Looking for the numbers

If you have been reading the posts, you will recognize the following two lists, and probably won’t need to look at them.  But if you are new to this blog, these lists contain the questions and directions of the thinking in writing the posts.

The Questions:

1. What is measurement?

2. How is it done?

3. How did it develop, when and why?

4  Why is it so important: what is the function of measuring?

5. Is it being used appropriately?

The Elements:

  • Measurement is a social activity, in that measurement is not usually for one person only
  • Measurement is done to something (object, process, performance) in order to capture some characteristic of that something, for comparison, communication or replication
  • There must be sufficient language to communicate the measurement of that something’s characteristic(s), not just the words but the concepts behind the words
  • A way to record or capture the measurement of that something beyond language, such as writing, symbolic marks, numerical system, etc.
  • Scales against which to compare the measurement, either previous measurements done in a similar fashion or perhaps, some standards

With this post, we will be looking at the third item in the list of elements, and trying to understand, briefly, where numbers started to show up in human discourse.  This will require some understanding of the history of language, the history of numbers and some of the latest thinking about the way that humans learn their numbers as children.

In an earlier post, A First Look Back, I mentioned three old artifacts, the three ‘decorated’ bones that have early dates: one known as the Lebombo (Lembombo) Bone, dated to approximately 37,000 to 43,000 years ago; the Wolf Bone, dated to approximately 30,000 years ago; and the Ishango Bone, approximately 20,000 to 25,000 years ago.  While these certainly look as if the person or persons incising the bones was counting, there is no way to know if they had “numbers” to match each of the scratches, as in number words, or if they merely put a scratch on without actually counting, as in thinking – well, that’s another one.  The oldest, the Lebombo bone has 29 scratches on it, which has been matched to the lunar cycle and to women’s menstrual cycle.  The Ishango bone (both of these bones are fibulas from baboons), to be discussed a little later in this post, has also been linked to the lunar cycle, which led one researcher, Claudia Zaslavsky, to proclaim “who but a woman keeping track of her cycles would need a lunar calendar?” and “women were undoubtedly the first mathematicians!”1.  That is fine with me: I couldn’t care less who was the first mathematician, though if women were first, they do not seem to have maintained their primary status.

I question how a bone with 29 marks would be used, either with or without words, though.  Yes, I can imagine a woman marking a scratch every day between flows, but that’s good for once.  I’m not sure how it might be used after the first set of markings: was there a thread tied around it that was moved across the next scratch every day?  Actually, the same questions would apply if it had been used to mark lunar cycles regardless of the sex of the user, how do you use it for the second month? And, were words for the numbers used to discuss the meaning of the scratches?

The Wolf Bone (shin bone of a wolf) evidently has 55 to 57 scratches on it: the only interpretation that I’ve found says “…probably a record kept by a hunter of the number of kills to his credit”2, which makes it appear to be a tally stick.  Again, the words “fifty-five” or “fifty-seven” are not necessary to understand a tally.

The Ishango Bone is more complex than these first two since it has three rows of notches.

Row (a) contains four groups of notches with 9, 19, 21 and 11 markings.  In Row (b) there are also four groups, of 19, 17, 13, 11 markings.  Row (c) has eight groups of notches in the following order: 7, 5, 5, 10, 8, 4, 6, 3.  The last pair (6, 3) is spaced closer together, as are (8,4) and (5, 5, 10), suggesting a deliberate arrangement in distinct sub-groups…

…the Ishango Bone appears to have been more than a simple tally.  Certain underlying numerical patterns may be observed within each of the rows…. The markings on rows (a) and (b) each add up to 60: 9 + 19 + 21 + 11 = 60 and 19 + 17 + 13 + 11 = 60, respectively.  Row (b) contains the prime numbers between 10 and 20.  Row (a) is quite consistent with a numeration system based on 10, since the notches are grouped as 20 + 1, 20 – 1, 10 + 1 and 10 – 1.  Finally, row (c), where sub-groups (5,5,10), (8,4) and (6,3) are clearly demarcated, has been interpreted as showing some appreciation of the concept of duplication or multiplying by 2.3

The original archeologist interpreted these markings as possibly being an arithmetical game (1962). Another researcher examined the bone with a microscope some years later (1972), found more marks visible only to the microscope, and concluded that the markings on the bone represented a lunar phase record.

I cannot tell if either of these interpretations is a “just-so” interpretation, but I do not have a better or even an alternative one that I can offer: 3 rows, 2 adding up to 60 and one to 42 for a total of 162?  More, if the microscope is to be believed.  Huh?

The conclusion of Joseph’s account has a wonderful quote: “A single bone may well collapse under the heavy weight of conjectures piled upon it.”4 Amen!

It should be clear that using sticks or bones or even cave walls to make marks for enumerations does not necessarily mean that those making the marks had names for numbers, nor does it mean that they didn’t have names for numbers.  Looking back at the evolution of language, even with the one language proposed by some and mentioned in an earlier post associated with John McWorter’s timeline of 200,000 to 150,000 years ago as a possible beginning, and 35,000 years ago as the beginning of

a cultural explosion which does not appear to have been possible without language, the actual words are not traceable.  The furthest back that linguists have been able to trace languages is to the beginning of writing around 5500 to 6000 years ago, and at best, the pronunciations are uncertain.  It is only with the beginning of writing, though, that number systems appear which seem to need names for numbers.

