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Origins of Math
It is difficult to imagine a world without mathematics. Even if we could easily dream of a history devoid of the inventions and learning applied mathematics has made possible, it strains the mind to imagine a world without even simple counting.
And yet this is the world in which the human race lived for hundreds of generations before beginning to distinguish quantity beyond broad concepts like one and many. Aboriginal cultures that have been largely sheltered from the influence of larger civilizations reveal this early history of quantification preserved in their language, such as the !Kung bushmen of the Kalahari who express ten as "two and two and two and two and two."
The most ancient evidence of counting comes to us by way of notches carved into lengths of bone, arranged in sets to track periodic events like the phases of the moon or the days between the seasonal equinoxes, or perhaps even the years of a person's life. This kind of tallying has been seen in artifacts as musty as 30,000 years old.
To progress from tallying -- the simple accruing of individual events -- to real counting required an abstract leap to understand the similarity between different kinds of tallied things: as one hunter is to three hunters, so is one spear to three spears. In other words there came a time when people noticed that there was an element of number that was purely conceptual; one could count things but one could also count the numbers themselves without any reference to objects or events in the real world.
Despite the fact that this is the kind of material covered on Sesame Street today, back then it was a fairly big deal. Bear in mind: these were the days when fire was still considered high technology, and entire cultures still hadn't quite figured out where babies come from.
Progress was made.
As human cultures became more advanced the need for accounting grew. What is the worth of one flock of sheep compared to another? How is a quantity of property to be divided among heirs? Is there a relationship between the number of days of rainfall and the height of crops? What's my share of the war spoils from this village our armies have just decimated? That sort of thing.
And like people of any age the ancients were curious about the true nature of the unpredictable world they saw around them. They yearned to make sense of it all, and in this light they could not ignore the transcendental nature of numbers. Counting was the same no matter what was being counted, or by whom. Counting was the same regardless the words used to name the numbers. Counting revealed an unchanging bond between all things, living or dead, great or small, hither or yon. In short, numbers were obviously mystical.
Strings of Symbols
Numbers were power, for they could bring order from confusion.
And the earthly powers-that-be had a tendency to compete with their dead predecessors, especially in an age of megaliths. Thus the kings of old decreed temples honouring their gods be built, and each king wanted his to be grander than his father's.
One of the oldest forms of temple in the world is the stepped pyramid, or ziggurat. They are common pan-culturally chiefly because their construction is so elementary: blocks upon blocks, with fewer blocks on each successive level until the apex (where there would usually be an altar for religious rites). They were little man-made mountains. Early ziggurats had no interior space -- they were a monument to be climbed, to be experienced exclusively from without.
Thus it was only natural that the easiest way for the kings of old to better their fathers was to build a new pyramid directly overtop of the old one, using the older ziggurat's steps as a foundation.
As time went on it became harder and harder to show off in this way. Ever larger pyramids tended to fall apart a lot, leading to expensive embarrassments and more often than not some hapless mason would forfeit his life for it. Wiser kings reasoned that the sages who could use numbers to predict the coming and going of the seasons might well apply their craft to problems of engineering.
Could they use mathematics to design a pyramid whose sides would not sheer off under their own weight?
The key to moving beyond rudimentary arithmetic was finding a way to encode the numbers representing quantities into symbols -- symbols that could be manipulated according to known procedures, allowing the computation of numbers too complex to be handled in the imagination alone. The earliest known attempt at such an encoding system evolved naturally enough from the earliest known form of writing: the cuneiform script of the Sumerian people in ancient Mesopotamia.
Mesopotamia features some of the world's earliest, grandest ziggurats.
The Sumerian culture was subsumed by the vibrant and aggressive Babylonian civilization, which inherited many Sumerian traditions including practices of law, clergy and accounting. Their still-primitive system of mathematics had its roots in the calendar and religious mysticism, likely stemming from Sumerian methods for determining eclipses and the solstice. The Babylonians rounded off the length of the solar year to 360 days, so it is not surprising that they chose the factor 60 as the basis of their counting, seeing as it was obviously a holy number around which the natural world was constructed. They employed a sexigesimal system based on units of six to express place value, similar to our modern decimal system based on units of ten. This allowed them to accurately manipulate fractions, and perform complex division and multiplication (though there is no evidence that they used it to shelve their library books).
Thus is was only natural that the pharaohs of Egypt looked to Babylonian learning to inform the construction of their mighty and enduring pyramids, culminating in the grand works at Giza that we can still admire today, over four thousand years later. They stand today thanks to the innovative architecture of Egyptian scholars who had studied Babylonian ways like the multi-talented Imhotep (2667-2648 BC) and the plump vizier Hemiunu.
