Tuesday, January 31, 2012

More Odd Numerals in Indus Script

Total Frequency
328 (14.5%)
661 (23.5%)
1043 (46.1%)
1089 (38.8%)
495 (21.8%)
530 (18.9%)
219 (7.8%)
66 (2.9%)
68 (2.4%)
41 (1.8%)
51 (1.8%)
76 (3.4%)
94 (3.3%)
7 (0.3%)
11 (0.4%)
3 (0.1%)
4 (0.1%)
1 (0.04%)
1 (0.03%)
70 (3.1%)
80 (2.8%)
2267 (100%)
2808 (100%)

Frequencies of Apparent Numerals in Indus Script (data on posts, quotes, and stacked forms from Korvink 2008: 60; ligatures and calculated percentages from author’s data). *Asterisk marks statistics that should not be considered number related.

Many of the peculiarities of Indus “numerals” have been noted by other researchers besides myself.  For example, in a presentation at the Fifth Harvard Indology Roundtable in 2003, Steve Farmer noted several (“Five Cases of ‘Dubious Writing’ in Indus Inscriptions,” available online at http://www.safarmer.com/downloads ).  He first notes the uneven distribution, which I present in the table above.  Briefly, Farmer observes the very high frequency of “two” and “three,” with rapidly declining frequencies above those two “numerals.”  This pattern is not unusual, as Dehaene and Mehler’s (1992) research demonstrates.  In seven modern languages, the frequency of number words in speech as well as writing was found to decrease with magnitude.  Exceptions to this rule typically appear at 10 and 12 (as well as some larger numbers that do not appear in Indus script). 
Detail from seal H-67 with inscription: SINGLE POST / CAGED MALLET.
Here are two types of possible numerals, a "post" and 4 in a ligature ("caging").

As the table above demonstrates, there still seem to be many more Indus “twoes” than expected – especially of the short variety that I term “quotes.”  Korvink’s analysis (2008) clears up this problem by showing that two short strokes (BI-QUOTES) is not a numeral.  He also shows SINGLE QUOTE to be non-numerical, so the columns labeled “Quotes Freq.” for the first two numbers (one and two) should be removed from the above table.  Once this is done, the frequencies of the apparent numerals largely follow the expected trend, frequencies are more in line with Dehaene and Mehler’s findings. 

Still, even if we remove SINGLE QUOTE and BI-QUOTES from consideration, anomalies remain.  First, TWO POSTS remains more frequent than ONE POST, against Dehaene and Mehler’s “rule.”  This discrepancy could be eliminated by adding the relatively frequent use of SINGLE POST in ligatures to its appearances as an independent sign, i.e., add all the occurrences of a sign “with attached post” (changing total frequency of “one” to 482, making up 24.7% of the numeral occurrences).  “Ones” would then outnumber “twoes.”  To be even-handed, we would need to add occurrences of a sign “between posts” also as ligatures with a possible “two” to that total (changing total frequency of “two” to 411, or 21.1%). 
Seal M-991 with inscription: MAN HOLDING SHISH KEBAB /
SINGLE POST / POT HATTED BEARER (if, as many scholars
believe, numerals must precede what they enumerate in the Indus
script, then the "post" seems to enumerate nothing, as it occurs
before a function sign, a terminal).

Unfortunately for such obvious attempts to norm the frequency data, adding the frequencies of all the “stacked” forms then creates a whole new problem.  There are now too many “threes,” with 530 occurrences of all forms making up 27.2% of the total (ligatures with “three” include THREE POSTS WITH ATTACHED TRI-FORK and THREE POSTS UNDER CHEVRON).  Still, it is not a stretch to consider “three” a special case, since the frequency of this numeral creates a local increase in the folklore and mythology of most European cultures and their descendants.  Note here that if we take instances of “caged” signs to be ligatures with “four,” the 219 occurrences of this numeral in all forms become 11.2% of the total, well below that of the smaller numerals.

There do not seem to be many ligatures with larger numerals to skew the frequency norm.  But there remains a local increase at “seven” that is not predicted by Dehaene and Mehler’s study.  The local increase at “twelve,” on the other hand, is expected.  But so is one at “ten” that does not occur.  In conclusion, then, compared with the norm established by Dehaene and Mehler, the apparent numerals in Indus script show too many “threes” and “sevens” and too few “tens.”
Tablet H-219 with inscription (right to left): BARBELL ON POST /
5 POSTS / RAKE.  The first two signs often occur together, but no
other apparent numeral appears with BARBELL ON POST like this.

