Zero
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Figure 20: Cartesian coordinates
Descartes quickly realized how powerful his coordinate system was. He used it to turn figures and shapes into equations and numbers; with Cartesian coordinates every geometric object—squares, triangles, wavy lines—could be represented by an equation, a mathematical relationship. For example, a circle at the origin can be represented by the set of all points where x2 + y2 – 1 = 0. A parabola might be y – x2 = 0. Descartes unified numbers and shapes. No longer were the Western art of geometry and the Eastern art of algebra separate domains. They were the same thing, as every shape could simply be expressed as an equation of the form f(x, y) = 0 (Figure 21). Zero was at the center of the coordinate system, and zero was implicit in each geometric shape.
To Descartes, zero was also implicit in God’s domain, as was the infinite. Since the old Aristotelian doctrine was crumbling, Descartes, true to his Jesuit training, tried to use nought and infinity to replace the old proof of God’s existence.
Like the ancients, Descartes assumed that nothing, not even knowledge, can be created out of nothing, which means that all ideas—all philosophies, all notions, all future discoveries—already exist in people’s brains when they are born. Learning is just the process of uncovering that previously imprinted code of laws about the workings of the universe. Since we have a concept of an infinite perfect being in our minds, Descartes then argued that this infinite and perfect being—God—must exist. All other beings are less than divine; they are finite. They all lie somewhere between God and nought. They are a combination of infinity and zero.
However, though zero appeared and reappeared throughout Descartes’s philosophy, Descartes insisted unto his death that the void—the ultimate zero—does not exist. A child of the Counter-Reformation, Descartes learned about Aristotle at the very moment when the church was relying upon his principles the most. As a result, Descartes, indoctrinated with the Aristotelian philosophy, denied the existence of the vacuum.
Figure 21: A parabola, a circle, and an elliptic curve
It was a difficult position to take; Descartes was certainly mindful of the metaphysical problems of rejecting the vacuum entirely. Later in his life he wrote about atoms and the vacuum: “About these things that involve a contradiction, it can absolutely be said that they cannot happen. However, one shouldn’t deny that they can be done by God, namely, if he were to change the laws of nature.” Yet, like the medieval scholars before him, Descartes believed that nothing truly moved in a straight line, for that would leave a vacuum behind it. Instead, everything in the universe moved in a circular path. It was a truly Aristotelian way of thinking—yet the void would soon unseat Aristotle once and for all.
Even today, children are taught “Nature abhors a vacuum,” while the teachers don’t really understand where that phrase came from. It was an extension of the Aristotelian philosophy: vacuums don’t exist. If someone would attempt to create a vacuum, nature would do anything in its power to prevent it from happening. It was Galileo’s secretary, Evangelista Torricelli, who proved that this wasn’t true—by creating the first vacuum.
In Italy workmen used a kind of pump, which worked more or less like a giant syringe, to raise water out of wells and canals. This pump had a piston that fitted snugly in a tube. The bottom of the tube was placed in the water, so when the piston was raised, the water level followed the plunger upward.
Galileo heard from a worker that these pumps had a problem: they could only lift water about 33 feet. After that, the plunger kept on going upward, yet the water level stayed the same. It was a curious phenomenon, and Galileo passed the problem on to his assistant, Torricelli, who started doing experiments, trying to figure out the reason for the pumps’ curious limitation. In 1643, Torricelli took a long tube that was closed at one end and filled it with mercury. He upended it, placing the open end in a dish also filled with mercury. Now, if Torricelli had upended the tube in air, everyone would expect the mercury to run out, because it would quickly be replaced by air; no vacuum would be created. But when it was upended in a dish of mercury, there was no air to replace the mercury in the tube. If nature truly abhorred a vacuum so much, the mercury in the tube would have to stay put so as not to create a void. The mercury didn’t stay put. It sank downward a bit, leaving a space at the top. What was in that space? Nothing. It was the first time in history anyone had created a sustained vacuum.
No matter the dimensions of the tube that Torricelli used, the mercury would sink down until its highest point was about 30 inches above the dish; or, looking at it another way, mercury could only rise 30 inches to combat the vacuum above it. Nature only abhorred a vacuum as far as 30 inches. It would take an anti-Descartes to explain why.
