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.

**I. General Philosophy**

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.

**II. The Theory of Schemes**

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.

**III. Theory of Toposes.**

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.

**IV. The Examined Life**

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.

Thank you for showing us non-mathematicians a glimpse of this world.

I was kind of hoping for a post regarding net neutrality.

Thanks for this great post. Just a very minor comment, Grothendieck lived in the Pyrenees, not the Alps.

Anonymous: Fixed. Thank you.

Thanks for writing this. :)

As with Landsburg, I never met or came within the presence of Grothendieck. Unlike Landsburg, I don’t know that it would have mattered if I had. I suspect whatever enrichment I would have received would pale in comparison to the enrichment I get from Landsburg’s posts.

This is

so damn fun, Landsburg! I suppose only people with a broader, deeper point of view – experts – would be able to comment on the accuracy of what Landsburg says. All I can comment on is the tantalizing joy I get from the idea that there’s a whole new frame of reference, brought just to the edge of my comprehension. The ability to convey this stuff is a rare talent unto itself. If it’s snake oil, it’sdamn goodsnake oil.Maybe it’s my 20th-century thinking, but I occasionally feel guilty that fun insights on substantive stuff get expressed and tossed aside on the web. Precisely because Landsburg translates esoteric stuff into easily-grasped metaphors, this is the kind of thing that authoritative sources would be unlikely to cite – yet precisely the most useful stuff for the general reader. (“Ugh — metaphor, poetry, how imprecise….”) Perhaps some version of these Grothendieck posts should become an External Link on the Grothendieck Wikipedia page?

As an aside, here is nobody.really’s nearly pointless conjecture: A (Euclidean?) point has one side.

What object has the fewest “flat” sides?

In 3 dimensions: A tetrahedron has 4 sides, each side formed by a triangle – a 3-sided object.

In 2 dimensions: A triangle has 3 sides, each side formed by a line segment – a 2-sided object.

In 1 dimension: A line segment has 2 sides, each side formed by a point. To maintain the pattern, a point would need to have one side.

Corollary conjecture: In 4 dimensions, there are 5-sided objects with each side formed by a tetrahedron. And so on.

Indeed, thanks for sharing your understanding.

nobody.really:

Corollary conjecture: In 4 dimensions, there are 5-sided objects with each side formed by a tetrahedron. And so on.Not only is this correct, but the sequence of spaces you’re talking about is the fundamental building block of all topology. One of the things Grothendieck worked on late in life was the question of why this building block works so well and whether some other building block would serve equally well — in other words, are we focused on this one because it’s the best or because it’s the first one we happened to think of?

Non-mathematicians reading his obituaries will likely get the impression that he was a disturbed man from an emotionally traumatic childhood who somehow got a cult following for his abstract nonsense.

“Grothendieck’s work was also a stepping stone to solutions of enigmas famous among mathematicians — the Poincare conjecture, for instance — but far more arcane.” NY Times, most emailed news story today, India Times

“His contributions to mathematics were often likened to those of Albert Einstein in physics. … Other scholars came to apply Mr. Grothendieck’s theoretical frameworks to such fields as computer programming, software development, satellite communications, classification systems and the study of biological data.” Wash. Post

“His techniques were a necessary tool in the work done by Maryam Mirzakhani that earned her the historic first Fields Medal awarded to a woman earlier this summer.” Bus. Insider/Yahoo

I do not see how his work is related to any of those things.

Thank you for doing this, Steve. This was very fun to read.

Beautiful, man!

Awesome post. Keep ‘em coming, there’s no one else out there explaining incredibly complex mathematics at just the perfect level of difficulty for interested non-experts to follow.

The article made it onto the recommendation-list of thebrowser.com, much to my satisfaction.

Another thank you for writing this post, and doing it so well. Until reading this, it was impossible for me to appreciate his work to any meaningful degree since I had no frame of reference for what it entailed or how important it was.

These types of posts (dumbing down things I couldn’t hope to understand to a point where I can catch a glimpse of them) are my favorite here.

Well done Steve, truly.

Great post, and a (perhaps naive) question. Do you believe that as new vistas open up in the mathematical universe, it makes it more likely that similar discoveries (inventions?) will occur in physics? In the past that seemed to be the case, but was it a coincidence? That is, do you think the physical universe is in some fundamental sense “about math,” i.e. the embodiment of math?

The Generalistimo!

I did my PhD at the University of Montpellier and though I never met Grothendiek (I was too late), he was always a legend there. When my colleagues first mentioned him to me, they spoke of “Alexander the Great” and when once I objected to a colleague about that unfounded analogy with the Greek warlord citing Grothendieck’s pacifism, he replied: “different in purpose, similar in courage”.

So reading your words about the courage of Grothendieck’s works put an ear-to-ear smile on my face.

Scott Sumner:

Do you believe that as new vistas open up in the mathematical universe, it makes it more likely that similar discoveries (inventions?) will occur in physics?This is clearly the case, and Grothendieck’s legacy is a superb example. Theoretical physicists now routinely make use of a vast conceptual apparatus that comes out of Grothendieck’s work, including modern algebraic geometry, algebraic K-theory, category theory, and even motives. Try browsing, for example, through John Baez‘s pages for a krazillion examples.

do you think the physical universe is in some fundamental sense “about math,” i.e. the embodiment of math?You are late to the party! :) This is one of the primary themes of my book The Big Questions, and something I’ve blogged about frequently. The answer is yes.

