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Enlightening Symbols

Enlightening Symbols: A Short History of Mathematical Notation and Its Hidden Powers

Joseph Mazur
Copyright Date: 2014
Pages: 240
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  • Book Info
    Enlightening Symbols
    Book Description:

    While all of us regularly use basic math symbols such as those for plus, minus, and equals, few of us know that many of these symbols weren't available before the sixteenth century. What did mathematicians rely on for their work before then? And how did mathematical notations evolve into what we know today? InEnlightening Symbols, popular math writer Joseph Mazur explains the fascinating history behind the development of our mathematical notation system. He shows how symbols were used initially, how one symbol replaced another over time, and how written math was conveyed before and after symbols became widely adopted.

    Traversing mathematical history and the foundations of numerals in different cultures, Mazur looks at how historians have disagreed over the origins of the numerical system for the past two centuries. He follows the transfigurations of algebra from a rhetorical style to a symbolic one, demonstrating that most algebra before the sixteenth century was written in prose or in verse employing the written names of numerals. Mazur also investigates the subconscious and psychological effects that mathematical symbols have had on mathematical thought, moods, meaning, communication, and comprehension. He considers how these symbols influence us (through similarity, association, identity, resemblance, and repeated imagery), how they lead to new ideas by subconscious associations, how they make connections between experience and the unknown, and how they contribute to the communication of basic mathematics.

    From words to abbreviations to symbols, this book shows how math evolved to the familiar forms we use today.

    eISBN: 978-1-4008-5011-2
    Subjects: Mathematics

Table of Contents

  1. Introduction
    (pp. ix-xx)

    A mathematician, a musician, and a psychologist walked into a bar . . .

    Several years ago, before I had any thoughts of writing a book on the history of symbols, I had a conversation with a few colleagues at the Cava Turacciolo, a little wine bar in the village of Bellagio on Lake Como. The psychologist declared that symbols had been around long before humans had a verbal language, and that they are at the roots of the most basic and primitive thoughts. The musician pointed out that modern musical notation is mostly attributed to one Benedictine monk Guido...

  2. Note on the Illustrations
    (pp. xxiii-xxiv)
  3. Part 1 Numerals

    • [Part 1 Introduction]
      (pp. 1-2)

      BAKHSHÂLÎ MANUSCRIPT (date in dispute: 400 – 700). Indian.

      Shows that the Indians had a place-value system in place before 700 AD.

      BISHOP SEVERUS SEBOKHT (ca. 575 – ca. 666). Syrian. Science and philosophy writer.

      Author of the earliest known extant reference to Hindu-Arabic numerals outside of India.

      BRAHMAGUPTA (598 – 668). Indian. Mathematician-astronomer.

      HisBrahmasphutasiddhanta(628) has the first known use of zero (a small black dot) as a number, not just as a placeholder.

      HARUN AL-RASHID (8th century). Persian. Caliph.

      Founded the House ofWisdom in Baghdad, a library and translation center that contained manuscripts of mathematics and philosophy translated into Arabic...

    • Chapter 1 Curious Beginnings
      (pp. 3-9)

      No one knows precisely when humans first began to deliberately leave marks for communication with others. Surely it was in that misty period of time, when herds of woolly mammoth freely wandered Europe, and all sorts of living creatures were following the northward spread of food and vegetation from the plains of Africa.¹ The ice of Europe had been receding for centuries in the slow ending of one of the great climate changes of all time. Most of the human population was still in southern Asia.²

      That was between fifty and thirty thousand years ago, when humans had to think...

    • Chapter 2 Certain Ancient Number Systems
      (pp. 10-25)

      Call them what you wish—Babylonians, Sumerians, or Akkadians. We have heard their stories before. Almost every history of early Western mathematics begins with the Babylonian conception of number, a so-called sexagesimal (base 60) system for writing large numbers, formulations of multiplication tables, and ideas for astronomy.

      But who were those Babylonians, and why were they the ones to first come up with human civilization, culture, art, and science? To answer, examine the geographical region of the Fertile Crescent, that crescent-shaped region between the Eastern Mediterranean and the Persian Gulf, and running through southeastern Turkey to Upper Egypt. It happens...

    • Chapter 3 Silk and Royal Roads
      (pp. 26-34)

      Topography and frequent pounding of hooves and shoes formed the East–West route connecting China to India and India to Persia. There were no road crews. The Silk Road was not one particular road, but rather a series of land and sea routes crisscrossing Eurasia, passing over 4,000 miles of wildly rugged terrain, and connecting to other routes traveled mostly by Indian merchants, agents, and explorers. Formed sometime around the second century BC, it connected to the Royal Road in the Zagros Mountains of Persia, where postal offices relayed mail and where one could find fresh horses for a journey...