Farmer’s second observation is that specific “numerals” appear alongside specific non-numerical signs in many cases.  For example, STACKED 3 occurs preferentially before OVERLAPPING CIRCLES, while 3 QUOTES appears in the company of FISH, POT HATTED BEARER, FORK, BI-FORK TOPPED HAIR PICK, and CEE.  THREE POSTS, in rather surprising contrast, typically stands next to CUP, SPEAR, or SLASHES IN OVERLAPPING CIRCLES (2003: 13).  Among these non-numerical signs, there are some that do not occur with any other “numeral,” including OVERLAPPING CIRCLES and BI-FORK TOPPED HAIR PICK.  While other “numerals” do precede FORK, FISH, and CEE, there are odd gaps:  only 3 and 5 precede CEE; while there are examples of 1, 2, 3, 4, and 6 FISH, but no 5.  Thus, if FORK, FISH, and CEE represented commodities – which is what one would expect from Near Eastern parallels – then every numeral should appear before each of these commodities at some point.  The fact that this is not the case strongly suggests that apparent numerals are not enumerating commodities after all.

Farmer notes that one explanation proposed for this peculiarity is that Indus “numerals” may sometimes function as phonetic elements based on the rebus principle.  Since he considers the Indus script not closely tied to language, he finds this explanation unlikely (2003: 14).  As I have said before, I agree with Farmer’s assessment.  But the phonetic explanation is worth examining in more detail.
Seal H-48 with inscription: 3 QUOTES / BI-FORK TOPPED HAIR PICK.
No other apparent numeral appears with this type of "hair pick," but
3 QUOTES pairs this way 14 times.

One might wonder, to begin with, whether number words are likely to be homophones of other words, a necessary precondition for using numerals in this way.  Those researchers who assume that Indus script is a fully developed writing system often take Dravidian to be the language of the inscriptions.  And in Dravidian, it is relatively easy to find homophones – or near homophones – for the basic numerals.  For example, “two” can be reconstructed in Proto-Dravidian as something like *iru, and a homophone (at least in Tamil) means “to exist, remain, live” (Burrow and Emeneau 1984, nos. 401 and 407).  The word for “four” was something like *nālu, while a near homophone means “day,” *nāļ(u).  Using the rebus principle in English, we could write “2” for “too” and “4” for “for.”

There are some parallels to this type of usage in other writing systems.  In Egyptian hieroglyphs the numeral “two,” in the form of two short strokes (often slanted), appears at or near the end of a word to indicate dual case, where it has a quasi-numerical meaning and is usually pronounced y.  As a result of this type of use, it occasionally represents the sound without dual meaning.  Similarly, “three” (three short vertical strokes) appears often to indicate plural meaning, where it usually is pronounced w, but occasionally represents the sound without plural meaning (Gardiner 1976: 536-7 on glyphs Z4 and Z2).  Other Egyptian numerals smaller than 10 are almost always strictly numerals.  In the case of larger numerals, the rebus principle seems to work in the opposite direction.  A hobble for cattle provides the rebus for “ten” (md), a coil for “hundred” (št), a lotus plant for “thousand” (h3), and so on.
The names of Egypt's King Tut in hieroglyphs, Tutankhamen
(far left & left); and his Horus name, Nebkheperure (right).
Three vertical strokes represent the "u" in the Horus name.

Chinese provides further examples.  In fact, virtually every word in Mandarin Chinese is a syllable with multiple meanings.  So, this is a language that could make extensive use of the rebus principle.  “Three” is pronounced san1, for example, written with three horizontal strokes.  The same written character can also be pronounced san3 (the same syllable pronounced with a different tone), with a possible meaning “to reiterate.”  All the numerals from 1 to 10 – with the possible exception of 2 – could be used in a similar way given the frequency of homophones. 

However, in practice, this is not what happens.  The word for “five” is wu3 in Mandarin, while “I, me” is wo3 – not quite the same.  As it happens, “I, me” can also be expressed with wu2 (again, the same syllable as the number but a different tone), which is written with the character for “five” over that for a mouth.  Combining the mouth character with another is one way of indicating that a word should be pronounced in a certain way but understood with a different meaning.  This is not quite the same as the rebus principle, but an extension of it.  The number “eight” is ba1; another ba1 meaning “open-mouthed” is written with the mouth character on the left and the numeral “eight” on the right.
Detail from Japanese koto music.  Large characters are numerals representing
particular strings of the koto, although only the "cross" in the next line from the
bottom and the "cursive r" on the bottom are numerals outside of music (10 & 9).

