To return to Table 2, found in the previous post, we can see that Indo-European cultures show a strong predilection for the number 3. In the English and Scottish ballads, in German and Russian folktales, the frequency of the number 3 is higher than than of 2. Those of us who grew up on stories of three pigs, three bears, three wishes, not to mention 3 strikes, 3 chances, counting to 3 before dire consequences, and so on do not find this at all surprising. In fact, this heavy emphasis on three pervades Western culture to such a degree that Axel Olrik stated that this was all so because “the organization of nature itself brings it forward” (1909: 140). But this is not true for the other cultures in the sample and this is not true for Dehaene and Mehler’s cross-cultural data. As folklorist Alan Dundes argues, it is just as likely that Western scholars find threes in nature (and in their data) because that is what their culture induces them to look for (1968).
|American board game demonstrating knowledge of the number 5|
For now, we will not worry about whether all of the ones and twoes in the Indus script represent numerals or not. None of the ones in Table 2 fit the profile. The frequency of 1 in each case seems too low. In large part this has to do with the English language. I did not count instances of the words “a” and “an” in the songs or tales, only occurrences of the actual number word “one.” But I read all of these things in English translation and in some of the original languages there is no such distinction. Thus, we should probably disregard the “one” row for all languages here. The “two” row, on the other hand, is very high in frequency for all languages and the fact that the Indus script shows a very high frequency here as well simply seems normal.
When we reach the row for the number four, we see that two columns depart from the expected pattern of declining frequency. Both of these are Native American, the Lakota (or Sioux) and the Navaho. Folktales of these and many other Native Americans abound with examples of four, as Dundes notes. Seven is the favored number in the Canaanite tales, as it probably would be in the Bible if I were to analyze that lengthy book. Consider the creation in seven days, seven apiece of the clean animals on the ark with Noah (versus the more familiar two each of the unclean), Egypt’s seven good years followed by seven lean years, Hebrew slaves freed after seven years, Jacob’s seven-year bride service, Jesus’ admonishment to forgive not just seven times but seven times seven times, etc.
In fact, I’ve often wondered whether the strong resemblance between the Indo-European words for “seven” and the Afro-Asiatic words for the same number – especially the Semitic words – cannot be traced to some early period of contact between these two language families (and throw “six” in, too, for good measure). This might be one of those areal characteristics, a subfield of research so despised in linguistics it might be considered the stinky armpit (as compared to historical linguistics, with its long, glorious, and nationalist upbringing). But I don’t want to press that feeble argument. Just make a note of the fact that 7 also makes a strong showing in the English and Scottish ballads, no doubt due to its Biblical presence. This strong showing generally fades out when the same ballads migrate to America. The sevens fade into threes, which are much easier to rhyme (three and me, thee, see, glee, fee, free, be, he, etc.). But before we leave the sea of sevens, note that the people of the Indus Valley also liked 7. This is the one apparent numeral that really stands out in the Indus column. It alone violates the rule of steadily decreasing frequency with increasing magnitude.
In China, 5 is considered the perfect number and the dictionary has a great list of things that come in fives. There are five elements, five virtues, five metals, five classics, five blessings, five grains, five senses, five social relations (all male), five colors, five viscera, five poisons, five flavors, five continents, five tones in the Mandarin dialect (although Mandarin speakers almost never use the fifth one), and even five directions because they count the center (Fenn and Tseng 1940: 607-608). For this reason, I fully expected the Japanese tales to be chock full of fives also. Didn't they get most of their culture from China? But no, to my astonishment, four and seven were more frequent than five. Seven was the one that really stood out, breaking the expected pattern of declining frequency.
|Egyptian board game (Senet) demonstrating fear of the number 4|
What is especially surprising about this is that in Japanese, both 4 and 7 are unlucky numbers. Still, they are only bad luck in the forms borrowed from China, where they are homophonous (or nearly so in the case of 7) with the word for “death.” Because of this scary connection (which should ultimately be Chinese), when the Japanese count with the Chinese loan words, they switch into their native Japanese words for 4, 7, 14, 17, 40, 70, 400, and 700 (Association for Japanese-Language Teaching 1984: 26 and 48). You can’t be too careful!
