27 Mar 2009

Pre-History and Early History of Digital Computation in Eves' An Introduction to the History of Mathematics

by Corry Shores
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[My hypothesis is that Deleuze's new image of thought can offer a new image of artificial intelligence. The following is part of that investigation.]

Howard Eves

An Introduction to the History of Mathematics

Selections from chapter 1 regarding the history of digital computation

Eves begins "with the emergence in primitive man of the concept of number and the process of counting." (9c)

Early Counting

Archaeological evidence shows that humans used counting as least as far back as 50,000 years ago. So it appeared long before recorded history. However, we can make some assumptions about how counting developed.

Primitive humans' earliest number sense was probably "recognizing more and less when some objects were added to or taken from a small group, for studies have shown that some animals possess such a sense." As societies evolved, counting developed. Tribes needed to know the number of its own members in comparison to enemy tribes. Most likely the earliest counting method was tallying, which draws one-to-one correspondences. (9d) Fingers and stick markings have long been common tally techniques. Counts could be preserved by using more enduring representations such as pebbles. Soon writing would allow for symbols to keep the count.

In the early stages of counting, qualitative differences were distinguished between quantities. [First take note of Bergson's theory that we experience numbers primarily as qualities: §75 and §77]. So consider all the words in English for two, used for qualitatively different things that are quantitatively the same: a team of horses, a span of mules, a yoke of oxen, a brace of partridge, a pair of shoes, a couple of days. It was not until much later that humans abstracted the common property two as a representation of an abstract quantity independent of any association to concrete qualitatively-distinguished objects. (11d)

Larger quantities required more systematic methods. Consider our decimal system. We have ten digits,
0 1 2 3 4 5 6 7 8 9.
When you add one to nine, then we begin making combinations with the given ten digits. This is the carry operation. Ten is considered the base or radix. So the decimal system is base ten.

This is probably from the fact that we have ten fingers. Philologists tell us that the word eleven comes from "ein lifon," which means "one left over." Twelve comes from "twe lif" or "two over ten." And thirteen "three and ten;" fourteen "four and ten." Twenty comes from "twe-tig," or "two tens." Also, hundred comes from "ten times." (12b)

Early Numerical Representation

These names were not original assigned to written symbols. More likely they were finger symbols. Hence the word "digit." (13b) [image credits listed below. click on image for an enlargement.]

Later the hand signs were assigned to written symbols or "numerals." (14c)

There are many ways that cultures have grouped their symbols. Our Hindu-Arabic numeral system uses positions. Such a positional numeral system selects a base number of digits. In our case ten. (19c)

Consider the number 52493. We could present it as:

5(10­­4) + 2(103) + 4(102) + 9(10­­1) + 3(10­­0)

So the formulation for number N would be:

N = anbn + an-1bn-1 + . . . + a2b2 + a1b + a0

where 0 ai < b, i = 0, 1, . . . n.

Then, we represent number N to the base b as a sequence of symbols:

anan-1 . . . a2a1a0


Early Computation

Most of the computing patterns we use today, such as long multiplication and division, were not developed until the fifteenth century. It took so long mostly on account of physical obstacles. For long there was not a plentiful and convenient writing method. These difficulties were overcome with the invention of the abacus (Greek abax, "sand tray."). (21c.d) It is considered the earliest mechanical computing device. We use objects to signify the quantities for each decimal place. When we add enough values to take us beyond ten, then we "carry over" that value as just one token in the next decimal place [we later will better detail the abacus.]

Our decimal numerals came from India through Arabia. Exactly when they arrived in Europe is not clear. But it seems to have been introduced into Spain by the Arabs when they invaded in 711 A.D, and the system was widely known by the twelfth century.

Then for the next 400 years there was the battle between the abacists and the algorists. (24d)

Around 1500 A.D., the algorists won supremacy. By the 18th century, there was almost no trace of the abacus to be found in Western Europe. (21-25)


Eves, Howard. An Introduction to the History of Mathematics.London: Brooks/Cole - Thomson Learning, 1990.

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