Sometimes a numeral functions as a phonetic element in a character that includes a radical (usually the element on the left) other than the mouth.  For example, si4 is “four”; another si4, meaning “mucus,” is written with the radical for water on the left and the numeral on the right.  Or there is “ten,” pronounced shi2; another shi2, meaning “a file of ten (people),” has a radical that resembles a chevron (a very schematic person).

But other numerals do not show this type of usage, even though they could, theoretically.  For example, qi1 is “seven,” written with a character somewhat resembling our “t.”  There is a homophonous word meaning “to steep (tea),” but it is written completely differently.  The numeral for “nine,” pronounced jiu3, looks a little like a cursive “r,” which is nothing like the character for the homophonous “scallions” which in turn is quite different from the equally homophonous “wine.”

Thus, while a numeral can be used to represent the sound of the number word rather than its numerical meaning, I know of no writing system that systematically makes use of all of its numerals this way.  In fact, examples are infrequent in all the writing systems with which I am familiar.  When it comes to the Indus script, we can point to no certain examples where a sign’s meaning is agreed upon, much less the ancient pronunciation.  We cannot then prove that the rebus principle explains any particular usage of that sign.  Even if we assume that (1) the script conveys phonetic information, and (2) this information is conveyed through a rebus, and (3) the language is Dravidian, it would still be quite unwarranted to assume that all the “numerals” are also phonetic elements.  It would be better to select just a few apparent numerals, ones whose frequency departs from the expected pattern, and focus only on their possible extended use as symbols for sounds.  Thus, the high frequencies of “one” and “two” are part of the expected pattern, so we would not need to invoke phonetic principles for them.  The unusual spikes in frequency are at “three” and “seven,” so this is where the rebus principle would most likely be in effect.
Seal M-179 with inscription: 3 QUOTES / TRI-FORK, one of several
"numeral" + FORK combinations that occurs often on seals.

As it happens, the Proto-Dravidian word for “seven,” something like *ēŗu, is the same as or similar to a word for a male animal in some Dravidian languages, ēŗu.  In some languages, too, there is more than one homophone (e.g., ēru, “to rise, ascend”).  If we are determined to show that the excess frequency of “seven” is due to its use as a phonetic symbol, how do we decide which homophone the Harappans intended?  The procedure adopted by Parpola and by Fairservis is essentially this:

1.       Identify a sign based on what it looks like;

2.       Find a Dravidian word with that meaning that has cognates in many modern Dravidian languages, from Burrow and Emeneau’s A Dravidian Etymological Dictionary (1984);

3.       Find homophones for the selected word that fit the context where the original word is not suitable.

Going through these steps, Parpola reads FISH as “star,” mīn, while Fairservis sees it as a loop of thread with diacritical marks, instead identifying FAT EX as “star,” cukka, (Parpola 1994: 275; Fairservis 1992: 50 and 83-4).  In other words, a single method leads to varying results.  FISH and FAT EX cannot both mean “star.”  And even if one of these researchers is correct in his identification, which Dravidian word is involved?  Both have cognates in many modern Dravidian languages (Burrow and Emeneau 1984, 12 languages for mīn, 3994; 10 languages for cukka, 2175). 

Farmer strongly disagrees with the whole process, since it is based on what he considers an unwarranted assumption.  The Indus “numerals” cannot be phonetic elements, in his view.  Instead, he suggests that they are “numerological symbols – as ‘The Three,’ ‘The Seven,’ ‘The Twelve,’ and so on – referring to divine, celestial, or mythological forces” (loc. cit.). 
A group of seven dots and a crescent, symbols of the Pleiades (The Seven) and the moon
as commonly found on Mesopotamian cylinder seals (detail, Collon 1987: 76, fig. 335).

One example of such a numerological symbol is a group of Mesopotamian deities called Sebittu, Akkadian for “The Seven” (Black and Green 1992: 162-3).  This term refers to the constellation of the Pleiades, seen as seven gods.  Another such group includes demons considered the sons of the god An, “heaven,” and goddess Ki, “earth.”  Yet another Seven are benevolent gods, possibly of Elamite origin, who oppose the seven demons.  The Babylonians also had seven apkallū, the Seven Sages, anthropomorphic deities who carry axes, knives, bows, and arrows, all to fight demons.  Earlier, too, the Sumerians knew “The Seven” (Iminbi in their language). 

