Positional Notation in the Indus Script?

Seal K-59 with inscription: DEE-SLASH / SINGLE POST / BI-QUOTES + SINGLE QUOTE //
STACKED 5 (according to Koskenniemi and Parpola; Wells sees the third sign as STACKED 3).

In a series of earlier posts, I addressed the issue of numerals in the Indus script.  After comparing the frequency of occurrence of every “numerical” sign, I concluded that the pattern suggested a folkloric rather than economic pattern.  However, there are researchers who do consider these signs numerals and see at least some of the inscriptions as economic records (e.g., Fairservis 1992: 60-66; Wells 2011: 124-5).  It is worth examining occurrences of each proposed numeral to further investigate this matter.  Assuming that Korvink is correct with his statistical analysis of Indus signs, SINGLE QUOTE and BI-QUOTES are not numerals (2008: 22).  Instead, they function as constant elements in the prefix (as does PINCH, which no one considers a numeral).  As such, I will not include these two signs in my discussion of possible numerals. 

ONE

With the removal of SINGLE QUOTE from the ranks of apparent numerals, the representation of the numeral “one” can only be as SINGLE POST.  If this is correct, then “one” seems to follow CHEVRON (14 occurrences) and SQUARE or DIAMOND (8 occurrences).  However, the same “one” precedes CUPPED SPOON (6 times) and BATTERY (2 times, as well as following BATTERY twice!).  It follows DEE-SLASH (4 times), but precedes MALLET (5 times, and CAGED MALLET twice).  It apparently precedes CUP on tablets (5 times). 

Tablet H-303A with inscription: ODD STACKED 7 / 2 POSTS // POT / COMB.

It is rather problematic to interpret SINGLE POST in each of these occurrences as enumerative, i.e., as “one x” (one chevron, one quadrangle, one cupped spoon, one mallet, one cup, etc.).  Normally, numerals either precede or follow the item enumerated, in a given language, not both.  But even if we ignore this difficulty, SINGLE POST remains odd for a numeral because it also precedes a terminal, without any apparent link to a prior sign (12 occurrences: 6 times before POT HATTED BEARER, once before SPEAR, 3 times before POT, and 3 times before TRI-FORK TOPPED POT + POT).  In these cases, there seems to be nothing for the “numeral” to enumerate.

We cannot conclude that SINGLE POST regularly precedes signs or that it regularly follows them; in addition, its occurrence before a terminal suggests that SINGLE POST conveys information on its own.  This suggests that the sign can only be a numeral part of the time, if at all.  As it happens, Fairservis posits two different functions for SINGLE POST, partly based on such peculiarities (1992: 174 and 180).  His sign K-3 represents a measuring stick, while O-1 is the number one (as an adjective).  Since K-3 and O-1 are identical, one might wonder whether such a division is possible.  But there is evidence of just such a division in other ancient scripts.  In Egyptian hieroglyphs, the reader may recall, a single vertical stroke can function as the numeral “one,” but it also functions as an indicator of an ideograph when it follows a non-numerical glyph.  In Mycenaean Linear B as well, a vertical stroke is sometimes a numeral but also appears in inscriptions as a kind of punctuation mark, separating items.  So, the idea that only some of the instances of the Indus SINGLE POST are numerical is quite plausible.  In fact, this seems quite likely since one of the relatively common “attachments” in ligatures is a post (e.g., MAN HOLDING POST; PRAWN WITH ATTACHED POST; FIGURE EIGHT WITH ATTACHED POST; GRID WITH ATTACHED POST; TRI-FORK WITH ATTACHED POST, etc.).

TWO

In a similar manner, the numerical meaning “two” would mostly likely be represented by TWO POSTS since BI-QUOTES is part of a prefix (STACKED TWO is theoretically possible but appears in only one fragmentary and thus equivocal inscription in the KP concordance, while I see it only on pot shards).  Like SINGLE POST, TWO POSTS appears capable of conveying a message all by itself in 7 instances (as the medial segment in M-1358 and KP 3914, in final position after OVERLAPPING CIRCLES BETWEEN DOUBLE POSTS, as a second medial element following a terminal in M-257, M-720, and KP 8909, as the sole variable in a prefix in M-866). 

There are also a few signs with which it pairs regularly.  It follows OVERLAPPING CIRCLES (26 times), also following ODD STACKED “numerals” (9 occurrences).  But it precedes FISH (65 times, plus 15 times preceding CAGED FISH), FORK (11 times), BLANKET WITH TICKS (10 times), MALLET (9 times, plus once before STRIPED MALLET).  It even precedes FAT EX IN DIAMOND (12 times, not distinguished in the KP concordance from CIRCLED FAT EX).  The KP concordance shows this “numeral” preceding CUP as well (59 times; I count a few differently and find some instances of the reverse sequence, CUP + 2 POSTS).  There is certainly more evidence for a preceding position here, but the pairing with OVERLAPPING CIRCLES is regular enough that this suggests there is no simple rule placing TWO POSTS before another sign in a pair. 

Fairservis does not suggest that TWO POSTS be broken into two categories as he does with SINGLE POST.  But such is the case with Egyptian hieroglyphs, where two strokes can mean “two” or, within a word, the same two strokes (or, often, slanted ones) indicate either dual meaning or the phonetic element y, which is the most common dual suffix.  In early Chinese writing, too, two horizontal strokes form the numeral “two,” but also function as an old form of shang4, “high, above” (with the top stroke a little shorter than the bottom one) or xia4, “low, below” (with the bottom stroke a little shorter) (Wieger 1965: 28-29).  In Cretan hieroglyphs, two strokes are thought to represent a divider, probably between words (Younger CITE).  Thus, it is possible that the Indus TWO POSTS sign may act as a numeral part of the time, but as something else in other contexts.