In 1623, Descartes was twenty-seven, and Blaise Pascal, who would become Descartes’s opponent, was zero years old. Pascal’s father, Étienne, was an accomplished scientist and mathematician; the young Blaise was a genius equal to his father. As a young man, Blaise invented a mechanical calculating machine, named the Pascaline, which is similar to some of the mechanical calculators that engineers used before the invention of the electronic calculator.
When Blaise was twenty-three, his father slipped on a patch of ice and broke his thigh. He was cared for by a group of Jansenists, Catholics who belonged to a sect based largely on a hatred of the Jesuit order. Soon the entire Pascal family was won over, and Blaise became an anti-Jesuit, a counter-counter-reformationist. Pascal’s newfound religion was not a comfortable fit for the young scientist. Bishop Jansen, the founder of the sect, had declared that science was sinful; curiosity about the natural world was akin to lust. Luckily, Pascal’s lust was greater than his religious fervor for a time, because he would use science to unravel the secret of the vacuum.
About the time of the Pascals’ conversion, a friend of Étienne’s—a military engineer—came to visit and repeated Torricelli’s experiment for the Pascals. Blaise Pascal was enthralled, and started performing experiments of his own, using water, wine, and other fluids. The result was New experiments concerning the vacuum, published in 1647. This work left the main question unanswered: why would mercury rise only 30 inches and water only 33 feet? The theories of the time tried to save a fragment of Aristotle’s philosophy by declaring that nature’s horror of the vacuum was “limited”; it could only destroy a finite amount of vacuum. Pascal had a different idea.
In the fall of 1648, acting on a hunch, Pascal sent his brother-in-law up a mountain with a mercury-filled tube. On top of the mountain, the mercury rose considerably less than 30 inches (Figure 22). Was nature somehow perturbed less by a vacuum on top of a mountain than by a vacuum in the valley?
Figure 22: Pascal’s experiment
To Pascal, this seemingly bizarre behavior proved that it wasn’t an abhorrence of the vacuum that drove the mercury up the tube. It was the weight of the atmosphere pressing down on the mercury exposed in the pan that makes the fluid shoot up the column. The atmospheric pressure bearing down on a pan of liquid—be it mercury, water, or wine—will make the level inside the tube rise, just as gently squeezing the bottom of a toothpaste tube will make the contents squirt out the top. Since the atmosphere cannot push infinitely hard, it can only drive mercury about 30 inches up the tube—and at the top of the mountain, there is less atmosphere pushing down, so the air can’t even push the mercury as high as 30 inches.
It is a subtle point: vacuums don’t suck; the atmosphere pushes. But Pascal’s simple experiment demolished Aristotle’s assertion that nature abhors a vacuum. Pascal wrote, “But until now one could find no one who took this…view, that nature has no repugnance for the vacuum, that it makes no effort to avoid it, and that it admits vacuum without difficulty and without resistance.” Aristotle was defeated, and scientists stopped fearing the void and began to study it.
It was also in zero and the infinite that Pascal, the devout Jansenist, sought to prove God’s existence. He did it in a very profane way.
The Divine Wager<
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What is man in nature? Nothing in relation to the infinite, everything in relation to nothing, a mean between nothing and everything.
—BLAISE PASCAL, PENSÉES
Pascal was a mathematician as well as a scientist. In science Pascal investigated the vacuum—the nature of the void. In mathematics Pascal helped invent a whole new branch of the field: probability theory. When Pascal combined probability theory with zero and with infinity, he found God.
Probability theory was invented to help rich aristocrats win more money with their gambling. Pascal’s theory was extremely successful, but his mathematical career was not to last. On November 23, 1654, Pascal had an intense spiritual experience. Perhaps it was the old Jansenist antiscience creed that was building up in him, but for whatever the reason, Pascal’s newfound devotion led him to abandon mathematics and science altogether. (He made an exception for a brief time four years later, when he was unable to sleep owing to illness. He started doing mathematics and the pain eased. Pascal believed that this was a sign that God was not displeased with his studies.) He became a theologian—but he could not escape his profane past. Even when it came to arguing about God’s existence, he kept coming back to those crass gambling Frenchmen. Pascal argued that it was best to believe in God, because it was a good bet. Literally.
