Counting on Your Body in Papua New Guinea

James Rauff
Milliken University

 

1. Counting.

Counting is usually the first thing that comes to mind when people think of mathematics.  Indeed, counting is the first bit of school mathematics learned by children. The NCTM's Principles and Standards for School Mathematics (2000) asserts that counting is "the foundation for students' early work with number" (p.79).  Counting is also regarded as one of the earliest manifestations of mathematics, but is usually passed by quickly in standard histories in order to get on to the meatier mathematics.  Katz (1998) devotes a mere four pages (out of  850) to counting while Burton (1999) ups the coverage to 6 pages (out of 671).  Neither of these authors discuss body-part counting.  Standard histories have a lot of mathematics to cover. Consequently, speculations on how counting began or how some non-Western cultures go about tallying  may seem less important than solving a cubic equation.  Nevertheless, there is value in examining the mathematical notions of people outside of the main currents of Western mathematical history.   Mathematical thinking is universal and the number sense is innate (Butterworth, 1999).  Mathematics is a part of culture and its manifestations across cultures are part of its story (D'Ambrosio, 1989, 1992).

 In that spirit of ethnomathematics, this paper is about a particular kind of counting that developed among many of the people now living in the country of Papua New Guinea (PNG).  Body-part counting is unique and fascinating.  It opens a window to other cultures and provides new narratives to the story of mathematics.   Too often, however, the mathematical notions of non-technological or non-literate people are deemed "primitive", not mathematics, or simple.  It is important at the onset to emphasize that the use of body-part tally-systems does not imply "primitive" mathematical ability.  Indeed, some of the people who use these tally systems have algebraically complex kinship systems (See Tjon Sie Fat, 1990).  Furthermore, despite its apparent  simplicity, counting requires a wide range of cognitive abilities.  Lakoff and Nunez  (2000, p.51) list nine "cognitive capacities needed in order to count on our fingers." Counting is a complex cognitive activity and is intricately interwoven with language and culture (See Tyler, 1969 for a discussion of culture and cognition). 

A detailed examination of the social, cultural, psychological, and mathematical dimensions of body-part tally-systems is well beyond the goal of this essay.  Instead, I hope to raise the awareness of body-part tally-systems within the mathematics community, exhibit some of the wonders and mysteries of the practice as a humanistic-mathematical phenomenon, and to offer avenues of investigation for those who are drawn to its magic.

2.  Papua New Guinea. 

Papua New Guinea (PNG) is a constitutional, parliamentary democracy of over 5 million people.   PNG encompasses a group of islands and  the eastern half of the island of New Guinea  between the Coral Sea and the South Pacific Ocean (See Map 1.)  PNG has over 5000 kilometers of coastline and includes more than 1400 islands.  The topography of Papua New Guinea varies dramatically from beaches and mangrove forests and swamps to rain forest to high mountains that can receive snow.  The rugged interior of PNG has enabled its indigenous populations to develop without contact with European culture until well into the twentieth century (in some cases as late as 1970).  There is some disagreement among linguists as to the number of languages spoken in PNG, but the estimates generally hover around 800 (e.g., Ethnologue, 2002; Wurm, 1982).

Map 1. Papua New Guinea (CIA, 2002)

In a collection of over 800 distinct languages one would expect to find wide variation in numerals and counting words.  (See Lancy (1983) for a classification scheme of 225 PNG languages.)  Indeed, Papua New Guinea has enough variety to keep students of numeral systems busy for decades  A couple of examples will serve to illustrate the complexity and variety of PNG numeral systems.

Consider Nasioi, a Papuan  language spoken on the island of Bougainville.  Nasioi possesses an extensive system of numerical classifiers.  Different numerals are used depending upon what is being counted.  Table 1 shows the variations in Nasioi counting.  The basic linguistic pattern of Nasioi numerals is clear.  The number morpheme is a prefix on the noun class. The number morphemes are  (1) na-,  (2) ke-, (3) be-  and  (4) kare-.  Why the noun classes are divided the way they are is an interesting question of ethnoscience that awaits a future investigator.

 

Table 1. Nasioi (Solomon Province) (Lean, 2002, Vol. 3, 23ff).

