A New York Times Notable Book.
The Babylonians invented it, the Greeks banned it, the Hindus worshiped it, and the Church used it to fend off heretics. Now it threatens the foundations of modern physics. For centuries the power of zero savored of the demonic; once harnessed, it became the most important tool in mathematics. For zero, infinity's twin, is not like other numbers. It is both nothing and everything.
In Zero, Science Journalist Charles Seife follows this innocent-looking number from its birth as an Eastern philosophical concept to its struggle for acceptance in Europe, its rise and transcendence in the West, and its ever-present threat to modern physics. Here are the legendary thinkers—from Pythagoras to Newton to Heisenberg, from the Kabalists to today's astrophysicists—who have tried to understand it and whose clashes shook the foundations of philosophy, science, mathematics, and religion. Zero has pitted East against West and faith against reason, and its intransigence persists in the dark core of a black hole and the brilliant flash of the Big Bang. Today, zero lies at the heart of one of the biggest scientific controversies of all time: the quest for a theory of everything.
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[ THE ORIGIN OF ZERO ]
There was neither non-existence nor existence then; there
was neither the realm of space nor the sky which is beyond.
What stirred? Where?
The Rig Veda
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 natural as zero seems to us today, for ancient peoples zero was a foreignand frighteningidea. 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 there is the power to shatter the framework of logic.
The beginnings of mathematical thought were found in the desire to count sheep and in the need to keep track of property and of the passage of time. None of these tasks requires zero; civilizations functioned perfectly well for millennia before its discovery. Indeed, zero was so abhorrent to some cultures that they chose to live without it.
Life without Zero
The point about zero is that we do not need to use it in the
operations of daily life. No one goes out to buy zero fish.
It is in a way the most civilized of all the cardinals, and
its use is only forced on us by the needs of cultivated
Alfred North Whitehead
It's difficult for a modern person to imagine a life without zero, just as it's hard to imagine life without the number seven or the number 31. However, there was a time where there was no zerojust as there was no seven and 31. It was before the beginning of history, so paleontologists have had to piece together the tale of the birth of mathematics from bits of stone and bone. From these fragments, researchers discovered that Stone Age mathematicians were a bit more rugged than modern ones. Instead of blackboards, they used wolves.
A key clue to the nature of Stone Age mathematics was unearthed in the late 1930s when archaeologist Karl Absolom, sifting through Czechoslovakian dirt, uncovered a 30,000-year-old wolf bone with a series of notches carved into it. Nobody knows whether Gog the caveman had used the bone to count the deer he killed, the paintings he drew, or the days he had gone without a bath, but it is pretty clear that early humans were counting something.
A wolf bone was the Stone Age equivalent of a supercomputer. Gog's ancestors couldn't even count up to two, and they certainly did not need zero. In the very beginning of mathematics, it seems that people could only distinguish between one and many. A caveman owned one spearhead or many spearheads; he had eaten one crushed lizard or many crushed lizards. There was no way to express any quantities other than one and many. Over time, primitive languages evolved to distinguish between one, two, and many, and eventually one, two, three, many, but didn't have terms for higher numbers. Some languages still have this shortcoming. 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."
Thanks to the very nature of numbersthey can be added together to create new onesthe number system didn't stop at three. After a while, clever tribesmen began to string number-words in a row to yield more numbers. 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.
Few people count by twos like the Bacairi and Bororo. The old wolf bone seems to be more typical of ancient counting systems. Gog's wolf bone had 55 little notches in it, arranged into groups of five; there was a second notch after the first 25 marks. It looks suspiciously as if Gog was counting by fives, and then tallied groups in bunches of five. This makes a lot of sense. It is a lot faster to tally the number of marks in groups than it is to count them one by one. Modern mathematicians would say that Gog, the wolf carver, used a five-based or quinary counting system.
