Two and a half millennia ago, from his perch in a Greek colony on Italy’s Terranean Sea, a deep thinker, Zeno, generated a series of pesky thought games. His parables challenged standard assumptions about space and time. His most famous brain teaser involving distance across an interval, suitably now still called Zeno’s Paradox, pitted a tortoise against the Trojan War hero of Homer’s Iliad, the semi-divine Achilles.
Achilles ran so fast that his long chestnut hair flowed behind him like a mane. (He was raised by Centaurs…) So, in an imagined footrace, with the tortoise given a modest head start, the smart drachmas would have been on Achilles.
The Nearer His Destination, the More It Slipped Away
But not so fast, Zeno reasoned. The greatest, fleetest warrior of all time could never overtake the ponderous reptile. (So held Aristotle who summarized Zeno’s lost writings.) Why not? Well, see, before Achilles could pull even with the tortoise, obviously he would need to cover half the distance. Obviously.
And having once covered half the distance? (“Wait for it!” we can almost hear Zeno saying after all these centuries.) Before he reached the tortoise, wouldn’t Achilles have needed to cover half the distance once again? And again? And then so on and on! Got it? (Cue the consternation.) Plainly, distance can be infinitely subdivided! Distance expands inexorably, by halves. Poor old Achilles, superpowers aside and stuck in slo-mo, the nearer his destination the more it was slip-sliding away.
Common sense finds the logical fault in Zeno’s argument; the demi-god could have covered the whole distance in one sprint, leaving the tortoise eating fiery dust. Mathematicians, a little more troubled, have a subtler answer: Operations expressed as fractions can be rounded up or in this case (½ ÷ ½ …), rounded down all the way to zero.
Why Should Zero Have Made Thinkers So Uncomfortable?
Zero lay at the heart of Zeno’s discomfort. For the early Greeks, conceptually, “nothing” equaled chaos. Chaos meant emptiness. Chaos was an abyss, a chasm. Roundabout, chaos is related to our words “gape” and “yawn,” words whose origins trace to Sanskrit. So the picture of a yawning chasm, dangerous and frightening in its dark, magnetic, swallowing depth, is an old image and an ancient notion. Chaos preceded creation in early religions. In both Hindu and Judaic/Christian/Muslim traditions, darkness preceded light literally and enlightenment figuratively. Before the light, there was only chaos. Nothing at all.
The Yawning Chasm of Hell, Detail from the fresco, Church of St. Nicholas, Raduil, Bulgaria, c. 10th Century.
Source: Anton Lefterov / Wikimedia
Making Something of Nothing
But there is a simpler, less mythopoeic explanation about the trouble with zero. It is very hard to think of nothing. Give it a try.
Time’s up. How’d you do? And then nothing, nothingness, hard enough to conceive, is even harder to represent. The reason is almost funny. To picture zero, to represent it, you must make something of nothing.
A World Without Zero Is Hard to Imagine
Zero is key to our mathematical reasoning now, so much so that it is difficult to believe that zero needed to be invented. In fact, zero was discovered more than once. Babylonian scribes and accountants needed an empty space that acted as a placeholder, handy for distinguishing 4 from 40 and 40 from 4,000. Around 500 B.C. they began to draw a wedge-shaped character, literally meaning nothing. For the Mesopotamians, though, zero did not exist on a number line, the way we think of it.
Without the zero, for their part, the Greeks had resorted to a dot for a placeholder. And the dot proved versatile. Two dots make a line. Three a triangle. Four a square. With five dots you can construct a pyramid. And so on up into more complex figures that enable spatial reasoning. Thus, the Greeks became exceptionally clever at geometry. But without a zero, alas, algebra laid out of their reach.
The invention of the zero had to wait until late in the ninth century; the figure itself was first inscribed on a temple in ancient India. Afterward, zero journeyed from the subcontinent to the Arab world. The Persian astronomer Muhammad ibn Musa al Khwarizmi, who synthesized Indian and Greek mathematics, coined the term “algebra” early in the flourishing, tolerant 9th-century “Islamic Golden Age.” He gave us the Hindu-Arabic numerical system.
But it was not until the 1200s that an Italian mathematician named Fibonacci picked up the concept of zero as he traveled through North Africa and brought it back to Pisa. The history is inspiring. After enabling algebra and calculus, zero gave rise to complex engineering, and eventually the computers we use today. Zero was a wonderful innovation.
Much Ado About Nothing in the West
Yet zero faced a rocky road in the West. Zero is disturbing. Add zero to or subtract zero from a number gives you that same number. Multiplying or dividing by zero yields zero. Zero is strange.
The city-state Florence banned the zero in the 14th century, partly out of reflexive conservatism of scribes and accountants, but more likely as part of the suspicion of Islam in the era of the medieval Crusades. More fundamentally, for Christian fathers, numbers vibrated with mystic significance. They deeply mistrusted the very concept of nothingness. Nothingness represented the abyss, and so zero, the “devils’ number,” wafted with a sulfurous aroma of the abysmal Christian hell.
Your Brain on Zero
Zero strains children’s developing brains as budding numeracy seems to recapitulate a halting intellectual history. Children begin to understand numbers from 1-10 around two years of age. Grasping nothing at all requires a further level of abstraction. Two years will pass before children even begin to name zero “0.” Neuroscientists have found that understanding nothing depends upon the development of a specific brain region, the posterior parietal cortex, which takes roughly a further two years.
Again, zero has both a numerical value (as a point on a number line) and a symbolic value—as the symbol “0.” Neuroscientists who examined activity at the micro-level of single neurons found that these circuits could respond either to an empty set (nothingness, again), or the numeral 0. But the neurons could not respond to (and thus lead to processing) both symbol and concept at the same time.
This new research at the tiniest dimensions helps us begin to see more clearly why zero is so devilish to fathom.