	1	2	3	4
People	narung(m) nani(f)	kenankara	benaura	karenaura
Quadrapeds (mammals)	navoro	kevoroka	bevoroi	karevoroi
Birds	nau	keura	bekuri	karekuri
Fruit	nau	keuka	bekupi	karekupi
Hollow objects	naro	keroka	beropi	kareropi
Tools	nari	kerika	bevintu	karevintu
Bag-like objects	nara	keraka	berapi	karerapi
Ropes	navin	kevinta	bevintu	karevintu
Leaf-shaped objects	nane	keneka	berapi	karerapi
Years & months	navera	keveraka	beverapi	kareverapi
Days	namung	kemunta	bemuntu	karemutu
Trees	nave	kevera	beveru	kareveru
Flat objects	namo	kemoka	bemopi	karemopi

 

The case of Enga, spoken in the Western Highlands Province on the island of New Guinea, is also intriguing.  Table 2  shows the Enga counting words from 1 to 32.

 

Table 2:  Enga (Western Highlands) (Lean, 2002, Vol.9, pp.5ff)

Number	Word	Analysis
1	me(n)dai	1
2	lapo	2
3	tepo	3
4	kitome(n)de	4
5	yungi	5
6	tokage	6
7	kalage	7
8	tukulapo	two arrows
9	tukutepon(ya) me(n)dai	two arrows + 1
10	tukutepon(ya) lapo	two arrows + 2
11	tukutepon(ya) tepo	two arrows + 3
12	tukutepon(ya) gato	two arrows + end
13	mapun (ya) me(n)dai	sweet potato + 1
14	mapun (ya) lapo	sweet potato + 2
15	mapun (ya) tepo	sweet potato + 3
16	mapun (ya) gato	sweet potato + end
17	yupun (ya) me(n)dai	ground + 1
18	yupun (ya) lapo	ground + 2
19	yupun (ya) tepo	ground + 3
20	yupun (ya) gato	ground + end
21	watakapun (ya) me(n)dai	wild + 1
22	watakapun (ya) lapo	wild + 2
23	watakapun (ya) tepo	wild + 3
24	watakapun (ya) gato	wild + end
25	paipan (ya) me(n)dai	come and go + 1
26	paipan (ya) lapo	come and go + 2
27	paipan (ya) tepo	come and go + 3
28	paipan (ya) gato	come and go + end
29	yanapun(ya) me(n)dai	dog + 1
30	yanapun(ya) lapo	dog + 2
31	yanapun(ya) tepo	dog + 3
32	yanapun(ya) gato	dog + end

 

The Enga counting system consists of a cycle of 60 subdivided into 15 cycles of four.  The first few counting words are numerals, but after that the counting words follow the pattern base+n where n takes on the values 1, 2, 3, and "end", and the base is a word or phrase with no numerical meaning.   The Enga count continues in like fashion until it ends at 60 with the appropriate kaeapalun(ya) gato ("I stop + end").  This is a fascinating pattern and begs for an ethnomathematical analysis.  Nevertheless, it is not my intention to offer such an analysis here, nor to provide a summary or analysis of PNG counting systems.  Instead, I focus solely on body-part tally-systems.

3.  Body-part tally-systems.

My discussion of PNG body-part tally-system relies heavily on the data compiled by Glendon Lean (2002).  Lean's extensive notes on counting systems survey 740 languages in Papuan New Guinea. I cross-checked Lean's data with other published work  (Biersack, 1982; Carrier, 1981; Cheetham, 1978;  Conant, 1896;  Franklin and Franklin, 1962; Hurford,1987;  Lancy, 1983; Levy-Bruhl, 1926; Panoff, 1970; Pumuye, 1978; Saxe, 1981; Smith, 1978). I included theYupno counting system as described by Wassmann & Dasen (1994) which is not included in Lean's notes.   Of these 741 languages, 38 have explicit  body-part tally-systems. 

Nearly all of languages investigated  begin counting on the little finger of one hand.  In some cases, the left hand is specified as the starting hand, but in most the count may start on either hand.  The count then proceeds through the fingers of the starting hand, then to the thumb and then up the adjoining arm to either the head or chest.  The count next proceeds down the other side of the body until it ends at the little finger of the other hand.  Figure 1 shows the count for Anggor, a language spoken in the West Sepik Province.

