Topics

• A recent book by David Berlinski, The Advent of the Algorithm (Harcourt, Inc. 2000), begins its account with Leibniz (1646 - 1716), failing to pay even a passing tribute to the long and illustrious history of this procedure. According to the author,

  an algorithm is
  a finite procedure,
  written in a fixed symbolic vocabulary,
  governed by precise instructions,
  moving in discrete steps, 1, 2, 3,...,
  whose execution requires no insight, cleverness,
  intuition, intelligence, or perspicuity,
  and that sooner or later comes to an end.

  In our ethnocentric view of the history of science and mathematics, we tend to think that the only debt we owe is to the Greeks. We study Euclid's geometry in school, paying due homage to this great mathematician, but when we study arithmetics and algebra we fail to recognize the enormously important role that the Arabic/Islamic world played in the development of mathematics. With the end of Greek science a period of stagnation began in Europe, the center of mathematical development being shifted to India, Central Asia and the Arabic countries. [A D Aleksandrov, A N Kolmogoroff, M A Lavrent'ev, Mathematics: Its Contents, Methods, and Meaning, MIT Press, 1956, 1963, Part I, p. 38].
  Fortunately, recent research paints a new picture of the debt that we owe to Arabic/Islamic mathematics. Certainly many of the ideas which were previously thought to have been brilliant new conceptions due to European mathematicians of the sixteenth, seventeenth and eighteenth centuries are now known to have been developed by Arabic/Islamic mathematicians around four centuries earlier. In many respects the mathematics studied today is far closer in style to that of the Arabic/Islamic contribution than to that of the Greeks. [...] The regions from which the 'Arab mathematicians' came was centred on Iran/Iraq but varied with military conquest during the period. At its greatest extent it stretched to the west through Turkey and North Africa to include most of Spain, and to the east as far as the borders of China. The background to the mathematical developments which began in Baghdad around 800 is not well understood. Certainly there was an important influence which came from the Hindu mathematicians whose earlier development of the decimal system and numerals was important. There began a remarkable period of mathematical progress with al-Khwarizmi's work and the translations of Greek texts. [Arabic Mathematics: Forgotten Brilliance?]
• Abu Ja'far Muhammad ibn Musa al-Khwarizmi lived in Baghdad between approximately 780 and 850. His greatest contributions are perhaps to be found in his book Hisab al-jabr w'al-muqabala. From the title, the word algebra entered into the European languages ("al-jabr" literally means "transposition," i.e. the transfer of quantities from one side to the other of an equation), and from his name we now have the word algorithm. It is interesting to note the motivation behind this work. In his own words:

  ...what is easiest and most useful in arithmetic, such as men constantly require in cases of inheritance, legacies, partition, lawsuits, and trade, and in all their dealings with one another, or where the measuring of lands, the digging of canals, geometrical computations, and other objects of various sorts and kinds are concerned.

  Such motivation is likely to have been the motor of most of the ancient history of mathematics. We should of course include astronomy. In fact, most of the Arab/Islamic mathematicians would have called themselves astronomers.
  Particularly for the purposes of our course, it is important to notice the role that computing had in al-Khwarizmi's work He wrote:

  When I consider what people generally want in calculating, I found that it always is a number. I also observed that every number is composed of units, and that any number may be divided into units. Moreover, I found that every number which may be expressed from one to ten, surpasses the preceding by one unit: afterwards the ten is doubled or tripled just as before the units were: thus arise twenty, thirty, etc. until a hundred: then the hundred is doubled and tripled in the same manner as the units and the tens, up to a thousand; ... so forth to the utmost limit of numeration.

• Here is an example of the algorithm used by al-Khwarizmi to solve the equation x² + 10 x = 39. In his own words:

  ... a square and 10 roots are equal to 39 units. The question therefore in this type of equation is about as follows: what is the square which combined with ten of its roots will give a sum total of 39? The manner of solving this type of equation is to take one-half of the roots just mentioned. Now the roots in the problem before us are 10. Therefore take 5, which multiplied by itself gives 25, an amount which you add to 39 giving 64. Having taken then the square root of this which is 8, subtract from it half the roots, 5 leaving 3. The number three therefore represents one root of this square...

  Compare this procedure with the formula you were taught in grade school: x = -(10/2) ± ½ √((10/2)² + 4 ⋅ 39). Notice that al-Khwarizmi writes that the solution represents one root of this equation. He knew that second-degree equations have two roots.
• This all too brief glimpse into the world of Arabic/Islamic mathematicians can not even begin to do justice to their fundamental contributions, and I urge you to read further the beautiful survey offered by the sources quoted above. We will now do an equally brief excursus into the complex world of Mayan mathematics. Once again I suggest you browse through the MacTutor History of Mathematics Archive at the University of St Andrews, Scotland.
• The so-called of the Maya civilisation ranges from about 250 to about 900. The Mayans occupied a fairly broad territory, and the Yucatán peninsula in particular, where the remains of the great city of Tikal can be found. The rulers were astronomer priests who lived in the cities who controlled the people with their religious instructions. Farming with sophisticated raised fields and irrigation systems provided the food to support the population. A common culture, calendar, and mythology held the civilisation together and astronomy played an important part in the religion which underlay the whole life of the people. Of course astronomy and calendar calculations require mathematics and indeed the Maya constructed a very sophisticated number system. [ibidem]

   

  

  The Dresden Codex

   

  The number system used by the Mayans was in base 20. Perhaps this number represented the total of a person's fingers and toes. This conjecture seems to be supported by the observation that the number 5 appears to have a special role. Notice also the presence of zero. Zero is crucial in a positional system, even though the Mayan did not follow it consistently.

   

  

  The Mayan Number System

   

  In the words of L F Rodriguez, quoted in the reference mentioned above,

  The Mayan concern for understanding the cycles of celestial bodies, particularly the Sun, the Moon and Venus, led them to accumulate a large set of highly accurate observations. An important aspect of their cosmology was the search for major cycles, in which the position of several objects repeated.

  It is likely, therefore, that the need to develop and refine their cosmology constituted a powerful incentive for inventing and improving, not only a complex number system, but also a variety of computing techniques, many of which can be found not only in the Dresden Codex, but are also carved on the walls of their temples. And viceversa: the invention and availability of computing techniques islikely to have stimulated renewed interest in a variety of areas. This is certainly true today, and I can not think of any reason why it may have not been generally true also in the past.
• Before concluding this very sketchy review of some of the prehistorical achievements in mathematics and computation, it is essential to be remember that I have barely scratched the surface. The term "prehistory" itself is used rather ambiguously here: Stonehenge is a prehistoric artifact, but the Dresden Codex is not. I have not said anything about China, and generally about the Far East, or about Africa, and so on. I do hope that some of you will investigate these topics in their essay.

 
Readings, Resources and Questions

• Browse the Arabic and Mayan pages in the MacTutor History of Mathematics Archive.
• How do archeologists and historian manage to decipher ancient documents, such the Dresden Codex? How reliable are their interpretations?
• Which areas, other than the ones explicitly mentioned in this lecture, do you think sparked the quest for mathematical symbols and algorithms in ancient cultures?