THREE

For the meaning “three,” there are three possible Indus signs: 3 QUOTES, 3 POSTS, and STACKED 3.  THREE POSTS can occur essentially alone, thus appearing to convey information by itself (in M-215 before a terminal; doubled before a terminal in K-22; alone before a prefix constant in H-92).  THREE POSTS also sometimes appears at the end of an inscription where it must be following another sign if it is part of a pair (Kd-9, H-911, M-139, doubled in K-4, and in M-734 doubled).  In fact, its most frequent pairing is following CUPPED SPOON (or the variant CUPPED POST), which occurs 103 times.  Still, there are 82 pairings with CUP in the KP concordance, where THREE POSTS precedes the other sign (again, I see a few in the reversed CUP + 3 POSTS sequence).  These two distinct patterns suggest that 3 POSTS might also have more than one function, only one of these being numerical.

Broken seal H-589 with inscription: 3 POSTS / 5 QUOTES / CEE // POT.

As for THREE QUOTES, it precedes FISH 16 times (also preceding CAGED FISH once); precedes CEE 15 times (also preceding BACK CEE once in an inscription that includes DOUBLE CEES as well); precedes FORK 15 times; BI-FORK TOPPED HAIR PICK 14 times; and CUP in KP 2604 (once in my database, in M-500).  Interestingly, 3 QUOTES occurs twice in the latter inscription, once in the prefix preceding SINGLE QUOTE, and once before CUP at the end.  In two instances, the inscription on a tablet contains both 3 QUOTES and 3 POSTS (H-239 and H-735, in each of which the “quotes” appear on side A and the “posts” on side B).  There are some other, less frequent pairings where 3 QUOTES precedes another sign as well (4 times before EX, 4 times before LOOP MAN HOLDING STICK, 3 times before one or another STOOL, twice before CHEVRON, etc.).  All this seems to indicate that, even if there is no clear-cut symbol of “one” or “two,” 3 QUOTES must surely mean “three,” appearing regularly before another sign. 

However, against this interpretation are (1) instances directly preceding a terminal (10 occurrences before POT HATTED BEARER, one instance before MAN), (2) instances directly preceding the constant in a prefix (2 occurrences before SINGLE QUOTE), and (3) occasions when it stands alone (completely alone in KP 3766; as the whole medial segment following a prefix in M-358).  There are also a few peculiar inscriptions where it occurs alongside other “numerals” (KP 4402: 3 QUOTES / 4 QUOTES / 3 QUOTES; KP 3717: 2 STACKED ROOFS / 3 QUOTES; KP 3715: ROOF / 3 QUOTES / ... [illegible]).  The third objection here, the occurrence of 3 QUOTES alongside another “numeral,” is evidence for Wells’ view that Indus script has positional notation (2011: 124-5).  Presumably, he reads KP 4402 as “three hundred and forty-three.”  I will return to this idea after examining other proposed numerals.

STACKED 3 is less common than the other apparent “threes,” but it regularly occurs preceding OVERLAPPING CIRCLES (18 occurrences).  In one interesting inscription, STACKED 3 precedes POTTED 3 (KP 9301).  Is this an indication that the short strokes inside the POT are also numerical?  Unfortunately, occurrences of apparent numerals before POTTED 1, POTTED 2, POTTED 3, POTTED 4 do not support such a function.  SINGLE POST occurs handily before POTTED 1 in two inscriptions (M-125 and KP 1106) but TWO POSTS precedes once also (H-80).  SINGLE POST also precedes POTTED 2 in two inscriptions (M-1151 and KP 654), while THREE POSTS precedes in one (H-776).  POTTED 3 is preceded by 3 POSTS once (M-411) and by STACKED 3 once (KP 9301), but also by SINGLE POST in one case (KP 563).  Thus, over half the time, the “numeral” preceding the POT does not correspond to the “numeral” inside it.

STACKED 3 makes a plausible numeral before OVERLAPPING CIRCLES, but not when it appears alone (on side C of L-66, where an inscription of 6 signs occurs on side A, and a single sign on each edge C, D, E, and F).  Another oddity is that OVERLAPPING CIRCLES has STACKED 3 before it as well as 2 POSTS after it in at least two cases (H-904, and H-340 or KP3420 x 2).  In these instances, if STACKED 3 represents “three,” then 2 POSTS must not represent “two” (or vice versa – if 2 POSTS is “two,” then STACKED 3 must not be “three”).

FOUR

The meaning “four” might plausibly be conveyed by FOUR QUOTES, FOUR POSTS, or STACKED FOUR.  Of these, FOUR POSTS is less frequent than the smaller “numerals.”  It precedes BIRD (C-1) in one instance at the beginning of an inscription.  In another case (KP 9702), it follows POTTED 3 while preceding CIRCLED VEE.  Its most frequent occurrence is before CUP (34 instances, plus one with FISH between the “numeral” and the CUP).  Thus, it could be a numeral.

FOUR QUOTES also makes a plausible numeral where it precedes FISH (4 occurrences) or FORK (27 occurrences).  It precedes CEE only once (KP 207), which often occurs alongside other “numerals.”  Plus, it occasionally comes before some other sign (once each before PRICKLY CORN HOLDER, STRIPED HORN, and CROSSROADS EX; twice before a DUBYA).  In one surprising instance, 4 QUOTES precedes CUP (KP 230 x 3) where one would expect “posts”; while in another inscription, it is STACKED 4 that precedes FISH, where one again expects “posts.”  These inscriptions suggest that 4 QUOTES = 4 POSTS = STACKED 4.  There are no cases where one of these “numerals” regularly follows another sign.  This evidence would seem to suggest that the varied ways of writing a numeral have no meaning.