Anggor includes the wrist (6 and 18), forearm (7 and 17), elbow (8 and 16), upper arm (9 and 15), shoulder (10 and 14), breast (11 and 13) and the sternum (12) in its tally-system.  The count ends at 23 and it is customary to call such a system a 23-cycle system.  For counts above 23, such systems would usually continue on with a phrase like "one man and " begin the count anew.  (Although see Saxe, 1981 for a count that proceeds back up the arm in the reverse direction.) 

The Faiwol body-part tally-system is an example of one that includes counting locations on the head.  This 27-cycle system is shown in Figure 2.  The inside of the elbow is counted 8 and 20 in Faiwol in contrast to the outside elbow in Anggor.  Eleven and  seventeen in Faiwol are counted at the collar bone.

There is great variation in the cycle length of PNG body-part tally-systems.  Cycles of length  12, 14, 18, 19, 22, 23, 25, 26, 27, 28, 29, 30, 31, 32, 33, 35, 37, 47, 68, and 74 have been reported.  When the cycle length is odd there is usually a unique midpoint to the count that is located along the person's line of symmetry (e.g., the sternum in Anggor and the nose in Faiwol).  Table  3 summarizes the cycle-lengths of PNG body-part tally-systems and gives the midpoint location for cycles of odd length.

Miriam is unique in having three count locations on the body's line of symmetry (Lean ,2002, Vol. 12, p.36).  Yupno differs from all other languages in several ways.  First, the count in Yupno proceeds back and forth across the body with 6 -10 being counted on the right hand after 1 -5 is counted on the left. Second, the toes are included in the count with 11-15 on the left foot and 16-20 on the right foot.  Finally, Yupno includes locations below the waist with 31 counted at the left testicle, 32 at the right testicle, and 33 at the penis. (Wassmann and Dasen, 1994, p.84).

 

Figure 1.  Anggor (West Sepik) (Lean, 2002, Vol. 13, p. 14)

 

 

Figure 2.  Faiwol (West Province) (Lean, 2002, Vol. 12, p. 1)

 

 

Table 3.  Cycles of PNG Body-Part Tally-Systems

Cycle Length	People	Province	Midpoint
12	Ama	ES
14	Duna	SH
18	Dabu	WP
	Anggor	WS
	Agob	WP
	Mabiuag	WP
19	Gizra	WP	sternum(10)
	Gidra	WP	sternum (10)
22	Baruga	OP	nose(12), mouth(13)
	Sakam	MO
23	Awin	WP	sternum(12)
	Anggor	WS	sternum (12)
	Yuri	WS	sternum (12)
	Purari	GP	chest (12)
	Kabon	MA	hole above breast bone (12)
	Kalam	MA	bone of neck (12)
	Murupi	MA	sternum (12)
25	Enga	WH	nose(13)
	Amanab	WS	heart(13)
26	Giri	MA
27	Faiwol	WP	nose(14)
	Mianmin	WS	nose (14)
	Telefol	WS	nose(14)
	Tifal	WS	nose(14)
	Oksapmin	WS	nose(14)
	Hewa	SH	nose(14)
	Orokola	GP	nose(14)
	Sanio	ES	nose(14)
28	Paiela	WH	nose(14)
29	Alambak	ES	nose(15)
	Miriam	WP	navel(14), top of chest(15), front of throat (16)
	Huli	SH	nose(15)
30	Ninggirum	WP
31	Gende	MA	nose(16)
32	S.Mende	SH
33	Yupno	MA	none-alternates sides
35	Fasu	SH	nose(18)
37	Fol	SH	centre of nose (19)
47	W. Kewa	SH	bridge of nose (24)
	E. Kewa	SH	between eyes (24)
68	Kewa	SH
74	Nagutman	WS

Guide to Provinces in Table 3:  ES (East Sepik), GP (Gulf Province),  MA (Madang), MO (Morobe),  OP (Oro Province), SH (Southern Highlands),  WH (Western Highlands),  WP (Western Province), WS (West Sepik),

 

4. Language.

The relationship between the words spoken while counting, and the body part referenced, is complex.  Lean (2002, Vol. 12, p.2)  observes that "body-part tally-systems tend to fall into two types: 1) those in which all tally-words have a body-part referent, and 2) those in which the first few number words are numerals with no body-part referent, the remaining tally-words all referring to body-parts."  

