Mathematics as a Cultural Force

Historian Amir Alexander on Euclidean geometry’s far-reaching effects.

Jessica Gross | Longreads | Sept. 2019 | 14 minutes (3,556 words)

In his new book, Proof!: How the World Became Geometrical, historian Amir Alexander advances an audacious claim: that Euclidean geometry profoundly influenced not just the history of mathematics, but also broader sociopolitical reality. In prose that makes his passion for the material both clear and catching, he describes how Euclid’s Elements present a vision of a perfectly rational order, but one that was viewed as purely theoretical: There was no place for geometrical ideals in messy reality. In the 1400s, Leon Battista Alberti, an Italian polymath, upended that understanding, countering that the world was, in fact, fundamentally geometrical. Other thinkers, from Copernicus to Galileo, followed. And, as Alexander argues, this sea change had profound implications: If the world was geometrical—not only rational, but also hierarchical and permanent—then that was the divinely ordained social order, too. Euclidean geometry, that is, was used to justify monarchy.

Explaining the interconnectedness between mathematics and culture—how mathematical principles aren’t separate from or even just born into a culture, but profoundly shape it—is nothing new for Alexander, whose previous books include Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World and Duel at Dawn: Heroes, Martyrs, and the Rise of Modern Mathematics. When we spoke by phone in August, we discussed French gardens’ geometrical designs as propaganda; how cities’ structures advance their ideals; and how Euclidean geometry’s decline had as deep an effect as its rise.

Because I struggled with history in school, I am always curious when people choose to make it their life’s work. So maybe we can start there: What do you love about studying, writing about and now, at UCLA, teaching history?

I do love history, and I think it has something to do with growing up in Israel, in Jerusalem. There, it’s not just the one history, but layer upon layer upon layer of history—different histories, competing histories. Every stone and every building there has its own story. You can go back 100 years, you can go back 1,000 years, sometimes thousands of years, and everybody is very much invested in their version of history, often to the exclusion of others.

Also, especially the years that I was growing up in Israel, archaeology was huge because it was seen through a Zionist perspective. That is, you’re digging up Biblical history, you’re digging up the connection of the Jewish people to the land of Israel. It was all around; the air was imbued with it. I think in some ways, whatever your politics—whether you’re a Zionist or an anti-Zionist, whatever your view of the occupation—in some ways, living there, you feel like it is just the latest chapter of a story that began a very long time ago.

So I think that was the origins of my fascination with history, although, as for my work, it went in a very different direction.

Right. Rather than pursue an archeological focus, you’ve carved out a niche as a historian of mathematics. How did that come about for you?

I try to look at everything as story. In some ways, the most challenging thing for me is to look at what is seen as the most ahistorical subject, mathematics, and to say, wait a minute, maybe in some ways it does have a history.

When I say that mathematics is seen as not having a history, I mean that mathematical truths are supposedly eternal, they’re unchanging, it doesn’t matter the context, it doesn’t matter the time, they’re always true, one and one equals two. It’s been true since the beginning of time, whether there have been humans or historical figures or mathematicians who knew it or not.

At first, in some ways, I just found it intellectually challenging to say, “Let’s try a history of this.” Let’s track the ways in which even the most abstract, seemingly ahistorical thing actually does have a history: Mathematics was done by people, created by people, people who had ideas, who had a life, who lived in a particular time. Let’s see how those things interacted, how the particular context shaped the mathematics and how the mathematics shaped the historical context.

So that’s one side of it. But there’s more of a personal angle to how this came to be. My father was a theoretical physicist in Israel, and a very successful one. He loved math, and was certainly a mathematical prodigy. He was a very modest man in most ways, and not given to many extreme pronouncements—except when he talked about math. Then, he would get this spark in his eyes, and he would speak in prophetic terms about how math had opened his eyes. I thought, “Wow. Maybe I should get a taste of that.”

So I studied math as an undergraduate. I was not as good as my father was, certainly, but I was good at math. I got my degree in mathematics from Hebrew University. But I did not have any mystical experiences as my father suggested. So then it became about: How do I take math and connect it not just to those higher spheres that my father somehow saw, but to life, to my life, to historical life?