Still, there are some peculiar instances where the “numeral” seems to enumerate nothing.  In KP 4420 (x 4) 4 QUOTES is the only sign; in KP 4434 the inscription reads 4 QUOTES / 4 QUOTES (or should this be interpreted as 8 QUOTES?); in KP 2799 the inscription includes 7 QUOTES / 4 QUOTES (or should this be seen as the rare-as-hen’s-teeth numeral “eleven”?); in KP 4341 we find ONE ROOF / 4 QUOTES; in KP 247 there is an equally odd pair of signs, STACKED TRIPLE CIRCLES / 4 QUOTES.  Like the smaller “numerals,” 4 QUOTES also occasionally precedes a terminal (once each with POT HATTED BEARER and with MAN).  In KP 2796 (or M-1384) the inscription reads: CIRCLED FORK / BEARER // 4 QUOTES / 5 QUOTES / ROOF (analyzable as Medial / Terminal // Numeral / numeral / numeral or MT + ###).  Compare with the latter KP 2800: 4 STACKED ROOFS / 4 QUOTES.  If we attempt to read 4 QUOTES + 5 QUOTES + ROOF as involving positional notation (451?), we are then troubled by the reverse sequence in KP 2800 (are the “roofs” ones or tens?).  If the number of “roofs” equals that of “quotes” because the two mean the same thing, then is this inscription essentially the same as 4 QUOTES / 4 QUOTES?  The Harappan numerical system remains quite obscure.

FIVE

Moving on to possible “fives,” 5 POSTS conveys a message by itself following a MT sequence (M-1465).  It follows BARBELL ON POST 14 times, its most frequent pairing.  In one additional case, the initial sign is illegible (L-12) before 5 POSTS, perhaps to be reconstructed as BARBELL ON POST.  In two other inscriptions, 5 POSTS begins the sequence, in one case preceding STRIPED FAT LEG LAMBDA, in the other preceding CUPPED SPOON.  FIVE QUOTES, on the other hand, never follows but only precedes a sign in regular pairs: 11 times before FORK, 8 times before CEE, as well as a handful of other cases that show no regular pairs.  Oddly enough, there is no recorded instance of 5 QUOTES + FISH. 

Then there are the odd occurrences of 5 QUOTES: two instances before a terminal (before POT HATTED BEARER in KP 2113 x 3, before POT in KP 1618); as a second medial segment following a terminal (KP 2753 x 2); alongside another apparent numeral in M-1384.  Then, STACKED 5 occurs after a terminal in KP 6043 (COMB / STACKED 5).  It might seem a bit less peculiar if we read these two signs in the opposite direction, as STACKED 5 / COMB, giving an MT sequence rather than the highly peculiar TM.  Even with this “correction,” it remains odd for a “numeral” to precede a terminal since it does not seem to enumerate here.  Further, in K-59, STACKED 5 alone forms the medial segment, after a prefix that is odd in itself (DEE-SLASH / SINGLE POST / BI-QUOTES / SINGLE QUOTE).  If we read the last two signs in the prefix as a single sign, instead (STACKED 3), we still have the rather odd sequence of DEE-SLASH + 135.  There is also the unusual sandwiching of the HORN between two “numerals” in M-331 (4 QUOTES / STRIPED HORN / STACKED 5).  Then, too, the same “numeral” follows FOOT at the end of one inscription (BM 123208 from the British Museum).

SIX

Koskenniemi and Parpola note no instances of 6 POSTS in their concordance (although I see at least one case of 6 POSTS / CUP).  They do show four occurrences of 6 QUOTES: once as the whole inscription (KP 4407); once as a second medial segment following a terminal (KP 2797); once beside another possible numeral (6 QUOTES / 7 STACKED ROOFS in KP 2798); and once following STRIPED TRIANGLE and preceding a terminal (M-678).  All of these are problematic for the interpretation of 6 QUOTES as a numeral.  Most instances of a possible “six” are in the form of STACKED 6.  In this form, one finds the expected pairings with FORK (8 instances) and FISH (15 instances).  There are also three inscriptions where this “numeral” precedes DOUBLE GRIDS (KP 2873, 4507, 2717). 

Tablet H-916 with inscription: ODD STACKED 6 / 2 POSTS // POT / COMB.

Nevertheless, STACKED 6 sometimes forms a medial section by itself: after the prefix in M-1341; before the terminal POT in KP 5063 (or C-3); and perhaps before the terminal COMB in KP 7027.  In the latter case, I see PINCH on the other side of STACKED SIX, although the KP concordance shows only an illegible sign.  Since COMB seems to begin the inscription in violation of its other occurrences as a terminal, perhaps this inscription should be read in the opposite direction.  COMB would then be the terminal in its usual place at the end, but STACKED 6 would remain a non-enumerative “numeral” forming the medial segment.

SEVEN

The numeral “seven” is particularly common in the folklore of some regions (parts of Europe and the Near East) and I have speculated that this was the case in the Indus Valley Civilization.  This numeral, if it appears in the script, is most often written as STACKED 7, although there is an occasional 7 QUOTES as well as one reported instance of 7 POSTS, not found in the KP concordance (see reference to article in The Hindu below).  STACKED 7 precedes FISH just once (H-9), but comes before FORK more frequently (6 times).  It also regularly precedes EF TOPPED EXIT (19 occurrences) and BATTERY (13 occurrences).  There are other occasional such pairings (twice before STRIPED VEST, once before DOUBLE CEES).  But one pairing shows a reversed sequence, with STACKED 7 following STRIPED TRIANGLE (9 occurrences, plus 3 instances of STRIPED TRIANGLE + 7 QUOTES).  The strength of this pattern makes it conceivable that the single instance of STRIPED TRIANGLE + STACKED 6 (M-678) is an error for STRIPED TRIANGLE + STACKED 7.  There are also two odd inscriptions: 7 QUOTES occurs alone once (KP 3768) and once alongside another apparent numeral (KP 2799).

EIGHT

The numeral “eight” can only occur in the Indus script in stacked form – there are no 8 QUOTES, and no 8 POSTS.  STACKED 8 occurs three times before FORK, an expected sequence, but is not found before FISH.  Once it also precedes DOUBLE STRIPED MALLETS.  More difficult to interpret is the single instance preceding a terminal (KP 7059: STACKED 8 / POT / COMB).

NINE

The KP concordance notes two instances of 9 QUOTES (both times as the whole inscription), although I have not seen it.  Mostly, “nine” would have to be in the form of STACKED 9.  This occurs once before FORK (never before FISH) in H-23.  It also occurs once as the medial segment, before a terminal (KP 3699).  There is one other instance (KP 3491), but here the sign following the “numeral” is illegible.

TEN

STACKED 10 is the only possible form for “ten,” a sign I have not seen and which the KP concordance notes only once, alone (KP 3734).  There is no version of “eleven,” so far as I can tell, although one inscription with two apparent numerals might be added together to create such an instance (KP 2799: 7 QUOTES / 4 QUOTES).

TWELVE

Finally, there is STACKED 12, considerably more frequent than most of the higher “numerals” previously discussed.  This occurs 4 times before one or another “fish” (once before FISH, one before MARKED FISH, twice before FISH UNDER CHEVRON).  Other rare pairings include 3 occurrences before PRAWN and two before FAT CHEVRON.  STACKED 12 occurs once each before CRAB, STRIPED FAT LEG LAMBDA, RAKE, FOOTED STOOL WITH TICK, DOUBLE GRIDS, DOUBLED AITCH, CUPPED SPOON + 3 POSTS, POTTED 3, SLASHES IN OVERLAPPING CIRCLES, CIRCLED FORK, and DOUBLE CARTWHEELS.  In at least two of these cases, it confuses matters to read it as a numeral (with CUPPED SPOON + 3 POSTS, if the last sign is a numeral, and with POTTED 3 if the internal marks represent a numeral). 

Against the numerical interpretation, STACKED 12 frequently appears before a terminal (12 times before POT; 3 times before CHEVRON HATTED BEARER and once before BEARER).  It also directly precedes the constant of the prefix twice (KP 2584 and KP 9602).  Thus, like the other apparent numerals, STACKED 12 seems to convey a message by itself a fair proportion of the time.

Tablet K-69 with inscription: DOUBLE MEN / DOUBLE STACKED 12 / BI-QUOTES //
RAKE OVER RAKE / DEE / DEE-SLASH (note the slight separation between the 12's).

Unlike most other “numerals,” a doubled STACKED 12 occurs as well.  The doubled form precedes RAKE (H-14); follows DOUBLE MEN before the prefix constant BI-QUOTES (K-69 through K-75); and follows STRIPED FAT LEG LAMBDA while preceding CUPPED SPOON (L-115).  Wells classifies doubled instances of a sign as separate signs; thus, STACKED 12 is his sign W201 while DOUBLE STACKED 12s is W207.  The doubled form appears the same in the above three inscriptions, in the KP concordance.  One STACKED 12 is clearly separated from the other in each case.  But in the photographs, this separation is not so clear.  The multiple copies from Kalibangan do show such a separation, but neither H-14 nor L-115 follows suit.  In the latter two inscriptions, the “numeral” appears as three stacked rows of eight “quotes.”  Thus, this could also be interpreted as STACKED 24.  The problem with doing that is that there are no clear instances of any numerals between 12 and 24.

ODD STACKED 6-9

There is also a problematic symbol that seems to contain an irregular number of short strokes, ODD STACKED.  There may be six, seven, eight, or nine strokes in this sign, according to my count.  Out of 15 occurrences of this sign (if it is a single sign), it precedes 2 POSTS in 10 inscriptions.  Once, the sign ends the inscription following a terminal (KP 601).  Twice it appears just before a terminal (KP 1415 x 3 and KP 8451).  In the last, it comprises the medial segment by itself.  In two other inscriptions, it might be interpreted as a numeral preceding another sign (once before CRAB in L-47; once before BUGS ON STRIPED LEAF in M-1078).  In one case, KP 1415, I count 12 slanting “quotes” in the ODD STACKED while KP show 9 (MARKED FISH / ODD STACKED / FLANGE TOPPED POT / POT).  It is tempting to interpret the ODD STACKED as “water,” following Fairservis, because of the peculiarities of its appearances.  However, as should now be clear, all the other supposed numerals demonstrate similar peculiarities.

Seal H-65 with inscription: ODD STACKED 8 / 2 POSTS // POT
(note the similarity between this seal's inscription and that of the
two tablets shown above, H-303 and H-916).

