Newton and Leibniz

The ancient Greek philosopher Zeno pointed out a great paradox that had mathematicians and philosophers perplexed for millennia.

Achilles, the fastest runner in the world, is facing off in a footrace against a tortoise. To make things more sporting, the tortoise is given a head start.

According to Zeno, every time Achilles reaches the point where the tortoise was, the tortoise has moved a little bit further ahead. For Achilles to reach the new position of the tortoise, he has to cover this new, albeit smaller, distance.

Again, by the time he does, the tortoise advances just a little more. Zeno argues that this process will continue infinitely, implying that Achilles will never be able to overtake the tortoise.

It’s tough to find any logical fault with this reasoning, and yet it’s just not how things appear to happen in the real world. What’s going on here?

This is the very question that two titanic thinkers of the 17th century answered around the same time, both roughly 2200 years after the original paradox was first written down. Doesn’t it seem like a coincidence that both guys would have solved the problem around the same time like this? Was one copying the other’s work?

Well, that’s exactly what both Isaac Newton and Gottfried Wilhelm Leibniz accused one another of doing, shortly after each had independently discovered calculus. Let’s back up a bit and talk about what that really means for a moment.

One way to think about this conundrum is with math. Achilles travels half the distance to the turtle in an instant, and then he travels half of that distance, and so on. So, the notation looks something like this:

1/2 + 1/4 + 1/8 +…

This is the approach that Leibniz took. This makes perfect sense: he was obsessed with symbols, and with making a complete mathematical framework from symbols that existed and which he helped to invent. Leibniz was approaching a longstanding logical issue from a mathematical and logical framework.

Newton, on the other hand, was obsessed with physical shapes. He famously thought a lot about how the moon kept orbiting the Earth, so it’s probably no surprise to you that he thought about abstract curved spaces and astronomical bodies a great deal.

Newton realized that there was an issue with this physical framework, though. Suppose you wanted to measure the length of a curved line with a long ruler. You’d quickly overshoot the curved line you were trying to measure with the ruler itself, and even if you got a smaller and smaller ruler out, you would always need a shorter ruler to measure the rest of the curve.

Can you see how these are really the same problem these two gentlemen were trying to solve?

This is how calculus came to be. On one side of the English Channel, Newton thought about how to solve the physical manifestation of Zeno’s paradox, while across that body of water in Hanover, Leibniz thought about the same idea, but through a mathematical framework.

Four and a half centuries later, the modern world is indebted to the work of both of these men, but back then, they simply hated one another. Each simply tried to claim that they had the original idea that a new form of math was needed to describe adding up ever-smaller infinities.

That both men should solve these problems at the same time can be explained without some kind of telepathic mysticism. Instead, we only need to invoke standing on the shoulders of giants, an apt phrase that Newton himself coined in a letter. Both Newton and Leibniz had been exposed to ideas that enabled them to solve Zeno’s paradox, but from completely different approaches.

“If I have seen further, it is by standing on the shoulders of giants.”

Isaac Newton in a 1675 letter to Robert Hooke

The human ego is a powerful thing. Here, it drove two titanic minds apart and certainly hurt the development of science and math for at least a generation as England and mainland Europe hid their homework from one another, seeking credit instead of progress.

Fortunately, science has gradually opened up more and more, and today we have a unified platform of discovery in the internet. Once an idea is published, it’s nearly instantly everywhere, read by a thousand minds in days and analyzed by today’s best tools in an instant.

Incidentally, Leibniz is credited with publishing his idea first, giving a lot of legitimacy to his claim of primacy, but Newton had street cred, as you can clearly see here: