Zero The Biography of a Dangerous Idea
The biggest questions in science and religion are about nothingness and eternity,
The Siriona Indians of Bolivia and the Brazilian Yanoama people don’t have words for anything larger than three; instead, these two tribes use the words for “many” or “much.”
The languages currently used by the Bacairi and the Bororo peoples of Brazil show this process in action; they have number systems that go “one,” “two,” “two and one,” “two and two,” “two and two and one,” and so forth. These people count by twos. Mathematicians call this a binary system.
The early Greeks, for instance, used the word “fiving” to describe the process of tallying.
It was a sin punishable by having his heart fed to a horrible beast called the devourer.
The new Greek system would need only two symbols: π for 80, and ζ for 7.
Babylonian style of counting. And thanks to this system, zero finally appeared in the East, in the Fertile Crescent of present-day Iraq.
Zero was born out of the need to give any given sequence of Babylonian digits a unique, permanent meaning.
Zero was a digit, not a number. It had no value.
For this reason, we are stuck with a troublesome, zero-free calendar. The
The reason: zero was dangerous.
Emptiness and disorder were the primeval, natural state of the cosmos, and there was always a nagging fear that at the end of time, disorder and void would reign once more. Zero represented that void.
There is a lot of power in this simple number. It was to become the most important tool in mathematics. But thanks to the odd mathematical and philosophical properties of zero, it would clash with the fundamental philosophy of the West.
Nothing can be created from nothing. —LUCRETIUS, DE RERUM NATURA
The Greeks had a very different attitude. To them, numbers and philosophy were inseparable, and they took both very
Hippasus of Metapontum
Because of this, he was a strict vegetarian.
In ancient Greece, Pythagoras was remembered for a different invention: the musical scale.
This is what Pythagoras meant when he insisted, “All is number.”
What shape, after all, could zero be?
Greeks would have had to revamp their entire way of doing mathematics. They chose not to.
One of the first mathematical proofs in history was about the incommensurability/irrationality of the square’s diagonal.
Zero conflicted with the fundamental philosophical beliefs of the West, for contained within zero are two ideas that were poisonous to Western doctrine. Indeed, these concepts would eventually destroy Aristotelian philosophy after its long reign. These dangerous ideas are the void and the infinite.
This is the biggest failure in Greek mathematics, and it is the only thing that kept them from discovering calculus.
was a member of the Eleatic school of thought,
the concept of dividing lines into infinite pieces—nobody could actually do it, so the infinite doesn’t exist in reality.
Aristotle just wished infinity away by stating that it is simply a construct of the human mind.
This line of reasoning had another consequence—and this is why Aristotle’s philosophy endured for so many years. His system proved the existence of God.
When Christianity swept through the West, it became closely tied to the Aristotelian view of the universe and the proof of God’s existence. Atomism became associated with atheism. Questioning the Aristotelian doctrine was tantamount to questioning God’s existence.
Archimedes, the eccentric mathematician of Syracuse. He was the only thinker of his day to glimpse the infinite.
Killing Archimedes was one of the biggest Roman contributions to mathematics.
We don’t have to worry about mixing up the value of the number—its cardinality—with the order in which it arrives—its ordinality—
God is omnipotent. There is nothing God cannot do. But God, the ultimate goodness, cannot do evil. Therefore evil is nothing. It made perfect sense to the medieval mind.
Contests between the abacists and the so-called algorists who used Indian numerals were the medieval equivalents of the Kasparov versus Deep Blue chess match (Figure 15
Abu Hamid al-Ghazali, declared that clinging to Aristotelian doctrine should be punishable by death. The debate ended shortly thereafter.
Maimonides’ argument was, indeed, a “proof” of God’s existence—something incredibly valuable in any theology.
With that stroke the void moved from sacrilege to holiness.
a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it lent to all computations put our arithmetic in the first rank of useful inventions. —PIERRE-SIMON LAPLACE
Fibonacci is best remembered for a silly little problem he posed in his book, Liber Abaci, which was published in 1202. Imagine that a farmer has a pair of baby rabbits. Babies take two months to reach maturity, and from then on they produce another pair of rabbits at the beginning of every month. As these rabbits mature and reproduce, and those rabbits mature and reproduce, and so on, how many pairs of rabbits do you have during any given month?
Before Arabic numerals came around, money counters had to make do with an abacus or a counting board. The Germans called the counting board a Rechenbank, which is why we call moneylenders banks.
Not only did they use counting boards, they used tally sticks to record loans: a money value was written along the stick’s side, and it was split in two (Figure 16). The lender kept the biggest piece, the stock. After all, he was the stockholder.*
But the advantages of zero and the other Arabic numerals were not so easily dispensed with; Italian merchants continued to use them, and even used them to send encrypted messages—which is how the word cipher came to mean “secret code.”
It was an Italian architect, Filippo Brunelleschi, who first demonstrated the power of an infinite zero: he created a realistic painting by using a vanishing point.
It is no coincidence that zero and infinity are linked in the vanishing point. Just as multiplying by zero causes the number line to collapse into a point, the vanishing point has caused most of the universe to sit in a tiny dot.