But why five? Deep down, it's an arbitrary decision. If Gog put his tallies in groups of four, and counted in groups of four and 16, his number system would have worked just as well, as would groups of six and 36. The groupings don't affect the number of marks on the bone; they only affect the way that Gog tallies them up in the endand he will always get the same answer no matter how he counts them. However, Gog preferred to count in groups of five rather than four, and people all over the world shared Gog's preference. It was an accident of nature that gave humans five fingers on each hand, and because of this accident, five seemed to be a favorite base system across many cultures. The early Greeks, for instance, used the word "fiving" to describe the process of tallying.
Even in the South American binary counting schemes, linguists see the beginnings of a quinary system. A different phrase in Bororo for "two and two and one" is "this is my hand all together." Apparently, ancient peoples liked to count with their body parts, and five (a hand), ten (both hands), and twenty (both hands and both feet) were the favorites. In English, eleven and twelve seem to be derived from "one over [ten]" and "two over [ten]," while thirteen, fourteen, fifteen, and so on are contractions of "three and ten," "four and ten," and "five and ten." From this, linguists conclude that ten was the basic unit in the Germanic protolanguages that English came from, and thus those people used a base-10 number system. On the other hand, in French, eighty is quatre-vingts (four twenties), and ninety is quatre-vingt-dix (four twenties and ten). This may mean that the people who lived in what is now France used a base-20 or vigesimal number system. Numbers like seven and 31 belonged to all of these systems, quinary, decimal, and vigesimal alike. However, none of these systems had a name for zero. The concept simply did not exist.
You never need to keep track of zero sheep or tally your zero children. Instead of "We have zero bananas," the grocer says, "We have no bananas." You don't have to have a number to express the lack of something, and it didn't occur to anybody to assign a symbol to the absence of objects. This is why people got along without zero for so long. It simply wasn't needed. Zero just never came up.
In fact, knowing about numbers at all was quite an ability in prehistoric times. Simply being able to count was considered a talent as mystical and arcane as casting spells and calling the gods by name. In the Egyptian Book of the Dead, when a dead soul is challenged by Aqen, the ferryman who conveys departed spirits across a river in the netherworld, Aqen refuses to allow anyone aboard "who does not know the number of his fingers." The soul must then recite a counting rhyme to tally his fingers, satisfying the ferryman. (The Greek ferryman, on the other hand, wanted money, which was stowed under the dead person's tongue.)
Though counting abilities were rare in the ancient world, numbers and the fundamentals of counting always developed before writing and reading. When early civilizations started pressing reeds to clay tablets, carving figures in stone, and daubing ink on parchment and on papyrus, number systems had already been well-established. Transcribing the oral number system into written form was a simple task: people just needed to figure out a coding method whereby scribes could set the numbers down in a more permanent form. (Some societies even found a way to do this before they discovered writing. The illiterate Incas, for one, used the quipu, a string of colored, knotted cords, to record calculations.)
The first scribes wrote down numbers in a way that matched their base system, and predictably, did it in the most concise way they could think of. Society had progressed since the time of Gog. Instead of making little groups of marks over and over, the scribes created symbols for each type of grouping; in a quinary system, a scribe might make a certain mark for one, a different symbol for a group of five, yet another mark for a group of 25, and so forth.
The Egyptians did just that. More than 5,000 years ago, before the time of the pyramids, the ancient Egyptians designed a system for transcribing their decimal system, where pictures stood for numbers. A single vertical mark represented a unit, while a heel bone represented 10, a swirly snare stood for 100, and so on. To write down a number with this scheme, all an Egyptian scribe had to do was record groups of these symbols. Instead of having to write down 123 tick marks to denote the number "one hundred and twenty-three," the scribe wrote six symbols: one snare, two heels, and three vertical marks. It was the typical way of doing mathematics in antiquity. And like most other civilizations Egypt did not haveor needa zero.
Yet the ancient Egyptians were quite sophisticated mathematicians. They were master astronomers and timekeepers, which meant that they had to use advanced math, thanks to the wandering nature of the calendar.