Our understanding of Egyptian mathematics comes sources like the Rhind Papyrus authored by the scribe Ahmes (1680-1620 BC), which records with examples a series of addition-based methods for solving pragmatic problems in measuration, geometry and proportion. Ahmes was not a mathematician himself but a secretary, though his devotion to the philosophy of mathematics is evident in his comments:
Accurate reckoning: the entrance into knowledge of all existing things and all obscure secrets.
Egyptian mathematics were often hazy when it came to distinguishing between exact reckoning and approximation, leading to a list of non-rational features like "preferred" fractions (fractions involving numbers or proportions considered holy) and a somewhat skewed notion of pi. Since Egyptian mathematics was based solidly on problems of application these approximations were often "good enough" for the purposes of the craftsmen making use of them. Where an error of a tenth of a degree may offend a pure mathematician, to a stone-mason whose cutting accuracy was limited by primitive tools such trivial deviations were irrelevant.
Despite this theoretical apathy the Egyptian approach did become more rigorous over the centuries as engineers learned from their mistakes, creating a solid and tested foundation for the amazing accomplishments of the Greeks.
Mathematic Superstars of Ancient Greece
Though the Greek (or Hellenic) civilization has its roots as far back as 3000 BC, the golden age of learning did not come until the Greeks turned their back on Egyptian-style hieroglyphic writing and embraced Phoenician-style alphabetic writing as the basis for their communications. Freed of the pictorial bonds of hieroglyphics the Greeks found a new facility in definitively expressing abstract concepts, leading to revolutions in philosophy, music, rhetoric, drama, poetry and mathematics whose reverberations would be felt down through the centuries to our present era.
The influence of Babylonia and Egypt was greatest in the Ionian city of Miletus, which many consider to be the birthplace of science. One of the earliest superstars of the Milesian math scene was Thales (c.643-546 BC), teacher many great mathematicians including Anazimander and Anaximenes. It is said that Thales' acumen first became apparent to the general public when he accurately predicted a solar eclipse in 585 BC, demonstrating that eclipses were periodic events determined by geometry and time, and not omens cast by the gods reflecting their evaluation of events taking place on Earth.
This was followed by a great celebration among the goats who would otherwise have been satisfied to appease the gods in meagre hope of staving off the next eclipse.
Thales' probable student Pythagoras (c.569-c.475 BC) travelled to Babylonia and Egypt to broaden his mathematical knowledge, eventually settling on the Greek island of Samos and founding two societies: the semicircle -- a municipal forum for exploring politics and philosophy -- and a secret society that met outside of the city, in the caves. This quasi-religious, quasi-scientific community sought to advance their understanding of and communion with the true nature of the universe by devout studies of the esoteric properties of geometry. This evolved into a religious school at Croton, whose inner-circle followers, called mathematikoi, observed the Spartan life of monks, giving up all worldly possessions and eschewing the eating of meat in order to better devote themselves to the pursuit of mathematical truth. Due to Pythagoras' love of secrecy and codes, none of these advances is available to us today; we know only of his skill through the admiration of his contemporaries.
Zeno of Elea (c.488-c.435 BC) was a bratty punk who at the age of forty took up serious study of the Athenian intelligentsia's pet theories and then challenged the city's brightest to solve a series of logical paradoxes. The brainiest thinkers of the West would spend centuries attempting to unravel these conundrae, including Kant, Hume, Hegel, Cauchy, Weierstrass, Cantor, Russell and Einstein. At the heart of Zeno's thinking was the difference between continuous and discrete data, a riddle we would not have the appropriate tools to tackle until the advent of differential calculus in the 17th century.
Zeno may have personally chatted with the famous philosophical layabout Socrates (469-399 BC), whose wife Xanthippe was known to dump various things (water, garbage) over his head in order to tear his attention away from chatting his way to wisdom. Socrates contributed nothing to mathematics directly apart from an example of dedication to the relentless quest for truth, regardless of whose feathers may get ruffled in the process.
Socrates, of course, was put to death by his peers.
Among the students of Socrates before he died was the philosophical dramatist Plato (427-327 BC), who would lend his greatest contribution to the advancement of mathematics by founding his school at Academos, a suburb of Athens. This institute of higher learning (from which we derive our English word academy) was like a modern university in many respects, and would foster advancements in mathematics for several centuries until being forcibly closed by the Roman Emperor Justinian as a source of dangerously non-Christian ideas.
