The History of Mathematics
Instructor: G. Donald Allen
Spring, 1997

The Origins of Mathematics

{The origins of mathematics accompanied the evolution of social systems. Many, many social needs require

• counting,
• calculations,
• measurement

• The worth of a herdsman cannot be known unless some basics facts of counting are known.
• An temple cannot be built unless certain facts about triangles, squares, and volumes are known.
• An inheritance cannot be distributed unless certain facts about division (fractions) are known.

From practical needs such as these, mathematics was born.

One view is that the core of early mathematics is based upon two simple questions.

• How many?
• How much?

This is the cardinal number viewpoint.

Ordinals

Another view is that mathematics may have an even earlier basis on ordinals used perhaps for rituals in religious practices or simply the pecking order for eating the fresh game. Such basic questions are thus:

• Who is first, etc?
• What comes first, etc?

We will take the cardinal numbers viewpoint in the following.

HOW MANY?

As indicated earlier, as society formed and organized, the need to express quantity emerged. Even at this early level, perhaps as early as 250,000 years ago, there must have begun a transition from sameness to similarity of numbers.

one wolf one sheep
two dogs wolf two rabbits
five warriors five spears

This abstraction of the concept of number was a major step toward modern mathematics.

HOW MANY?

From artifacts even more than 5,000 years old, notches on bones have been noted. Were these to count seasons, kills, children? We don't know. But the need to denote quantity must have been significant.

The English language, as others, has quantifier to indicate plurality

school of fish

pack of wolves

flock of geese

HOW MANY?

Other examples of counting and enumerations reveal just how enumeration began and proceeded.

1. The Indians of the Tamanaca on the Orinoco River.
   

   HOW MANY?

2. The Dammara tribe in Africa (19th century). Trading of tobacco sticks for sheep. The tradesman knew the equivalence:

   2 sticks = 1 sheep
   However, he was unable to cipher correctly the formula:

   4 sticks = 2 sheep
   So, at the very early stage of counting numerical equivalences there is no such fact as two times two equals four.

3. Certain Australian aboriginal trives counted to two only, with any number larger than two called simply as much or many.

4. Other South American Indians on the tributaries of the Amazon were equally lacking in number words. They count count to six, but had no words for three, four, five, or six. For example, three was expressed as two-one.

5. The bushmen of Africa could count to ten with just two words.

   ten=2+2+2+2+2+2

   For larger numbers the descriptive phrases became too long.

   Even earlier records

6. The earliest records of counting do not come from words but from physical evidence -- scratches on sticks or stones. Old stone age peoples had devised a system of tallyign by groups as early as 30,000 B.C. There is an example of the shinbone from a young wolf found in Czechoslovakia in 1937. It is about 7 inches long, and is engraved with 55 deeply cut notches, of about equal length, arranged in groups of 5. (Modern systems!!!)

7. There is other evidence dating from 8500 B.C on the shores of Lake Edward (in the Queen Elizabeth National Park in Uganda). An incised bone fossil contains groups of notches in three definite columns. Odd and unbalanced, it does not appear decorative. One set of is arranged in groups of 11, 21, 19 and 9 notches. Another is arranged in groups of 3, 6, 4, 8, 10, 5, 5, 7 notches. Many have conjectured on the meaning of these groups. (Lunar months, doubling, halving, ...)

Some etymology.

• The word tally comes from the French verb tailler, ``to cut", like the English word tailor. The root Latin word is tailler, talea, ``to cut". Note also the English word write comes from the Anglo-saxon writan, ``to scratch".

• Our word calculate comes from the Latin calculus, pebble.
• The English thrice, just like the Latic ter, has the double meaning: three times, and many. There is a plausible connection between the Latin tres, three, and trans, beyond. The same can be said regarding the French trés, very, and trois, three.
• Some primitive languages have words for every color but have no word for color. Others have all the number words but no word for number. The same is true for other words.

• English is very rich in native expressions for types of collections: flock, herd, set, lot, bunch, to name a few. Yet the words collection and aggregate are of foreign extraction. From Bertrand Russell we have the quote, ``It must have required many ages to discover that a brace of pheasants and a couple of days were both instances of the number two." Today we have many terms to describe the idea of two: couple, set, team, twin, brace, etc.

Tallying

Tally sticks have been used since the beginning of counting. But it was not limited to ``primitive" peoples. The acceptance of tally sticks as promissory notes or bills of exchange reached all levels of development in the British Exchequer tallies. (12 century onwards.) It took an act of parliament in 1846 to abolish the practice.

An anecdote: The double tally stick was used by the Bank of England. If someone lent the Bank money, the amount was cut on a stick and the stick was then cut in half. The piece retained by the Bank was called the foil, and the other half was called the stock. It was the receipt issued by the Bank. The holder of said became a ``stockholder" and owned ``bank stock". When the holder would return the stock was carefully checked agained the foil; if they agreed, the owner would be paid the correct amount in kind or currency. A written certificate that was presented for remittance and checked against its security later became a ``check".

Tallying on a bone or stick is both ancient and modern. A more ancient form of counting was done by means of knots tied in a cord -- though counting is carried out to this day by knots or beads. Both objects and days were so tallied. From King Darius of Persia, we have this command given to the Ionians:

The King took a leather thong and tying sixty knots in it called together the Ionian tyrants and spoke thus to them: ``Untie every day one of the knots; if I do not return before the last day to which the knots will hold out, then leave your station and return to your several homes."

Knotted cords, called quipus were also used by the Incas of Peru. The conquering Spaniards noted that each village and an official of the knots, who maintained complex accounts on knotted cords of several colors and thicknesses, and performed a function similar to today's city treasurer.

HOW MANY?

Systems of enumeration.

Primitive:

notches, sticks, stones

Egyptians:

symbols for 1, 10, 100, 1,000, ... 1,0000,000.

Babylonians:

two symbols only--cunieform

Greeks:

alphabetical denotations, plus special symbols

Roman:

Roman numerals, I,V,X,L,C,D,M..

Arabs:

Ten special symbols for numbers.

Modern:

Ten special symbols for numbers.

Methods of ciphering.

Devices:

Abacus, counting boards.

Symbolic:

Arithmetic.

HOW MANY?

Bases for numbering systems

• binary -- early
• ternary -- early
• quinary -- early
• decimal
• vigesimal
• sexigesimal
• combinations of several

A study among American Indians showed that about one third used a decimal scheme; one third used a quinary/decimal scheme; fewer than a third used a binary scheme; and about one fifth used a vegesimal system. and a ternary scheme was used by only one percent.

HOW MUCH?

When counting or asking how many, we can limit discussions to whole positive integers. When asking how much, integers no longer suffice. Examples:

Given 17 seedlings, how can they be planted in five rows?

Given 20 talons of gold, how can they be distribution to three persons?

Given 12 pounds of salt, how can it be divided into five equal containers?

When asking how much we are led directly to the need of fractions.

HOW MUCH?

Another how much question is connected with measurement.
Where?

• Construction. To build graneries, or ovens to bake bread, or pyramids, or temples we need formulas for quantity, or area or volume.
• Planting. To divide arable plots we need formulas for plane area and those for seasons.
• Astronomy. To study the motions of stars we need angular and temporal measurment.
• Taxes and commerce. To properly assess taxes, we need ways to compute percentages (fractions).

To consider questions of how much we need more advanced numbers and arithmetic; we also need concepts of geometry.

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