The Sumerians also had groups of a very different number, the 50 giants of Eridu and the 50 lahama of the underworld, both groups being satellites of the water god, Enki (1992: 76).  In addition, some individual deities were assigned a numeral that could be used instead of a name.  The Akkadian god Ellil (Sumerian Enlil) is sometimes represented with the numeral 50, the goddess Ištar (Sumerian Inanna) as the numeral 15 (Black, George, and Postgate 2000: 70 and 135).

Further west, the Egyptians also used numerological expressions for certain groups of deities.  The primary gods of the city Heliolopolis were collectively termed “The Nine,” usually given in the Greek form Ennead (Egyptian psdt) (www.philae.nu/akhet/ennead.html ).  These include the creator god Atum, his son and daughter Shu and Tefnut, their son and daughter Geb and Nut, and the four children of the latter couple, Osiris, Isis, Set, and Nephthys.  There was a rather different group of eight gods worshipped at Hermopolis known as the Ogdoad, that is, “The Eight” (including the couples Nu and Naunet, Heh and Hauhet, Kek and Kauket, and Amun and Amaunet) (www.philae.nu/akhet/Ogdoad.html ).

One final example of numerological usage comes from much further afield.  The Mongols of Genghis Khan’s day held the number “nine” to be especially lucky, with some individuals using it as their name.  His father’s name, Yesugei, is based on the word for this number (yesün in the modern language) (Sanders and Bat-Ireedui 1999: 239).  Two of the khan’s wives also used this number in their names, namely, Yesui and Yesugen (adjectival endings differentiate these three).  Other numbers do not receive this type of emphasis in Mongolian.
Seal Ad-6 with inscription: STACKED 3 / OVERLAPPING CIRCLES / POT //
BARBELL BETWEEN POSTS / ANKH.  The first 2 signs are a common pair,
which occurs in 18 inscriptions.  No other apparent numeral pairs with the
OVERLAPPING CIRCLES as consistently.

Thus, a numeral in the Indus script might be based on a number, but used in one of these ways: to designate a particular group of deities or legendary characters, to identify an individual deity or person, or as part of a name or the name itself.  This is an attractive explanation for the apparent numerals that do not clearly quantify anything.  For example, where a “numeral” appears before nothing but a terminal, one might read “of The Three” (for M-215: 3 POSTS / BEARER), assuming one sees the BEARER as a genitive marker, following the Finnish researchers Parpola, Koskenniemi, Parpola and Aalto 1969, as described in Possehl 1996: 124).  The appearance of a stereotyped “numeral” + sign sequence might also be an indicator of a numerological symbol.  Thus, STACKED 3 + OVERLAPPING CIRCLES might represent something like “The Three (deities) of (the town of) ‘Overlapping Circles.’”  That is, like the Egyptian ennead and ogdoad, the Indus STACKED 3 might indicate the number of principal deities and the place where they were worshipped.  Farmer wonders whether EF TOPPED EXIT represents a hearth with a fire (2003: 15).  If it is, then the common pair STACKED 7 + EXIT might indicate a particular place with seven special hearths.  “Seven Hearths” might be the name of a real place, the designation of a city or region.  Or it might be a place in myth or legend.

Finally, Farmer presents one possible mythological explanation for “seven.”  Let us say that the various FORKS in Indus script represent feathers.  If that is correct, then perhaps the inscriptions from Banawali that pair a FORK with an apparent numeral seven depict something like “seven feathers” (e.g., B-10: 7 QUOTES / TRI-FORK; B-12: FORK / STACKED 7).  Then, on at least two seals there are seven anthropomorphs with feathers (H-97 and M-1186), perhaps the significance of the inscriptions could be amended to “the seven feathered ones.”  These could be deities or they could be humans engaging in cultic, i.e., religious acts.
Broken seal H-97 with (partial?) inscription: DOUBLE ESSES WITH EAR /
Note the seven figures beside the inscription, perhaps a depiction of "the
seven feathered ones," a human or divine version of STACKED 7 + FORK?

I mentioned in a previous post the fact that certain apparent numerals typically precede one non-numerical sign but just as typically follow some other non-numerical sign (e.g., 3 POSTS precedes CUP and FISH, but follows CUPPED SPOON).  It is interesting to note that in Sumerian, cardinal numerals usually precede the object numbered, while ordinals occur after the object numbered (Langdon 1911: 122).  If at least some of the Indus “numerals” really are based on numbers, this potential parallel suggests one hypothetical explanation for the co-existence of two different positions.