(In case you want to know what not to say, the Japanese loan word for 4 is shi, so don’t ever say that! The native word is yon or yottsu, but we don’t have time to discuss when to use which form. It gets complicated. And besides I’ve forgotten the details. I studied Japanese a very long time ago and probably contracted a little Mad Cow while in England in the meantime. The corresponding loan for 7 is shichi, and don’t you ever let me catch you saying that either. The native word is nana or nanatsu. You can say those words all you like. The unlucky Chinese numbers are si4 and qi1 respectively. The word for “death” -- which you must promise not to read aloud – is si3 in Chinese, shi in Japanese. Note that the bad luck must be worse in Japan than in China because “death” is exactly the same as the Japanese loan word for “four,” while it’s at least a different tone in Chinese. The Chinese “seven” does not sound like “death” at all, so right ahead and say it. No, I take that back. There's a nasty homophone, “grief” qi1, something the Japanese apparently don’t know about. Still, bad luck is bad luck. One mustn’t quibble about its source. My Japanese professor was adamant about that!)
As an aside here, let us note that the Yurok, also Native Americans but in California and so over the mountains from the folks who were so fond of 4, shared the Far Eastern regard for 5 from way back. In other words, they totally ignored their American neighbors’ love of 4 and did their own thing, setting a precedent that Californians follow to this day. There is a moral in this: there is no cultural law that says one must do something just because one’s neighbors do it. That should be on a poster in fluorescent letters. We might even need to nail that poster to a door somewhere.
Why is that? Because one of the oft-repeated arguments for the existence of a fully developed writing system in the Indus Valley rests on this very Cultural Law. The Harappans’ neighbors had a fully developed writing system, you see. That means the Sumerians and Babylonians, possibly the Egyptians, although we don't necessarily want to talk about them (it depends on the season). People who make this argument forget, of course, that the Harappans had neighbors in other directions (north, south, east) who lacked such writing systems, but we shan’t talk about that, shall we?
Anyway, the argument continues, having observed their western neighbors writing, writing, writing, and noting how wondrously useful this writing was, the Harappans could not have failed to adopt such a system themselves! I dare say I have often done things that my neighbors have not done. I have also failed to do things that my neighbors have done, because I thought my neighbors were stark raving loony. Societies can be the same. It is entirely possible for one society to fail to copy the practices of neighboring societies. It actually happens all the time, even when it would seem to make good sense to some outside observer (because sometimes people don't give a flip what some outside observer thinks). Some societies are quite open to cultural influences from outside. Some societies are quite closed to outside influences. Most are somewhere in between. All in all, this is a very silly argument. Now, let’s move on.
Some Harappan inscriptions contain only “numerals,” e.g., H-370, SEVEN POSTS. These strokes appear on a pot shard where I take them to be the long strokes or “posts.” Another viewer might take them to be the short strokes or “quotes.” Where there are no other symbols, one cannot prove it either way. However one classifies the symbol, it seems a bit odd to have nothing but numerals. If the seven strokes mean “seven,” the symbol still leaves one wondering, “Seven what?”
Nevertheless, we know of similar inscriptions from the Near East. Many of the earliest clay tablets from southern Iraq and from neighboring Iran (later known as Elam) are of an economic nature. They can often be read as essentially of this type, six of something, 30 of something, 25 of something. One cannot always tell exactly what is being enumerated. Still, usually the type of numeral gives a clue to the general nature of the item being enumerated in the Near East (Schmandt-Besserat 1992 & 1996). To find parallels to this extremely vague type of record, nothing but strokes, one has to go farther back in time or farther afield. This type of record most resembles that on tally sticks on which a simple mark is made to enumerate each item or person (Menninger 1969: 226-229).
But the single “numeral” that appears by itself is not the only oddity in the Harappan script. There is also the peculiar inscription H-303, which contains an unusual STACKED SEVEN comprised of two short strokes over three over two, in which the lowest strokes are diagonal. The following sign is the TWO POSTS. After these two apparent numerals there are two common terminals, POT and COMB (the latter showing six diagonal teeth). One must ask why one inscription would contain two numerals next to each other, apparently 7 and 2. The first thing that apologists for the script as a full writing system want to suggest is that one type of “numeral” represents ones and the other type represents tens and this is like our place value notation. Thus, this would be read as either 72 or 27, depending on whether the short strokes represent ones or tens. However, we must remember that these inscriptions date to the Bronze Age, after all, a time when our type of place value notation had not yet developed.
