gruber.

12) Once he was elected, Obama routinely spoke outrageous and obvious falsehoods about health care policy, most famously “If you like your coverage you can keep it”. I’ve argued elsewhere that this was more of an absurdity than a lie, because he can’t possibly have expected anyone to believe it. After all, if you’re doing nothing to increase the total health care resources available, and if you’re making more of those resources available to some people, then you’ve **got** to be making fewer of those resources available to some other people — and surely this is obvious to anyone over the age of five. But then maybe I’m underestimating the inattentiveness (or stupidity?) of the American voter.

13) Re the Cadillac tax then, I do wish Gruber had behaved differently, but I cannot say he behaved particularly badly. The (outrageously) bad behavior was all on Obama’s part.

14) But this addresses only one of the many areas where Gruber has boasted about misleading people. I haven’t completely thought through how I feel about all the other instances, though my gut feeling is that some of them are considerably more troubling — the exploitation of quirks in the CBO scoring process, for example. That, perhaps, will be fodder for another blog post.

]]>I never met Alexander Grothendieck. I was never in the same room with him. I never even saw him from a distance. But whenever I think about math — which is to say, pretty much every day — I feel him hovering over my shoulder. I’ve strived to read the mind of Grothendieck as others strive to read the mind of God.

Those who did know him tend to describe him as a man of indescribable charisma, with a Christ-like ability to inspire followers. I’ve heard it said that when Grothendieck walked into a room, you might have had no idea who he was or what he did, but you definitely knew you wanted to devote your life to him.

And people did. In 1958, when Grothendieck (aged 30) announced a massive program to rewrite the foundations of geometry, he assembled a coterie of brilliant followers and conducted a seminar that met 10 hours a day, 5 days a week, for over a decade. Grothendieck talked; others took notes, went home, filled in details, expanded on his ideas, wrote final drafts, and returned the next day for more. Jean Dieudonne, a mathematician of quite considerable prominence in his own right, subjugated himself entirely to the project and was at his desk every morning at 5AM so that he could do three hours of editing before Grothendieck arrived and started talking again at 8:00. (Here and elsewhere I am reporting history as I’ve heard it from the participants and others who followed developments closely as they were happening. If I’ve got some details wrong, I’m happy to be corrected.) The resulting volumes filled almost 10,000 pages and rocked the mathematical world. (You can see some of those pages here).

I want to try to give something of the flavor of the revolution that unfolded in that room, and I want to do it for an audience with little mathematical background. This might require stretching some analogies almost to the breaking point. I’ll try to be as honest as I can. In the first part, I’ll talk about Grothendieck’s radical approach to mathematics generally; after that, I’ll talk (in a necessarily vague way) about some of his most radical and important ideas.

Imagine a clockmaker, who somehow has been oblivious all his life to many of the simple rules of physics. One day he accidentally drops a clock, which, to his surprise, falls to the ground. Curious, he tries it again—this time on purpose. He drops another clock. It falls to the ground. And another.

Well, this is a wondrous thing indeed. What is it about clocks, he wonders, that makes them fall to the ground? He had thought he’d understood quite a bit about the workings of clocks, but apparently he doesn’t understand them quite as well as he thought he did, because he’s quite unable to explain this whole falling thing. So he plunges himself into a deeper study of the minutiae of gears, springs and winding mechanisms, looking for the key feature that causes clocks to fall.

It should go without saying that our clockmaker is on the wrong track. A better strategy, for this problem anyway, would be to **forget all about** the inner workings of clocks and ask “What **else** falls when you drop it?”. A little observation will then reveal that the answer is “pretty much everything”, or better yet “everything that’s heavier than air”. Armed with this knowledge, our clockmaker is poised to discover something about the laws of gravity.

Now imagine a mathematician who stumbles on the curious fact that if you double a prime number and then halve the result, you get back the number you started with. It works for the prime number 2, for 3, for 5, for 7, for 11…. . What is it about primes, the mathematician wonders, that yields this pattern? He begins delving deeper into the properties of prime numbers…

Like our clockmaker, the mathematician is zooming in when he should be zooming out. The right question is not “Why do primes behave this way?” but “What other numbers behave this way?”. Once you notice that the answer is **all** numbers, you’ve got a good chance of figuring out **why** they behave this way. As long as you’re focused on the red herring of primeness, you’ve got no chance.

