The late arrival of negative numbers

 Here is a remarkable line from Pascal’s Pensées: “I know people who cannot understand that when you subtract four from zero what is left is zero.”

Do you hear the tone? It’s the exasperated sound of a man who’s frustrated that he has friends who can’t grasp the simplest ideas in mathematics.

What we notice: This learned man had no concept of negative numbers. I’m pretty sure I learned about them from Miss Fisher in third grade.

Leidy Klotz pointed out Pascal’s difficulty in his book Subtract (Aug. 27, 2022).

We tend to think — well, at least I do — that our conceptual foundations develop incrementally, in roughly the order we learn them in school. (We’re ready for negative numbers in third grade but have to wait on geometry until later.)

Pascal’s line is evidence that’s just not so.

Pascal did some interesting work on probability theory. Soon after Pascal died, Leibnitz and Newton developed calculus in the late 1600s. Meanwhile, the leading lights of Western Civilization would stumble along without negative numbers, which didn’t come into common usage until the 19^th century.

• Source: Leidy Klotz, Subtract: The Untapped Science of Less; New York: Flatiron Books, 2021.