As noted earlier, Wells argues that instances where two or more numerals stand next to each other represent positional notation.  This seems unlikely on the face of it, at least as far as the type of positional notation we have with Arabic numerals.  In the first place, this is an anachronistic proposal.  Positional notation was a late development, historically speaking.  In the Bronze Age, other known numeral systems basically use repetition of some type of mark to distinguish numerals, with distinct marks indicating different types of numerals.  We see this with Mycenaean Linear B, where vertical strokes represent ones, horizontal strokes represent tens, and circles represent hundreds.  Ancient Egyptian also used such a system, with vertical strokes again denoting units, the medj (resembling the Indus ROOF) indicating tens, a coil for hundreds, and so on (Gardiner 1976: 259).  In such systems, scribes have a tendency to group larger numerical signs on one side, smaller ones on the other.  So, in this sense, position means something.  In Linear B, for example, the ones are usually on the far right, with tens to their left, and hundreds to the left of the tens.  The Egyptians were a bit more erratic, partly because hieroglyphs sometimes read from right to left and sometimes from left to right, but also because they would readily stack groups of signs to fill the available space in an esthetically pleasing way.  But again, the larger numerals tend to precede the smaller ones.

In proto-cuneiform and proto-Elamite, as I have mentioned before, there is something similar to positional notation, where impressed wedges and circles of various sizes (along with some more complex signs) are repeated for different measures (Damerow and Englund 1989: 76).  These are generally not decimal systems.  For example, in the sexagesimal system S, a wedge represents “one,” with 10 of these equal to one small circle; 6 small circles equal one large wedge; 10 large wedges equal one large wedge with an impressed circle inside it; and six of the last equal one large circle.  A single text may, then, include more than one type of numeral.  For example, the obverse of proto-Elamite tablet TY 23 includes a heading (one of the “prickly triangles”), an entry resembling the Indus VEST (representing grain), and a numeral composed of 2 small circles, 5 small wedges, and 3 rotated wedges.  There is probably one more circle on a line beneath the other numerals (5 rotated wedges equal one regular wedge; 6 regular wedges equal one small circle, in this Š system).  Here, again, large numerals or measures tend to occur before smaller ones.

Elamite cuneiform tablet, obverse: "15 minas of copper; // for edging (?); //
of large horn(s)" (Stolper 1984: 72).  First sign upper left is the numeral 15,
a horizontal wedge (more or less) for 10 plus 5 vertical wedges, stacked.

Let us compare these patterns from deciphered numerical systems with what occurs in the Indus script.  Wells provides a few examples that he thinks demonstrate positional notation (2011: 125).  On tablet H-303, we find: ODD STACKED 7 / 2 POSTS // POT / COMB.  His explanation would make the “numeral” here 72 (or conceivably 27).  But his table shows POT + COMB as indicating what is counted with this “numeral,” an untenable position if Korvink correctly identifies these as terminals.  Seal H-589 provides: 3 POSTS / 5 QUOTES / CEE // POT.  Wells would take the “posts” and “quotes” as 35 (or 53), with CEE + POT the item counted.  Removing the POT following Korvink’s analysis, we are left with 35 of CEE.  But in the first example, the short strokes come first (ODD STACKED 7), followed by long strokes (2 POSTS), while in the second, long strokes come first (3 POSTS) and short strokes follow (5 QUOTES).  If there is a system as in Linear B and Egyptian here, it is not obvious.  We must ignore the type of strokes to maintain Wells’ positional notation.  It is easy enough to assume that the ancient Harappan “scribes” used “stacked” forms simply to save space and that these mean nothing different from “posts.”  But what would explain the presence of both “posts” and “quotes” in the same inscription, if the difference is meaningless?  It is hard to come up with an explanation other than whim.

To really resolve such an issue, we need more data than Wells provides.  For this purpose, I have compiled a list of inscriptions that include more than one apparent numeral.  Inscriptions are my transcriptions from the KP concordance, giving the KP numerical code where I do not find the inscription myself (because it is probably in Volume 3 of the Corpus) or I am uncertain (because I probably misread some part of the inscription).  Where I can identify the KP inscription in the Corpus, I give the standard identification (letter indicating location, plus artifact number).  In each inscription, I have placed doubled slashes at the end of the prefix, separating it from the medial section, and at the end of the latter, separating it from the terminal.  Where something follows the terminal, double slashes further indicate the end of the terminal.  Double backslashes indicate where part of an inscription may be separate because it is on another line, but this cannot be demonstrated unequivocally.  I have highlighted the “numerals” by putting them in bold; the signs whose status as numerals is somewhat doubtful are in bold italics.  Pairs are underlined.  I have added an asterisk before a few inscriptions that I find particularly interesting or puzzling.

KP 2427                 DEE-SLASH / SINGLE POST / BI-QUOTES // SINGLE POST / STACKED 12 / DOUBLE CARTWHEELS / STRIPED FAT LEG LAMBDA.

*M-399                  FISH UNDER CHEVRON / CUPPED SPOON / 3 POSTS / STACKED 12 // POT

H-131                     FEATHERED DUCK HEAD / RAYED CIRCLE / BATTERY / STRIPED MALLET / BI-QUOTES // STACKED 12 / CUPPED SPOON / 3 POSTS / WHISKERED FISH / MARKED FISH / OVERLAPPING CIRCLES / 2 POSTS // POT

KP 1079                 CARTWHEEL / BI-QUOTES // 2 POSTS / FISH / WHISKERED FISH / STACKED 12 // CHEVRON HATTED BEARER

M-90                      CIRCLED VEE / BI-QUOTES // SINGLE POST / STACKED 3 / OVERLAPPING CIRCLES // POT

KP 3512 (A)           STACKED 3 / OVERLAPPING CIRCLES // POT // (B) 3 POSTS / CUP; (KP 4043 B) 4 POSTS / CUP; (KP 3891 B) 2 POSTS / CUP; (KP 4309 B) SINGLE POST / CUP)

KP 3420                 STACKED 3 / OVERLAPPING CIRCLES / 2 POSTS / DOTTED WINDOW / ...

