Zero - Charles Seife [40]
But we made an assumption here—that there is a 50-50 chance that God exists. What happens if there is only a 1/1000 chance? The value of being a Christian would be:
It’s still the same: infinite, and the value of being an atheist is still negative infinity. It’s still much better to be a Christian. If the probability is 1/10,000 or 1/1,000,000 or one in a gazillion, it comes out the same. The only exception is zero.
If there is no chance that God exists, Pascal’s wager—as it came to be known—makes no sense. The expected value of being a Christian would then be 0 ×?, and that was gibberish. Nobody was willing to say that there was zero chance that God exists. No matter what your outlook, it is always better to believe in God, thanks to the magic of zero and infinity. Certainly Pascal knew which way to wager, even though he gave up mathematics to win his bet.
Chapter 5
Infinite Zeros and Infidel Mathematicians
[ZERO AND THE SCIENTIFIC REVOLUTION]
With the introduction of…the infinitely small and infinitely large, mathematics, usually so strictly ethical, fell from grace…. The virgin state of absolute validity and irrefutable proof of everything mathematical was gone forever; the realm of controversy was inaugurated, and we have reached the point where most people differentiate and integrate not because they understand what they are doing but from pure faith, because up to now it has always come out right.
—FRIEDRICH ENGELS, ANTI-DUHRING
Zero and infinity had destroyed the Arisotelian philosophy; the void and the infinite cosmos had eliminated the nutshell universe and the idea of nature’s abhorrence of the vacuum. The ancient wisdom was discarded, and scientists began to divine the laws that governed the workings of nature. However, there was a problem with the scientific revolution: zero.
Deep within the scientific world’s powerful new tool—calculus—was a paradox. The inventors of calculus, Isaac Newton and Gottfried Wilhelm Leibniz, created the most powerful mathematical method ever by dividing by zero and adding an infinite number of zeros together. Both acts were as illogical as adding 1 + 1 to get 3. Calculus, at its core, defied the logic of mathematics. Accepting it was a leap of faith. Scientists took that leap, for calculus is the language of nature. To understand that language completely, science had to conquer the infinite zeros.
The Infinite Zeros
When, after a thousand-year stupor, European thought shook off the effect of the sleeping powders so skilfully administered by the Christian Fathers, the problem of infinity was one of the first to be revived.
—TOBIAS DANZIG, NUMBER: THE LANGUAGE OF SCIENCE
Zeno’s curse hung over mathematics for two millennia. Achilles seemed doomed to chase the tortoise forever, never catching up. Infinity lurked in Zeno’s simple riddle. The Greeks were stumped by Achilles’ infinite steps. They never considered adding infinite parts together even though Achilles’ strides approach zero size; the Greeks could hardly add steps of zero size together without the concept of zero. However, once the West embraced zero, mathematicians began to tame the infinite and ended Achilles’ race.
Even though Zeno’s sequence has infinite parts, we can add all of the steps together and still stay within the realm of the finite: 1 + ½ + ¼ + 1/8 + 1/16 +…= 2. The first person to do this sort of trick—adding infinite terms to get a finite result—was the fourteenth-century British logician Richard Suiseth. Suiseth took an infinite sequence of numbers: ¼, 2/4, 3/8, 4/16,…, n/2n,…, and added them all together, yielding two. After all, the numbers in the sequence get closer and closer to zero; naively, one would guess that this would ensure that the sum remains finite. Alas, the infinite is not quite that simple.
At about the same time Suiseth was writing, Nicholas Oresme, a French mathematician, tried his hand at adding together another infinite sequence of numbers—the so-called harmonic series:
½ + 1/3 + ¼ + 1/5 + 1/6 +…
Like the Zeno sequence