Something akin to place value notation appears in ancient scripts where distinct types of symbols were used for numerals of different values. For example, the Egyptians used the upside-down “U” shape to represent tens. This symbol, like so many others in the hieroglyphic system, is based on the rebus principle. The original glyph represented a hobble for cattle, used as a determinative for the word for a cattle stall, mdt. This word sounded like the word for “ten” in Egyptian, mdw, so the same symbol was used in writing out the number word phonetically. Eventually the glyph was used as an abbreviation for the whole word, in essence as the numeral 10. At this point, vertical strokes representing the ones and medjes, representing tens, appear next to one another. Other symbols, derived in a similar manner, appear in the same line to represent hundreds, thousands, and so on.
Chinese numerals, on the other hand, closely reflect speech. The first three numbers (1, 2, 3) are still written with strokes, although these are horizontal rather than vertical in normal writing. From “four” on, the characters are more complex. The number 11 is written as it is spoken, shi2-yi1, that is, “ten-one.” “Twelve” is shi2-er4 or “ten-two.” “Twenty,” on the other hand, is er4-shi2 or “two-ten(s)” (since Chinese does not mark the plural for nouns, the number word does not need to change, either). There is a symbol for hundred, one for thousand, one for ten thousand, and one for hundred thousand. These all existed long ago, although I am not sufficiently familiar with the history of Chinese writing to say when each arose. There was no symbol for zero in this system, originally, or in Egyptian hieroglyphs. In either system, when writing a number such as 102, the symbol(s) for “two” appears immediately alongside that for “hundred.” In both systems, there is a distinction between the way 7 is written and 70. So it isn’t place value notation in either Egyptian hieroglyphs or Chinese. It is not very likely to be place value notation in the Indus script either.
Another objection to separating long and short strokes into tens and one in Harappan is that these types of signs do not appear for all of the numerals. Fairservis noted that the frequency drops precipitously for both “posts” and “quotes” after 7, concluding from this that the Harappans must have had a base-8 number system (1992: 62). Neither of these hypotheses fits well with some of the facts. Although not as common as the lower “numerals,” there are instances of an apparent “eight,” some of “nine,” some reports of “ten” (not observed by all scholars), and many occurrences of “twelve” (counted in various ways by the different authors). In a base-8 system, we would expect the tally marks to end at “seven,” with a new symbol for “eight,” in the Egyptian manner (i.e., the medj appears at “ten”). But this is not what happens.
Fairservis suggests that the OVERLAPPING CIRCLES sign represents “eight” (1992: 62). With some hedging, he makes the MALLET (a square surmounted by a vertical line) “nine,” and the item I term the BED (shaped like an “H” with extra feet and two horizontal lines) “ten,” sometimes with a SINGLE POST attached by a short backslash on the left (1992: 63-64). “Eleven” becomes QUOTES IN OVERLAPPING CIRCLES, which he sees as a combination of THREE QUOTES and the symbol for “eight” (1992: 65). While his description confuses me, he seems to be saying that the MAN HOLDING DEE-SLASH (or a man with a bow and arrow) is “twelve” (1992:-65; see also 155). Alternatively, he may only be claiming that these last symbols – which are clearly not tally marks – are numerical only with reference to the calendar. That is, OVERLAPPING CIRCLES is not “eight” but “eighth month.”
Fairservis also suggests that the Indus “fingernail” is a numeral, by which he means the symbols I term CEE/BACK CEE and ROOF (1992: 67-69). He cites Chinese: “when two strokes follow the sign for ten they equal the number 12, when they precede the sign this indicates multiplication and equals 20” (1992: 69). As noted earlier, Chinese numerals are characters just like other, based on the language, which is not quite the same thing as either place value notation or true addition and multiplication.