*H-904                   STACKED 3 / OVERLAPPING CIRCLES / 2 POSTS // POT // VEE IN DIAMOND / STRIPED FAT LEG LAMBDA / CROSSROADS EX / 3 POSTS / CUP

KP 2796                 TRI-FORK UNDER CHEVRON / STRIPED MALLET / DIAMOND / CHEVRON / SINGLE POST / DOUBLE CARTWHEELS // CIRCLED FORK // BEARER // 4 QUOTES / 5 QUOTES / ROOF

KP 3457                 ... / STACKED 7 / BATTERY // POT / COMB // 3 POSTS / CUP

KP 1113                 2 POSTS / ... / ... [or POTTED 1] / SINGLE POST / STACKED 7 / BATTERY // POT

M-776                    CIRCLED VEE / BI-QUOTES // STACKED 7 / EF TOPPED EXIT / OVERLAPPING CIRCLES / 2 POSTS // POT

*H-268                   BOAT / SPACESHIP / FISH / BI-QUOTES // CIRCLED FORK / CUPPED SPOON / 3 POSTS / STACKED 7 / EF TOPPED EXIT // POT

*H-383                   DEE-SLASH / BI-QUOTES // CUPPED SPOON / 3 POSTS / STACKED 7 / EF TOPPED EXIT // POT

*M-50                    ... / BI-QUOTES // CRAB / WHISKERED FISH / CUPPED SPOON / 3 POSTS / STACKED 7 / EF TOPPED EXIT // POT

*M-714                  DOUBLE BELTED RECTANGLE / VEE IN DIAMOND / BI-QUOTES // CIRCLED FORK / CRAB / MARKED FISH / CUPPED SPOON / 3 POSTS / STACKED 7 / EF TOPPED EXIT // POT

H-272                     DOUBLE CEES WITH EAR / 2 POSTS / PANTS / BI-QUOTES // WHISKERED FISH / STACKED 7 / EF TOPPED EXIT // POT

**M-668               PRAWN / SKEWERED CHEVRON / POTTED 3 / STACKED 7 / EF TOPPED EXIT / CUPPED SPOON / 3 POSTS / ...

*KP 1020               CORN HOLDER / TRIPLE TRIANGLES / POTTED 1 / CUPPED SPOON / 3 POSTS / STACKED 7 / EF TOPPED EXIT / HAIRY HUNCHBACK // POT

M-1555-60            DOT IN FISH / STACKED 7 / EF TOPPED EXIT / MAN HOLDING QUOTE // SPEAR // 2 POSTS / QUAD-FORK // CRAB IN BUD TOPPED DUBYA (M-547-9 are duplicates except that the final sign is not present)

*M-494-5              POTTED 1 / DOUBLE CEES WITH EAR / CARTWHEEL BETWEEN DOUBLE POSTS / STACKED 7 / DOUBLE CEES / RAKE / WINGED MAN // POT // SNOWFLAKE / ANKH / POTTED 1 // BEARER WITH SHOULDER YOKE // BIRD BETWEEN PARENS / OVERLAPPING CIRCLES / 2 POSTS \\ CIRCLED FORK / CRAB / HAIRY HUNCHBACK // POT / BEARER // TRIPLE CUPS / TRI-FORK

KP 2799                 7 QUOTES / 4 QUOTES

KP 4507                 STACKED 6 / DOUBLE GRIDS // 3 POSTS / CUP

M-1384                  TRI-FORK UNDER CHEVRON / STRIPED MALLET / DIAMOND / CHEVRON / SINGLE POST / DOUBLE CARTWHEELS // 6 QUOTES / 7 STACKED ROOFS

M-136                    STRIPED BOWTIE WITH EXTRA RIBBON / 5 CEES / SINGLE POST / STACKED 6 / FISH // POT / MAN

H-789                     STACKED 6 / FISH // SPEAR / COMB // 3 POSTS / CUP

KP 3356                 STACKED 6 / FISH // SPEAR / COMB // SKEWERED DONUT / VEE IN DIAMOND / CUPPED EX / 3 POSTS / CUP

H-942                     STACKED 6 / FISH // SPEAR // 3 POSTS / CUP

C-40                        FOOTED STOOL / PINCH // FISH UNDER CHEVRON / WHISKERED FISH / BLANKET / CROSSROADS EX // POT // STACKED 6 / FISH / BUD TOPPED DUBYA / CROSS-HATCHED CIRCLE / 3 STACKED ROOFS / 5 QUOTES / MAN HOLDING QUOTE / TRI-FORK

KP 751                   ... / SINGLE POST / STACKED SIX [I think there may be POT at the end, M-1597?]

KP 2893                 2 POSTS / 5 QUOTES / QUAD-FORK

*M-20                    STRIPED BATTERY WITH BENT LEGS / TRIPLE TRIANGLES / 3 POSTS / 3 POSTS / VEE IN DIAMOND / BI-QUOTES // 5 QUOTES / TRI-FORK

*KP 1266               5 QUOTES / LOOP MAN HOLDING STICK / CARTWHEEL / 2 POSTS

C-30                        STACKED 5 / LOOP MAN HOLDING STICK / 2 POSTS / CARTWHEEL / 2 POSTS (CARTWHEEL BETWEEN DOUBLE POSTS?)