No single explanation easily accommodates all the data on “numerals.”  This may indicate there is more than one explanation – i.e., more than one function for these signs.  For example, it may be that the apparent numerals are just that only when several different ones commonly appear before a particular sign.  That would mean that there are actual numerals only preceding FORK, FISH, and CUP.  Where only one particular “numeral” pairs with a particular sign, the group may represent a person, a deity, or a group of mythological beings, perhaps associated with a specific place.  And where the “numeral” seems to convey information on its own (where the inscription includes only the “numeral” or a “numeral” before a terminal sign), such a mythological or numerological explanation seems quite likely.
Tablet H-945A and B with inscriptions: CAGED MARKED FISH / COMB //
CUP / 4 POSTS (a crack seems to join two of the "posts," one of which may
be only more of the crack since this appears as 3 POSTS in the KP concordance).
CUP occurs many times with SINGLE POST, 2 POSTS, 3 POSTS, and 4 POSTS,
only once with 6 POSTS, never with any other number of "posts."

The signs that appear with various “numerals” before them are the most likely to represent enumerated objects.  These primarily include CUP, FISH, FORK.  CUP follows SINGLE POST (5 occurrences), 2 POSTS (59 occurrences), 3 POSTS (82 occurrences plus 2 additional ambiguous instances), 4 POSTS (34 occurrences), and 6 POSTS (1 instance).  It is still a bit odd that no form of “five” occurs in combination with CUP, though. 

FISH follows SINGLE POST (12 occurrences: 4 with FISH, 1 with MARKED FISH, 1 with FISH UNDER CHEVRON, 5 with WHISKERED FISH, 1 with CAGED WHISKERED FISH), as well as after 2 POSTS (66 occurrences; plus 15 instances before CAGED FISH = 81 occurrences of “two” before one or another “fish”).  Larger “numerals” are made up of short strokes: 3 QUOTES + FISH (16 occurrences); 4 QUOTES + FISH (5 occurrences); STACKED 6 + FISH (15 occurrences); STACKED 7 + FISH (1 occurrence), STACKED 8 + “FISH” (4 occurrences, of which 1 is with FISH, 1 with MARKED FISH, and 2 with FISH UNDER CHEVRON).  Again, it is odd that there is no form of “five” alongside a FISH.  There are also 2 slightly anomalous occurrences of 3 POSTS + FISH and possibly 1 of 4 POSTS + FISH.
Bar seal H-151 with inscription: FEATHERED DUCK HEAD / PINCH //
STACKED SEVEN / QUINT-FORK.  Do the last two signs represent the
seven figures on H-97 and M-1186?

The various types of FORK (TRI-FORK, QUAD-FORK, QUINT-FORK) occur in essentially the same contexts and may, thus, be meaningless variants of one another.  On the other hand, it is also possible that the difference in stroke count is meaningful but the concepts symbolized are closely related.  For example, if all of the FORKS mean “grain,” each might still depict a particular type: TRI-FORK could be wheat, QUAD-FORK barley, and QUINT-FORK millet.  In any case, FORK follows various apparent numerals:  2 POSTS (11 occurrences); 3 QUOTES (15 occurrences); 4 QUOTES (27 occurrences); 5 QUOTES (11 occurrences), STACKED 6 (8 occurrences), STACKED 7 (6 occurrences), STACKED 8 (3 occurrences).  Here we finally see a combination with “five,” none with “nine,” “ten,” or “twelve.”  Note that the appearance of “eight” with a FORK only three times would not seem significant except as part of this larger pattern of “numeral” + FORK.

If CEE represents a crescent moon, then its meaning might be calendrical, with numerals indicating which month: 3 QUOTES (16 occurrences), 4 QUOTES (1 occurrence), 5 QUOTES (8 occurrences).  Note that here there are relatively few “numerical” combinations, which would indicate that only a few “months” are mentioned.  In various cuneiform archives, such a restricted range of time periods does sometimes occur as, for example, at Tall-I Malyan (Stolper 1984: 14-15, where names of 11 months appear, with most texts dated to one of four contiguous months).  But in such cases, it is likely that destruction of the archive location, often through burning, has artificially preserved economic accounts from a single year.  The occurrence of only two (or three, if the instance with “four” is not an error) month names in the Indus script would have to be due to some other – probably sociocultural – reason.  In any case, there are too few examples of “numeral” + CEE to verify my very hypothetical guess that CEE might be a moon.
Seal H-512 with inscription: 5 QUOTES / CEE / POT.
CEE appears with this "numeral" and with 3 QUOTES
but not with most apparent numerals, so it probably is
not a depiction of the moon.  But with "five" apparently
beside it, CEE is unlikely to represent "five" either.

