作者：Joseph Mazur
出版社：Princeton University Press
副标题：A Short History of Mathematical Notation and Its Hidden Powers
发行时间：March 23rd 2014
来源：下载的 epub 版本
Goodreads：3.6 (68 Ratings)
豆瓣：无
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 d’Arezzo, who lived at the turn of the first millennium, but that a more primitive form of symbol notation goes almost as far back as Phoenician writing. I, the mathematician, astonished my friends by revealing that, other than numerals, mathematical symbols—even algebraic equations—are relatively recent creations, and that almost all mathematical expressions were rhetorical before the end of the fifteenth century.
“What?!” the psychologist snapped. “What about multiplication? You mean to tell us that there was no symbol for ‘times’?”
“Not before the sixteenth…maybe even seventeenth century.”
“And equality? What about ‘equals’? the musician asked.
“Not before…oh…the sixteenth century.”
“But surely Euclid must have had a symbol for addition,” said the psychologist.
“What about the Pythagorean theorem, that thing about adding the squares of the sides of a right triangle?”
“Nope, …no symbol for ‘plus’ before the twelfth century!”
A contemplative silence followed as we sniffed and sipped expensive Barolo.
As it turned out, I was not correct. And far, far back in the eighteenth century BC, the Egyptians had their hieroglyphical indications of addition and subtraction in glyphs of men running toward or away from amounts to be respectively added or subtracted. And from time to time, writers of mathematical texts had ventured into symbolic expression. So there are instances when they experimented with graphic marks to represent words or even whole phrases. The Bakhshâlî manuscript of the second century BC records negative numbers indicated by a symbol that looks like our plus sign. In the third century, Diophantus of Alexandria used a Greek letter to designate the unknown and an arrow-like figure pointing upward to indicate subtraction. In the seventh century, the Indian mathematician Brahmagupta used a small black dot to introduce the new number we now call “zero.” And symbols were timidly beginning to find their way into mathematics by the second half of the fifteenth century. Of course, for ages, there have been the symbols that we use to designate whole positive numbers.
That night at the enoteca, I didn’t know that my estimate for the adoption of symbols was premature by several centuries. Sure, Diophantus in the third century had his few designations; however, before the twelfth century, symbols were not used for operational manipulation at the symbolic level—not, that is, for purely symbolic operations on equations. Perhaps I should have pushed the edge of astonishment to claim, correctly, that mostmathematical expressions were rhetorical before the sixteenth century.
Ever since that conversation, I have found that most people are amazed to learn that mathematics notation did not become really symbolic before the sixteenth century. We must also wonder: What was gained by algebra taking on a symbolic form? What was lost?
Traced to their roots, symbols are a means of perceiving, recognizing, and creating meaning out of patterns and configurations drawn from material appearance or communication.
The word “symbol” comes from the Greek word for “token,” or “token of identity,” which is a combination of two word-roots,sum (“together”) and the verb ballo (“to throw”). A more relaxed interpretation would be “to put together.” Its etymology comes from an ancient way of proving one’s identity or one’s relationship to another. A stick or bone would be broken in two, and each person in the relationship would be given one piece. To verify the relationship, the pieces would have to fit together perfectly.
On a deeper level, the word “symbol” suggests that, when the familiar is thrown together with the unfamiliar, something new is created. Or, to put it another way, when an unconscious idea fits a conscious one, a new meaning emerges. The symbol is exactly that: meaning derived from connections of conscious and unconscious thoughts.
Can mathematical symbols do that? Are they meant to do that? Perhaps there should be a distinction between symbols and notation. Notations come from shorthand, abbreviations of terms. If symbols are notations that provide us with subconscious thoughts, consider “+.” Alone, it is a notation, born simply from the shorthand for the Latin word et. Yes, it comes from the “t” in et. We find it in 1489 when Johannes Widmann wrote Behende und hubsche Rechenung auff allen Kauffmanscha (Nimble and neat calculation in all trades). It was meant to denote a mathematical operation as well as the word “and.”
Diophantus of Alexandria was born more than five hundred years after Euclid. His great work, Arithmetica, gave us something closer to algebraic solutions of special linear equations in two unknowns, such as x + y = 100, x − y = 40. He did this not by using the full power of symbols, but rather by syncopated notation—that is, by the relatively common practice of the time: omitting letters from the middle of words. So his work never fully escaped from verbal exposition. It was the first step away from expressing mathematics in ordinary language.
There is that old joke about joke tellers: A guy walks into a bar and hears some old-timers sitting around telling jokes. One of them calls out, “Fifty-seven!” and the others roar with laughter. Another yells, “Eighty-two!” and again, they all laugh.
So the guy asks the bartender, “What’s going on?”
The bartender answers, “Oh, they’ve been hanging around here together telling jokes for so long that they catalogued all their jokes by number. All they need to do to tell a joke is to call out the number. It saves time.”
The new fellow says, “That’s clever! I’ll try that.”
So the guy turns to the old-timers and yells, “Twenty-two!”
Everybody just looks at him. Nobody laughs.