Creating a stable calendar was a problem for most ancient peoples, because they generally started out with a lunar calendar: the length of a month was the time between successive full moons. It was a natural choice; the waxing and waning of the moon in the heavens was hard to overlook, and it offered a convenient way of marking periodic cycles of time. But the lunar month is between 29 and 30 days long. No matter how you arrange it, 12 lunar months only add up to about 354 daysroughly 11 short of the solar year's length. Thirteen lunar months yield roughly 19 days too many. Since it is the solar year, not the lunar year, that determines the time for harvest and planting, the seasons seem to drift when you reckon by an uncorrected lunar year.
Correcting the lunar calendar is a complicated undertaking. A number of modern-day nations, like Israel and Saudi Arabia, still use a modified lunar calendar, but 6,000 years ago the Egyptians came up with a better system. Their method was a much simpler way of keeping track of the passage of the days, producing a calendar that stayed in sync with the seasons for many years. Instead of using the moon to keep track of the passage of time, the Egyptians used the sun, just as most nations do today.
The Egyptian calendar had 12 months, like the lunar one, but each month was 30 days long. (Being base-10 sort of people, their week, the decade, was 10 days long.) At the end of the year, there were an extra five days, bringing the total up to 365. This calendar was the ancestor of our own calendar; the Egyptian system was adopted by Greece and then by Rome, where it was modified by adding leap years, and then became the standard calendar of the Western world. However, since the Egyptians, the Greeks, and the Romans did not have zero, the Western calendar does not have any zerosan oversight that would cause problems millennia later.
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 the volumes of objectslike 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. Yet despite the Egyptians' brilliant geometric work, zero was nowhere to be found within Egypt.
This was, in part, because the Egyptians were of a practical bent. They never progressed beyond measuring volumes and counting days and hours. Mathematics wasn't used for anything impractical, except their system of astrology. As a result, their best mathematicians were unable to use the principles of geometry for anything unrelated to real world problemsthey did not take their system of mathematics and turn it into an abstract system of logic. They were also not inclined to put math into their philosophy. The Greeks were different; they embraced the abstract and the philosophical, and brought mathematics to its highest point in ancient times. Yet it was not the Greeks who discovered zero. Zero came from the East, not the West.
The Birth of Zero
In the history of culture the discovery of zero will always
stand out as one of the greatest single achievements of the
Tobias Danzig, Number:]BRK The Language of Science
The Greeks understood mathematics better than the Egyptians did; once they mastered the Egyptian art of geometry, Greek mathematicians quickly surpassed their teachers.
At first the Greek system of numbers was quite similar to the Egyptians'. Greeks also had a base-10 style of counting, and there was very little difference in the ways the two cultures wrote down their numbers. Instead of using pictures to represent numbers as the Egyptians did, the Greeks used letters. H (eta) stood for hekaton: 100. M (mu) stood for myriori: 10,000the myriad, the biggest grouping in the Greek system. They also had a symbol for five, indicating a mixed quinary-decimal system, but overall the Greek and Egyptian systems of writing numbers were almost identicalfor a time. Unlike the Egyptians, the Greeks outgrew this primitive way of writing numbers and developed a more sophisticated system.
Instead of using two strokes to represent 2, or three Hs to represent 300 as the Egyptian style of counting did, a newer Greek system of writing, appearing before 500 BC, had distinct letters for 2, 3, 300, and many other numbers (Figure 1). In this way the Greeks avoided repeated letters. For instance, writing the number 87 in the Egyptian system would require 15 symbols: eight heels and seven vertical marks. The new Greek system would need only two symbols: [Pi] for 80, and [Zeta] for 7. (The Roman system, which supplanted Greek numbers, was a step backward toward the less sophisticated Egyptian system. The Roman 87, LXXXVII, requires seven symbols, with several repeats.)
Though the Greek number system was more sophisticated than the Egyptian system, it was not the most advanced way of writing numbers in the ancient world. That title was held by another Eastern invention: the Babylonian style of counting. And thanks to this system, zero finally appeared in the East, in the Fertile Crescent of present-day Iraq.
