ARUN RATH, HOST:

Now, we're going to talk about one of the weirdest paradoxes in math. And don't worry if you hate math. Here's the weirdness in a nutshell. Say you have a line in space. It's 1-inch long and you want to divide it. Easy. You cut it in half, cut that half in half and so on and so on. But at some point you get to something so small you can't cut anymore. All those infinitely small things make up your line. They're so small they have a length of zero, because if it were any more than that, you could just keep on dividing.

But hang on, that's impossible because even if you add up a billion zeroes, you get nothing. What happened to our line? That is what used to blow math minds in the days before quantum physics and LSD. Now, if you're willing to ignore those troubling questions, you could actually get some pretty useful math, like calculus.

Amir Alexander says the fight over how to resolve the paradox informed real fighting, 500 years of wars and dispute that produced modern Europe. In his new book, "Infinitesimal," Alexander says not everyone was willing to go along, starting with the Jesuits.

AMIR ALEXANDER: They were coming really out of this crisis of the reformation where the old order and the church had been splintered apart. And their mission was to reestablish a proper hierarchical order. There was a Jesuit by the name of Christopher Clavius. And he believed that the answer is in mathematics, in particularly geometry because geometry is orderly. It is absolutely certain. And once you get results in geometry, nobody can argue with you.

RATH: They're provable.

ALEXANDER: Everything's absolutely provable. And no sane person can ever dispute something like the Pythagorean theorem. He said, if we could have theology like that and we get rid of all those pesky Protestants who keep arguing with us, right, because we could prove things.

RATH: So to understand what was troubling about this paradox for the Jesuits and other representatives of this point of view, it was the fact that this could not be explained or was there something about the idea of something being so small it was indivisible?

ALEXANDER: It really was the fact that there is a paradox. There was basically a mistake at the heart of mathematics, because the whole point of studying mathematics is because it is absolutely certain and everything is known, and everything has its place. And there is a very orderly hierarchy of results there. And now in the middle of that you throw this paradox and you can get all those strange results. That basically means that mathematics can't be trusted. And if mathematics can't be trusted, what else can?

And they, the Jesuits, fought a long and hard campaign to try and wipe out this whole mathematical approach of infinitesimals.

RATH: Well, people have been arguing about it since the Greeks. Did they really think they could wipe this concept off the face of the earth?

ALEXANDER: They really did their best, and they succeeded very well in Italy. Italy was, before the 17th century and into the 17th century, it was really the mathematical capital of Europe. It had the greatest mathematicians, the greatest mathematical tradition. It was the leader in mathematics...

RATH: ...greatest astronomer.

ALEXANDER: Yeah, Galileo and mathematicians like Cardano and Commandino and others. It was a very, very impressive tradition of mathematics, the best in Europe. And by the time the Jesuits were done, that was gone, all of it. By the 1670s, Italy was a complete backwater in mathematics and the sciences.

RATH: Let's jump to England, because this is another wild aspect of this, that you tie this into basically the English Civil War.

ALEXANDER: Absolutely. Because in England what happened was that during the civil war, the whole social order was suddenly under threat, those revolutionaries, those radical sects, all of them just bubbled and threatened the established order that was dominated by the propertied classes. And again, there were two approaches to this. One of them was represented by Thomas Hobbs, who's known today of course as a political philosopher. But at the time he also considered himself a mathematician and was widely acknowledged as a mathematician.

And he thought the only way to reestablish order was, much like the Jesuits, just wipe off any possibility of dissent. Establish a state that is absolutely logical, where the laws of the sovereign have the force of a geometrical proof. On the other side, his personal enemy and rival for almost three decades was John Wallis. And he believed that the chance for that kind of order is gone.

What you have to build now is some space where dissent can be allowed, within limits at least, where not everything can be known at any given time, and build a society and build a social order from the ground up, rather than imposing it by one single law.

RATH: So if the Jesuits and Thomas Hobbs had won, had been able to shut the door on the infinitesimal, what would the Western world look like now, do you think?

ALEXANDER: Well, I think things would've been very different. I think if they had won then it would've been a much more hierarchical society. There was certainly, in a world like that, there would not be room for democracy, there would not be room for dissent. Most likely we would not have also all this great accomplishments that mathematics is at the heart of today. Modern science, modern technology, and everything from your cell phone to this radio station to airplanes, to cars and trains, it is all fundamentally dependent on this technique of infinitesimals.

RATH: Amir Alexander, fascinating speaking with you. Thank you so much.

ALEXANDER: Thank you so much for inviting me.

RATH: Amir Alexander teaches history at UCLA. His new book "Infinitesimal" is out now. Again, thanks for listening. This is ALL THINGS CONSIDERED from NPR West. I'm Arun Rath.

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