Chapter 2

The social life of numbers (I): Number Classes and Counting

Numbers

Why is counting so complex and why does it deserve nearly a full chapter of its own?

Numbering ontology defines how unique from or similar to a language is to other languages and in a way, is a powerful tool in determining its relatedness. A polylingual European, or a mathematician, for example, might not initially appreciate there being any difference in way numbers are defined; they might claim, for example, that "mathematics defies culture." The flip-side of this statement is that the way in which commonplace mathematical relations are defined, viewed and communicated, often go a long way in defining the way we look at the word.

To appreciate varying ontologies we need to step out of a European understanding to Indigenous and Native²⁵ understandings of the world, by stepping into the mechanics of their languages, and in this case, numerical ontologisms.

Reference:

Gary Urton. The Social Life of Numbers: A Quechua Ontology of Numbers and Philosophy of Arithmetic. University of Texas Press, 1997.
Basque Numbering...Séamas ó Direáin

There are four orders of number representation: cardinal number not followed by a noun (maoluimhreacha), primary cardinal (bunuimhreacha) personal cardinal (pearsanta) and ordinal numbers (orduimhreacha). These are simple to relate to because they have an equivalent in most other languages.

Number class	English	Irish
Primary numerals* (Primary cardinals)	Zero, one, two...	Maoluimhreacha M F
Cardinal	Zero (things), one (thing), two (things)...	Bunuimhreacha M F
Personal/associative cardinals	Zero (persons), one (person), two (people)...	Uimhreacha Pearsanta M F
Ordinals	Zeroth, first, second, third...	Orduimhreacha M F
*	Primary cardinal (maoluimhreacha) are distinct from personal/associative cardinals in that they are spoken differently

Primary numerals have no attached noun - as in simple counting.

Náid, aon, dó, trí...

Primary numerals become cardinal numbers when counting things or animals. (Animals are sometimes counted using personal numbers as a form of endearment).

Ceann ar bith, ceann amháin, péire, trí cinn, ceithre cinn, cúig cinn...

Personal cardinal numbers, as the name suggests, are used for counting people.

Duine ar bith (nobody, 0 people), duine (1 person), beirt (2 people), triúr, ceathrar, cúigear...

Ordinal numbers are also followed by a thing/animal/person but assigns orders, for example:

Náidiú (zeroth), céad (first), dara (second), tríú, ceathrú, cúigiú...

Primary numerals

Placing "a" before the number as in "a sé" or "fiche a seacht," to the letter of the Standard is completely up to the discretion of the writer of Irish; although it is the Standard written form, it is, nonetheless, rarely found organically in speech. Standard does not apply to speech when a canonical form exists, as in this case.

The primary cardinal numerals are as spoken as follows:

One to hundred²⁶
0	Náid M F or Neamhní M F	7	seacht M F	14	ceathair déag M F
1	haon M F	8	hocht M F	15	cúig déag M F
2	dó M F	9	naoi M F	16	sé déag M F
3	trí M F	10	deich M F	17	seacht déag M F
4	ceathair M F	11	haon déag M F	18	hocht déag M F
5	cúig M F	12	dó dhéag M F	19	naoi déag M F
6	sé M F	13	trí déag M F	20	fiche M F
Twenty to one hundred...
21	fiche haon M F	29	fiche naoi M F	64	seasca ceathair M F
22	fiche dó M F	30	tríocha M F	70	seachtó M F
23	fiche trí M F	31	tríocha haon M F	75	seachtó cúig M F
24	fiche ceathair M F	40	ceathracha M F	80	ochtó M F
25	fiche cúig M F	42	ceathracha dó M F	85	ochtó cúig M F
26	fiche sé M F	50	caoga M F	90	nócha M F
27	fiche seacht M F	53	caoga trí M F	96	nócha sé M F
28	fiche hocht M F	60	seasca M F	100	céad M F