Now, not all problems are like that. Some problems benefit from zooming in, others from zooming out. Grothendieck was the messiah of zooming out — zooming out farther and faster and grander than anyone else would have dared to, always and everywhere. And by luck or by shrewdness, the problems he threw himself into were, time after time, precisely the problems where the zooming-out strategy, pursued apparently past the point of ridiculousness, led to spectacular, unprecedented, indescribable success. As a result, mathematicians today routinely zoom out farther and faster than anyone prior to Grothendieck would have deemed sensible. And sometimes it pays off big.

There are, of course, times when it **does** pay to examine the inner workings of things. Jean-Pierre Serre, another titan of modern mathematics with whom Grothendieck had an intimate working relationship, was often, in Grothendieck’s words “the yang to my yin”. If there was a nut to be opened, Grothendieck suggested, Serre would find just the right spot to insert a chisel, he’d strike hard and deftly, and if necessary, he’d repeat the process until the nut cracked open. Grothendieck, by contrast, preferred to immerse the nut in the ocean and let time pass. “The shell becomes more flexible through weeks and months — when the time is ripe, hand pressure is enough.”

In other words, the philosophy was this: If a phenomenon seems hard to explain, it’s because you haven’t fully understood how general it is. Once you figure out how general it is, the explanation will stare you in the face.

It is, I believe, just plain impossible, without trying to teach you a **lot** of mathematics, to convey the extremes to which Grothendieck carried this philosophy, or the magnificence of its success. Of course it might still have had its limits. In a 1986 letter to Grothendieck, Serre raised the question of why Grothendieck had largely stopped working on mathematics:

One might ask oneself, for example, if there is not a deeper explanation than simply being tired of having to bear the burden of so many thousands of pages. Somewhere, you describe your approach to mathematics, in which one does not attack a problem head-on, but one envelops and dissolves it in a rising tide of general theories. Very good: this is your way of working, and what you have done proves that it does indeed work. For topological vector spaces or algebraic geometry at least…. It is not so clear for number theory….Whence this question: Did you not come, around 1968-1970, to realize that the “rising tide” method was powerless against this type of question, and that a different style would be necessary—which you did not like?

As far as I know, Grothendieck never responded to this letter.

My friend Bob Thomason once told me that the reason Grothendieck succeeded so often where others had failed was that while everyone else was out to prove a theorem, Grothendieck was out to **understand** geometry. So when Grothendieck set out to attack the notoriously difficult Weil Conjectures, the goal wasn’t so much to solve the problems as to use them as a test for the philosophy that if you generalize sufficiently, all difficult problems become easy.

In that sense, the Grothendieck seminar was both a magnificent success and in the end, to Grothendieck, a disappointment The last and hardest of the Weil conjectures was settled in 1974 by Grothendieck’s student Pierre Deligne, who solved the problem largely by immersing it in the vast Grothendieckian sea, but then, at the last minute, bringing out the mathematical equivalent of a chisel. Grothendieck never forgave him.

What are the Weil conjectures? [Feel free to skip this and the next paragraph if you don't care!] Very very roughly: Start with an equation (or, if you prefer, a family of equations) — say, for example, Y^{2}=x^{3}+10x-3. One solution is (X=2,Y=5). How many others are there? Well, it depends on what you count as a solution. Do you only allow integers? Or are you willing to consider solutions like (X=1,Y=√8) ? The more liberal you are about what kind of numbers you allow, the more solutions (or approximate solutions) you’re going to have. The Weil conjectures make some very precise quantitative claims about how the **number** of (approximate) solutions grows as you become increasingly liberal.

As we all know from high school, an equation itself defines a curve. Counting “allowable” solutions means counting the “allowable” points on this curve. That, in turn, turns out to require a very subtle understanding of the curve’s geometric properties — does it, for example, have any sharp kinks? Does it surround any “holes”? How many? Et cetera. The Weil conjectures say (in a very precise way) that these purely **geometric** properties control the answer to our purely **arithmetic** questions about the growth rate of the number of solutions.