However, I think there are sufficient examples with CEE to demonstrate that it cannot be a numeral, which is what Fairservis and Wells suggest (Fairservis 1992: 67-69; Wells 2011: 126-128).  Fairservis conflates CEE with ROOF, seeing both as originating from a fingernail mark in clay.  But these two signs do not occur in very similar contexts, making this untenable.  Wells posits the meaning “five” for CEE despite the existence of 5 QUOTES, 5 POSTS, and STACKED 5.  For this hypothesis, he cites several inscriptions containing the sequence STACKED 7 + EF TOPPED EXIT (M-644, M-50, M-714, H-383, H-268, H-272, K-13, M-1138, H-3, K-2, M-776, and M-98).  The last one in this group, M-98, also contains CEE (CEE / STACKED 7 / EF TOPPED EXIT // POT).  After all of these, Wells cites H-472, where there is again a CEE, but no STACKED 7 (CARTWHEEL / BI-QUOTES (HIGH) // BI-QUOTES (MID) / CEE / EF TOPPED EXIT // POT).  Here, Wells interprets BI-QUOTES (MID) + CEE as the equivalent of STACKED SEVEN, i.e., as “two” (BI-QUOTES) + “five” (CEE). 
Tablet M-578A and B with inscription (from right): STACKED 7 / EF TOPPED EXIT /
BUGS ON (STRIPED?) LEAF / POT (M-1534, and M-579 through M-581 are duplicates).

This is weak evidence for two reasons.  Signs that commonly appear in combination with one or more “numerals” can generally also be found without the accompanying “numeral.”  For example, although CUPPED SPOON is followed by 3 POSTS more than a hundred times, it also occurs without any following posts (H-5, M-656, Rgr 2, L-191[A1], etc).  Various FORKS commonly appear following several different “numerals,” but they, too, occur at the end of inscriptions without any “numerals” (M-403, H-657, L-29, etc.).  Thus, the occurrence of EF TOPPED EXIT without its accompanying STACKED 7 in H-472 follows a general pattern in which signs that often occur in pairs also appear singly. 

Besides this, CEE also occurs in M-98 where STACKED 7 does appear in its usual position preceding the EXIT.  If CEE is “five,” adding this to STACKED 7 equals 12.  There is an apparent “twelve” in the Indus script, but we have no reason to expect it in this position since STACKED 12 does not combine regularly with any other sign.  Thus, we have no good reason to add CEE as ”five” and STACKED 7 rather than interpreting the inscription as [CEE + (STACKED 7 + EXIT)] + [POT].  As a result, the appearance of a single CEE alongside an apparent numeral is insufficient support for Wells’ hypothesis. 
Seal M-98 with inscription: CEE / STACKED 7 / EF TOPPED EXIT / POT.
Wells considers CEE to be "five," which seems to make this inscription add
up to twelve, but it does not show STACKED TWELVE, the best candidate
for that numeral in the Indus script.

The other reason I find Wells’ hypothesis unlikely is due to his use of BI-QUOTES as a numeral in H-472.  It is relatively unusual to find two prefix constants in a single inscription, but this is by no means the only example (BI-QUOTES over BI-QUOTES resembling STACKED FOUR appears in M-658 also; PINCH + BI-QUOTES in M-1369; BI-QUOTES over SINGLE QUOTE resembling STACKED 3 in M-39, M-45, M-266, M-400, H-154, K-59, and Dlp-1; PINCH + SINGLE QUOTE in H-61 and C-11).  Korvink’s careful statistical study has established BI-QUOTES as a non-numerical symbol, so I follow his approach.

A final note on the unlikely proposition that CEE means “five” concerns the two “numerals” that regularly appear with it.  These “numerals” are “three” (if that is the correct interpretation of 3 QUOTES) and “five” (if that is the correct interpretation of 5 QUOTES).  If CEE were “five,” one would hardly expect to find it paired often with another type of “five,” the very pattern we find.  In fact, if CEE were any type of numeral, one would expect a variety of numerals to combine with it.
PRAWN / ZEE / CROSSROADS EX // POT (note the presence of two prefix constants, the 2nd and 3rd signs).