Above one hundred...
116	céad sé déag** (céad a sé déag)
125	céad sé déag** (céad fiche a cúig)
200	dhá chéad
300	trí chéad
400	cheithre chéad
500	cúig chéad
600	sé chéad
700	seacht gcéad
800	ocht gcéad
900	naoi gcéad
1000	míle
15,000	cúig déag míle ** (cúig míle dhéag)
a million	milliún
two million	dhá mhilliún
a billion	billiún
two billion	dhá billiún*
Not lenited since, if lenited, dhá bhilliún* would be indiscernible in speech from dhá mhilliún

** This form is not Standard, but is widely used, which is why it is included here.

Cardinal Numbers with things and people

Whereas primary cardinal numbers count simply numbers without any associations, personal cardinal numbers enumerate things, usually nouns.

#	Counting words that begin with a consonant		English equivalent	Counting words that begin with a vowel		English equivalent
0	rud ar bith F M		not a thing (nothing)	eala ar bith F M		no swan
1	aon chat amháin F M / cat amháin F M		a (single) cat	asal amháin F M / aon asal amháin F M		one donkey / one single donkey
2	dhá mhaide F M		two sticks	dhá eas F M		two waterfalls
3	trí chóta F M		three coats	trí úll F M		three apples
4	ceithre fhuiseog M		four skylarks	ceithre aisling F M		four daydreams
5	cúig nead M		five nests	cúig imeacht F M		five departures
6	sé bhuidéal F M		six bottles	sé úrscéal F M		six novels
7	seacht gcat F M		seven cats	seacht n-eala F M		seven swans
8	ocht mbuidéal F M		eight bottles	ocht n-asal F M		eight donkeys
9	naoi nead F M		nine nests	naoi n-eas M		nine waterfalls
10	deich gcluiche F M		ten games	deich n-imeacht F M		ten departures
11	aon túr déag / haon déag túr		eleven towers
12	dhá ghníomh déag / dó dhéag gníomh M		twelve actions
At this stage, cardinals are resembling primary numerals, as described above, but let's keep going to refresh...
Cardinal Numeral		Gaeilge			English
1,000		míle bliain			a thousand years
€ 2.5 M		dó punca cúig milliún euro dhá mhilliún go leith euro			two point five million euro two and a half million euro
3 10⁹ € 3,000 M		trí fá deich i gcumhacht a 9 trí mhilliún milliún / trí billiún			three times ten-to-the-(power of)-nine three thousand million / three billion
1684 CR		sé déag ochtó ceathair Comh-Ré			sixteen eighty four CE (common era) / AD
800 RCR		ocht gcéad Roimh Chomh-Ré			eight hundred BCE (before common era) / BC
45		ceathracha cúig céim			forty five degrees

Notice that nouns preceding consonants 1—6 are lenited when they begin with a lenitable consonant (as we know already, nouns beginning with vowels do not take lenition). Nouns that number 7—10 are eclipsed, whether they begin with a vowel or consonant. For example:

dhá chat...sé chat

seacht gcat...deich gcat

Personal cardinal numbers

Exactly like normal cardinal numbers, except specific to counting humans.

Zero to twelve...
0	duine* ar bith F M / bán F M
1	duine F M / duine amháin F M
2	beirt F M
3	triúr F M
4	ceathrar F M
5	cúigear F M
6	seisear F M
7	seachtar F M
8	ochtar F M
9	naonúir F M
10	deichniúr F M
11	aon duine dhéag F M / duine dhéag F M
12	dháréag F M / beirt duine dhéag F M
* duine incorporates any human noun such as fear, bean, buachaill, cailín, gearrchaile, comhluadair and so on

After the 12^th personal cardinal, counting can revert to primary numerals or remain cardinal until 99, as laid out in the table below. Both forms are found in speech.