Baruga, spoken in Oro Province, is of particular interest here because of the generality of the spoken words.  The Baruga body-part tally-system is a 22-cycle system that begins with the little finger of the right hand .  Baruga counting is shown in Table 4.   Notice that the counting must be visual since the word "doro" stands for 3,4,5,19,20, and 21.  Baruga counting also is unique in including the left eye before the central nose and mouth, thus breaking the vertical axis symmetry pattern.

 

Table 4.  Baruga (Oro Province) (Lean, 2002, Vol. 5, p.9)

Number	Word	Meaning
1	angusi	little finger-right hand
2	doro	ring finger - right hand
3	doro	middle finger-right hand
4	doro	index finger-right hand
5	ubei	thumb-right hand
6	tama	wrist- right
7	ungubo	elbow-right
8	visa	shoulder-right
9	dengoro	ear-right
10	diti	eye-right
11	diti	eye-left!
12	mendo	nose
13	bee	mouth
14	dengoro	ear-left
15	visa	shoulder-left
16	ungubo	elbow-left
17	tama	wrist- left
18	ubei	thumb-left hand
19	doro	index finger-left hand
20	doro	middle finger-left hand
21	doro	ring finger- left hand
22	angusi	little finger - keft hand

 

Fasu  (Table 5) and  Kewa (Table 6) are examples of counting systems that use a specific body-part word for each number.   In each of these languages the counting words for the numbers on the other side of the midpoint (after 18 in Fasu and after 24 in Kewa) are marked lexically with the word other or another.   This is typical of body-part tally-systems with or without a midpoint on the line of vertical symmetry.  Fasu is representative of several langauges that give the last number in the cycle a designation of closure.

 

Table 5.  Fasu (Southern Highlands) (Lean, 2002, Vol. 10, p.8)

Number	Word	Translation
1	meno	little finger
2	teta	ring finger
3	isia	middle finger
4	kitafa	index finger
5	kakorea	thumb
6	nama	palm of hand
7	yatipinu	inside wrist
8	kari	forearm
9	tokona	inside elbow
10	kaeyako	upper arm
11	kinu	shoulder
12	keno	collar bone
13	fufu	neck
14	senaki	ear
15	pare	cheek bone
16	hi	eye
17	no	side of nostril
18	terayia	ridge of nose
19	taku no	other side of nostril
20	taku hi	other eye
33	taku isia	other middle finger
34	taku teta	other ring finger
35	kenake urutae	last one counted

 

Table 6.  East Kewa (Southern Highlands) (Franklin & Franklin, 1962)

Number	Word	Translation
1	kegali	little finger
2	kegali yame	ring finger
3	ada ki	middle finger
4	mala	index finger
5	su	thumb
6	su midi	heel of palm
7	wara	palm
8	kerepo	wrist
9	pala ki	forearm
10	noe	large arm bone
11	noe luabu	small arm bone
12	koma	above elbow
13	winya ropa	lower upper arm
14	ali ropa	upper upper arm
15	pasa	shoulder
16	pasa kuli	shoulder bone
17	pasa midi	neck muscle
18	ma	neck
19	yaga	jaw
20	kale	ear
21	pae	cheek
22	le	eye
23	paki	inside corner of eye
24	riga	between the eyes
25	paki meda	inside corner of another eye
26	le meda	another eye
47	kegali meda	another little finger

 

Table 7 gives the counting system for Amanab, a language spoken in West Sepik Province.  Amanab is an example of a language in which the body-part tally-system coexists with a pure numeral system.   In a pure numeral system, the words designating a number have no other semantic content. Amanab counters may choose to name body parts for the numbers  1 to 4 or they may use number words.