M-331                    RECTANGLE / BOAT WITH OAR & STERN POST / SKEWERED CHEVRON BETWEEN BACK CEE & CEE // 4 QUOTES / STRIPED HORN / STACKED 5 // STRIPED FAT CHEVRON & DOWN HEART BETWEEN CEES // 3 QUOTES // SINGLE QUOTE / CEE

K-59                        DEE-SLASH / SINGLE POST / BI-QUOTES / SINGLE QUOTE // STACKED 5

H-982                     3 POSTS / BUGS ON STRIPED LEAF // POT // 4 POSTS / CUP

*H-771                   OVERLAPPING CIRCLES / SQUIRREL / 2 POSTS // 4 POSTS / CUP

KP 2800                 4 QUOTES / STACKED 4 ROOFS

KP 4434                 4 QUOTES / 4 QUOTES

KP 4402                 3 QUOTES / 4 QUOTES / 3 QUOTES

M-577                    ... / ... / STRIPED FAT LEG LAMBDA / BI-QUOTES // 4 QUOTES / QUAD-FORK // DOUBLE CEES WITH EAR / 2 POSTS / STRIPED MALLET CAGED BY DOUBLE QUOTES / MAN HOLDING POST / DOUBLE GRIDS

KP 2697                 4 QUOTES / QUAD-FORK / SINGLE QUOTE // ODD STACKED / 2 POSTS // POT

H-914                     CIRCLED VEE / 4 QUOTES / TRI-FORK // 3 POSTS / CUP

*M-64                    4 QUOTES / FISH / PINCH // CUPPED SPOON / 3 POSTS // SPEAR

KP 4341                 ROOF / 4 QUOTES

KP 2799                 7 QUOTES / 4 QUOTES

KP 3715                 ROOF / 3 QUOTES

M-1426                  VEE IN DIAMOND / BI-QUOTES // CRAB / RAKE / FISH / 3 POSTS / 3 POSTS / CUP // POT

M-512-3                 2 POSTS / FISH / HAIRY HUNCHBACK // POT // 3 POSTS / CUP / CEE / COIL

KP 3356                 STACKED 6 / FISH // SPEAR / COMB // SKEWERED DONUT / VEE IN DIAMOND / CUPPED EX / 3 POSTS / CUP

*H-774                   WHISKERED FISH / CUPPED SPOON / 3 POSTS / FOOTED STOOL // PINWHEEL / COMB // SHISH KEBAB / BATTERY / VEST / CUPPED EX / 3 POSTS / CUP

M-500                    LOOP MAN HOLDING STICK / CAGED MALLET // COMB // 3 QUOTES / SINGLE QUOTE // CROSSROADS EX / 3 QUOTES / CUP

H-177                     CIRCLED VEE / BI-QUOTES // SINGLE POST / CUPPED SPOON // BEARER // 3 POSTS / CUP

KP 3350                 FOOT / 3 POSTS / POTTED 2 / CIRCLED FORK // POT // 3 POSTS / CUP

KP 4126                 OVERLAPPING CIRCLES / 2 POSTS // POT // 3 POSTS / CUP

KP 4507                 STACKED 6 / DOUBLE GRIDS // 3 POSTS / CUP

*H-369                   POTTED 1 / TRI-FORK / POTTED 1 / SINGLE POST / CUPPED SPOON / 3 POSTS // SPEAR // 3 POSTS / CUP

KP 3490                 STACKED 6 / FISH // SPEAR // 3 POSTS / CUP

KP 4178                 2 POSTS / BOAT / STRIPED HORN // 3 POSTS / CUP

H-930                     5 POSTS / STRIPED FAT LEG LAMBDA // 3 POSTS / CUP

H-799                     RAKE / WINGED MAN / CARTWHEEL / 2 POSTS // 3 POSTS / CUP

KP 3376                 STACKED 7 / STRIPED VEST // POT / COMB // 3 POSTS / CUP

KP 3457                 ... / STACKED 7 / BATTERY // POT / COMB // 3 POSTS / CUP

H-789                     STACKED 6 / FISH // SPEAR / COMB // 3 POSTS / CUP

KP 3556                 2 POSTS / CAGED FISH // COMB // 3 POSTS / CUP

KP 4113                 BARBELL ON POST / 5 POSTS / RAKE // 3 POSTS / CUP

*H-914                   CIRCLED VEE / 4 QUOTES / TRI-FORK // 3 POSTS / CUP

*H-735                   BOAT / PINCH // POTTED 1 / 3 QUOTES / QUAD-FORK // 3 POSTS / CUP

*H-239                   DOUBLE MEN / BISECTED TRIANGLE // POT // 3 QUOTES / TRI-FORK // 3 POSTS / CUP

M-366                    ... / STACKED 12 / POTTED 3 / MAN BY CHEVRON / BIRD / 3 QUOTES / BI-FORK TOPPED HAIR PICK

M-326                    BOAT / PINCH // POTTED 1 / FISH UNDER CHEVRON / DOUBLE CEES // POT // CARTWHEEL / 3 QUOTES / EX / STRIPED BISECTED MALLET \\ BARBELL ON POST / 2 POSTS + SLASH / BARBELL ON POST / 5 POSTS // BARBELL // TEETH / OVERLAPPING CIRCLES // VEE IN DIAMOND / MAN HOLDING DEE-SLASH (I see CUP) / TRI-FORK

H-941                     RAKE / FISH // SPEAR // 3 QUOTES / EX / 2 POSTS / MALLET

*KP 3651               STACKED 3 / OVERLAPPING CIRCLES // POT / COMB // 3 QUOTES / FOOTED STOOL / DOUBLE GRIDS

KP 9903                 3 QUOTES / CHEVRON / ... / FISH / 3 QUOTES // MAN

*M-1316               3 QUOTES / CEE WITH SLASH / 2 POSTS / MALLET / PANTS / DOUBLE DOWN HEARTS / MAN HOLDING DOUBLE DEE-SLASHES

*H-25                     STACKED 12 / FISH UNDER CHEVRON / CUPPED SPOON / 3 POSTS / CIRCLED FORK / 3 QUOTES / CEE // POT

*M-658                  CIRCLED VEE / BI-QUOTES (HIGH) / BI-QUOTES (MID) // SKEWERED CHEVRON / 3 QUOTES / CEE / 2 POSTS / QUAD-FORK

*H-141                   FISH UNDER CHEVRON / MARKED FISH / BI-QUOTES // SKEWERED CHEVRON / 3 QUOTES / CEE / 2 POSTS / QUAD-FORK

*KP 4402                3 QUOTES / 4 QUOTES / 3 QUOTES [= 10?]

