Counting systems

1. Counting systems • Much loved at IOL (Eastern European tradition of “mathematical linguistics”) • Typically every other year (last time 2017) • Often involve some simple arithmetic  • And some semantics • Decimal system is very familiar, but you can be absolutely certain that, whatever it is, it won’t be base-10

2. Features of counting system • Most are base-10 for obvious reasons • But many have other bases, often mixed • Words may not be in the most logical order • Cf German etcdreiundvierzig 3+40 = 43 • Often, a different base is used for counting different things • Look out for “counting back” • Cf Roman numerals IV, XC etc. • Or German clock time, “halbvier”, “dreiviertalacht”

3. Some familiar examples • English has special words for 11, 12, has one system for teens (U+T), another for the rest (T+U), but is quite regular • French is decimal until it gets to 70 (soixante-dix, soixante-et-onze, etc.) =60+10, 60+11, then 80 (quatre vingts ie 4x20), and 90 (quatre-vingt-dix ie 4x20+10) • Counting in 20s (“vigesimal”) is quite widespread, • even English has “four score and seven”

4. Example 1: Danish 1 en 2 to 3 tre 4 fire 5 fem 6 seks 7 syv 8 otte 9 ni 10 ti 20 tyve 30 tredive 40 fyrre 50 halvtreds 60 tres 70 halvfjerds 80 firs 90 halvfems 23 treogtyve 47 syvogfyrre 71 enoghalvfjerds ?? seksoghalvtreds 95 ???? 100 ???? 100 hundrede 

5. Dozens • “Dozenal” systems • Old English/Irish monetary system was 12- and 20-based • Makes calculation easier (divide by 2,3,4,5,6,8,10,12,16,20,25,etc) • In Nimbia (Nigeria) multiples of 12 are the basic number words around which everything else is built. The number 29 is gume bi nibiyar((12x2)+5), and 95 is gumebo'onikwada((12x7)+11). • This and other examples from http://mentalfloss.com/article/31879/12-mind-blowing-number-systems-other-languages

6. Example 2: Welsh (trad) 20 ugain 25 pump ar hugain 30 deg ar hugain 31 un ar ddeg ar hugain 40 deugain 41 deugain ac un 50 hanner cant 60 trigain 70 deg a thrigain 11 un ar ddeg 12 deuddeg 13 tri ar ddeg 14 pedwar ar ddeg 15 pymtheg 16 un ar bymtheg 17 dau ar bymtheg 18 deunaw 19 pedwar ar bymtheg

7. Example 3: Yoruba 11 ọ̀kanlá 12 èjìlá etc 15 aárúndíógún 16 ẹẹ́rìndílógún 17 eétàdílógún 18 eéjìdílógún 19 oókàndílógún 1 ọ̀kan 2 èjì 3 ẹ̀ta 4 ẹ̀rin 5 àrún 10 ẹ̀wá 20 ogún 30 ọgbọ̀n 25 = 30-5 35 = 20x2-5 130 = 20x7-10 Some other complexities Also, different words for counting objects

8. Weird bases • Ndom (Papua New Guinea), has a base-6, number system. • It has basic words for 6, 18, and 36 and other numbers are built with reference to those. The number 25 is tondor abo mer abo sas (18+6+1), and 90 is nif thef abo tondor ((36x2)+18) • Huli (also PNG) uses a base-15 system. • Numbers which are multiples of 15 are simple words. Cf the English word for 225 and the Huli word ngui ngui, 15 15s. 80 in Huli is ngui dau, ngui waragane-gonaga duria ((15x5)+the 5th member of the 6th 15).

9. Mixed systems • In Bukiyip, another PNG language, there are two counting systems, depending on what you are counting. • Coconuts, days, and fish counted in base-3. • Betel nuts, bananas, and shields counted in base-4. • The word anauwip means 6 in the base-3 system and 24 in the base-4 system.

10. Oksapmin (NewGuinea)

11. How to solve counting system questions • Forget base-10, it will be something weird • They are often posed in the form of equations • Use your mathematical skills • Eg one question involved (some) sums of squares, but all number names were less than 10 – so you knew the words which were squared must be <4 • Also, some morphophonemics maybe involved • Expect the unexpected

12. Writing systems • Representation of spoken language • Always derivative • All have pros and cons • Often borrowed from somewhere else • A few examples are purpose-built by linguists • Following typology a bit misleading: most writing systems are hybrid

13. Types of writing system • Pictographic • Elements are pictures, combined in graphically-interpretable patterns (e.g. temporal sequence or spatial relationship) • Ideographic • Elements denote ideas, combined in a logical fashion • Logographic • Elements denote words or morphemes, combined morphosyntactically • Syllabic • Elements denote syllables, combined phonologically

14. Types of writing system • Moraic • Like syllabic elements, but units are a bit smaller • Alphabetic • Elements denote phonemes (more or less), combined phonologically • Featural • Elements denote distinctive features of phonemes (such as voicing or place of articulation), combined phonologically • Also worth mentioning: direction?

15. Issues • Logographic • Typical example Chinese • Logographs typically contain a semantic element and a phonetic element • eg 枘 枚 枝 枯 枰 柙 all names of trees (木) • Some characters used phonetically (esp eg in transliterations of proper names), or as grammatical markers

16. Issues • Alphabetic • Often borrowed from another language, so compromises are necessary • Typical example: English uses Latin alphabet but has (many) sounds that Latin lacks, and so has to use letters inventively (sh, th, ch, wh, ee, ow, ...) • Very rarely truly phonetic, 1:1 correspondences • Same letter/sequence has different pronunciation • A given sound can be written in different ways

17. Brief digression • How logical (or not) is the English writing system?

18. Issues • Syllabic • May be systematic, eg Inuktitut

19. Issues • Syllabic • Or not, eg Japanese

20. Issues • Featural • Usually the result of linguistically sophisticated planning • Examples: Korean, Visible speech

21. How to solve writing system questions • Depends on what information you are given • Always question your assumptions, eg is it alphabetic, LtoR, etc. • Look for singletons • Look for high-frequency symbols or sequences • Look for “minimal pairs” • Bear in mind general “phonotactics”