    • Chapter 4 The Indian Gift
      (pp. 35-50)

      Some low Brahmi numbers (figure 4.1) graphically resemble our low modern numbers. The Brahmi system, however, was conceptually very different. It was not a positional system of powers of ten. Rather, it was closer to an alphabet-based numerical system that requires long concatenated strings to represent even relatively low numbers.

      At one time, there was speculation that the figures past 4 had come from either the forms of initial letters or syllables of number words of the third century BC Brahmi alphabet. But they may have come from older, untraceable numerical symbols.¹ A more fitting origin is theDevanagariscript...

    • Chapter 5 Arrival in Europe
      (pp. 51-59)

      Curiously, few centuries have passed since our wonderful current number system was brought to Europe. There is a dispute over whether or not the person most responsible was Leonardo Pisano Bigollo (ca. 1170 – ca. 1250), one of the great mathematicians of his time, whose fame comes mostly from that celebrated problem of how rabbits multiply, a man more memorably known to us as Fibonacci. He was certainly not the discoverer of the answer to the rabbit question, which was asked in ancient India since about the turn of the first millennium to describe the metrical structure and underlying rhythm found...

    • Chapter 6 The Arab Gift
      (pp. 60-63)

      Abu Jafar Muhammad ibn Musa al-Khwārizmī’s portrait (figure 6.1), popularized by a 1983 Soviet Union postage stamp commemorating the twelve-hundredth anniversary of his birth, shows a bearded man with furrowed brow and dreaming eyes. Isn’t it extraordinary that we can know. . . hmm. . . what a particular ninth-century person looks like with little knowledge of his biography? The truth is that we hardly know what he really looked like. Al-Khwārizmī, who was the greatest Arab mathematician of his day, learned of the new Indian numbers from the Arabic translation of Brahmagupta’sBrahmasphutasiddhanta, and wrote a textbook on arithmetic...

    • Chapter 7 Liber Abbaci
      (pp. 64-72)

      In the ninth century, the Indian figures were still too new and too weird to spread far from the monasteries and scholarly hubbubs. After all, Europe did not know of a numeral system with azero, that single symbol that could be used to write an infinite range of numbers and at the same time represent nothing. The Babylonian system didn’t have one; neither did the Greek, nor the Roman.

      It was not as if there were no commerce and travelers to bring the numerals to Europe. There were plenty. It was that beast, zero—the stranger that caused enough...

    • Chapter 8 Refuting Origins
      (pp. 73-80)

      Suffering a need for documents he could not validly collect, Denis Vrain-Lucas resorted to stealing antique paper from several libraries in Paris by cutting the endpapers of old books. Using special self-made inks, he carefully imitated diverse hand-writings, and sold forgeries (letters and documents) to unsuspecting manuscript collectors.

      He was a law clerk and amateur historian with a genuine passion for collecting manuscripts of great historical importance. Over a sixteen-year period starting in about 1855, Vrain-Lucas sold over 27,000 autographed forgeries, many to his favored mark, Michel Chasles, who paid hundreds of thousands of francs over a nine-year period beginning...

  4. Part 2 Algebra

    • [Part 2 Introduction]
      (pp. 81-84)

      Going back in time once again, before Indian numerals were brought to Europe.

      Many of these initiators were either the first or best known for putting symbols into print:

      DIOPHANTUS (205 ± 15–290 ± 15). Alexandrian Greek. Mathematician.

      Wrote theArithmeticain ca. 250 AD. First to use symbol for minus (⋔) and unknown (ㄣ).

      HYPATIA (ca. 350–370). Greek. Mathematician.

      First notable woman mathematician, and commentator of theArithmetica.

      ARYABHATTA (476–550). Indian. Mathematician-astronomer.

      Used letters to represent unknowns.

      BRAHMAGUPTA (598–668), Indian. Mathematician-astronomer.

      Possibly the first writer to use zero (a small black dot) as...

    • Chapter 9 Sans Symbols
      (pp. 85-92)

      Many years ago, I had a few rare moments of being permitted to flip through the oldest surviving copy of Euclid’sElements, MS D’Orville 301. It was the privilege of a favored few, a privilege no easier than getting permission to visit the queen in her drawing room. First I had to obtain a reference from a respected professor of mathematics. Perhaps it wasn’t fully necessary to have it from a knighted professor, but his is what I got. Then, on the day of appointment, a man greeted me in the lobby outside the Special Collections room of the Bodleian...