	Cardinal Descriptor*
Number	Primary numeral description	Cardinal description
13	trí déag duine F M	trí dhuine dhéag
20	fiche duine F M	fiche duine
21	fiche a haon duine F M	duine is fiche
30	tríocha duine F M	? same as
32	tríocha a dó duine F M	beirt dhuine is tríocha
40	ceathracha duine F M	? same as
43	ceathracha trí duine F M	triúr dhuine is ceathracha
50	caoga duine F M	? same as
54	caoga a ceathair duine F M	ceathrar duine is caoga
60	seasca duine F M	? same as
65	seasca cúig duine F M	cúigear duine is seasca
70	seachtó duine F M	? same as
76	seachtó sé duine F M	seisear duine is seachtó
80	ochtó duine F M	? same as
87	ochtó a seacht duine F M	seachtar duine is ochtó
90	nócha duine F M	? same as
98	nócha a hocht duine F M	ochtar duine is nócha
100	céad duine F M
revert completely to primary numerals after this point...
109	céad is a naoi duine F M
560	cúig chéad seasca duine F M
1,333	míle trí chéad tríocha is a trí duine M
a million	milliún duine M
two million	dhá mhilliún duine M
a billion	billiún duine M
two billion	dhá billiún duine M
* Both of these descriptors for personal numbers above 13 are valid because the semi-cardinal description is more common in speech and formal writing. However, the cardinal description is increasingly common in speech and is more amenable to reading alpha-numeric text (such as "...13 duine...").

Exceptions

There are a few—albeit very commonly used—words that are exceptions to the ordinal system: bliain, ceann, cloigeann, seachtain, uair. (Less important are the archaic troigh (foot length), orlach (inch), pingin (penny)). For the sake of simplicity (a necessity, I think, at the end of a complicated number system), these discrepancies have been deferred to the Appendix.

Ordinal numbers

Ordinal numbers indicate ordering. Ordering things/animals and people differ in one instance

#	Spoken
0^th	an náidiú + F M
1^st	an aonú + F M / an chéad + F M
2^nd	an dara F M / an dóú + F M
3^rd	an tríú + F M
4^th	an ceathrú + F M
5^th	an cúigiú + F M
6^th	an séú + F M
7^th	an seachtú + F M
8^th	an t-ochtú + F M
9^th	an naoú + F M
10^th	deichiú + F M
11^th	an aonú + déag F M / an aondéagú + F M
12^th	an dara + déag F M / an dódhéagú + F M
100^th	an céadú + F M
124^th	an céad fiche ceathrú + F M
1,000^th	an míliú + F M
1,076^th	an míle, seachtú séú + F M
1,000,000^th	an milliúnú + F M
1,000,000,000^th	an billiúnú + F M
Cardinal personal cardinal	Inserting a noun/subject ( +) after the order, as in the table, changes the number from cardinal to personal cardinal i.e. an naoú duine déag or an naoi déagú duine, an céadú duine

From 11 upwards, it is acceptable to revert to the cardinal system and add a ú to indicate the number is ordinal. One can also use the semi-cardinal system as laid out in Table[dexcriptor] except ú is added to the end of the lowest prime(? As in 1,2,3...) number to indicate ordinal. The table below is an exact reproduction of the personal cardinal numbers table for numbers above 13, except with only ú added to change the character of the number from personal to ordinal. Of course, unlike personal numbers, 1 and 2 will retain the specific ordinal characteristics (aonú/céad, dara/dóú), which are unique to the ordinal system.