At first glance the Babylonian system seems perverse. For one thing the system is sexagesimalbased on the number 60. This is an odd-looking choice, especially since most human societies chose 5, 10, or 20 as their base number. Also, the Babylonians used only two marks to represent their numbers: a wedge that represented 1 and a double wedge that represented 10. Groups of these marks, arranged in clumps that summed to 59 or less, were the basic symbols of the counting system, just as the Greek system was based on letters and the Egyptian system was based on pictures. But the really odd feature of the Babylonian system was that, instead of having a different symbol for each number like the Egyptian and Greek systems, each Babylonian symbol could represent a multitude of different numbers. A single wedge, for instance, could stand for 1; 60; 3,600; or countless others.
As strange as this system seems to modern eyes, it made perfect sense to ancient peoples. It was the Bronze Age equivalent of computer code. The Babylonians, like many different cultures, had invented machines that helped them count. The most famous was the abacus. Known as the soroban in Japan, the suan-pan in China, the s'choty in Russia, the coulba in Turkey, the choreb in Armenia, and by a variety of other names in different cultures, the abacus relies upon sliding stones to keep track of amounts. (The words calculate, calculus, and calcium all come from the Latin word for pebble: calculus.)
Adding numbers on an abacus is as simple as moving the stones up and down. Stones in different columns have different values, and by manipulating them a skilled user can add large numbers with great speed. When a calculation is complete, all the user has to do is look at the final position of the stones and translate that into a numbera pretty straightforward operation.
The Babylonian system of numbering was like an abacus inscribed symbolically onto a clay tablet. Each grouping of symbols represented a certain number of stones that had been moved on the abacus, and like each column of the abacus, each grouping had a different value, depending on its position. In this way the Babylonian system was not so different from the system we use today. Each 1 in the number 111 stands for a different value; from right to left, they stand for "one," "ten," and "one hundred," respectively. Similarly, the symbol ?? in ?? stood for "one," "sixty," or "thirty-six hundred" in the three different positions. It was just like an abacus, except for one problem. How would a Babylonian write the number 60? The number 1 was easy to write: ??. Unfortunately, 60 was also written as ??; the only difference was that ?? was in the second position rather than the first. With the abacus it's easy to tell which number is represented. A single stone in the first column is easy to distinguish from a single stone in the second column. The same isn't true for writing. The Babylonians had no way to denote which column a written symbol was in; ?? could represent 1, 60, or 3,600. It got worse when they mixed numbers. The symbol ?? could mean 61; 3,601; 3,660; or even greater values.
Zero was the solution to the problem. By around 300 BC the Babylonians had started using two slanted wedges, ??, to represent an empty space, an empty column on the abacus. This placeholder mark made it easy to tell which position a symbol was in. Before the advent of zero, ?? could be interpreted as 61 or 3,601. But with zero, ?? meant 61; 3,601 was written as ?? (Figure 2). Zero was born out of the need to give any given sequence of Babylonian digits a unique, permanent meaning.
Though zero was useful, it was only a placeholder. It was merely a symbol for a blank place in the abacus, a column where all the stones were at the bottom. It did little more than make sure digits fell in the right places; it didn't really have a numerical value of its own. After all, 000,002,148 means exactly the same thing as 2,148. A zero in a string of digits takes its meaning from some other digit to its left. On its own, it meant ... nothing. Zero was a digit, not a number. It had no value.
A number's value comes from its place on the number line from its position compared with other numbers. For instance, the number two comes before the number three and after the number one; nowhere else makes any sense. However, the 0 mark didn't have a spot on the number line at first. It was just a symbol; it didn't have a place in the hierarchy of numbers. Even today, we sometimes treat zero as a nonnumber even though we all know that zero has a numerical value of its own, using the digit 0 as a placeholder without connecting it to the number zero. Look at a telephone or the top of a computer keyboard. The 0 comes after the 9, not before the 1 where it belongs. It doesn't matter where the placeholder 0 sits; it can be anywhere in the number sequence. But nowadays everybody knows that zero can't really sit anywhere on the number line, because it has a definite numerical value of its own. It is the number that separates the positive numbers from the negative numbers. It is an even number, and it is the integer that precedes one. Zero must sit in its rightful place on the number line, before one and after negative one. Nowhere else makes any sense. Yet zero sits at the end of the computer and at the bottom of the telephone because we always start counting with one.