Aside from the three signs, CUP, FORK, and CEE, most combinations of an Indus sign and an apparent numeral are more stereotyped.  For example, there are 14 occurrences of BARBELL ON POST + 5 POSTS and no instances of any other “numeral” following BARBELL ON POST.  I find this particularly intriguing because of the fact that “five” is not found with CUP or FISH.  If BARBELL ON POST represents a place, perhaps the pair indicates five clans from there; if it is a rank or status in society, the pair could represent the five people who form that rank (i.e., something like “the five judges” or “the five chieftains”).

The only apparent numeral that commonly precedes EF TOPPED EXIT is STACKED 7 (19 occurrences) and the same is true of BATTERY (13 occurrences).  The same STACKED 7 follows STRIPED TRIANGLE (9 occurrences, plus 3 more where 7 QUOTES follow the triangle).  These pairs are good candidates for a cultural or mythological meaning for “seven.”  There may be a similar reason for the appearances of various forms of “three” (14 occurrences of 3 QUOTES + BI-FORK TOPPED HAIR PICK; 19 occurrences of STACKED 3 + OVERLAPPING CIRCLES; 101 occurrences of CUPPED SPOON + 3POSTS).  If this is the case – where “three” perhaps means “lucky” – there is no ready explanation for the distinction in form.  Why should “quotes” precede one sign while “posts” follow a second and invariably STACKED 3 with the circles?  My own guess is that this exclusivity indicates STACKED 3 is not synonymous.  That is, while some instances of 3 QUOTES or 3 POSTS might actually mean “three,” STACKED 3 probably does not.
Inscription on pot shard Ch-2 (from right): DUBYA / 7 POSTS BETWEEN PARENTHESES / POT
(my clumsy reproduction has inadvertently deleted one of the "posts").  The central sign is one of
the ligatures with a possible numeral 7.

The most popular “numeral” is 2 POSTS (2P).  There are 5 occurrences of 2P + STRIPED LEAF; 6 of 2P + ANKH (or SKEWERED CHEVRON, apparently a variant); 5 of 2P + DOUBLE CEES and 3 with DOUBLE ESSES (which Koskenniemi and Parpola group as variants of one sign); 9 of 2P + MALLET (plus one more with STRIPED MALLET); 10 of 2P + BLANKET; and 11 of 2P + FAT EX IN DIAMOND.  These are in addition to the occurrences with the FISH and FORK.  There are also a couple of combinations where the “numeral” follows the other sign in the pair: 7 occurrences of ODD STACKED + 2P; 26 of OVERLAPPING CIRCLES + 2P.  A much weaker combination involves SINGLE POST, where it precedes CUPPED SPOON 5 times.  In these case, as in the case of STACKED 3 + OVERLAPPING CIRCLES, it seems especially unlikely that the “numeral” functions numerically.

The combinations that best support Farmer’s numerological hypothesis are those with an apparent numeral followed by a terminal or prefix constant.  These include several combinations with one of the “bearers”:  SINGLE POST + POT HATTED BEARER (6 instances); 3 QUOTES + POT HATTED BEARER (10 occurrences); 4 QUOTES + POT HATTED BEARER (1 example); 5 QUOTES + POT HATTED BEARER (1 example); STACKED 12 + BEARER (1 example); STACKED 12 + CHEVRON HATTED BEARER (3 instances); 3 POSTS + BEARER (2 examples, of which one is actually 3 POSTS + 3 POSTS + BEARER).  There are other instances where “numerals” precede a terminal or a prefix constant (where they do not occur as the second member of an identifiable pair).  But not all potential numerals make such an appearance: SINGLE POST + SINGLE QUOTE (M-1151, M-993), SINGLE POST + SPEAR (H-922-3), SINGLE POST + TRI-FORK TOPPED POT (KP 1385, KP 2785); 2 POSTS + MAN (KP 2705), 2 POSTS + SINGLE QUOTE (M-575 – M-577, H-6); 3 QUOTES + SINGLE QUOTE (M-331, M-1197, M-500); 5 QUOTES + POT (M-614); STACKED 7 + MAN (or the reverse in H-160); STACKED 8 + POT (KP 7059); STACKED 9 + POT (KP 3699); STACKED 12 + BI-QUOTES (M-43, K-69-75); STACKED 12 + PINCH (KP 9602); STACKED 12 + POT (M-381, KP 2669, M-847, M-1265, M-29, H-386, M-1273, M-399, H-661, M-18, KP 5074, M-1053). 
Broken seal H-131 with inscription: CUP ON PRONGS (?) / RAYED CIRCLED / BATTERY / STRIPED MALLET /
CIRCLES / 2 POSTS / POT (the KP concordance shows the first sign as FEATHERED DUCK HEAD).  Here, the
medial segment shows STACKED 12 before a common pair (CUPPED SPOON + 3 POSTS).  Thus, this does not
seem to indicate twelve mortars and pestles (one proposed meaning for CUPPED SPOON). 