	Ordinal Descriptor*
Number	Primary numeral description**	Cardinal-ordinal description
13	trí déagú duine M	tríú duine déag
20^th	fichiú duine M	fichiú duine
21^st	fiche a haonú duine M	an aonú duine is fiche***
30^th	tríochú duine M	? same as
32^nd	tríocha a dóú duine M	dara (dóú) duine is tríocha
40^th	ceathrachú duine M	? same as
43^d	ceathracha a tríú duine M	triú dhuine is ceathracha
50^th	caogú duine M	? same as
54^th	caoga a ceathrú duine M	ceathrú duine is caoga
60^th	seascú duine M	? same as
65^th	seasca a cúigú duine M	cúigiú duine is seasca
70^th	seachtú duine M	? same as
76^th	seachtó a séú duine M	séú duine is seachtó
80^th	ochtú duine M	? same as
87^th	ochtó a seachtú duine M	seachtú duine is ochtó
90^th	nóchú duine M	? same as
98^th	nócha a hochtú duine M	ochtú duine is nócha
100^th	céadú duine M
109^th	céad is a naoú duine M	? same as
119^th	an céad naoi déagú duine M	an céad is naoú duine déag
560^th	cúig chéad seascú duine M	? same as
1,333^th	an míle trí chéad tríocha a tríú duine M	? same as
millionth	a milliúnú duine M	? same as
second millionth	dara milliúnú duine M	? same as
billionth	an billiúnú duine M	? same as
second billionth	dara billiúnú duine M	? same as

* Again, both of these descriptors for ordinal numbers above 13 are valid, because, although the semi-cardinal description is more common in speech and formal writing, the cardinal description is increasingly common in speech and is more amenable to reading alpha-numeric text (such as "...13^ú duine..."). The cardinal description simply means adding ú to the cardinal number to indicate number ˙order * aonú duine is fiche: [is] is not the linking verb. Is, sometimes used in Irish as in this instance, is an abbreviation of agus

When writing ordinal numbers using digits, for example a date, the number is written without any modification. For example, the 4^th of January is written 4 Eanáir. Even in a sentence, we write:

Beidh mé as láthair ar an 23 agus 24 Feabhra

although, this would be spoken ordinally:

an tríú lá fichead agus an ceathrú lá fichead Feabhra / an fiche tríú agus fiche ceathrú Feabhra

The social life of numbers (II)

Arithmetic Relations: Orientation, Association...and Mathematical Characters

Mathematical Characters

Basic Mathematical Characters	English	Gaeilge
#	Hash symbol	haise F M
@	at symbol	comhartha ag F M
%	percentage symbol	céadchodán F M
=	equals sign	cothrom le F M
^	hat	hata F M
+	plus	móide (agus) F M
-	minus	lúide (minus) F M
x (*)	times (multiplied by)	méadaithe fá F M
&	ampersand	amparsan F M
(	open parentheses	oscail lúibíní F M
)	close parentheses	dún lúibíní F M

Basic Keyboard Characters	English	Gaeilge
!	exclamation mark	comhartha uaillbhreasa F M
?	question mark	comhartha ceiste F M
/	forward slash	tuslais F M
\	backslash	cúlslais F M
[	open square bracket	oscail lúibín cearnach F M
]	close square bracket	dún lúibín cearnach F M
"	open quotation	comhartha athfhriotail tosaigh F M
"	close quotation	comhartha athfhriotail deiridh F M
~	tilde	tilde F M
*	star symbol	réiltín F M
ABC...	uppercase letters	ceannlitreacha F M
abc...	lowercase letters	litreacha cás íochtair F M

Computer Keyboard Characters	English	Gaeilge
SHIFT	shift function	imlaoid F M
FN	function key	feidhmeochair F M
COMMAND	command key	eochair ordaithe F M
ALT/ALT Gr	Alt key/Alt Gr key	eochair Alt/ eochair Alt Graif F M
ENTER	enter	iontráil
ESC	escape	éalaigh
TAB	tab	táb
DEL	delete	scrios

Chirality (symmetricality)

Deiseal, tuathal (ciotach)

Tuathal: An taobh tuathail ("leftfield"), ina thír tuathail (chaotic, crazy): can be used positively and negatively, for example,

Bhí an cóisir ina thír tuathail, a mhac (the party was mad, man)

Good foot and goofy foot--terms used to describe left and right footedness in surfing and snowboarding: deiseal (good/right footed), tuathal (goofy footed)

To go around something (tháinig mé thimpeall air) this can also mean I solved a problem

To stay inside something (d'fhan mé taobh istigh de)

In sailing terms, to go inside something or around it is faoi (inside) and os a chionn (around). [map]

Clockwise: bealach na gréine (since the sun travels clockwise around the earth in the northern hemisphere), deiseal

Anti-clockwise: in aghaidh na gréine, tuathal

Describing consanguinity

Direct mathematical relationships—the degree of separation—are used to describe relationships of parallel generations (cousins). In English, degree of relatedness is denoted by ordinal description, that is, first cousin second cousin, and so on.