One seems like the appropriate place to start counting, but doing so forces us to put zero in an unnatural place. To other cultures, like the Mayan people of Mexico and Central America, starting with one didn't seem like the rational thing to do. In fact, the Mayans had a number systemand a calendarthat made more sense than ours does. Like the Babylonians, the Mayans had a place-value system of digits and places. The only real difference was that instead of basing their numbers on 60 as the Babylonians did, the Mayans had a vigesimal, base-20 system that had the remnants of an earlier base-10 system in it. And like the Babylonians, they needed a zero to keep track of what each digit meant. Just to make things interesting, the Mayans had two types of digits. The simple type was based on dots and lines, while the complicated type was based on glyphsgrotesque faces. To a modern eye, Mayan glyph writing is about as alien-looking as you can get (Figure 3).
Like the Egyptians, the Mayans also had an excellent solar calendar. Because their system of counting was based on the number 20, the Mayans naturally divided their year into 18 months of 20 days each, totaling 360 days. A special period of five days at the end, called Uayeb, brought the count to 365. Unlike the Egyptians, though, the Mayans had a zero in their counting system, so they did the obvious thing: they started numbering days with the number zero. The first day of the month of Zip, for example, was usually called the "installation" or "seating" of Zip. The next day was 1 Zip, the following day was 2 Zip, and so forth, until they reached 19 Zip. The next day was the seating of Zotz'0 Zotz' followed by 1 Zotz' and so forth. Each month had 20 days, numbered 0 through 19, not numbered 1 through 20 as we do today. (The Mayan calendar was wonderfully complicated. Along with this solar calendar, there was a ritual calendar that had 20 weeks, each of 13 days. Combined with the solar year, this created a calendar round that had a different name for every day in a 52-year cycle.)
The Mayan system made more sense than the Western system does. Since the Western calendar was created at a time when there was no zero, we never see a day zero, or a year zero. This apparently insignificant omission caused a great deal of trouble; it kindled the controversy over the start of the millennium. The Mayans would never have argued about whether 2000 or 2001 was the first year in the twenty-first century. But it was not the Mayans who formed our calendar; it was the Egyptians and, later, the Romans. For this reason, we are stuck with a troublesome, zero-free calendar.
The Egyptian civilization's lack of zero was bad for the calendar and bad for the future of Western mathematics. In fact, Egyptian civilization was bad for math in more ways than one; it was not just the absence of a zero that caused future difficulties. The Egyptians had an extremely cumbersome way of handling fractions. They didn't think of 3/4 as a ratio of three to four as we do today; they saw it as the sum of 1/2 and 1/4. With the sole exception of 2/3, all Egyptian fractions were written as a sum of numbers in the form of 1/n (where n is a counting number)the so-called unit fractions. Long chains of these unit fractions made ratios extremely difficult to handle in the Egyptian (and Greek) number systems.
Zero makes this cumbersome system obsolete. In the Babylonian systemwith zero in itit's easy to write fractions. Just as we can write 0.5 for 1/2 and 0.75 for 3/4, the Babylonians used the numbers 0;30 for 1/2 and 0;45 for 3/4. (In fact, the Babylonian base-60 system is even better suited to writing down fractions than our modern-day base-10 system.)
Unfortunately, the Greeks and Romans hated zero so much that they clung to their own Egyptian-like notation rather than convert to the Babylonian system, even though the Babylonian system was easier to use. For intricate calculations, like those needed to create astronomical tables, the Greek system was so cumbersome that the mathematicians converted the unit fractions to the Babylonian sexagesimal system, did the calculations, and then translated the answers back into the Greek style. They could have saved many time-consuming steps. (We all know how fun it is to convert fractions back and forth!) However, the Greeks so despised zero that they refused to admit it into their writings, even though they saw how useful it was. The reason: zero was dangerous.
The Fearsome Properties of Nothing
In earliest times did Ymir live:
was nor sea nor land nor salty waves,
neither earth was there nor upper heaven,
but a gaping nothing, and green things nowhere.