Bottom line: To study the Weil conjectures, you have to think very hard about the subtle properties of curves, surfaces and higher-dimensional objects. When you do this, you find yourself mentally “moving around” the curve, trying to hop from point to point. And — in a sense that I cannot hope to make precise here — you sometimes find that your mental exploration is hampered by the fact that there somehow aren’t **enough** points to hop to.

So life would be easier if these curves (and other objects) had more points. A normal person might say “Well, life’s not always easy. We’ll just have to get by somehow with the points we’ve got”. But Grothendieck lived by the conviction that **everything** is easy if you look at it right — which means there have **got** to be enough points. And if we think there aren’t, it must be because we haven’t yet figured out what a point is.

So — what **is** a point? Grothendieck’s insight was roughly that **a point is a landscape with only one place to stand** — or, a little more precisely, a point is a **space where all functions are constant**.

But wait a minute — what’s a function? It’s something with a domain — which in this case is our point — and a range, which is — what? Sometimes the range of a function is the rational numbers. Sometimes it’s the real numbers. Sometimes it’s the complex numbers.

This means, then, that there are many different **kinds** of points, depending on what kinds of (constant) functions they support. There are “real points”, where every function is equal to some constant real number. There are “complex points”, where every function is equal to some constant complex number. There are even points where every function is equal to some expression like (3x^{2}+1)/(7x^{3}+4). You might object that that’s not a constant — but it **is**, because the x in that expression is not a variable; it’s just a symbol, and that symbol always remains just x.

To Euclid, a point was just a point. Post-Grothendieck, a point has a great deal of internal structure, determined by the sorts of (always constant) functions that live on that point.

When you look at, say, the ordinary Euclidean plane, the points you see — the points that stretch out to infinity in all directions, the ones you familiarized yourself with in high school — are just the **real** points. But from a Grothendieckian perspective, that’s not the whole plane. There are also plenty of (invisible) complex points, (and those points, incidentally, can “spin in place”, ultimately because the complex numbers contain two square roots of minus one, which can be interchanged.) And there are plenty of far more complicated points besides. The plane is **teeming** with points you never learned about in high school.

It turns out that when you have all those extra points to work with, a lot of technical problems melt away, and you can solve a lot of problems you couldn’t solve before. Generalize sufficiently — allow the possibility that your notion of a point was always too specific and too cramped — and hard problems suddenly get easy.

Grothendieck rewrote the foundations of geometry not just once, but three times, first replacing classical geometry with his “theory of schemes” — the material we just touched on in Section II — and then going on to the “theory of toposes”, and finally the great unfinished “theory of motives”. One of the great goals of contemporary mathematics is to complete that final theory, which, if our expectations are correct, would give us the tools to settle many of the hardest outstanding problems in several areas of mathematics.

In this section, I’ll talk (once again in a vague sort of way) about the theory of toposes.

Let’s start over: What’s a point? Answer: A point is a landscape from which there is only one point of view. What’s a curve? It’s a landscape from which there are **many** points of view. What’s a surface? A landscape from which there are even **more** points of view. But what are we viewing?

The theory of schemes posits that we’re viewing the values of functions, which are constant on points, but can vary on curves. The theory of toposes posits that we’re viewing **entire mathematical universes**.

In classical mathematics, questions have unambiguous answers. Is 7 a prime number? Yes, unambiguously. Must the angles of a euclidean triangle add up to 180 degrees? Yes, unambiguously. How many fundamentally different kinds of symmetry are possible in a 2-dimensional design? Exactly 17, unambiguously. How many possible configurations are there for a Rubik’s cube? Exactly 43,252,003,274,489,856,000 as a matter of fact. Unambiguously.

That’s how the world looks when you’re standing on a point and therefore have only one point of view. That, then, tells us what a point is: It’s a place where the mathematical universe looks the way we expect it to. In some sense, we might as well say that a point **is** the classical mathematical universe.

(One little glitch: Points themselves belong to the mathematical universe, so if a point **is** the mathematical universe, then we’re in danger of something very like circularity. Grothendieck sidestepped this problem by defining the mathematical universe to contain all mathematical structures up to some unfathomably large size; the universe itself, being larger than that unfathomable size, does not count as a member of itself.)

