Far From 'Infinitesimal': A Mathematical Paradox's Role In History
Here's a stumper: How many parts can you divide a line into?
It seems like a simple question. You can cut it in half. Then you can cut those lines in half, then cut those lines in half again. Just how many parts can you make? A hundred? A billion? Why not more?
You can keep on dividing forever, so every line has an infinite amount of parts. But how long are those parts? If they're anything greater than zero, then the line would seem to be infinitely long. And if they're zero, well, then no matter how many parts there are, the length of the line would still be zero.
That's the paradox lurking behind calculus. The fight over how to resolve it had a surprisingly large role in the wars and disputes that produced modern Europe, according to a new book called Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, by UCLA historian Amir Alexander.
The Jesuits: Warriors Of Geometry
Today, mathematicians have found ways to answer that question so that modern calculus is rigorous and reliable. But in the 17th century, those questions didn't yet have satisfying answers — and worse, the results of early calculus were sometimes wrong, Alexander tells NPR's Arun Rath. That was a sharp contrast with the dependable outcomes of geometry.
"Geometry is orderly. It is absolutely certain. And once you get results in geometry, nobody can argue with you," Alexander says. "Everything is absolutely provable. No sane person can ever dispute something like the Pythagorean theorem."
That orderliness had captured the attention of the Jesuits, who had been trying to cope with the crisis of the Reformation.
"If we could have theology like that," Alexander explains, "then we could get rid of all those pesky Protestants who keep arguing with us, because we could prove things."
But the debate over infinitesimals threw a wrench into that thinking.
The whole point of mathematics was to be certain, Alexander says. "Everything is known, and everything has its place, and there's 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?"
So the Jesuits waged a war of letters, threats and intimidation against the supporters of the infinitesimal, a group that included some of Italy's greatest thinkers — Galileo, Gerolamo Cardano, Federico Commandino and others. In Italy, the Jesuits' victory was complete.
"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," Alexander says. "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."
An Infinitesimal Victory For The People
Meanwhile a similar situation was playing out in England, where civil war was also threatening upheaval. The aristocracy and propertied classes were desperate to hold onto their traditional power while lower class dissent fermented underneath. Thomas Hobbes, remembered today for his works of political philosophy like Leviathan, was also acknowledged at the time as a mathematician.
"He thought the only way to re-establish 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," Alexander says.
Part of Hobbes' strategy included a campaign against the infinitesimal, championed in England by Hobbes' greatest rival, a mathematician named John Wallis. Today, he's remembered best for introducing the familiar ∞ symbol, and he helped found the Royal Society of London. Over three decades of correspondence between the two, Wallis argued vehemently for the infinitesimal — and for democracy.
Wallis argued, "What you have to build now is some space where dissent can be allowed, within limits at least," Alexander says. "Build a society and build a social order from the ground up rather than imposing it by one single law."
Wallis' ideas eventually prevailed; Hobbes' opinions proved too unpopular, and fearing retribution from the rebels after they executed King Charles I, he pledged his allegiance to their new government in the 1650s.
A World Without Calculus: Would It Add Up?
What might have happened if the Jesuits and Hobbes had won out? What if the infinitesimal had been successfully stamped out everywhere?
"I think things would have been very different," Alexander muses. "I think if they had won, then it would have been a much more hierarchical society. In a world like that, there would not be room for democracy, there would not be room for dissent."
And more materially, he says, we might not have all the modern fruits of this kind of math. "Modern science, modern technology, and everything from your cell phone to this radio station to airplanes and cars and trains — it is all fundamentally dependent on this technique of infinitesimals."
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. Transcript provided by NPR, Copyright NPR.