The Elder Edda
It is hard to imagine being afraid of a number. Yet zero was inexorably linked with the voidwith nothing. There was a primal fear of void and chaos. There was also a fear of zero.
Most ancient peoples believed that only emptiness and chaos were present before the universe came to be. The Greeks claimed that at first Darkness was the mother of all things, and from Darkness sprang Chaos. Darkness and Chaos then spawned the rest of creation. The Hebrew creation myths say that the earth was chaotic and void before God showered it with light and formed its features. (The Hebrew phrase is tohu v'bohu. Robert Graves linked these tohu to Tehomot, a primal Semitic dragon that was present at the birth of the universe and whose body became the sky and earth. Bohu was linked to Behomot, the famed Behemoth monster of Hebrew legend.) The older Hindu tradition tells of a creator who churns the butter of chaos into the earth, and the Norse myth tells a tale of an open void that gets covered with ice, and from the chaos caused by the mingling of fire and ice was born the primal Giant. 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.
But the fear of zero went deeper than unease about the void. To the ancients, zero's mathematical properties were inexplicable, as shrouded in mystery as the birth of the universe. This is because zero is different from the other numbers. Unlike the other digits in the Babylonian system, zero never was allowed to stand alonefor good reason. A lone zero always misbehaves. At the very least it does not behave the way other numbers do.
Add a number to itself and it changes. One and one is not oneit's two. Two and two is four. But zero and zero is zero. This violates a basic principle of numbers called the axiom of Archimedes, which says that if you add something to itself enough times, it will exceed any other number in magnitude. (The axiom of Archimedes was phrased in terms of areas; a number was viewed as the difference of two unequal areas.) Zero refuses to get bigger. It also refuses to make any other number bigger. Add two and zero and you get two; it is as if you never bothered to add the numbers in the first place. The same thing happens with subtraction. Take zero away from two and you get two. Zero has no substance. Yet this substanceless number threatens to undermine the simplest operations in mathematics, like multiplication and division.
Table of ContentsChapter 0: Null and Void
Chapter 1: Nothing Doing: The Origin of Zero
Chapter 2: Nothing Comes of Nothing: The West Rejects Zero
Chapter 3: Nothing Ventured: Zero Goes East
Chapter 4: The Infinite God of Nothing: The Theology of Zero
Chapter 5: Infinite Zeroes and Infidel Mathematicians: Zero and the Scientific Revolution
Chapter 6: Infinity's Twin: The Infinite Nature of Zero
Chapter 7: Absolute Zeroes: The Physics of Zero
Chapter 8: Zero Hour at Ground Zero: Zero at the Edge of Space and Time
Chapter Infinity: Zero's Final Victory: End Time
Appendix A: Animal, Vegetable, or Minister?
Appendix B: The Golden Ratio
Appendix C: The Modern Definition of a Derivative
Appendix D: Cantor Enumerats the Rational Numbers
Appendix E: Make Your Own Wormhole Time Machine
What People are Saying About This
From Author of The End of Science
Charles Seife has made a marvelously entertaining something out of nothing. By simply telling the tale of zero, Seife provides a fresh and fascinating history not only of mathematics but also of science, philosophy, theology, and even art. An impressive debut for a promising young science writer.
From John Rennie, Editor in chief of Scientific American
Zero: The Biography Of A Dangerous Idea describes with good humor and wonder how one digit has bedeviled and fascinated thinkers from ancient Athens to Los Alamos. Charles Seife deftly argues that the concept of nothingness and its show-off twin, infinity, have repeatedly revolutionized the foundations of civilization and philosophical thought. If you're already a fan of mathematics or science, you will enjoy this book; if you're not, you will be by the time you finish it.
The universe begins and ends with zero.' So does Seife's book, but his readers, after finishing, will feel they've experienced a considerable something
Most Helpful Customer Reviews
Fantastic book, though most definitely written by a mathematician. Very interesting and concise. Now to find a good biography of Pythagoras..