Earth was no longer at the center of the universe. Yet Nicholas was not declared a heretic, and the church didn’t react to the new idea.
On the Infinite Universe and Worlds, where he suggested, like Nicholas of Cusa, that the earth was not the center of the universe and that there were infinite worlds like our own. In
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).
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.
“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.
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
Probability theory was invented to help rich aristocrats win more money with their gambling.
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.
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.
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:
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
Kepler chopped up the barrels—in his mind—into an infinite number of infinitely tiny pieces, and then added them back together again to yield their volumes. This may seem a backward way of going about measuring barrels, but it was a brilliant idea.
Volume-Measurement of Barrels,
Zero and infinity made the simple acts of taking tangents and finding areas appear to be self-contradictory.
The rules of mathematics were built around counting sheep and surveying property, yet these very rules govern the way the universe works.
The more general version of Newton’s law is F = ṗ, where p is an object’s momentum. Of course, Newton’s equations were eventually refined further by Einstein.)
As long as this flaw remained, calculus would be based upon faith rather than logic.
(In fact, faith was very much on Leibniz’s mind when he derived new mathematics, such as the binary numbers. Any number can be written as a string of zeros and ones; to Leibniz, this was the creation ex nihilo, the creation of the universe out of nothing more than God/1 and void/0. Leibniz even tried to get the Jesuits to use this knowledge to convert the Chinese to Christianity.)
l’Hôpital’s Analyse des infiniment petits
All of these expressions, but especially 0/0, could take on any value you desire them to have, depending on the functions you put in the numerator and denominator. This is why 0/0 is dubbed indeterminate.
George Berkeley, wrote a book entitled The Analyst, Or a Discourse Addressed to an Infidel Mathematician. (The mathematician in question was most likely Edmund Halley, always a supporter of Newton.)
“he who can digest a second or third fluxion, a second or third difference, need not, methinks, be squeamish about any point in divinity.”
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
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.
God made integers; all else is the work of man. —LEOPOLD KRONECKER
Descartes thought that these numbers were even worse than negative numbers; he came up with a scornful name for the square roots of negatives: imaginary numbers.
This is the fundamental theorem of algebra.
This is the definition of the infinite: it is something that can stay the same size even when you subtract from it.
The infinity of the rationals is nothing more than a zero.
In thermodynamics a zero became an uncrossable barrier: the coldest temperature possible. In Einstein’s theory of general relativity, a zero became a black hole, a monstrous star that swallows entire suns. In quantum mechanics, a zero is responsible for a bizarre source of energy—infinite and ubiquitous, present even in the deepest vacuum—and a phantom force exerted by nothing at all.
Kelvin’s discovery of absolute zero told physicists what they couldn’t do.
Thermodynamics is worse than a casino; you can’t win, no matter how much you work at it.
This is because the very act of measuring destroys some of the information we are trying to gather.
As you get into greatly curved regions of space, bodies’ masses effectively increase, a phenomenon known as mass inflation.
In the direction of the constellation Sagittarius, at the very center of our galaxy, sits a supermassive black hole that weighs as much as two-and-a-half million suns.
There’s no such thing as a free lunch. —“THE SECOND LAW OF THERMODYNAMICS
In 1998, NASA held a symposium entitled Physics for the Third Millennium, where scientists debated the merits of wormholes, warp drives, vacuum-energy engines, and other far-out ideas.
the zero-point energy might be the ultimate fuel. It is here that the mainstream of physics ends and the fringe begins.
Alien they seemed to be: No mortal eye could see The intimate welding of their later history . . . —THOMAS HARDY, “THE CONVERGENCE OF THE TWAIN”
A black hole is a zero in the equations of general relativity; the energy of the vacuum is a zero in the mathematics of quantum theory.
The big bang, the most puzzling event in the history of the universe, is a zero in both theories. The universe came from nothing—and both theories break down when they try to explain the origin of the cosmos.
This is a process called renormalization. “It is what I would call a dippy process,” wrote physicist Richard Feynman, even though Feynman won his Nobel Prize for perfecting the art of renormalization.
Hubble’s observations suggested that there was a time, called the big bang, when the universe was infinitesimally small and infinitely dense. Under such conditions all the laws of science, and therefore all ability to predict the future, would break down. —STEPHEN HAWKING, A BRIEF HISTORY OF TIME
Aristotle, the only possible way out of this quandary was to assume that the universe was eternal. It had always existed in the past, and would always exist.
“I have . . . again perpetrated something about gravitation theory which somewhat exposes me to the danger of being confined in a madhouse,” wrote Einstein,
Einstein’s hope for a steady, eternal universe was all but dead.
However, if we do discover a complete theory, it should in time be understandable in broad principle by everyone, not just a few scientists. Then we shall all, philosophers, scientists, and just ordinary people, be able to take part in the discussion of the question of why it is that we and the universe exist. If we find the answer to that, it would be the ultimate triumph of human reason—for we would know the mind of God. —STEPHEN HAWKING
The universe begins and ends with zero.