I received A History of Mathematics in the post last week, and have been reading through the opening four chapters. I have been interested in etymology since my history tutor outlined in my first year at university that defining a word etymologically can open up a much wider understanding than by researching a dictionary definition. Reading through the opening sections of this volume gives a similar idea, looking at the origin of the development of mathematics provides an overview of the context of our use of number. So far, I have read the first four chapters, and I like the late Prof Carl Boyer’s style and presentation, which was revised by Prof. Uta Merzbach. You can feel the energy, excitement and wonder about numbers in their writing.
The opening chapter begins with a quote from The Book of the Dead,
Did you bring me a man who cannot count his fingers?
It is a chapter which deals with ‘origins’ – the origin of number and the origin of ‘mathematics’. It discusses the initial approach to counting. The use of a hand (four fingers, and a thumb) could have provided us with a base five (5) system of counting. But the use of both hands, this could have provided us with the origin of a base ten (10) system of counting. The base ten of course gives us the familiar decimal system.
If we think about the use of both hands and feet, it may have given us the origin of the vigesimal system (base twenty (20)), perhaps quatre-vingt (French for eighty, literally four-twenty) is a remnant of that system? In the base ten model, anything over ten, would be one or two over ten, i.e. 11, 12, 13 etc.
Whereas, in other developments of counting eight (8) might have be viewed as the dual form of four (cf. Indo-German). While the Latin novem for nine (9), might point to novus (new), in that it was the beginning of a new (novus) sequence.
The opening chapter of the book questions the origin of numbers and wonders were numbers invented initially because of religion ( for instance, due to the construction of temples), or did numbers arise for secular land use. It is conjectured as to which may have came first.
Numbers, we are told, existed long before writing. So it is difficult to point to written fragments/ artifacts to construe the real origin. The chapter also deals with abstraction of types according to sets as to distinguish like with like and different. Finally discussion of geometry comes to the fore, as geometry refers to shapes, whilst arithmetic refers to the ‘adding’ of numbers. Here Boyer conjectures:
The concern of prehistoric man for spatial designs and relationship may have stemmed from his aesthetic feeling and the enjoyment of beauty of form, motives that often actuate the mathematician of today. We would like to think that at least some of the early geometers pursued their work for the sheer joy of doing mathematics, rather than as a practical aid in mensuration; but there are other alternatives. One of these is that geometry, like counting, had an origin in primitive ritualistic practice. The earliest geometric results found in India constituted what were called the Sulvasutras, or ‘rules of the chord’. These were simple relationships [apparently] that were applied in the construction of altars and temples” (Ibid. p. 7).
This practice of ‘rope stretching’ gave rise to the word for surveyors. But it may be the case that surveying arose from practical secular needs, or maybe due to the aesthetic feeling for design and order (p. 7).
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In chapter two (Egypt) – we move to consider written documents that have come down to us form the ‘history of mathematics’. The Egyptians’ approach to mathematics is according to Boyer, based on arithmetic, by doubling the addition of numbers we get a feeling of ‘multiplication’, but it is not true multiplication. Boyer takes us through the use of hieroglyphic notation, and with his help, we can see what each hieroglyph represents. As he outlines:
A single vertical stroke represented a unit, an inverted wicket or heel bone was used for 10, a snare somewhat representing a capital C stood for 100, a lotus flower for 1000, a bent finger for 10,000, a burbot fish resembling a polywog for 100,000, and a kneeling figure (perhaps God of the Unending) for 1,000,000 (p. 10).
For instance, have a look at the ‘Egyptian Mathematics’ page hosted at the University of Chicago:
http://cuip.uchicago.edu/wit/99/teams/egyptmath/numbers.htm
The Egyptians have left behind various documents including the Rhind Papyrus, or the Ahmes Papyrus (after the scribe who wrote it in c. 1650 BCE). This Papyrus uses the hieratic script (‘sacred’) as it was easier to write in hieratic as compared to hieroglyphs. In this work, discussion of fractions is outlined. They seem to indicate practical need for the use of fraction, although,
in some places the scribe seems to have had puzzles, or mathematical recreations in mind (p. 16).
Turning to geometric problems, the Egyptians appeared to solve geometric issues due to the flooding of the Nile. Herodutus in his writings records and holds this to be the reason for the development:
if the river carried away any portion of a man’s lot … the king sent persons to examine, and determine by measure the exact extent of the loss (quoted in Boyer, p. 8)
The Egyptians compared the area of a circle of 9 units diameter, to that of a square of 8 units on the side, and held them to be equivalent. If this is considered in terms of the modern formula for the area of a circular – Pi.R^2, the Egyptian rule leads Pi to be about 3 1/6. Which is a close approximation but not exact. Boyer points out,
here again we miss any hint that Ahmes was aware that the areas of his circle and square were not exactly equal. (p. 17)
The question of the Egyptians being the source of Pythagoras’ theorem also seems to be unfounded (see p. 16, 17).
In concluding the chapter, Boyer highlights that for a long period it was held that the Greeks had taken the rudiments of geometry from the Egyptians. Aristotle had also argued that geometry arose in the Nile Valley due to the leisure of the priests and so with such time they could give to explore mathematics. But Boyer believes that Egyptian geometry:
turns out to have been mainly a branch of applied arithmetic. Where elementary congruence relations enter, the motive seems to be to provide mensurational devices rather than to gain insight. (Ibid. p. 21 ff.)
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So to Mesopotamia – and cuneiform records (c. 4,000 BCE) in chapter three.
In this culture of Ancient ‘Babylon’, the wheel, writing and metals develop. The writing of these civilizations was probably antecedent to the Egyptian hieroglyphs. The sexagesimal system (base 60) which they utilized, still has its remnants in contemporary times. For instance, we use it for ‘time’, and angle measurements.
This system based on 60, allows for a lot of division, we can divide 60 into half, into thirds, fourths, fifths, sixths, tenths, twelfths, fifteenths, twentieths, thirtieths, allowing quite the range of sub-division. We continue to use this system although we are decimal based.
Some other interesting findings in this chapter, with the use of symbols for numbers – this civilization reduced the symbol for numbers to ‘two’ elements, so they did not need a broad range of hieroglyphs for a particular number.
The Mesopotamians did not use zero (0) – but did at times input symbols for spaces between numbers so as to indicate their position in thousands, hundrends, tens, units etc. Sometimes it was still problematic as to know what unit was being referred to. Also they did not use ‘zero’ at the end of their numbers, so it was not a ‘positional system absolutely’ (see Ibid. chapter three).
Chapter three also discusses the development of quadratic equations, cubic equations, Pythagorean triads, polygonal areas, geometry as applied arithmetic’. The question of the practical use of mathematics as compared to the development of mathematics for its own sake comes to the fore again at the end of this chapter. Boyer indicates that:
Pre-Hellenic cultures have been stigmatized […] as entirely utilitarian, with little or no interest in mathematics for its own sake. Here, too, a matter of judgement rather than incontrovertible evidence, is involved. Then, as now, the vast majority of mankind were preoccupied with immediate problems of survival. Leisure was far scarcer [but there were] in Egypt and Babylonia problems that have the earmarks of recreational mathematics.’ (p. 42).
And onwards we go to Ionia and the Pythagoreans!
(*source: Carl B. Boyer, A History of Mathematics, rev. Uta c. Merzbach (Oxford: Wiley, 1991)).