M-500                    LOOP MAN HOLDING SLASH / CAGED MALLET // COMB // 3 QUOTES / SINGLE QUOTE // CROSSROADS EX / 3 QUOTES / CUP

*H-735                   BOAT / PINCH // POTTED 1 / 3 QUOTES / QUAD-FORK // 3 POSTS / CUP

*M-1364               CRAB / CUPPED SPOON / 3 POSTS / FAT CEE // POT // CARTWHEEL / PINCH // 3 QUOTES / QUAD-FORK

M-318                    BATTERY / SINGLE POST / BI-QUOTES // 3 QUOTES / TRI-FORK

KP 8503                 3 QUOTES / MAN HOLDING CUP // SINGLE QUOTE // CIRCLED DOT WITH EAR / POTTED 1 / 2 POSTS / FISH // SPEAR

KP 9903                 3 QUOTES / CHEVRON / ... / FISH / 3 QUOTES // MAN

*KP 2587               DIAMOND / SINGLE POST / VEE IN DIAMOND / BI-QUOTES // WHISKERED FISH / DOT IN FISH // SPEAR // SINGLE POST / 3 QUOTES // POT HATTED BEARER

KP 3717                 2 STACKED ROOFS / 3 QUOTES

K-4                          BOAT / PINCH // POTTED 1 / BI-QUOTES // LOOP MAN HOLDING SLASH / BIRD BETWEEN PARENS / 3 POSTS / 3 POSTS.

In his discussion, Wells only considers the few cases where “numerals” appear next to other “numerals.”  I have also included the more numerous instances where two or more “numerals” occur in non-adjacent positions in the same inscription.  I find it interesting, for example, to note that in H-735, both 3 QUOTES and 3 POSTS occur.  These two signs are not beside each other; rather, the first is on side A, the second on side B.  In both cases, the “numeral” occurs as the first member of a relatively frequent pair.  There are many examples of “numerals” preceding a FORK and many preceding a CUP.  So here we would not be averse to interpreting each pair as a numeral followed by the item enumerated.  Except that we find two different types of numeral, one with strokes that are notably shorter than the other signs (3 QUOTES) and the other with strokes of the same height as the other sign (3 POSTS).  Why should this be?  I cannot find an explanation in positional notation following the pattern found in Linear B – there is no reason for interpreting one as 30 and the other as 3.  But it would seem risky to simply ignore some of the evidence – the fact that the two look different – and interpret both signs in the same way.  The very same problem occurs with interpreting M-1364.

In M-399, there is a different type of problem: FISH UNDER CHEVRON / CUPPED SPOON / 3 POSTS / STACKED 12 // POT.  There is a pair here, CUPPED SPOON + 3 POSTS, which is somewhat troubling in that it positions the “numeral” after the enumerated rather than the reverse.  But, a bigger problem is posed by the occurrence of STACKED 12 after this pair.  It is unlikely that the terminal is enumerated, since we would expect terminals to indicate some sort of function according to various researchers (a case marker according to Meriggi, Hunter, Heras, Pande, the Soviet scholars Gurov and Knorozov, and the Finnish team of Parpola, Koskenniemi, Parpola and Aalto, all cited in Possehl 1996: 163; an honorific suffix according to Fairservis 1992: 173; or even as a priestly title, according to Mahadevan, cited in Possehl 1996: 130).  If we wish to use Wells’ notion of positional notation to group the two “numerals” together, we have 3 POSTS + STACKED 12.  What sort of notation is this?  If the “scribe” wanted to represent the number 15, why separate 3 POSTS from the short, stacked strokes of 12?  

Thus, positional notation does not explain all the peculiarities of the apparent numerals in Indus script any more than keeping them separate does.  

REFERENCES

Damerow, P. and R. Englund. 1989. The Proto-Elamite Texts from Tepe Yahya. Cambridge, Mass.: Harvard University Press.

Fairservis, W. 1992. The Harappan Civilization and Its Writing: A Model for the Decipherment of the Indus Script. Leiden: E.J. Brill.

Korvink, M.P. 2008. The Indus Script: A Positional Statistical Approach. Gilund Press.

Koskenniemi, K. and A. Parpola. 1982.  A Concordance to the Texts in the Indus Script. Helsinki: Department of Asian and African Studies, University of Helsinki.

Possehl, G.L. 1996. Indus Age: The Writing System. Philadelphia: University of Pennsylvania Press.

Stolper, M.W. 1984. Texts from Tall-I Malyan. I. Elamite Administrative Texts (1972-1974). Philadelphia: Occasional Publications of the Babylonian Fund, 6, of the University of Pennsylvania.

Wells, B.K. 2011. Epigraphic Approaches to Indus Writing. Oxford and Oakville: Oxbow Books.

Wieger, Dr. L. 1965. Chinese Characters (originally 1915). New York: Dover.

Younger, J.G. 2011 on Cretan hieroglyphs: http://people.ku.edu/~jyounger/Hiero/Hgrids.html