Friday, October 15, 2010

A Handful of Harappan Fives: Apparent Numerals in the Indus Script

The first of the apparent numerals for “five” in the Indus script is made with five short vertical strokes set side by side.  I term this FIVE QUOTES and enumerate it V1.  It appears in the lists of other authors as KP125(a), W202, and Fs O-6.  That is, the last of these is actually for FIVE POSTS, as Fairservis does not list FIVE QUOTES separately.  Wells gives the frequency of this apparent numeral as 21, with 15 occurring at Mohenjo daro, three at Harappa, two at Lothal, and one at Kalibangan.

Proto-Elamite tablet with wedge-shaped numerals, including stacked six (handmade replica).

Both proto-cuneiform and proto-Elamite include true numerals made with five wedge-shaped marks in a horizontal row.  For proto-Elamite, the coding for five of these is M379~d.  But the use of numerical symbols was complex in both cases, with distinct types used for different classes of commodities rather than a single numeral for all items.  This is unlike modern numerals.  Thus, even if the apparent numerals in the Indus script could be shown to have enumerative functions, they might function in this same limited fashion.
Rows of short “quotation” marks appear in the rock art of North America, as a parallel case.  A set of five appears in Texas above a large, squarish, backward “C” shape (Newcomb 1996: 155, Pl. 108, no. 7).  Accompanying these short marks is a set of taller verticals to the right of the square cee.  In the corpus of rock art from the Far West, there are two groups of five “quotes” in one panel (Heizer and Baumhoff 1984: 170, fig. 107f).  In the second and lower set, positioned over a large circle, the first three strokes are taller than the last two.  Hence, it is not clear that these five strokes are intended as one grouping despite their proximity to one another.
Indus sign V2 is FIVE POSTS, a row of taller verticals.  This is also known as KP151, W205, and does appear in Fairservis as O-6, defined as the adjectival form of the number five.  Proto-cuneiform includes two different forms of a numeral made with five lines, one with five verticals (N58), another with five horizontals (N57).  The same vertical posts appear as a motif in Old Europe (DS 93).  An additional parallel occurs in the rock art of North America, where five posts are grouped above four pairs of strokes in Texas (Newcomb 1996: 159, Pl. 111).  Further west, five posts appear between two skewed grids (Heizer and Baumhoff 1984: 167, fig. 104h).
Wells states that there are 15 occurrences of this sign, with nine at Mohenjo daro, three at Harappa, one at Lothal, and two at Kalibangan.  Still, it is not always a simple matter to distinguish between 5 POSTS and 5 QUOTES.  Thus, where Wells describes 5 POSTS in M-614, I would term them 5 QUOTES.  In this inscription the vertical strokes are noticeably shorter than the single other sign, a POT.  Again, in K-94, Wells considers the five strokes to be 5 POSTS, whereas I see 5 QUOTES, because the strokes are a good deal shorter than the other sign, a GRAIN EAR.  Finally, the copper object K-122B has nothing on it but five strokes.  These might be classified as either 5 POSTS or 5 QUOTES, as one pleases.  Without another sign to compare them to, they are unclassifiable.  My inclination would be to place them in both categories, with a question mark. 
In any case, out of the 15 instances that Wells cites for V2, 10 have FIVE POSTS alongside BARBELLS ON POST.  This kind of predilection for the company of a particular sign is common among the apparent numerals of the Indus script.  It is just one of the oddities that makes me consider these “numerals” to be essentially non-enumerative in function.
A third apparent numeral is the STACKED FIVE, V3.  In each case, three short strokes appear over two (a feature which I encode as 3 x 2).  Sometimes the top row is centered over the bottom row (“a” variant), while at other times the bottom row is displaced to the right so that the rightmost marks in each row are lined up (“b” variant).  This sign was formerly KP125(b), W211, and Fs O-7 (defined as the number five, again).  Wells notes a total of four occurrences of this sign, one each at Mohenjo daro, Lothal, Kalibangan, and Chanhujo daro.

Mycenaean Greek decorative motifs of cupped "U" shapes and rosettes.

In proto-cuneiform, there are a number of numerals with five elements stacked up in a particular order.  As noted earlier, the different forms of numerals enumerate different types of commodity and do so at different levels.  It is a system somewhat similar to our own kitchen measures, with 8 ounces in a cup, 2 cups in a pint, 2 pints in a quart, and 4 quarts in a gallon.  In our measurement system, this is only for liquids.  When it comes to flour, sugar, and other such things, we measure by weight, with 16 ounces in a pound, and so on.  Still another set of measures exists for distance (inches, feet, yards, miles), another for temperature, another for time.  While we use the same numbers for each system, the ancients did not.  They would have used one set of numerical markers for, say, the ounces, a second set for the cups, a third set for the pints, a fourth set for the quarts, and a fifth set for the gallons.  And so it would go on. 
In proto-cuneiform, then, there are at least 18 different types of “stacked five” for this reason.  Similarly, in proto-Elamite, there are three different types of “stacked five.”  Even so, in neither case is there an exact match for the Indus type.  The closest is a variant of N14 in which there is a row of three round impressions over two, where the second row lines up with the top on the left.
In the rock art of North America, once again, I find a grouping of five small dots stacked together, both in Texas and in the Far West (Newcomb 1996: 157, Pl. 110, no. 14; Heizer and Baumhoff 1984, 170, fig. 107e).  In the Texas example, there are four in a grouping similar to that found on our dice, with the fifth centered but above the others.  These five dots are positioned above a single large dot, to the right of three long posts.  The group of four appears to go together, but the single dot above them may not actually be part of this same group.  It may be a separate entity, given its separation.  In the second group, from the Far West, the set is more loosely grouped.  There are two on the right, one over the other with a space to the left.  There is then a second group of two, one not quite over the other, with a larger space to the left.  Finally, a single dot is not quite even with the top dot on the far left.  Again, the first four dots are reasonably close to each other and do appear to make a single group.  But the last dot, making five, seems further away, less similar, and less clearly part of the same composition.
Among the motifs of Old Europe there is a different form of “stacked five.”  In this group, the top consists of two horizontal dashes.  The bottom comprises three vertical “quotes.”  Thus, it is almost the reverse of the Indus sign (DS 106).
The last of the apparent numerals drawn with five strokes is V4, QUINTUPLE CEES.  I place this in the list because it is KP177, although it does not appear in Wells or Fairservis.  Its presence in Koskenniemi and Parpola’s list is probably due to one seal, M-136.  On this, however, I see four CEES and a CEE-BOAT, not QUINTUPLE CEES.  Still, it is possible that I have overlooked some other instance inadvertently.
In proto-cuneiform there is a somewhat similar sign comprising three nested curves that are backward compared to our “C.”  This is ZATU637, meaning unknown.  Similarly, in Old Europe, a motif consists of three nested curves in the other direction, similar to our “C” (DS 115).  Five nested curves appear in the rock art of Nevada and eastern California as well, although they are turned downward, like the Indus ROOF (Heizer and Baumhoff 1984: 138, fig. 75b).  Texas provides no examples of five repetitions of this shape, but there are five elements similar to our number “7” to the right of a circle (Newcomb 1996: 156, Pl. 109, no. 7).  Thus, the motif of quintuple repetition does appear there.  Finally, in Australian rock art there is an engraving of “C”-like curves with four-fold repetition at Sundown Point, northwest Tasmania (Flood 1997: 233).
Nested curves are among the entoptic forms people “see” and which David Lewis-Williams posits as the inspiration for the earliest art (2002).  I never see as many as five at a time, but perhaps I am unusual.

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