Number systems most likely have evolved multiple times, appearing or having been captured differently in writing in multiple cultures.  While the vision of a single language branching is nice, there is no certainty that’s what happened, and so the speculation that the names for the first numbers were passed down all through the years, taking on slight changes or great changes as they were used by different people with different language groups is not supportable.  Possible? maybe, but provable, no.  What is clear from the linguists’ work is that every language has at least a few number words, and that as far back as proto-Indo-European has been reconstructed, there were number words up to ten.

What evidence there is shows different “scripts” or representations for the same numbers in the earliest number writing in each of the earliest numbering systems.  (Does it really need to be pointed out that however represented in script or sound, one, the number, has the same value for all systems, likewise, two has the same value, etc.  Regardless of how represented, what in English is called “two” plus the same value equals “four”, however represented.  Manipulation of these values is invariable across systems, regardless of the specific processes.  Probably doesn’t need to be stated?)  The systems on which I will focus were centered in Egypt, Mesopotamia, China, and the Indus Valley.  There were other places where numbering systems were developed, notably in Africa; North, Central and South America; and among Pacific Islanders, all of which led to alternative measuring systems.  These measuring systems are impressive in their own ways and will be discussed in more detail in subsequent posts, but for the current discussion, I want to focus on the writing down of measurements that appears to have occurred in conjunction with the beginning of written languages in these four places, which, because of their geographic locations, could have influenced each other, if not early in their respective histories, perhaps provided some cross-pollenization to each other.

In each of these four cultures, among the earliest written symbols that have been deciphered from each are representations of numbers.  The timelines for each date back to at least 3000 BC, or 5000 years ago.  There are tokens in clay from Mesopotamia that date back as far as 8000 years ago that are probably counters for objects for enumeration or accounting purposes.  In the descriptions that I’ve found, the tokens are described as being found often? sometimes? in “envelopes”, ball shaped containers with markings on the outsides.  From my reading, it is not clear whether all tokens have been found in these envelopes, or only some of the tokens, or whether the envelopes date back to 8000 years ago or only the tokens, and at some point, the envelopes appear.  Regardless of the sequence, the marks on the envelopes seem to be used to guarantee the enclosed counts via tokens.

When the first clay tablets appear, in about 3300 BC, they appear to be a substitute for the tokens and envelopes, and the number of these latter apparently diminishes after about 3000 BC.  The first clay tablets have numbers and symbols or pictograms, but no language: written accounts precede written language in Mesopotamia.  This does not appear to be the case in Egypt, where language in the form of early pictographic hieroglyphs was written from the beginning of the writing, nor does this appear to be the case in the Indus Valley, though there is more uncertainty about that, since the early Indus Valley script has yet to be satisfactorily deciphered.  The earliest of Chinese writing seems to be on oracle bones, and seems to contain both language and numbers, but numbers related to auspicious dates for events to occur.

The early number systems have words for one through ten.  The distinctions come in two forms: the number words and the symbols used to record them.  Groupings of five seem fairly common for the symbols, but symbols for 10, 20, 50, 100 and even larger numbers vary widely in whether they are present and if so, how they are used.  In our language, we have words that are compounds for numbers above thirteen (three-ten), and there are names that do not seem to be compounds, such as eleven and twelve.  Both Chinese and Indian numbers appear to be spoken in a similar manner, though as far as I have found, the Indian numbers do not include the words for the “rank”, so where English and Chinese use the equivalent words of two hundred twenty – two for 222, the Indian words would be two-two-two, with the position in the string indicating the rank.  The numbers that can be written are able to express large numbers (10,000 to 1,000,000) early in their use, which one could not expect without a fairly substantial time of use and evolution.  Unfortunately, there is no way to be more precise.

So, based on my research, it looks as if numbers were being used more than 5000 years ago, and may have been under development any time after the end of the last ice disappeared, roughly 12,000 years ago, which correlates to the time when agriculture developed.  Is one of these important for the development of the other: is agriculture necessary to the development of numbers, or the other way around, is the development of numbers necessary to the growth of agriculture?  Or are they separate developments, joined only by when they both developed?

I suspect that they did rely on each other: the concepts of agriculture would lead to being able to enumerate, whether the amount of grain harvested, the amount of seed needed to for the next crop, how many of us are there that need to be fed, how many goats or sheep or cattle does that mean we need to butcher, how long until the next flooding of the river, when should we commence the planting, etc.  But by the time that human intercourse became complicated enough to begin writing records of their abstractions, among the earliest abstractions to be captured in symbols were numbers and their related concepts.  The proliferation of measurement in the current way that we live has developed from there.

1 Zaslavsky, Claudia (1991b): Women as the first mathematicians, in: Women in Mathematics Education Newsletter, Vol.XIV, No.1, 4 – quoted in: African Mathematical Union Commission on the History of Mathematics in Africa, AMUCHMA-Newsletter – 9, http://www.math.buffalo.edu/mad/AMU/amu_chma_09.html#beginnings

2 Joseph, George Gheverghese, The Crest of the Peacock, Non-European Roots of Mathematics, Princeton University Press, Princeton, N.J., 2000.  P. 24

3 Ibid. P. 24 – 25.

4 Ibid. P. 27.

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