Well, I’ve owned the book for about 2 years–I guess it’s time I read it! I will do so.

One of my mathematical mentors, David Harrison, once said to me that there are two kinds of mathematicians, describing the types using a mining metaphor: prospectors and a hard-rock miners. Both are required for the progress of the mathematics in general, but as an individual you need to decide which kind you what to be. Do you strive for the big picture or use the chisel to crack the hard rocks. Grothendieck was a prospector of the first order – perhaps the best ever.

@10. I do not think Grothendieck had much (if any) influence on the solution of Poincare conjecture.

Though I met Grothendieck a few times in 1968 and several people around were influenced by his work, I seem to have been untouched (I am not proud to say this, but my mathematical interests were different) by his work. But I had slight glimpse of his methods in his Grothendieck-Reiman-Roch Thorem (again outside my area)”The significance of Grothendieck’s approach rests on several points. First, Grothendieck changed the statement itself: the theorem was, at the time, understood to be a theorem about a variety, whereas Grothendieck saw it as a theorem about a morphism between varieties. By finding the right generalization, the proof became simpler while the conclusion became more general. In short, Grothendieck applied a strong categorical approach to a hard piece of analysis. Moreover, Grothendieck introduced K-groups, as discussed above, which paved the way for algebraic K-theory.” from http://en.wikipedia.org/…/Grothendieck%E2%80%93Riemann…

gaddeswarup: Yes, of course Grothendieck-Riemann-Roch is yet another of the many many spectacular achievements that I skipped over in the interest of keeping this blog post at a sub-Grothendieckian length of less than, say, 10,000 pages.

But G-R-R is a great example, as you say, of several things I did touch on: First, while everyone else wanted to talk about varieties (note to non-mathematicians: “varieties” in this context means curves, surfaces, etc.), Grothendieck insisted on talking about the

relations betweenvarieties, and that changed everything. (Cf. my comment near the end of the post about a line or a circle being boring, while the fact that the line wraps around the circle is fundamental — this was meant to give just a bare glimpse of the tip of the tip of the iceberg of Grothendieck-Riemann-Roch).Second, the phenomenally general context in which everything is placed.

And finally, of course, they payoff, where you get a much deeper understanding, and a much better theorem, and all not because you zoomed in on the details, but because you zoomed out and found exactly the right generalization.

And then of course, this all leads to K-theory, which becomes by itself a major research area with fantastic applications to topology, arithmetic, geometry, and ultimately theoretical physics etc. etc. etc.

I come here to discover how little I know.

Thank you for the wonderful post on Grothendieck and his way of viewing things. I am a structural biologist who came to know only after Grothendieck died about him because of a facebook post of my son who is completing his masters in maths. Increasingly I have been reading more of Grothendieck fascinated that such a man existed and worked in this way uncovering vistas of beauty. I wish we have more such persons in science. Biology today seems to sorely lack such giants who are generalists.

This post is tantalizing but not really satisfying in the end. It sounds great at times, it gives you the feeling that you are glimpsing something fascinating, but I suspect its value is more for someone who already understands Grothendieck’s work and can take pleasure in these crystallized, literary nuggets you have made of it. But for the rest of us, the nuggets are still opaque. Incidentally, I think the real task is not to distill Grothendieck for the comprehension of a smart anthropology major. He will never get enough of it. The real task is to aim for someone like a smart engineer. As evidenced by the lack of successful explanations of things like algebraic geometry at such a level, this is a very hard task.

Your response to Scott Sumner, claiming that algebraic geometry etc. has proven useful in physics is wrong. Certainly other seemingly whimsical ideas from mathematics, like differential geometry or Lie groups, have done so, but not the subject matter of Grothendieck’s work. Perhaps it will be useful in physics someday, but to date it has not been part of any physical theory that has made successful predictions or that we have reason to believe describes the physical world. Many people in physics departments, who are really closer to mathematicians, have looked in this direction. But that is different from actually succeeding with it.

Not a mathematician but I got a lot out of this article. Truly excellent writing.

Very well written, and gives a glimpse of what he was doing, which is almost imopssible when looking up “Étale cohomology” “Grothendieck–Teichmüller group” “Schemes” “Grothendieck–Riemann–Roch theorem” etc. on the Internet. A great mathematician and even also philosopher.

Eric Dennis, the current understanding of quantization directly involves sheaves, and thus Grothendieck’s work. It’s true that much of the current theory on using toposes/category theory in quantum information theory is not standard working practice yet in physics, but it will be.

@Roger, at 10, toposes and categories are used in understanding (especially functional) programming languages. There are several textbooks on this. Goguen probably started the work applying toposes to software verification, or at least he worked on it very early, “Classification systems” might refer to concept lattices, which are certainly related to category theory; I’m not sure if that is what they had in mind. Satellites seem like a stretch, but if there *is* a connection to Grothendieck, I don’t think he would have been very happy about it… Grothendieck made important contributions to Teichmuller theory, which is probably the connection to Mirzakhani.