    • Chapter 10 Diophantus’s Arithmetica
      (pp. 93-108)

      The earliest works of the ancients that could be called algebra date back to the early Pythagoreans, or at least perhaps the Pythagorean Thymaridas of Paros, who, according to the Syrian philosopher Iamblichus, gave a rule for solving a certain set ofnsimple simultaneous equations in n unknowns. For three unknowns, the rule simplifies to:Given a sum of three quantities and also the sums of every pair containing one of those specified quantities, then that specified quantity is equal to the difference between the sums of those pairs and the total sum of the original three quantities.


    • Chapter 11 The Great Art
      (pp. 109-115)

      The art of algebra may have come from the Greeks or from the Hindus. However, the Brahmins of northern India had some idea of algebra long before the Arabians learned it, contributed to it and brought that art to Spain in the late eleventh century. The Indian mathematician Brahmagupta wrote theBrahmasphutasiddhantain 1,008 metered verses “for the pleasure of good mathematicians and astronomers.” Completed in 628, it not only advanced the mathematical role of zero but also introduced rules for manipulating negative and positive numbers, methods for computing square roots, and systematic methods of solving linear and limited types...

    • Chapter 12 Symbol Infancy
      (pp. 116-126)

      Algebra was not always called algebra. In the mid-fifteenth century some Italian and Latin writers called itRegula rei e census(Ruling Out of the Thing and Product). Mathematicians prefer short names for their fields—arithmetic, geometry, calculus, analysis, number theory, logic, and so on.

      François Viète first called it the “analytical art.” John Wallis gave it the English name “specious arithmetic.” Most likely, his word for it came from the Greek word εῐδος, whichmeant “species,” as well as the particular,specialpower of the unknown. The word “specious” was used to suggest that the species—monads, squares, cubes, century English,...

    • Chapter 13 The Timid Symbol
      (pp. 127-132)

      In Nuremberg,Germany, a year before Cardano’sArsMagnaappeared in print, people were studying Michael Stifel’sArithmetica Integra, a treatise on arithmetic and algebra. Stifel included several symbols that were already in use, such as +, −, and √, which he actually called “plus,” “minus,” and “radix”; still, there was no sign for “equals.”

      Symbols were beginning to appear in European manuscripts on algebra in two different styles: one from the Italians, the other from the Germans. The Italians used the word cosa (“what,” or “thing”) when referring to the unknown root of an equation. And since algebra was after all...

    • Chapter 14 Hierarchies of Dignity
      (pp. 133-140)

      “I feel obliged to speak of the supremacy, among all themathematical disciplines, of the subject that is nowadays called algebra by the common people,” Rafael Bombelli wrote in 1572.¹ Bombelli was an engineer whose work involved something to do with reclaiming marshlands and building bridges. HisL’Algebrawas published in 1579, but he began working on it twenty years earlier, when he had a break from his work at draining the Val di Chianamarshes in central Tuscany. InL’Algebra, we meet a new kind of notation for the unknown and its powers. Our modern notation for the polynomial x²−3x+2, for...

    • Chapter 15 Vowels and Consonants
      (pp. 141-149)

      François Viète, who wrote under the Latin name Franciscus Vieta, was a French mathematician having the great advantage of working in an era of abundant mathematical contributions from the Italians, Germans, and English.

      He expressed his famous computation for π entirely in six paragraphs of 139 words in proposition II of hisIsagoge

      His proposition II tells us how to approximate π by first inscribing a square in a circle, projecting the bisection of each side out to the circle to get an octagon, and repeating the process, first with the octagon, and then again and again with each resulting...

    • Chapter 16 The Explosion
      (pp. 150-159)

      René Descartes’sGeometriawas published just thirty-four years after Viète died. It had a new idea for notation, a rule: beginning letters of the alphabet were to be reserved for fixed known quantities and latter letters (pastp) were to represent variables or unknowns that could take on a succession of values. Descartes seems to have followed Thomas Harriot’s practice of using lowercase letters, though he denied ever having seen Harriot’s writings. To this day, this division of the alphabet at premains the loose standard rule.

      The German philosopher Daniel Lipstropius, a contemporary and biographer of Descartes, told us that...