Just as he analyzed the value—or expectation—of a gamble, Pascal analyzed the value of accepting Christ as savior. Thanks to the mathematics of zero and infinity, Pascal concluded that one should assume that God exists.
Before considering the wager itself, it is easy to analyze a slightly different game. Imagine that there are two envelopes, marked A and B. Before you are shown the envelopes, a flip of the coin determined which envelope has money in it. If the coin toss was a heads, A has a brand-new $100 bill inside. If the coin came up tails, B has the money—but this time, it’s $1,000,000. Which envelope should you choose?
B, obviously! Its value is much greater. It is not difficult to show this using a tool from probability theory called an expectation, which is a measure of how much we expect each envelope to be worth.
Envelope A might or might not have a $100 bill in it; it has some value, because it might have money in it, but it isn’t worth as much as $100, because you’re not absolutely sure that it contains anything. In fact a mathematician would add up all of the possible contents of envelope A and then multiply by the probability of each outcome:
The mathematician would conclude that the expected value of the envelope is $50. At the same time, the expected value of envelope B is:
So the expected value of B is $500,000—10,000 times as much as the expected value of envelope A. Clearly, if you are offered a choice between the two envelopes, the smart thing to do is to choose B.
Pascal’s wager is exactly like this game, except that it uses a different set of envelopes: Christian and atheist. (Actually, Pascal only analyzed the Christian case, but the atheist case is the logical extension.) For the sake of argument, imagine for the moment that there’s a 50-50 chance that God exists. (Pascal assumed that it would be the Christian God, of course.) Now, choosing the Christian envelope is equivalent to choosing to be a devout Christian. If you happen to choose this path, there are two possibilities. If you are a faithful Christian and there is no God, you just fade into nothingness when you die. But if there is a God, you go to heaven and live for eternity in bliss: infinity. So the expected value of being a Christian is:
After all, half of infinity is still infinity. Thus, the value of being a Christian is infinite. Now what happens if you are an atheist? If you are correct—there is no God—you gain nothing from being right. After all, if there is no God, there is no heaven. But if you are wrong and there is a God, you go to hell for an eternity: negative infinity. So the expected value of being an atheist is:
Negative infinity. The value is as bad as you can possibly get. The wise person would clearly choose Christianity instead of atheism.
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 and Suiseth’s sequence, all the terms get closer and closer to zero. However, when Oresme tried to sum the term
s in the sequence, he realized that the sums got larger and larger and larger. Even though the individual terms go to zero, the sum goes off to infinity. Oresme showed this by clumping the terms together: ½ + (1/3 + ¼) + (1/5 + 1/6 + 1/7 + 1/8) +…. The first group clearly equals ½; the second group is greater than (¼ + ¼), or ½. The third group is greater than (1/8 + 1/8 + 1/8 + 1/8), or ½. And so forth. You keep adding ½ after ½ after ½, and the sum gets bigger and bigger, and off to infinity. Even though the terms themselves go to zero, they don’t approach zero fast enough. An infinite sum of numbers can be infinite, even if the numbers themselves approach zero. Yet this isn’t the strangest aspect of infinite sums. Zero itself is not immune to the bizarre nature of infinity.
Consider the following series: 1 – 1 + 1 – 1 + 1 – 1 + 1 – 1 + 1 –…. It’s not so hard to show that this series sums to zero. After all,
(1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) + (1 – 1) +…
is the same thing as
0 + 0 + 0 + 0 + 0 + 0 +…
which clearly sums to zero. But beware! Group the series in a different way:
1 + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) + (-1 + 1) +…
is the same thing as
1 + 0 + 0 + 0 + 0 + 0 +…
which clearly sums to one. The same infinite sum of zeros can equal 0 and 1 at the same time. An Italian priest, Father Guido Grandi, even used this series to prove that God could create the universe (1) out of nothing (0). In fact, the sequence can be set to equal anything at all. To make the sum equal 5, start with 5s and -5s instead of 1s and -1s, and we can show that 0 + 0 + 0 + 0 +…equals 5.