What, then, is a curve? A curve is a place where you can move around, and look at things from many points of view. Is 7 a prime number? It might depend on where you’re standing. So we can identify a curve with a **different** mathematical universe, a universe that admits a certain amount of ambiguity — not at all the same as the classical universe we’re used to, but still a perfectly valid object of study.

And of course a **different** curve gives us still **another** mathematical universe, and a surface is yet another…

What’s the point (pardon the pun)? Two quite extraordinary things fall out of this viewpoint:

First, it turns out, miraculously enough, that when we see points and curves and surfaces as the homes for entire mathematical universes, we are able to **use** that insight to solve hard problems in classical geometry and arithmetic. We’re still studying the same old points and curves, but by **recognizing** that each of these points and curves supports an entire Universe, and by making **good use** of that insight, we can (rather incredibly) learn new things about the ordinary geometry of those points and curves.

Second, and quite independently of that, we now have a whole new family of mathematical universes to explore. Not only is that cool in its own right, but it sheds a lot of new light on the good old classical universe. For example, all of these universes satisfy many of the same basic axioms. So whenever we find a statement that’s true in one universe and false in another, we can conclude that our axioms are not enough to settle it. This yields vast new insights into the relative power of various axiom systems.

What I want to stress, of course, is not just the **success** of this viewpoint, but how **daring** it was. Who, before Grothendieck, would have dared to redefine a point, let alone identify it with the entire universe of classical mathematics? Yet this turned out to be exactly what was needed, both for solving old questions and for raising new ones.

There are at least three major themes running through all of Grothendieck’s mathematics. One, as we’ve seen, is his commitment to the principle that all problems become easy if only you can find the right generalizations. Another, as we’ve also seen, is his willingness to redefine classical objects like points and curves in order to make them more susceptible to being generalized. The third, which is equally central, is Grothendieck’s lifelong insistence that mathematical objects are intrinsically uninteresting — instead it’s the relations **between** mathematical objects that matter. The internal structure of a line or a circle is boring; the fact that you can wrap a line around a circle is fundamental.

Perhaps consistent with that philosophy — or, depending on how you look at things, perhaps in direct contradiction to it — Grothendieck seems to have devoted vast energy to meditating on his own place in the Universe, his role in history, his relations with other mathematicians, and the influences that made him the extaordinary man he was. Long after his retirement (and his disappearance to a remote village in the Pyrenees), he was producing 1000-page autobiographies, transcribing his dreams, and writing long letters, some of them devoted to visionary new mathematics, and others to the sort of rambling that led some old friends to believe he’d completely lost his mind. There are, apparently, another 20,000 pages of writings in a locked box at the University of Montpelier that might soon be opened.

If we can extrapolate from the fact that the unexamined life is not worth living, then Grothendieck led the most worthwhile life in history. In his early twenties, he wrote a doctoral thesis in the subject of “functional analysis” that provided new tools so powerful that they left almost no problems in that field left to solve. He then went off searching for a new field vast enough to contain his talent, and found it in algebraic geometry. In the days of the Seminaire de Geometrie Algebrique, he reportedly worked 18 hours a day, 7 days a week, 10 years per decade. And when he needed an even bigger subject, he brought that same vast energy to studying himself.

It has been the great privilege of my life to understand some small fraction of the mind of Grothendieck. The world is an infinitely more beautiful place because he lived.

Around 1970, Alexander Grothendieck, the greatest of all modern mathematicians and arguably the greatest mathematician of all time, announced — at the age of 42 — the official end of his research career. Another great mathematician once told me that he thought he knew why. Following two decades of discoveries and insights that, one after the other, stunned the mathematical world, Grothendieck had, for the first time, achieved an insight so unexpected and so consequential that he himself was stunned. Grothendieck had discovered his own mortality.

I am told that just a few hours ago, his vision proved accurate. But the notion of Grothendieck as a mortal seems hard to swallow. He dominated pure mathematics not just through the force of his ideas — ideas that seemed eons ahead of everyone else’s — but through the force of his personality. When, around 1960, he announced his audacious plan to solve the notoriously difficult Weil conjectures by first rewriting the foundations of geometry, dozens of superb mathematicians put the rest of their careers on hold to do their parts. The project’s final page count, including the twelve volumes known as SGA (Seminaire de Geometrie Algebrique) and the eight known as EGA (Elements de Geometrie Algebrique) approached 10,000 pages. The force and clarity of Grothendieck’s unique vision scream forth from nearly every one of those pages, demanding that the reader see the mathematical world in a new and completely original way — a perspective that has proved not just compelling, but unspeakably powerful.