In these cases, parallels from the folklore and mythology of other times and places may provide suggestions of meaning.  In a previous post, I mentioned Fairservis’ idea that SINGLE POST is a numeral only part of the time, symbolizing a measuring stick in other cases (O-1 versus K-3; 1992: 174 & 180).  In fact, he also considers the long, vertical stroke to have a third significance when attached in a ligature (H-8, 1992: 169).  Here, he sees it as a staff or rod meaning “eminence.”  This definition may well be incorrect, but the idea that it is distinct from a numeral and from the independent SINGLE POST could be correct. 
Fairservis also identifies STACKED TWELVE – as well as STACKED EIGHT, STACKED NINE, STACKED TEN, and ODD STACKED – as representations of water rather than as numerals (1992: 70-72).  Again, “water” or “rain” may be inaccurate as a definition, but at least some of these signs may indeed be something other than numerals.  STACKED TWELVE in particular behaves a bit differently from the smaller apparent numerals.  It does not regularly pair with another sign, for one thing.  For another, it appears more often just before a terminal than the others (12 instances of STACKED 18 + terminal versus 9 for SINGLE POST + T; 1 for 2 POSTS + T; 10 for 3 QUOTES + T and 3 for 3 POSTS + T; 1 for 4 QUOTES + T; 2 for 5 QUOTES + T; 1 each for STACKED 7, 8, 9 + T).  Plus, STACKED 12 occurs before two different prefix constants (PINCH, BI-QUOTES) rather than the single one that occasionally appears following other “numerals” (SINGLE QUOTE).  Potentially, then, STACKED 12 may be a depiction of something rather an a numeral, even if the other apparent numerals represent numbers.


Black, J. and A. Green. 1992. Gods, Demons and Symbols of Ancient Mesopotamia. Austin: University of Texas.

Black, J., A. George, and N. Postgate. 2000. A Concise Dictionary of Akkadian. Wiesbaden: Harrassowitz Verlag.

Burrow, T. and M.B. Emeneau. 1984. A Dravidian Etymological Dictionary. Oxford: Oxford University Press (Indian edition 1998 reprinted by Munshiram Manoharlal Publishers, New Delhi).

Collon, D. 1987 & 2005. First Impressions: Cylinder Seals in the Ancient Near East. London: The British Museum.

Dehaene, S. and J. Mehler. 1992. "Cross-linguistic regularities in the frequency of number words," in Cognition (43): 1-29.

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

Farmer, Steve. 2003. "Five Cases of 'Dubious Writing' in Indus Inscriptions: Parallels with Vinca Symbols and Cretan Hieroglyphic Seals. The Emblematic and Magical Nature of Indus Symbols." Notes/Handout from Fifth Harvard Indology Roundtable, May 10, 2003.  Available at http://www.safarmer.com/downloads .

Gardiner, A. 1976. Egyptian Grammar: Being an Introduction to the Study of Hieroglyphs. Oxford: Griffith Institute and Ashmolean Museum.

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

Koskenniemi, K. and A. Parpola. 1982. A Concordance to the Text in the Indus Script. Helsinki: Dept. of Asian & African Studies, University of Helsinki. of the Principal Roots in Sumerian. Paris: Librairie Paul Geuthner.

Langdon, S. 1911. A Sumerian Grammar and Chrestomathy with a Vocabulary of the Principal Roots in Sumerian. Paris: Librairie Paul Geuthner (reprint from the collection of the University of Michigan Library).

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

Sanders, A.J.K. and J. Bat-Ireedui. 1999. Colloquial Mongolian: The Complete Course for Beginners. London & New York: Routledge.

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

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



    Please find my two papers below and circulate amongst the skeptics, particularly!

    To state the obvious, the Indus script was a logo-syllabic script and a lost corpus did exist.


    Published in the ICFAI journal of history and culture, January 2011


    Published in International journal of philosophy and journal sciences , November 2012

    I am also introducing logo-syllabic thesis B in this paper

    The paper is very self-explanatory! does anybody still beg to differ?

    Sujay Rao Mandavilli

    1. If you want to advertise your papers, why don't you start your own blog instead of cluttering up mine?