 

Table 7.  Amanab (West Sepik) (Lean, 2002,, Vol. 13, p.13)

Number	Word	Translation
1	angik   or  munggo	little finger  or one
2	angangik or sambaga	ring finger or two
3	nibwari or sambaga munggo	middle finger or two-one
4	figinik or sambaga sambaga	index finger or two-two
5	afa wa goho-k	thumb
6	onuweso	wrist
7	enemsakak	mid-forearm
8	erengeg	inner elbow
9	enembuguk	mid-upper arm
10	guk	shoulder
11	hwanifuk	armpit
12	tut	breast
13	oruk	heart
14	mink tut	other breast
15	mink hwanifuk	other armpit
16	mink guk	other shoulder
17	mink enembuguk	other mid-upper arm
25	mink angik	other little finger

 

In Mianmin  (Table 8), the option for body-part words is not available.  Mianmin counters use number words to count up to six and then shift to body-part words for the numbers thereafter.  Neither Lean nor his sources give information for Mianmin counting beyond 23.  However, we would expect to see body-part words for these numerals.  It is also notable that Mianmin begins counting on the left thumb.

 

Table 8. Mianmin  (West Sepik) (Lean, 2002,, Vol. 13, p.35)

Number	Word	Translation
1	ele-yem	this-alone
2	asu	two
3	asu-matna	two - one more
4	asu-ke asu-ke	two-and two-and
5	asu-ke asu-ke mak-e	two-and two-and other
6	asu-ke asu-ke asu-ke	two-and two-and two-and
7	ban-lim	forearm on
8	hetlefab	inner elbow
9	tumin	shoulder joint
10	nakal	shoulder
11	tam-lim	side of face-on
12	klon-lim	ear-on
13	kin-lim	eye-on
14	munung-lim	nose-on
15	kin-milim	eye-other side
16	klon-milim	ear-other side
17	tam-milim	side of face-other side
18	nakal-milim	shoulder-other side
19	tum-milim	shoulder-joint-other side
20	hetlefab-milim	inner-elbow-other side
21	ban-milim	forearm-other side
22	gong-milim	wrist-other side
23	kweit-awok-milim	hand-thumb-other side

 

Although not a body-part tally-system, Yahang counting  (see Table 9) illustrates the blend of pure numerals, number words with no other semantic value, and body-part referents.

Yahang counting uses pure numerals for one, two, and perhaps three.  Four is an additive combination of two two's.  Five has the same meaning as hand.  The semantic correlation of five with hand  is widespread among PNG languages that do not use body-part tallies.  This correlation is present in many of the languages of the world  (Butterworth, 1999; Conant, 1896; Hurford, 1987).

The patterns of counting seen in these examples provide evidence for Hurford's (1987, p.82) assertion that  "The normal process of language-change over a long period would lead to words which originally had body part associations losing those associations and becoming pure numeral, or at least counting, words."

 

Table 9. Yahang (West Sepik) (Lean, 2002,, Vol. 13, p.32)

Number	Word	Translation
1	numusuku	one
2	kolou	two
3	kololomu	three (?? 1 + 2)
4	kolikolou	2 + 2
5	wom nateng	hand
6	nor nateng numu	hand and one
7	nor nateng kolou	hand and two
8	nor nateng kololomu	hand and three
9	nor nateng kolikolou	hand and four
10	kolou nateng	two hands
20	kolou nateng kolou inding	two hand; two feet

 

5.  Cultural context.

The semioticist Umberto Eco asserts that "usually a single sign-vehicle conveys many intertwined contents and therefore what is commonly called a 'message' is in fact a text whose content is  a multileveled discourse." (1976, p.57)   Viewed as semiotic systems, body-part tally-systems do more than count.  They reflect culture and communicate.

Consider the case of the Kewa pig-kill calendar (Pumuye, 1978).  The body count terms correspond to the months of preparation for the pig killing festivals that occur every five to seven years.

Table 10 was compiled from the information given by Pumuye (1978) and should be compared to the Kewa counting system shown in Table 6 which is slightly different.  Pumuye (1978) reports a 68-cycle body-part tally-system for East Kewa while Franklin and Franklin (1962) report a 47-cycle system.  This kind of variation is not uncommon in reports. It exhibits either the dynamic nature of the systems or differences in memory and understanding among informants. 

