Zero - The Biography of a Dangerous Idea

High-Level Thoughts

Math, philosophy, and theology have been intertwined for thousands of years. The concept of 0 was initially rejected by Western society because it clashed with tenants central to the Greek society, and later represented Satan to Medieval scholars. Also its why calendars are messed up.

Summary Notes

The story of zero is an ancient one. Its roots stretch back to the dawn of mathematics, in the time thousands of years before the first civilization, long before humans could read and write. But as naturally as zero seems to us today, for ancient peoples zero was a foreign - and frightening - idea. An Eastern concept, born in the Fertile Crescent a few centuries before the birth of Christ, zero not only evoked images of a primal void, it also had dangerous mathematical properties. Within zero is the power to shatter the framework of logic. (Page 5)

The Egyptians' innovation of the solar calendar was a breakthrough, but they made an even more important mark on history: the invention of the art of geometry. Even without a zero, the Egyptians had quickly become masters of mathematics. They had to, thanks to an angry river. Every year the Nile would overflow its banks and flood the delta. The good news was that the flooding deposited rich, alluvial silt all over the fields, making the Nile delta the richest farmland in the ancient world. The bad news was that the river destroyed many of the boundary markers erasing all of the landmarks that told farmers which land was theirs to cultivate. (The Egyptians took property rights very seriously, In the Egyptian Book of the Dead, a newly deceased person must swear to the gods that he hasn't cheated his neighbor by stealing his land. It was a sin punishable by having his heart fed to a horrible beast called the devourer. In Egypt, filching your neighbor's land was considered as grave an offense as breaking an oath, murdering somebody, or masturbating in a temple.)
The ancient pharaohs assigned surveyors to assess the damage and reset the boundary markers, and thus geometry was born. These surveyors, or rope stretchers (named for their measuring devices and knotted ropes designed to mark right angles), eventually learned to determine the areas of plots of land by dividing them into rectangles and triangles. The Egyptians also learned how to measure volumes of objects - like pyramids. Egyptian mathematics was famed throughout the Mediterranean, and it is likely that the early Greek mathematicians, masters of geometry like Thales and Pythagoras, studied in Egypt. (Page 11)

Zero clashed with one of the central tenets of Western philosophy, a dictum whose roots were in the number philosophy of Pythagoras and whose importance came from the paradoxes of Zeno. The whole Greek universe rested upon this pillar: there is no void.
The Greek universe, created by Pythagoras, Aristotle, and Ptolemy, survived long after the collapse of Greek civilization. In that universe there is no such thing as nothing. There is no zero. Because of this, the West could not accept zero for nearly two millennia. The consequences were dire. Zero's absence would stunt the growth of mathematics, stifle innovation in science, and, incidentally make a mess of the calendar. (Page 25)

The Greeks had inherited their numbers from the geometric Egyptians. As a result, in Greek mathematics there was no significant distinction between shapes and numbers. To the Greek philosopher-mathematicians they were pretty much the same thing. (Even today, we have square numbers and triangular numbers thanks to their influence.) In those days, proving a mathematical theorem was often as simple as drawing an elegant picture; the tools of ancient Greek mathematics weren't pencil and paper - they were a straightedge and compasses. And to Pythagoras the connection between shapes and numbers was deep and mystical. Every number-shape had a hidden meaning, and the most beautiful number-shapes were sacred. (Page 27)

Aristotle simply declared that mathematicians "do not need the infinite, or use it." Though "potentially" infinities could exist in the minds of mathematicians - like the concept of dividing lines into infinite pieces - nobody could actually do it, so the infinite doesn't exist in reality. Achilles runs smoothly past the tortoise because the infinite points are simply a figment of Zeno's imagination, rather than a real-world construct. Aristotle just wished infinity away by stating that is is simply a construct of the human mind.
From that concept comes a startling revelation. Based upon the Pythagorean universe, the Aristotelian cosmos (and its later refinement by the astronomer Ptolemy) had the planets moving in crystalline orbs. However, since there is no infinity, there can't be an endless number of spheres; there must be a last one. This outermost sphere was a midnight blue globe encrusted with tiny glowing points of light - the stars. There was no such thing as "beyond" the outermost sphere; the universe ended abruptly with that outermost layer. The universe was contained in a nutshell, ensconced comfortably within the sphere of fixed stars; the cosmos was finite in extent, and entirely filled with matter. There was no infinite; there was no void; there was no zero.
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. (Page 46)

Medieval scholars branded the void as evil - and evil as void. Satan was quite literally nothing. Boethius made the argument as follows: God is omnipotent. There is nothing God cannot do. But God, the ultimate goodness, cannot do evil. Therefore evil is nothingness. It made perfect sense to the medieval mind.
Lurking underneath the veil of medieval philosophy, however, was a conflict. The Aristotelian system was Greek, but the Judeo-Christian story of creation was Semitic - and Semites didn't have such a fear of the void. The very act of creation was out of a chaotic void, and theologians like Saint Augustine, who lived in the fourth century, tried to explain it away by referring to the state before creation as "a nothing something" that is empty of form but yet "falls short of utter nothingness." This fear was so great that Christian scholars tried to fix the Bible to match Aristotle rather than vice versa. (Page 61)