I once went to a therapist and he said, “Oh, it’s obvious, it’s all about connecting with your father.” Well, maybe. [Laughs] I don’t know. But I certainly wanted to connect things that I learned from him with things that I loved myself.


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It’s really interesting that you say you don’t have this mystical relationship to math that your father seemed to have, because your excitement and passion for math is so obvious in your writing. You start this book with the way that geometry transformed painting through the discovery of perspective, and there was this little nugget there that blew my mind. You point out that mirrors produce a perfect three-dimensional image on a flat surface. I don’t know why but, even having looked in millions of mirrors over the course of my life, it has never occurred to me as anything other than obvious that we can see ourselves three-dimensionally from a flat surface. When you pointed out how unbelievable a feat this is, I was stunned. Did that stun you, too, when you were researching this?

Yeah, it really did. We are so, so used to it that I think we miss the magic of it. I’m the same as you, a mirror is a mirror. You just stand in front of it and everything seems self-evident. But when you step back and think, “Well, what is actually going on here?,” maybe you can recapture some of the astonishment of people when mirrors were first introduced in the Middle Ages. How does this flat surface, without any intervention, any artifice, produce the most beautiful three-dimensional images?

That played a major role in the discovery in Florence that, actually, there is a geometrical structure to the world—that this wasn’t just an abstract concept, but that the world itself was fundamentally and deeply geometrical. So it started with mirrors and perspective and then, as I tell in the story, it spread out from there.

With other fields, people can argue and argue and talk and talk and talk, but geometry lets you prove something is absolutely certain, fixed, and eternal. It’s like the mind of God.

You write about the inflection point between this discovery of geometry as a natural order, rather than a theoretical ideal, and then the wielding of this geometrical order as a justification for subjugation, essentially. You start by describing French gardens and the way that their construction was used to basically inculcate ideas of submission in French people. This is fascinating to me: the suggestion that constructing a geometrical garden is not just symbolic of the governmental order, but actually a way of enforcing power.

I point to the discovery of linear perspective as the point where, in a sense, geometry was brought into the world. That was the genesis of the notion that the world is structured according to geometrical principles. And that notion has a lot of implications.

Geometry is shorthand for Euclidean geometry: the one that was formulated by Euclid in 300 BC, and which was seen as perfectly rational, perfectly ordered, with truths that are absolutely certain and unshakeable and immutable—that have always been true, that are forever true. With other fields, people can argue and argue and talk and talk and talk, but geometry lets you prove something is absolutely certain, fixed, and eternal. It’s like the mind of God.

And suddenly, if you start thinking about the world itself in that way, then you start thinking that the world is orderly, that there is an order there that is fixed, that is rational, that is immutable. So what does that do to our political ideas, to political order, to state power, to culture, even to art? How does that change how we see the world, the world around us, and our place within it?

It was Charles VIII who brought to France, from Italy, this new style of garden. These geometrical French gardens, which ultimately culminated in Versailles, were used to present the rules of the French king not as something arbitrary—he has a big army and he’s going to do bad things to you if you deny his power—but as an inescapable part of the geometry of the universe. That is, the world is geometrical, and what you see in those gardens reflects the geometry of the world, and all this geometry culminates in and points to the power of the king. Because geometry, apart from being orderly and perfectly true and systematic and so on, is also hierarchical. So in some ways these gardens were a tool of propaganda, and an expression of a deeply ingrained belief in a geometrical world and in the place of French royalty within it.

Ultimately, what you see at Versailles is this perfect geometrical garden, perfectly structured, perfectly hierarchical, with the palace at the top of the hill. All straight lines lead to this magnificent palace: the king’s seat. So the king is the culmination of the order of the universe. That’s what you learn in Versailles.

What I found really interesting about Versailles in particular is your argument that the overwhelm of being there was not a mistake but rather an intentional choice. That overpowering the visitor through the scale of the garden and how much stuff is in it further advances what you just described: the concept of the ultimate and divine authority of the king.