 

Table 10. Kewa Pig-Killing Calendar (Pumuye, 1978)

Month Number	Month Name with Number Word Italicized	Gesture	Activity
1	Ki-kegali eke mere	Touch or bend little finger.	Prepare raw materials to build long rows of houses for the festival.
2	Ki-kegali-yame eke mere	Touch or bend ring finger.	Assign festival participants a room in row of houses.
3	Ki-anda-ki eke mere	Point middle finger.	Make new gardens to produce enough food for the festival
4	Ki mala eke mere	Touch the index finger.	Weed the taro gardens and make the vegetable gardens.
5	Ki  su eke mere	Point to the thumb.	Judge and comment upon progress with the gardens.
etc.
14	Ki wina ropa eke mere	Touch the region known as the woman's armlet.	Slaughter all the pigs

 

Beyond calendars, the body-part tally-systems merge with the idiosyncrasies of individual cultures in a variety of ways.  What is actually counted varies widely.  The whole spectrum of objects to be counted seems to be represented.  Examples range from the Huli, who count everything (Cheetham, 1978) to the Loboda, who seem to count nothing but money and use number rhetorically (Thune, 1978).  The Yupno represent a position somewhere in between (Wassmann & Dasen, 1994)   The Yupno count string bags,  grass skirts, pigs , traditional string money, and modern PNG money. They do not count days, people, sweet potatoes, nor betel nuts.

The most detailed account of the cultural use of body-part tally-systems is Biersack's (1982) investigation of the Paiela, who live in the Western Highlands Province.  The Paiela use a 27-cycle body-part tally-system (with the nose as the midpoint) augmented by an additional count at 28 called pondo (two clenched hands knocked together).   Biersack argues that Paiela counting is an "element in a communication process and that its logic is therefore communicational."  (Biersack, 1982 p.816)  She explains that the body tallying of  pigs is actually a two-stage communication process where the body-tallying is the first or private stage and the enumerating of the pigs is the second or public stage.   She says that the logic of Paiela counting is understood not as arithmetic but as a semiotic system.  That is, "the concrete elements -- the vocabulary, tallying procedures, procedures of actual enumeration, and the linkage of these to transactional processes -- are so organized as to generate two levels of signification. At the first level messages of friendship are signified (and counted). But at the second level a statement is made, using a binary metacode, about the certainty or uncertainty of the message and, by implication, the stage of the communication process itself." (Biersack, 1982, p.823)

Paiela counting should be viewed not as primitive counting, but as sophisticated communication.

"Paiela counting behavior thus reveals that the Paiela are conversant with a number of principles associated with pattern and information: that information must be borne on a marker; that it necessarily alleviates uncertainty; that pattern is a relationship between things; and that pattern can be created and destroyed, that pattern is not preserved."  (Biersack, 1982, p.825)

7. Fingers and toes and elbows and brains.

Body-part tally-systems are an important part of the cultural history of mathematics.  As PNG becomes more Westernized , the body-part tally-systems are dying out (Lancy, 1983; Lean, 2002).  Indeed, PNG has a modern educational system and mathematics education follows the western model including mathematics competitions and computer assisted instruction (Sukthankar, 1999). We can hope to record traditional body-part tally-systems and understand them as best we can before they are gone.  But, we should not view body-part tallies as an aberration of mathematical development.  These systems are  natural extensions of counting on one's fingers and toes.  Finger and toe counting is widespread in PNG (Lean, 2002; Mimica, 1992).   There are also traditions of finger counting in mercantile trade in Africa,  in Europe and in Asia  (Ifrah, 2000; Menninger, 1969; Zaslavsky, 1973).  The Venerable Bede even wrote a manual on the process in the 8^th century.  Occasional finger arithmetic techniques appear with the promise of making us all better at mathematics (Chisanbop, 1980).  The neuropsychologist Brian Butterworth (1999, p.226) hypothesizes that "as a child grows and develops, the subitizing circuits in the inferior parietal link up with the finger circuits in the intraparietal sulcus. The fingers therefore gain an extended representation by this link: they come to represent the numerosities."  Body-part counting is, therefore, part of our nature.

 

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