In Grothendieck, modesty would have been ridiculous, and he was never ridiculous. Here, in his own words — words that ring utterly true — is Grothendieck’s own assessment of how he stood apart (translated from French by Roy Lisker):

Most mathematicians take refuge within a specific conceptual framework, in a “Universe” which seemingly has been fixed for all time – basically the one they encountered “ready-made” at the time when they did their studies. They may be compared to the heirs of a beautiful and capacious mansion in which all the installations and interior decorating have already been done, with its living-rooms , its kitchens, its studios, its cookery and cutlery, with everything in short, one needs to make or cook whatever one wishes. How this mansion has been constructed, laboriously over generations, and how and why this or that tool has been invented (as opposed to others which were not), why the rooms are disposed in just this fashion and not another – these are the kinds of questions which the heirs don’t dream of asking . It’s their “Universe”, it’s been given once and for all! It impresses one by virtue of its greatness, (even though one rarely makes the tour of all the rooms) yet at the same time by its familiarity, and, above all, with its immutability.

When they concern themselves with it at all, it is only to maintain or perhaps embellish their inheritance: strengthen the rickety legs of a piece of furniture, fix up the appearance of a facade, replace the parts of some instrument, even, for the more enterprising, construct, in one of its workshops, a brand new piece of furniture. Putting their heart into it, they may fabricate a beautiful object, which will serve to embellish the house still further.

Much more infrequently, one of them will dream of effecting some modification of some of the tools themselves, even, according to the demand, to the extent of making a new one. Once this is done, it is not unusual for them make all sorts of apologies, like a pious genuflection to traditional family values, which they appear to have affronted by some far-fetched innovation.

The windows and blinds are all closed in most of the rooms of this mansion, no doubt from fear of being engulfed by winds blowing from no-one knows where. And, when the beautiful new furnishings, one after another with no regard for their provenance, begin to encumber and crowd out the space of their rooms even to the extent of pouring into the corridors, not one of these heirs wish to consider the possibility that their cozy, comforting universe may be cracking at the seams. Rather than facing the matter squarely, each in his own way tries to find some way of accommodating himself, one squeezing himself in between a Louis XV chest of drawers and a rattan rocking chair, another between a moldy grotesque statue and an Egyptian sarcophagus, yet another who, driven to desperation climbs, as best he can, a huge heterogeneous collapsing pile of chairs and benches!

The little picture I’ve just sketched is not restricted to the world of the mathematicians. It can serve to illustrate certain inveterate and timeless situations to be found in every milieu and every sphere of human activity, and (as far as I know) in every society and every period of human history. I made reference to it before , and I am the last to exempt myself: quite to the contrary, as this testament well demonstrates. However I maintain that, in the relatively restricted domain of intellectual creativity, I’ve not been affected by this conditioning process, which could be considered a kind of ‘cultural blindness’ – an incapacity to see (or move outside) the “Universe” determined by the surrounding culture.

I consider myself to be in the distinguished line of mathematicians whose spontaneous and joyful vocation it has been to be ceaseless building new mansions.

We are the sort who, along the way, can’t be prevented from fashioning, as needed, all the tools, cutlery, furnishings and instruments used in building the new mansion, right from the foundations up to the rooftops, leaving enough room for installing future kitchens and future workshops, and whatever is needed to make it habitable and comfortable. However once everything has been set in place, down to the gutters and the footstools, we aren’t the kind of worker who will hang around, although every stone and every rafter carries the stamp of the hand that conceived it and put it in its place.