    • Chapter 17 A Catalogue of Symbols
      (pp. 160-164)

      William Oughtred died on Sunday, the thirteenth of June 1660, at the age of eighty-eight. John Aubrey tells us, “He was a little man, had black haire, and blacke eies (with a great deal of spirit). His head was always working. He would drawe lines and diagrams on the dust. He had burned all his papers, claiming that, ‘the world was not worthy of them.’ He was so superb. He burned also several printed books, and would not stirre, till they were consumed.”¹ If you examine the engraved portrait of him by the Czech engraver Wenzel Hollar, you will find...

    • Chapter 18 The Symbol Master
      (pp. 165-168)

      Gottfried Leibniz, a man “of middle size and slim figure, with brown hair, and small but dark and penetrating eyes,” was the genius of symbol creation.¹ Alert to the advantages of proper symbols, he worked them, altered them, and tossed them whenever he felt the looming possibility that some poorly devised symbol might someday unnecessarily complicate mathematical exposition. He had studied Bombelli and Viète, and foresaw how symbols for polynomials could not possibly continue into algebra’s generalizations at the turn of the seventeenth century. He knew how inconvenient symbols trapped the advancement of algebra in the fifteenth and sixteenth centuries....

    • Chapter 19 The Last of the Magicians
      (pp. 169-176)

      Isaac Newton, a man “rather languid in his look and manner, which did not raise any great expectation in those who did not know him,” gave figurative credit to those giants on whose shoulders he stood.¹ Popular accounts of Newton recall his famous line, “If I have seen further it is by standing on ye shoulders of Giants,” which goes back to the twelfth century when the French Neoplatonist philosopher Bernard of Chartres compared his generation “to [puny] dwarfs perched on the shoulders of giants.” Bernard pointed out that we see more and farther than our predecessors, not because we...

  5. Part 3 The Power of Symbols

    • Chapter 20 Rendezvous in the Mind
      (pp. 179-188)

      Before the sixteenth century, almost anyone with enough determination could comprehend the elements of almost any mathematical writing. With quill and parchment, a quiet room, an open window with refreshing breezes, enough tallow to keep candles burning through the night, and an inordinate amount of mind-contorting labor, it was still possible to write mathematics in natural language words. Mathematics was readable to anyone who wished to parse its language, its springs, its gears, and its logic.

      “Jabberwocky,” theThrough the Looking Glassverse that begins “Twas bryllyg, and the slythy toves” gives an impression of what sensible language sounds like...

    • Chapter 21 The Good Symbol
      (pp. 189-191)

      The first appearance of the symbol π came in 1706. William Jones (how many of us have ever heard of him?) used the Greek letter π to denote the ratio of the circumference to the diameter of a circle.¹ How simple. “No lengthy introduction prepares the reader for the bringing upon the stage of mathematical history this distinguished visitor from the field of Greek letters. It simply came, unheralded.”² But for the next thirty years, it was not used again until Euler used it in his correspondence with Stirling.

      We could accuse π of not being a real symbol. It...

    • Chapter 22 Invisible Gorillas
      (pp. 192-209)

      A frog easily catches insects in motion, but will not bother a most appetizing fat housefly sitting directly in front of him. A fly could safely crawl onto the frog’s back without any worry of being gobbled up. Place a plate of dead files in front of the frog and he will sit there like a stone garden ornament. The poor frog would starve to death rather than attack something that is not moving.

      The pond in my yard is filled with frogs of all sizes. I see one, but he does not see me—not really. His eyes don’t...

    • Chapter 23 Mental Pictures
      (pp. 210-215)

      Years ago, during some summers on Cape Cod, I would jog down a dirt road, passing a giant German shepherd tethered by chain who would bark one long “grrrrrhoff” as I passed—just one. I would answer with a very quiet “whoof,” my “hello” in what I thought might be dog language. After a few days into the season, the dog stopped “grrrrrhoffing,” and just watched me pass by.

      What was going on in his head? He must have learned that it was me passing, that I was friendly and that I could make sounds just like he could. So...

    • Chapter 24 Conclusion
      (pp. 216-220)

      We think in fuzzy pictures, cloudy symbols—there, yet not there—senses and impressions that allow us to go about our daily business. In literature, the conscious track has a lag time. Read Dostoyevsky’sCrime and Punishmentand come to the point when Raskolnikov crushes the old woman’s head with a swing of an ax. What role does the ax play as we read further? Why did Dostoyevsky decide that the old woman should be killed by an ax and not by a gun nor bludgeoned to death with a poker? How would our psyches have responded if another weapon...

  6. Appendix A Leibniz’s Notation
    (pp. 221-222)
  7. Appendix B Newton’s Fluxion of xn
    (pp. 223-223)
  8. Appendix D Visualizing Complex Numbers
    (pp. 228-229)