The incomparable Aristotle (384-322 BC) was a student at Plato's Academy (Class of '61, voted "most likely to dominate Western thought for millennia"), and so was the amazing Euclid (c.325-265 BC) whose phenomenal work The Elements would become the cornerstone of mathematical education for next two thousand years (talk about a shelf life!). The Euclidean school was concerned chiefly with replacing assumptions with irreducible axioms -- that is, boiling down any logical argument to its bare elements, so basic they cannot be refuted. One such general axiom ran like so:
Things which are equal to the same thing are equal to each other....Which is indeed hard to argue against. Building from this kind of elemental foundation, Euclid and his Alexandrian followers categorically reshaped the study of mathematics in general and geometry in particular, giving us a language and logical syntax for attacking even the most complex problems.
One famous student of Euclid's school was the brilliant Archimedes (287-212 BC), whose youthful invention of a pump for extracting water from the Earth is still in use in some parts of the world today. A mechanical genius, he also invented numerous engines of war employed to defend the city of Syracuse against Roman invaders. Obsessed with geometry Archimedes made many advances in the easy computation of volume, area and perimeter. He is perhaps most famous for running through the streets of Syracuse naked while crying out, "Eureka! Eureka!" (Greek for "I've found it!") after discovering the key to volumetric displacement while having a bath one day and noting how the water splashed over the sides when he got in.
Christianity became the official religion of the Roman Empire after the death of the first publicly Christian emperor, Flavius Valerius Constantinus (306-337 AD), and for various sticky theological reasons this made pure research very difficult for any scientist or academic who was averse to being burned alive for spreading heresy.
Thus the challenge of nurturing the growth of mathematics had to be taken up outside of Christendom, and so we must turn our attention east, to the lands conquered by the Islamic armies of Mohammed (571-632 AD). Muslim doctrine did not forbid scholars from studying foreign experts no matter their religious persuasion, which fostered an open exchange of ideas across borders.
A ninth century caliph called Al-Ma'mum established in Baghdad a research institute very much like Plato's Academy, known as Bayt Al-Hikma ("House of Wisdom"), dedicated to the preservation and translation of learning from all over the world, securing by treaty the right to borrow renowned works from Greece, Babylonia and even India.
Great Arabian scholars like Abu'l-Hasan al-Uqlidisi (born c.940 AD) created tomes like the famous Kitab al-fusul fi-l-hisab al-Hindii ("The Book of Chapters on Hindu Arithmetic"), amalgamating Hindu arithmetic with Arabic advances in notation to demonstrate the first use of decimal fractions and an astonishingly modern algorithm for solving multiplication problems.
The word algorithm itself is derived a mispronunciation of the name of another man with a fabulously long epithet, Abu Ja'far Muhammad ibn Musa Al-Khwarizmi (c.790-c.850 AD), called by many "the father of algebra." Al-Khwarizmi wrote the great Hisab al-jabr w'al-muqabala ("The Compendious Book on Calculation by Completion and Balancing") written in 830, from which we get the word algebra). The tome detailed solutions to quadratic equations and complex questions of geometry. Another of his works, Al-jabr wal Magabala ("On Algebra") was eventually translated into Latin and would be studied in Europe for centuries.
Let's not forget Abu'l-Hasan Thabit ibn Qurra (836-901 AD) whose family wealth enabled him to journey from his native Turkey to the centre of learning of Baghdad, where he learned too much. When he returned to his hometown of Harran he was accused of evil magic and forced to flee into permanent exile in Baghdad. Thabit is personally responsible for translating and preserving dozens of ancient texts that would have been lost without his efforts -- texts by Euclid, Archimedes, Ptolemy and Eutocius. Thabit worked as a court astronomer and is credited with the development of non-Euclidean geometry and spherical trigonometry.
Also a fan of trigonometry was Mohammad Abu'l-Wafa al'Buzjani (940-998 AD), who translated the works of Diophantus and worked tireless to refine our understanding of the orbit of the holy moon, bringing unprecedented accuracy to the field of astronomy.
Mathematics was imported to the mediaeval Muslim world as an art, and exported as a science.
Perchance to Dream
Like many forms of higher learning, the pursuit of advanced mathematics took a bit of a nap during the Dark Ages. Thanks to the efforts of centres of Islamic academia the learning of the ancients would survive the book-burning pyres in Europe. Armed with efficient systems of notation derived from the Hindus, European scholars would be able to build on the tradition of Greek breakthroughs when the light of the Renaissance finally arrived.
While it is the later, more advanced mathematical constructs (like the calculus or quantum mechanics) that tend to garner much attention, we must sometimes pause to appreciate the foundation upon which they are built: the works of Archimedes and Euclid and Abu'l-Hasan al-Uqlidisi, and the patien tallying of forward-thinking neolithic people whose names we will never know.