Zero was a fascinating journey. I read it in two sittings. I'm a high school senior in a college-level intro calculus course though, and I wonder how the less-initiated reader finds Zero. I would caution those who lack a patience for higher order mathematics, or a familiarity with physics and calculus to think twice before delving into Zero. You will undoubtedly enjoy it, but I wonder if you will understand the intricacies of the latter half of the book.
Who would have thought that a book about zero would be so interesting? But it is - and then some. Easily readable, even for mathophobes - and lots of fun.
Seife's book is overwrought, heavy on style and woefully weak (even inaccurate) on substance. His writing style is hyperbolic and filled with inane puns about the number zero (as can be seen in the chapter names). He goes way overboard in estimating the Greeks' attitude toward the number zero; as quite a bit of scholarship has shown, the number was not some kind of bete noire to the Greeks, rather, they simply found no need to incorporate it as a placeholder because of their geometric mathematical focus and the counting systems they used in commerce. Seife gets confused about the history of Aristotelianism in Europe-- he states that it basically kept Europe in the Dark Ages, while the Arab civilization (which imparted the numeric system containing zero to Europe in the late Middle Ages) rejected Aristotelian thought. In fact, as any middle school history student could point out, Europe was in the Dark Ages in large part because it almost totally forgot Aristotle's work. Though clearly many of Aristotle's ideas would turn out to be incorrect, his observational and scientific approach to things was crucial to eventually beginning the Age of Reason. In fact, one of the Arabs' greatest contributions to the history of thought was that they *translated* Aristotle into Arabic and studied it thoroughly, then transmitted this new learning and way of thought into Europe in about the 1200s. Seife mixes this up entirely. His descriptions of Newtonian and Leibnizian calculus are not bad, but when he gets into the cosmology and the physics he's way out of his league. One of the most fascinating things about 20th century science is the way in which Einstein's work and quantum physics have both totally revised the notion of the vacuum, filling it with activity of many stripes-- but Seife glosses this over in a few poorly written pages, and misses the whole point of what modern work has shown about the field. This is not the book to read about this topic.
Fascinating take on the history of ideas from the perspective of a mathematician. Argues that the exclusion of zero/infinity from the Western thought process (since Aristotle) limited and formed our philosophical and religious ideas and ultimately our early development in science. Seems like a stretch, but the author makes a convincing case. Also tracks the use of zero/infinity from India through the Islamic countries and ultimately into the West. The real joy of the book is in its sweep through history from this unique perspective. Lots of fascinating, little-known historical trivia tidbits throughout.
I found the first part of this book interesting, as I had never realized how radical the idea of zero was considered. The rest of it I loved; it brought back many pleasant memories of my Philosophy of Science and Vistas in Astronomy courses. The only complaint I had was that I wanted more details about the topics (and that¿s not bad for an introductory book)!
An intriguing look at the history of a special number, zero. Written in an easily digestible form it gives a great overview over the clash of philosophies which happened around zero and what effects of them we still face today. Not only for science and math buffs!
An interesting little book which gives you an extraordinarily insightful look at the History of Mathematics, how it progressed in societies and how societies progressed with it. It is also a nice way to gage just how well you know the subject. I know mathematics at about the level of the Renaissance.
It's, err, about Math. Why on earth I'd pick this up considering I barely understand division is beyond me. It's not bad for a "layperson" to follow, really, though a lot of the examples were over my head (i.e. I still don't understand the concept of Zero being infinity...)It's an interesting look at the evolution of math, and interesting in that I hadn't realized that Zero is a relatively new understanding for humankind. (I wonder if the aliens gave it to us?)Anyway, it's good for what it is, but I won't be investing any time in mathematical research anytime soon... math is hard!
Zero is organized well enough, moving along history from its start as a placeholder to its linchpin in calculus. Following his explanation of calculus, though, the book descends into a rehashing of, say, A Brief History of Time and the narrative becomes diffuse. Throughout, Seife's data and research are compelling, even if his arguments (connecting Aristotle, faith, and the notion of zero) are not always convincingly conveyed.