Fairservis reads the Indus inscriptions from right to left, as do most others, seeing the following uses of the “fingernail” symbol (1992: 69):
(1) ) )) ))) )))) multiples of crescents up through 7
(2) ||^ crescent plus number up through 7 [where the chevron represents the ROOF]
(3) |||^/^ crescent added to crescent plus number to an unknown limit [^/^ representing the ROOF stacked upon another ROOF, which I cannot enter on the keyboard]
(4) ^||| number plus crescent
Examples of (2) include H-802, H-804, and C-306. Examples of (4) include M-923 and M-925. Stacked crescents alone occur on K-302 (1?). Crescent shapes do appear on pot shards, alone, where their orientation (CEE? ROOF? CUP?) is anyone’s guess. Since they also occur on pots side by side and touching, in a way reminiscent of a simple drawing of a flying bird (as very low, rounded “M” or “W” shapes), this may not be the same symbol as the very deep upside-down “U” shape found on copper objects.
In addition to the difficulties of interpreting what I term the ROOF (or “fingernail marking”), the oddities of frequency distribution of the “numerals,” and the possibility that some other Indus signs could be numerals, there is the odd STACKED TWELVE. This apparent numeral only appears in this one form, four short strokes over four over four. What does Fairservis make of this? Since he posits a base-8 number system, it cannot mean “twelve” or it ruins his hypothesis. So, he says it means “water,” more specifically in the form of rain (1992: 71). He notes the similarity between the three rows of four strokes in this sign and the three rows of three strokes in one form of the STACKED NINE, plus one form of the STACKED EIGHT (containing strokes in the configuration 2 x 3 x 3). These are all water, he says, the last two being on the ground in the form of streams and rivers rather than falling from the sky (op cit.).
This explanation also handily deals with the STACKED NINE UNDER TABLE. This last sign happily resembles the Egyptian glyph of the sky with rain lines coming down and the Chinese symbol for rain, a horizontal line over something similar to the Indus TABLE bisected by a vertical line. Between the vertical line and the sides of the “table,” there are short strokes, two on each side. Fairservis has drawn his Chinese rain rather inaccurately, making these short strokes diagonal in the wrong directions, it seems to me. The rain lines on the Egyptian glyph seem uncharacteristically short and the sky itself anemically thin. There is a third symbol, too, blandly labeled “American Southwest” which covers a lot of territory. Since I live in part of this Southwest, I find myself wondering where he saw this symbol since I don’t recognize it. It doesn’t look Navaho and I haven’t seen it in the collection of Texas rock art or the collection of rock art from Nevada and California. That would seem to leave New Mexico and Arizona. Or was he, perhaps, thinking of Oklahoma? This is one of those times when the word “universal” seems to be implied. Since “everybody” depicts rain like this, why shouldn’t the Harappans have done so? It’s so obvious!
Except that it isn’t quite as obvious as all that, the symbols aren’t quite right, the universe is actually a bit more varied than that, and the Harappans just might not have intended anything wet by stacking twelve little strokes in that way. They just might have meant “twelve” something. Twelve is one of those numbers that makes a spike in Dehaene and Mehler’s chart. It would in mine if I had included it all those years ago, but I chose to stop at “ten” because at the time I was studying toddlers. They were stopping at 10 so I did too (and counting occurrences of number words is a dreadfully tedious business, so I haven’t been moved to repeat the study, even though I still have most of the sources). What a pity! But I rather suspect that a stack of twelve little strokes did not mean quite the same thing as a stack of nine or eight little strokes. It’s only a suspicion, but the stacks don’t look the same. The forms of 8 seem to vary more than most other apparent numerals, so perhaps some of these have meanings having to do with water. But it’s hard to believe that the “twelve” is water. It might be calendrical. It might be something else though. There were Twelve Tribes in ancient Israel. They might be signs of an ancient zodiac, deities on a Harappan Olympus, elders in a council who decided the fate of a person accused of a crime (like our modern jury).