[Need map here of notional family tree showing generations in layers and relations of col ceathrar, col cúigear]

Colors

Colors are no more complicated than in other languages, if you ignore the fact that some colors have subjective, or mood associations. For this reason, colors, as we would see on a color palette are described under the column, "Scientific Colors." The "Subjective Color/Mood" column describes a concept that is associated with the color, giving the word a separate meaning if used in the context of that concept. An English-language equivalent would be "I'm feeling blue," or "a blue note," are doing the same thing: associating a mood or a movement to the color, but on their own, the colors mean just the colors, in most European languages. Le Gaeilge, certain colors are dual-functional, but never intone more than one meaning in a single usage

Color	Scientific Color	Subjective Color/Mood	Color can describe skin/hair/complexion
Uaithne	green	none	no
Glas	green (can denote grey too)	"pseudo" i.e. Glasoileán (tidal island) glasaimsir (undefinable season in weather)	no
Liath	gray	none	hair
Dubh	black	depression	hair only
Gorm	blue	black (skin)*	Sub-Saharan African (black) skin color
Rua	red	none	red hair rust color dark beer
Dearg	red	none	no
Geal	relative brightness, daylight	bright light cheerful	skin, hair and demeanor (gealgháireach)
Dorcha	dark	none	hair
Bán	white	empty	no
Color Tone
Breac	speckled	fresh (new) brackish (semi-saline)	speckled complexion or general coloring, such as a salmon, for example
Some Common Colors With No Duality
Dath	Color
Buí	yellow
Oráiste	orange
Bán dearg	pink
Donn	brown
Corcra	purple
Maigeanta	magenta
Cian	cyan
*	In contemporary speech, people have starting using dubh to describe black skin. This has no usefulness, as it causes ambiguity between, skin color and hair color, as well as assigning a word that has historically negative connotation to a race.

Describing direction

Direction can be described from both stationary and non-stationary (dynamic) positions/points of view.

In Sailing terms (ancient to modern), there are two very important directional orientations. The first is where you are traveling to (Course to Steer) and the second is where you are coming from (Estimated Position). Estimated Position requires that we calculate the direction we are coming from. In English it is usually only common use the direction you are coming from when describing wind direction (said differently, north westerly wind comes from the northwest, it is not travelling in a north-westerly direction it is travelling in the opposite southeasterly direction). In Maritime terms, understanding both these concepts are very important to passage planning as both determine differently which vecor varibles you must calculate. In Irish, directionality of travel is equally important in both perspectives, that is, where you are headed and where you are coming from. Does this point to a strong maritime-oriented culture of which the language is a left-over? Who knows, we'll leave that question to someone who has more time for that kind of question, like Bob Quinn [Quinn 2005], perhaps! Hopefully, the illustrations below will show you exactly how this works in practise.

Bearing (path taken or already followed):

In the image above, imagine our little man in the middle, Maitias, being the center of focus. For a person outside the circle describing his movements from the point of view of his origin and not his end point²⁷, his position when he reaches the edge of the circle is related as a function of his starting point (exactly as wind direction, that is which direction Maitias came from).

So, if you are standing outside the circle, at the following positions, these will be Maitias's transits or bearing:

00:00 hrs (12 o'clock): Aneas: Tá Maitias ag teacht aneas/ tháinig sé aneas/Tháinig mise aneas

04:30 hrs (4:30 o'clock): aniar aduaidh: Tá Maitias ag teacht aniar aduaidh/ tháinig sé aniar aduaidh/Tháinig mise aniar aduaidh

Heading/bearing (destination):