Yeah, exactly. I mean, it’s not supposed to be nice. It is supposed to overwhelm. And I found it fascinating when I found that Louis XIV himself wrote an itinerary of how you’re supposed to walk through the garden—where you’re supposed to stop, where you’re supposed to look, what view you are supposed to see—basically integrating the visitors themselves into this geometrical order.

Everything in Versailles has its place. Every tree, every lake, every pond, every fountain, every blade of grass, every flower has its place—and so do the people. The people also have their precise place in that perfect, unchallengeable hierarchy that is Versailles.

The gardens are really, I think, the most startling, magnificent and dazzling example of how the ideal of a geometrical world manifested directly as political power.

What’s so effective about these gardens, too, it seems to me, is that a visitor isn’t conscious of the way it’s working. As you said, it’s propaganda; to be consciously aware of it would almost undermine it.

Yes. The power of the geometrical gardens is precisely in the fact that you look at that geometrical order and you understand that there is no other way. There is no alternative. This is the true natural order of the world. You walk there, and it’s not that you think, “Oh, they designed a garden that made me feel that way.” It’s simply the reality all around you. Who can deny that when you walk those trails, everywhere you look, you see perfectly straight lines that all converge on this magnificent, incomparable palace at the end? The king is in his place not because of some whim, but because of a deep, unchallengeable, irresistible geometrical order. And the implication is that this deep geometrical order is true everywhere: all over France and, implicitly, all over the world. An entire ideology and worldview is encoded in that landscape.

I love Paris, it’s maybe my favorite place I’ve ever been, and I love walking through the gardens. Maybe I’m dense, but it hadn’t occurred to me so consciously until I read your book how geometrical the design of the gardens and city itself is. So I was thinking about what it means that I am so drawn to this aesthetic. I’m a sort of rigid person who really likes order and I think there is something calming about being in a place where everything is orderly. But it was alarming to have that uncovered to me through your prose. We obviously don’t live in the same political reality in which these gardens were constructed, but I wonder how you would explain attraction to these designs in terms of individual psychology?

Versailles is now a museum, really. We don’t read it as if it’s speaking to us, we read it more like, that was how it was 300 years ago. But a city like Paris is different. It’s a living city, with those grand, arrow-straight boulevards with monuments that speak the same geometrical language. It doesn’t say there is one king at the top of it all, but it very much talks about the deep, irrefutable, unquestionable order of the French state and perhaps the famous French bureaucracy and the French civil service. And it still speaks to that.

The only American city that compares to that is Washington, D.C., which was designed by a Frenchman who grew up in Versailles. And it also has that implication. Of course it’s not a royalist ideology, it’s a republican one, but it is also one that presents the political order as necessary and grounded in the deepest order of the universe that is geometry.

Where do you live?

I live in Brooklyn, New York.

New York is the opposite of Washington, D.C. I find Washington very reassuring in these days, I have to say. With the current political situation and the polarization and not to mention the mass shootings, it seems like all our cherished institutions, our republican institutions, are under attack. When I visit Washington, D.C., I find something very reassuring about it precisely because of that geometrical order: You go there and you see Capitol Hill and the houses of Congress with the great American flags. Okay, that’s still there. And you see the White House at the other side of Capitol Avenue and regardless of what you think of its occupant, you think, “This will outlive any particular president.” The deep order that is implied there, I do find it reassuring. I go there and I look around and I say, “We will survive this, it will be okay.”

So I’m kind of like you, I’m also attracted to cities like that, to architecture like that. To the message that there is a deeper, fundamental order that is not just a matter of the whim of one president or one congress or even one generation, that there is a deeper order here that will prevail. So although geometry was used very powerfully by autocrats from Louis XIV to Hitler in his redesign of Berlin, this is not the only way in which it can be used. In Washington, D.C., it is used to establish a republican political order on absolute, eternal geometrical foundations.

You cannot take for granted anymore that there is one single necessary order.

In the book, you describe how in 1823, a Hungarian mathematician named János Bolyai proved that there are infinite alternate geometries existing alongside each other, which made it impossible to continue upholding this one, single, unified order. You argue that this discovery coincided with or helped prompt the decline of widespread investment in supreme leaders.