When it comes to determining the meaning of non-enumerative “numerals,” we can only guess, as I was just doing with the “twelve.” We do have the anthropological practice of using ethnographic parallels as one possibility. This cannot prove anything but it can give us clues. “Two” is something obvious in the natural world because of bilateral symmetry. We have two eyes, two ears, two nostrils, two arms, and two legs. My own toddlers loved to play a little game in which I asked them to point out a body part or facial feature. I might say, “Where’s Mama’s ear?” Boobeleh points to one of Mama’s ears. I then ask, “Where’s Mama’s knee?” But Boobeleh is not listening. She is bound and determined to point out Mama’s other ear. This behavior, which is contrary to Mama’s planning and direction and which is entirely consistent from one time to the next, demonstrates Boobeleh’s concept of “two” prior to the onset of speech. In the same way, a year later, I ask little Bubba, “Where’s Mama’s nose?” He not only points, he runs his thumb up one of Mama’s nostrils. I quickly ask, “Where’s Mama’s toe?” But, like his sister a year earlier, he will not be distracted from the very important task of running his grubby little thumb up Mama’s other nostril. By the time he’s done this, chances are Boobeleh will have grabbed his favorite toy and the body parts game will have come to an abrupt end with a bout of screaming. Nevertheless, Bubba will have satisfactorily demonstrated that he, too, has a prelinguistic concept of the number two.
In the social world, we find pairs such as male and female, day and night, hot and cold, wet and dry, life and death, domestic animals versus wild animals, and many other such oppositions. In the Far East, many such oppositions are systematized into the general two-fold division of Yin and Yang. Yang represents the male principle, which is active, hot, and light. Yin represents the female principle, which is passive, cool, and dark. But where other cultures added value judgments to such divisions, making one good and one evil, the easterner sees the two as complementary. In addition, each contains a seed of the other within itself. One wonders whether the Harappans had a binary view of the world and, if so, whether they had yet some other way of handling such oppositions.
The significance of three appears to be due to what is called subitizing. It is the largest number that we can accurately grasp at a glance, without counting. If you have a toddler handy, you can demonstrate this with a little experiment. Put three strawberries in a dish for the tot, or some other highly coveted item. Put your pointing finger right in front of your boobeleh’s face. It has to be right there or the highly coveted item will prevent boobeleh from noticing your existence. “Look!” you cry, staring off into space, “a baby wolf!” When boobeleh looks away from the strawberries, you surreptitiously swipe one, hiding it. When boobeleh looks back at the dish, what is the reaction? My boobeleh notices immediately and raises the roof. But let there be more strawberries and boobeleh is only suspicious, staring for a long time at the dish, looking over at me, back at the dish, back at me, repeating this motion several times. There is no screaming if the number of strawberries is originally four or more. If the number of strawberries is originally more than six or so, she registers no reaction and simply starts eating. She hasn’t noticed anything is amiss. Babies seem to be born with this subitizing ability, many animals possess it to one degree or another, and even birds and some insects share it (Dehaene 1997).
Four is often considered the number of cardinal directions, north, south, east, and west (though not necessarily stated in that order). I say “often” because the Chinese add the center to this, the Mongols sometimes add “up” and “down” and other peoples may add still other directions. The Navaho myth which I read is filled with fours, all of which go back to these four directions, based on the rising of the sun in the east and the setting of the sun in the west. The four points of the Lakota are slightly different, focusing on the solstice points. Thus, although the number is the same, the actual directions are skewed to the northwest or southeast, compared to those of the Navaho. In any case, compass directions of one type or the other may underlie the presence of the apparent numeral 4 in the Indus script. This is an attractive hypothesis for the short strokes placed in four corners around another sign, one common type of ligature termed “caging.”
As seen in Table 2, 5 is the culturally significant number for the Yurok. Of course, this obviously relates to the number of digits on the average human hand and foot. In some languages the word for “five” actually means “hand.” For example, Hawaiian lima means both “five” and “hand” (Judd, Pukui, and Stokes 1945: 94 and 104). This is generally true in languages of Polynesia. Unfortunately, I know no Yurok, so I cannot say whether it is the case in that language as well. A drawing of a hand might readily be used for the numeral 5 in such a language. However, I do not know of any examples of this use of the hand unless the hand stencils in the Cosquer cave prove to be such.
It seems conceivable that a few of the Indus signs might be intended to replicate either the hand or foot with such an idea in mind. There is one which resembles a fringed flag. It might be intended as a squared-off image of a hand. The number of lines is correct for fingers, although the post which holds it up is rather long to be a thumb. The POTTED THREE has the correct number of strokes to be a foot, if one considers the outside strokes to include the big and little toes. Still, I would not want to press either of these identifications. Calling the occurrences of 5 POSTS /5 QUOTES/ STACKED 5 “hand” is probably going too far as it is.