I recommend this book not because you will enjoy reading it. In fact, I guarantee you will not. But it is a book that everyone should read. Don¿t get me wrong, it is a wonderfully well-written book and it will fly by from the moment you read the first chapter, but it is very painful have to experience even a moment of what Abdulrahman Zeitoun experienced during and after Hurricane Katrina. Eggers is able to masterfully tell this story that touches on family, faith and the responsibility of a government, all while giving the readers a chance to step into the shoes of one who lived it, if only for a moment.
This is a most interesting book about the number zero. I don't believe I ever realized before what a radical idea it was, and what havoc it played with number theory, such as it was in Aristotle's day and for centuries to come in the West. Seife's ability to explain the role of zero, and its twin, infinity in conjunction with imaginary numbers and Riemann geometry was like an epiphany. Is it too late to become a mathematician? Highly recommended, especially if you are not a mathematician.
Starting with the Egyptian and Greek geometricians Seife relates the history of a number with very peculiar properties and its polar opposite, infinity. That makes this a book about nothing and everything. He uses it to mathematically prove that Sir Winston Churchill was a carrot and includes instructions on how to ¿make your own wormhole time machine.¿ For the most part he uses drawings rather than mathematical formulae to illustrate concepts, making this a very accessible book for the non-mathematicians among us.
I was initially skeptical about this book, because in the early chapters I felt like Seife sometimes presented questionable anecdotes as fact. Plus, for the chapters that I knew the most about, I lamented the lack of footnotes, which isn't really a fair criticism of a popular science work. The book quickly won me over, though, and I often found myself reluctant to put it down. The story presented here is about much more than math: there's history, philosophy, religion, and modern physics too. Much of the material was already familiar to me, but Seife brought it all together into a satisfying overview of the evolution of western thought. I would recommend this even to people who don't particularly like math; it's not very technical and is full of information that would be interesting to anyone.
0 + ( It's a book about math. And I read it. ) - ( It took me nine months. )= 0For three weeks after I finished Zero: The Biography of a Dangerous Idea, its central figure looked out ominously at me. In that way, Charles Seife was entirely successful in this piece of pop-nonfiction, weaving together the creation of the "zero", its place in history of mathematical theory, its religious controversies, its philosophical significance and ultimately, its . It's to Seife's credit that he manages to weave out of these eclectic approaches a coherent story that borders at times upon the epic... while never being too important not to include an irreverent tangent about Pythagoras's acute dislike of beans.If anything, Seife trends too sprightly at times. Though I admire his stance in neither dumbing down the material nor making it intimidating for the casual reader, at some point, no matter how breezily one explains black holes or the Casimir effect- there's no disguising that there are some vast concepts being covered. As it is, I believe you definitely have to at least of heard of some of these ideas (particularly in the last third) to enjoy the new contexts he weaves for them in his narrative. Myself, I sort of managed alright with some first year Calculus and Physics schooling. I can't say I ever turned down the chance for more trivia, and Zero delivered in spades. Also, know this: the first appendix details a mathematical proof on why Winston Churchill is a carrot.
Not only has zero not always existed, numbers aren't quite as concrete as our math teachers would have us believe. Seife presents the entire history of counting and numbers before getting into the history, philosophy and theology surrounding the number zero (and frequently, infinity). It helps to be somewhat comfortable with mathematical concepts, but it is not mandatory at all. Nor is it mandatory to know much about Greek philosophy-and the two get about as much attention. This is an excellent and sweeping history of how religion has had to change itself because of the immutable idea of nothingness. This also goes into the history of physics, particularly quantum physics and string theory, and astronomy. This is because, in almost all situations, mathematical theorums work beautifully and explain nature and the cosmos-until you have to account for zero.Well writen and researched. Highly recommended for any level reader-layman or expert.
There's an adage of TV newswriting: tell 'em, tell 'em, and tell 'em again. Mr. Seife apparently belongs to this school. He tells us uncountably times (>0 but less than infinity) that zero was dangerous. Still, the book covers a lot of ground in an approachable manner. I just wished he would stop repeating himself.
Confusing at times but really great read if you're into this kind of book
We read this book in trig class through the year. Was fun to find out the history of such a simple yet complex idea. I highly recomend this book if you love learning.