Zero - Charles Seife [48]
Though mathematicians of the day sniped at Berkeley’s logic, the good bishop was entirely correct. In those days calculus was very different from other realms of mathematics. Every theorem in geometry had been rigorously proved; by taking a few rules from Euclid and proceeding, very carefully, step by step, a mathematician could show how a triangle’s angles sum to 180 degrees, or any other geometric fact. On the other hand, calculus was based on faith.
Nobody could explain how those infinitesimals disappeared when squared; they just accepted the fact because making them vanish at the right time gave the correct answer. Nobody worried about dividing by zero when conveniently ignoring the rules of mathematics explained everything from the fall of an apple to the orbits of the planets in the sky. Though it gave the right answer, using calculus was as much an act of faith as declaring a belief in God.
The End of Mysticism
A quantity is something or nothing; if it is something, it has not yet vanished; if it is nothing, it has literally vanished. The supposition that there is an intermediate state between these two is a chimera.
—JEAN LE ROND D’ALEMBERT
In the shadow of the French Revolution, the mystical was driven out of calculus.
Despite calculus’s shaky foundations, by the end of the eighteenth century, mathematicians all over Europe were having stunning successes with the new tool. Colin Maclaurin and Brook Taylor, perhaps the best British mathematicians in the era of isolation from the Continent, discovered how to use calculus to rewrite functions in a totally different form. For instance, after using some tricks in calculus, mathematicians realized that the function 1/(1 – x) can be written as
1 + x + x2 + x3 + x4 + x5 +…
Though the two expressions look dramatically different, they are (with some caveats) exactly the same.
Those caveats, which stem from the properties of zero and infinity, can become very important, however. The Swiss mathematician Leonhard Euler, inspired by calculus’s easy manipulation of zeros and infinities, used similar reasoning as Taylor and Maclaurin and “proved” that the sum
…1/x3 + 1/x2 + 1/x + 1 + x + x2 + x3…
equals zero. (To convince yourself that something fishy is going on, plug in the number 1 for x and see what happens.) Euler was an excellent mathematician—in fact, he was one of the most prolific and influential in history—but in this case the careless manipulation of zero and infinity led him astray.
It was a foundling who finally tamed the zeros and infinities in calculus and rid mathematics of its mysticism. In 1717 an infant was found on the steps of the church of Saint Jean Baptiste le Rond in Paris. In memory of that occasion, the child was named Jean Le Rond, and he eventually took the surname d’Alembert. Though he was raised by an impoverished working-class couple—his foster father was a glazier—it turns out that his birth father was a general and his mother was an aristocrat.
D’Alembert is best known for his collaboration on the famed Encyclopédie of human knowledge—a 20-year effort with coauthor Denis Diderot. But d’Alembert was more than an encyclopedist. It was d’Alembert who realized that it was important to consider the journey as well as the destination. He was the one who hatched the idea of limit and solved calculus’s problems with zeros.
Once again, let us consider the story of Achilles and the tortoise, which is an infinite sum of steps that get closer and closer to zero. Manipulating an infinite sum—whether it is in the Achilles problem or in finding the area underneath a curve or finding an alternate form for a mathematical function—caused mathematicians to come up with contradictory results.
D’Alembert realized that the Achilles problem vanishes if you