Parpola favors an interpretation of STACKED SIX / FISH as referring to the Pleiades (2009: 275). STACKED SEVEN / FISH is the Big Dipper (Ursa Major, known as the Seven Sages in India). If there were some proof of the proposed identification of the “fish” sign as representing a star and/or deity, this would go far toward supporting the hypothesis that the Harappans spoke a Dravidian language. But there is no proof. While the “fish” resembles the Christian ICHTHYS symbol, it also resembles the proto-cuneiform beer container and seems to stand on its back fins unlike a real fish, even where it is not crowded by other symbols. It might actually be fat person, a pregnant woman or a big man, not a fish at all. Those 6 FISH and 7 FISH inscriptions might actually refer to merchants from the Near East who were excessively fond of sixes and sevens. The Harappans might even have picked up the words for six or seven from those neighbors who are supposed to have given them the idea of writing in the first place, the folks in the Near East.
“Eight” normally appears stacked as four over four, so it might not really mean 8 at all. It could refer to two of whatever “four” means. The Navaho tale is full of this sort of thing. Each direction is associated with a yei (supernatural being) or a pair of beings. There might have been a god and goddess of the north, a god and goddess of the south, such a divine pair for the east and another divine pair for the west. This well known group of eight deities might be symbolized by the STACKED EIGHT. This apparent numeral has more than one form, though, so there might be more than one meaning.
“Nine” shows up in more than one form, but the “stacked” type is the most common. This is usually five short strokes over four, also appearing as three over three over three (all slanting in a form which Fairservis considered a stream or river). We should take time out here to note that in Russian fairy tales 3 is so popular that its multiples also make little spikes, overcoming the steadily downward sloping trend of the graph (which unfortunately I haven’t figured out how to show you). This occurs because in the ever popular story starring three brothers, each brother has three difficult tasks and when brother number three – always the youngest – gets his turn, he has to kill a three-headed dragon, then a six-headed one, and finally a nine-headed one, or some such. The multiples of three appear in similar ways repeatedly. This predilection for the multiples of 3 as well as 3 itself was even more pronounced in earlier folklore among Indo-Europeans, as part of my study showed (not shown here). Nines are very big in the Iliad, for example, where they greatly outshine the measly sevens. As a mere footnote, we’ll note in passing that the Mongols also were excessively fond of threes and nines. Genghis Khan’s dad was named “Nine” (Yesugei) in Mongolian as were two of his wives (Yesui and Yesugen) also in Mongolian, and one of his daughters-in-law in Turkish (Tokuz) (Kahn 1998: 206).
When we reach 10, which ought to rise to in prominence, we find it hardly present at all. It is rare as hen’s teeth as my grandma would say, so rare that some of those who list the Indus signs don’t list it. When it does shows up in a list, it is in stacked form, five strokes over five. But in the folklore table, those who like fives also like tens. Not a surprise.
Eleven appears nowhere in the folklore table and it does not appear in the Indus script. This is interesting. The next number, 12, is big in the Dehaene and Mehler table, in folklore, and in the Indus script. The fact that there are twelve lunar months in a solar year (more or less) probably has a lot to do with this, but it isn’t the whole story. As we saw in part I of the story of 3, not everyone follows a lunar calendar with twelve months. The Mixtecs did not. They had other mystical, magical numbers.
What can we glean from all this? I have wandered hither and yon excessively, I realize. I have been pondering the nature of numbers for a long time. My master’s thesis focused on the acquisition of language in the small child, so I shall end this with a little conversation between such a small child and myself, on the nature of a particular number. It may or may not be instructive.
Bubby: What’s the last number?
Mama: There is no last number. Numbers go on and on forever. That’s called infinity.
Bubby: Do they go all day?
Bubby: Can you count to infinity?
Mama: No, nobody can.
Bubby: Not even Papa?
Mama: No, nobody can.
Bubby: What’s after infinity?
Mama: Nothing comes after infinity.
Bubby: Then infinity is the last number.
Mama: No, the numbers just go on and on...that’s infinity.
Bubby: Then does it go infinity-one, infinity-two, infinity-three...?
Here’s to infinity-three!