Embarrassed, he sits down, and asks the bartender, “Why didn’t anyone laugh?” The bartender says, “Well, son, you just didn’t tell it right …”
Examine the left column, reading from top to bottom. Knowing nothing of the ancient script, we can guess that the column represents the numbers from 1 to 12. What about the second column from the left? If our first guess is correct (and how could it not be?), we would know that the first symbol in that column represents the number 9. What could the next number down be? It must be a juxtaposition of the symbol for 10 and the symbol for 8. Could it be 18? By the same reasoning, the third symbol seems to represent 27. Hmm …could the second column be the multiples of 9? That seems to be true until the sixth line down—that is, 6 × 9 = 54. At the seventh line down, something strange seems to happen. The symbol looks as if it is a 4. But is it? If it is, then that second column is not a list of multiples of 9. So what could it be?
We notice that there is a space between the first wedge mark and the other three. If there is a hope that the second column is a list of multiples of 9, then the seventh symbol should be 63. Perhaps the space indicates that we should multiply by 60 before adding the three wedges. That would give the correct multiple of 9.
Note: In the classical commentaries, these symbols are capitalized. See Diophanti Alexandrini, Opera Omnia. Also see Heath, A History of Greek Mathematics, 448.
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 the Brahmasphutasiddhanta in 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 of quadratic equations.
Algebra was not always called algebra. In the mid-fifteenth century some Italian and Latin writers called it Regula 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 εὶ∼δος, which meant “species,” as well as the particular, special power of the unknown. The word “specious” was used to suggest that the species—monads, squares, cubes, and so on—generally represented all known and unknown quantities. In fifteenth-century English, the word “specious” meant pleasing to the eye in form, yet deceptive. The word was still in use with that meaning in the eighteenth century and could be found in Samuel Johnson’s Dictionary of the English Language printed in 1785. Newton called algebra “universal arithmetic,” presumably because it embodied all the universal laws of arithmetic to be used on general equations. Petrus Ramus thought the Arabic name “algebra” was a vulgar name for “an art of [such] admirable subtlety.” For Descartes, the Arabic name was “barbarous.”
The German philosopher Daniel Lipstropius, a contemporary and biographer of Descartes, told us that Descartes’s most brilliant idea came to him while watching a fly crawl along a curved path. It was a fable of course, implying that the Cartesian coordinate system owes its origin to Descartes describing the path in terms of its distance from the walls, that a fly was responsible for one of math’s most radical shi s: a relatively early marriage of algebra and geometry. It was a fable because Descartes’s coordinate system looked nothing like our modern one with its horizontal and vertical axes indicating related variables. The story later inflated into a more sweeping fiction of how Descartes, because of his poor health, would lie in bed late each morning meditating on how all of science could be made as certain as mathematics.
All these operations are constructible with straightedge and compass, and therefore provable from Euclid’s axioms. And any problem that can be expressed through a geometric construction that uses straightedge and compass alone can also be expressed by a polynomial equation of degree one or two.
The Cartesian coordinate system is more than just an orienting system, more than just a way to get from here to there. It is a way to see geometry through the lens of algebra. Descartes (and Fermat too) gave us something incredibly special. He showed us that thinking itself has optional modes. He taught us that we have optional modes for conceptualizing problems. We may wish to attack the question of whether or not an arbitrary angle can be trisected using straightedge and compass alone, an ancient problem of geometry. It may be naturally expressed geometrically with words such as “line” and “angle.” But sometimes we are fooled by what we think is natural, entrapped and limited by unnecessary contortions of conceptualization.
To see how the Cartesian system gives us the link between geometry and algebra, we briefly remind ourselves what we either learned or missed in high school math. Keep in mind that the coordinate system invented by Descartes (as well as by Fermat) was not quite the system we use today. In fact, it was not the first idea of a coordinate system: the fourteenth-century cleric Nicole Oresme had a similar idea. But Descartes’s idea was a kick-starter for today’s more developed concept that was introduced later.
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. 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.
By the last half of the seventeenth century, mathematics manuscripts were aflame with symbols, largely due to Leibniz, along with Oughtred, Hérigone, Descartes, and Newton. Textbook writers and lesser-known mathematicians generated hundreds of new symbols. Symbol creation at that time was in vogue, but with little understanding of the unanticipated messes that would sooner or later corner creative thought.
As we have seen, Descartes borrowed most symbols and tweaked them to improvement. Oughtred introduced hundreds of potential symbols without much concern for their merit. Even when some were clearly problematical, he continued to use them for the sake of unflappable consistency. This was also true of Hèrigone.
Leibniz, on the other hand, made symbols a priority in his attempts at clear writing. He was convinced that excellent notation was key to comprehension in all matters of human thought. “The true method,” he wrote, “should further us with an Ariadne’s thread, that is to say, with a certain sensible and palpable medium, which will guide the mind as do the lines drawn in geometry and the formulas for operations, which are laid down for the learner in arithmetic.”
Tools of the infinite and infinitesimal, along with an intuitive grasp of the continuum, were being created and accepted. Imaginary numbers were in the vocabulary. Algebra and its astute use of symbols had prepared mathematics for the calculus revolution. Physics was boosted to a science. And Newton had—in the words of Albert Einstein—“the greatest advance in thought that a single individual was ever privileged to make.”
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