The rightful place of such a worker is not in a ready-made universe, however accommodating it may be, whether one that he’s built with his own hands, or by those of his predecessors. New tasks forever call him to new scaffoldings, driven as he is by a need that he is perhaps alone to fully respond to. He belongs out in the open. He is the companion of the winds and isn’t afraid of being entirely alone in his task, for months or even years or, if it should be necessary, his whole life, if no-one arrives to relieve him of his burden. He, like the rest of the world, hasn’t more than two hands – yet two hands which, at every moment, know what they’re doing, which do not shrink from the most arduous tasks, nor despise the most delicate, and are never resistant to learning to perform the innumerable list of things they may be called upon to do. Two hands, it isn’t much, considering how the world is infinite. Yet, all the same, two hands, they are a lot ….

Sometime in the near future, I hope to fashion a blog post or two that will convey at least a scintilla of the Grothendieckian worldview to a non-mathematical audience. For now, I point you to a few good biographical articles and some of Grothendieck’s own most influential writing, which I’ve posted here. I particularly recommend the piece by Colin McLarty, the first few pages of which, at least, are mostly non-technical.

**Edited to add:** See my followup post here.

So apparently there was this pumpkin……

A colleague spotted it on the floor in front of my office door on Sunday afternoon and was intrigued enough (or weirded out enough) to snap a couple of pictures:

Unfortunately, by the time I came into work on Monday, the pumpkin had mysteriously disappeared. And I didn’t cross paths with my colleague until late this afternoon, which is when I first learned that there had ever **been** a pumpkin.

Obviously I owe someone a note of thanks, particularly as this pumpkin appears to have come with a gift card attached. But I can’t even begin to guess the identity of the giver, the reason for the gift, or the contents of the note inside.

I’m hoping that anyone who feels motivated to carve a pumpkin for me will also feel motivated to take an occasional peek at my blog, and will therefore see this post and come forward. Anyone?

]]>It was the election of 1994 that knocked the idealism out of me. Republicans ran on a national platform of reform, they won — and nothing happened. My recollection (someone correct me if I have this wrong) is that a series of substantial reform bills passed the Republican house in short order, and all of them died in the Republican senate. My guess (without having thought too hard about it) is that this is the natural order of things because Senate campaigns are so expensive that no matter what legislation the House sends up, there’s always some committee chairman with a large donor who opposes it.

There is no reassurance to be had from the identities of the likely new chairmen-to-be: Thad Cochran at Appropriations, Pat Roberts at Agriculture, Jeff Sessions at Budget, Orrin Hatch at Finance. Even aside from the question of what you can or can’t get past the White House, these are not the sort of people I want rewriting the tax code; they are not the people I want setting agricultural policy; they are not the people I want in charge of immigration reform.

So color me cynical about whether this election will make much difference. On the other hand, it was pretty nice to see voters in a variety of states repudiate vicious attacks on candidates who dared to defend outsourcing, candidates who dared to cut education spending, candidates who dared to cut the size of government, and candidates who dared to resist the demands of public employee unions. For the most part, I was unhappy with the Republican candidates for their failure to articulate anything positive. But by and large, it seemed to be the Democrats who sank to the lowest forms of demagogy, and I was glad to see those candidates lose.

Besides, I don’t follow sports, which means there’s only one night every two years when I get to engage in recreational exuberance, rooting with wild abandon for “my” team without regard to the fact that it’s only a game. In that light, I had a very good night indeed.

]]>From Katharine Q. Seelye of the New York Times, writing with no apparent sense of irony about Rhode Island gubernatorial candidate Serena Mancini:

She favors raising the minimum wage and indexing it to inflation, for example, and opposes making Rhode Island a “right to work” state. Her chief focus is creating jobs.

If you doubt the existence or direction of bias at the New York Times, ask yourself when you’re next likely to read a Times piece that says something like:

She favors widespread deregulation, for example, and opposes all taxes on capital income. Her chief focus is alleviating poverty.

Wait, that’s an imperfect analogy, since (unlike the passage from Ms. Seelye) it actually makes sense. Let me try again:

She favors prison terms for adulterers, for example, and opposes legalized contraception. Her chief focus is freedom of sexual expression.

Soon we will have blessed relief from the nonsense of the campaigners, but alas the nonsense of journalists will continue year-round.

]]>That was the day Father had told the Burdens that Cash Benbow would never be elected Marshal in Jefferson. I don’t reckon the women paid any more attention to it than if all the men had decided that the day after tomorrow all the clocks in Jefferson were to be set back or up an hour.