You cannot take for granted anymore that there is one single necessary order. In fact, this never went uncontested: Louis XIV might say that he was the one true king and an expression of an unchanging, universal order, the order of the universe, but his political rivals, especially England, said, “No, you’re not, we don’t buy any of that, and we’ll create gardens that are the exact opposite of Versailles.” There was a completely different order to the world than Louis XIV’s. So it was always contested. But it was never more profoundly contested than with non-Euclidean geometry, which said there can be an infinity of truths. They are all absolutely true, but they are incompatible with each other. That was a profound challenge to the notion of a single, necessary order.

But I still think, nevertheless, this idea is not dead. I think we still feel its power. There’s a lot of pushback against the idea that everything is relative and all truths can be equally true, one next to the other. I think we still find something very reassuring about this idea that reason is reason and we should follow rationality systematically and dispassionately, and that will give us a true answer both scientifically and politically. I think that is expressed in those geometrical designs and I think we still find it very powerful and reassuring.

You present so many examples of urban planning that reflects the structure of that place’s government at the time that it was constructed. Is that always the case, or are there ever seats of power that are not designed to reflect the structure of the government at the time in this conscious way?

Well, I think governments generally like geometry. It all comes from Versailles. But that doesn’t mean that all seats of power are always designed in this geometrical fashion. New York is not the political capital of the United States, but it’s probably the primary seat of economic and cultural power. In many ways, outside the political, New York is the capital of the United States. And if you think about Manhattan, it’s the complete antithesis of a design based on a single, necessary, geometrical order. What you have instead is a grid in which every point is the intersection of two coordinates, which is to say that every point is as good as any other. If you look at Manhattan on a map, it’s just kind of a blank geometrical space in which you can do anything at all. As I see it, it emphasizes, rather than a single necessary political or social order, infinite possibility. And that’s a completely different design.

Yes, and of course that is exactly why many people move to New York. You live in Los Angeles, whose “design” is often mocked.

Yes. [Laughs]

Can you talk a bit about how L.A. is structured and what that might be communicating to its inhabitants, whether consciously or not?

Well, Washington, D.C., is totally the exception among American cities. The dominant pattern by far is the grid. I think the interesting thing about Los Angeles is really its unboundedness. San Francisco is a peninsula. It’s surrounded on three sides by water and then it spreads in various directions, but the city itself is clearly bounded. And Manhattan of course has clear boundaries. Los Angeles is maybe now reaching some limits, but it seems, at least, completely unbounded. You take this open grid in which every place is equal to any other and you just keep expanding and expanding the city in that way without any clear center or any clear periphery or hierarchy or particular order. It’s just this abstract open space. It’s the extreme opposite of Versailles. In Versailles, there’s a center, and everything looks to one place. In Los Angeles, everything is this surface grid. There is a suggestion of boundlessness.

Since you look at how developments in math throughout history and how they track with what’s happening sociopolitically, I’m curious if you could talk about what’s happening today. Do you still see such a strong connection between mathematical debates and our current sociopolitical reality?

Well, we spoke before about this turning point in mathematics away from the concept of one single necessary order of the world. With non-Euclidean geometry, I think that is fundamentally shattered. I’m not saying non-Euclidean geometry is responsible for all of our fractured politics, but I do think that the notion that there is not just one truth is still reverberating. It has become a commonplace of mathematics that there is no privileged truth, that any kind of truth is simply dependent on your assumptions. And if that’s true for mathematics, one of the most perhaps rigorous, rational, universal systems, then it implicitly will have to be true about everything else as well.

The power of mathematics to shape the world decisively, I think, really reached its peak in those centuries between, say, 1400 and 1800. Beginning in the 19th century there was a movement in mathematics to once again separate mathematics from the world. It doesn’t mean that the world is not mathematical, you can still describe the world mathematically, but you cannot assume that the world is an expression of mathematics. So I think it is harder, now, to point to particular developments in mathematics that would have the kind of power that they once had.

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Jessica Gross is a writer based in New York City.