—William Faulkner, The Unvanquished |

In early 20th century China, goods were frequently transported by barges pulled by teams of six men. The men were paid only if they delivered their goods on time. Therefore they all agreed to pull as hard as possible.

This is a classic example of what economists call a Prisoner’s Dilemma — a situation where everyone wants to cheat, regardless of whether he believes the others are cheating. Any bargeman might reason that “If the others are pulling hard, we’re going to make it anyway, so I might as well relax. And if the others are **not** pulling hard, we’re **not** going to make it anyway — so I **still** might as well relax .” Therefore they all relax and nobody gets paid.

According to my late and much lamented colleague Walter Oi, the bargemen frequently solved this problem by hiring a seventh man to whip them whenever they appeared to be giving less than 100%. You might suppose, at least if you’re a person of ordinary tastes, that hiring a man to whip you is never a good idea. There’s a sense in which you’d be right. But hiring a man to whip your **colleagues** can be a very good idea indeed, and if that requires getting whipped yourself, it might prove to be an excellent bargain.

If I’d lived in China a hundred years ago, I believe I’d have gone out of my way to buy goods from the teams with whipmasters — partly because that’s where I’d expect the best service, but also partly because I’d feel a certain combination of admiration and loyalty for the teams who were working so hard to earn my business.

That’s how I feel about the folks at Amazon. Based on the fabulous service I’ve been getting, I’m confident these people are knocking themselves out to do a good job for me. In fact, it’s been widely (and perhaps accurately) reported that during a heat spell a couple of summers ago, workers in an un-airconditioned Pennsylvania warehouse continued to fill orders even as several were being treated for heat sickness.

There’s a narrative going around that tries to paint these workers as victims, though I’ve heard no version of that narrative that makes clear who, exactly, is supposed to have victimized them — the stockholders? the management? the customers? the do-nothing Congress? But there’s little point in trying to make sense of this narrative, since it’s so obviously wrong to begin with.

Imagine a team of ambitious but relatively low-skilled workers. They know that if they all push themselves to the limit, they’ll all be more productive and therefore earn higher wages. They also know that if they all **promise** to push themselves to the limit, they’ll all break their promises, figuring that success or failure depends almost entirely on what the others do.

What these workers need, and presumably want, is an enforcer to prevent the breakdown of the agreement. Fortunately for those Amazon warehouse workers, they found that enforcer, presumably in the person of a good warehouse foreman. And even more fortunately, the foreman did his job. A warehouse foreman who sends you home in a heat wave might be no more useful than a whipman who puts down his whip when the going gets tough.

Surely there are some circumstances on the barge trail when the going gets **so** tough that you’d want the whipman to ease up on you, and surely if, for example, the warehouse is burning down around you, you’d want the warehouse manager to send you home. But equally surely, the whole point of a whipman — or a warehouse foreman — is that for the most part, you want him to be a stern taskmaster.

Now it’s easy to say that the world would be a better place if nobody had to work to the point of heat prostration in order to earn a living. That’s true, and we should work toward creating such a world. But it’s **also** true that we do not currently live in such a world, that people will therefore try to do the best they can in **this** world, and that we do those people no favor by trying to hamper their efforts. Poor but ambitious people will seek and find ways to be part of productive teams. Sometimes those productive teams will require a whipmaster or a harsh foreman as an additional partner. The teams will be glad to find those partners, and I for one will be sufficiently impressed with their ambition that I’ll seek to reward them by buying their products.

In 1992, pressure from the U.S. government led to the closure of Bangladeshi factories that employed 50,000 children in conditions that you and I would surely consider harsh. An interviewer seeking a few words of appreciation from those children instead heard this from a ten-year-old girl named Moyna:

They loathe us, don’t they? We are poor and not well educated, so they simply despise us. That is why they shut the factories down.

Think about that the next time someone suggests pressuring Amazon into changing its labor standards. Once we’ve succeeded in denying those workers the opportunity to earn the rewards that can only come from high productivity, they’ll have every right to suspect that we did it because we despised them.

I don’t despise them. I admire the hell out of them. And that’s one of the reasons why I buy from Amazon.

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