Re: Types of numerals

--- In conlang@yahoogroups.com, John Vertical <johnvertical@H...>
> wrote:
> I've been thinking about numerals lately. Particularily, of all the
> possible different types of them. So here goes loads of rambling on
> the topic. Feel free to steal and/or shoot down any ideas
> contained. Commenting on them I even welcome. :)
> .:DEFINITIONS:.
> First off, I confess that I not sure if you'll recognize "numeral"
> as a word for the class of "number words" (never seen it used in
> that way in English; only Finnish.)
Yes, linguists writing in English use "numeral" to mean
"words for numbers", to distinguish from "number" usually short for
"grammatical number" (singular, plural, etc.)

> If it actually is something else instead, do tell. Also, whenever
> I'm talking about "series", I mean an ordered infinite series where
> each member is related in meaning to the respective natural number.
> By my definition, all numerals must belong in some series (won't be
> much of a numeral otherwise).
Including fraction-words and/or mixed-fraction-words?

> .:BASIC SERIES:.
> Every language probably has the two basic series - the natural
> cardinals (one, two, three...) and the natural ordinals (first,
> second, third...) But is it always the former which is the open
> lexical category?
Both of them technically have to be "open" in order to accommodate
numbers that have never before been used; but in some languages one or
the other is not, in fact, and "open" (sub-)category.  In these
languages there is a highest number one can count to.

> Does any language have ordinals as the unmarked series instead?
I don't know of any language which has _all_ ordinals "marked"
(especially in contrast to cardinals); but as I understand it some
languages do have some ordinals that are just as "primitive" as their
corresponding cardinals.

(In set theory, as opposed to language, it is, in fact, the ordinals
that are primitive.  Ordinality is intrinsically defined, whereas
cardinality is extrinsically defined; it is possible to have two
different standard transitive models of set theory which will
(perforce) have exactly the same ordinals, but an ordinal which
happens to be uncountable in one, happens to be countable in the
other, because the function matching it with "lowercase omega" is
available in the second model, but not in the first.)

> I presume that another universal feature is that while numerals are
> an open class (theoretically more open than any other word class -
> but lets not go there now), after a certain threshold, all words
> relating to a certain number will be derived similarily.
This is covered in some book either written by or edited by Greenberg.

What you say is true for those languages which "can handle"
(i.e. have words for) the biggest numbers.

In other words; up to a point, the statistical implicational
universal is, the higher numbers a language "can handle", the more
regular its lexicogeny (rhematopoeisis) for numerals will be.

> Typically there is a system to derive infinitely many cardinals and
> a system to derive other series from them.
That seems right.

> However... isn't it theoretically possible to have more than
> one "root series"?
_I_ don't see why not.

> This would probably need a base of 5 or less, given that languages
> usually have only very few non-cardinal numerals which are
> unrelated to the corresponding cardinal words.
I do not understand why you think your conclusion follows from your
premise.

> In fact, all languages I know of have 2 per series tops,
What does that mean, exactly?  I don't understand.
(*)

> but I imagine languages with
> trial as a lexical number
Do you mean, "trial as a grammatical number", i.e., on a par with
singular and dual and plural (and, possibly, paucal)?

> might have 3?
Why would the two phenomena have anything to do with each other?

> Does this happen?
1) Since I didn't understand the question marked (*) above, I don't
understand this question either.
2) I suspect that if I understood this question, my answer would
be "I don't know", since I don't know enough 'weird' natlangs.
(**)

> And in a conlang, would a little more, maybe 5, be plausible?
Plausible?  I don't know (see (*) and (**) above).
Feasible?  _I_ see no reason why not.

> Hm, what I'm proposing might be a little hard to grasp from the
> previous paragraph, so I'll construct an example using English and
> base 2. So suppose the cardinal series goes
> "one, two, onety-one...";
> but meanwhile, the ordinal series goes
> "first, second, firsty-first..." rather than "...onety-first...".
> That is, NO ordinals would be derived from the corresponding
> cardinals - but rather simpler ordinals in a way similar, but
> perhaps not identical, to how more complex cardinals are derived
> from simpler cardinals.
1) I have never heard of a natlang for which _all_ ordinals were
thus-independent from the cardinals; but
2) I see no reason to prevent it in a conlang.

> One could then split the class of numerals into
> "cardinal-derived" vs. "ordinal-derived"
> - maybe even contrasting other series purely by their roots.
> This is almost trivial to extend into mathematical series
> (half vs. halfth),
1) What does "halfth" mean?
2) Many languages derive all but finitely many unit fractions from
ordinals.  Thus, a language might have 1/2 (every language I know),
1/3 (Ancient Egyptian?), and 1/4 (English) may have lexically-
independent forms (in fact 1/2 usually does, and not only 1/3 and
1/4, but often 2/3 (Ancient Egyptian?) and 3/4 (no example comes to
mind), do as well); but "1/n" for "n" > 4 might denoted by a word
which is a short form of the phrase "the nth part of" (English, and
every other language I know how to express fractions in).  If that
were the case for your language, how would a speaker speak of "the
qth member of a series" when "q" is a fraction?  You'd have to be
able to derive an ordinal from an ordinal-derived numeral.

> but it might be possible to carry it over to grammatical series
> too - eg. contrasting the (cardinal-derived) word "trio" with an
> (ordinal-derived) word meaning maybe something along the lines
> of "third member of a trio".
Interesting and useful idea!

You should know, however, (as I suspect you do), that English, for
example, has many words for "a collection of two (somethings)"; the
words depend largely on what the "somethings" are, but for
many "somethings" there is more than one word denoting "two
(somethings)".
These words are often morphologically independent both of the noun
for the singular and of any other form of the numerals "two"
or "second".  In other words, they are totally suppletive.
English also has many words for "a collection of (many somethings)";
these are called "collective nouns", and are likewise completely
suppletive.  But the words for pairs, braces, yolks, etc. are
morphologically independent of the collective nouns, as well.
I believe English has a few words meaning "a set of three X" for just
a few particular nouns "X".  I can't think of them at the moment.
Other English words for "n Xes" are regularly derived from source-
language borrowed words for "n"; duet, trio, quartet, quintet,
sextet, octet, etc.

> .:A MATHEMATICAL P.O.V.:.
> "Mathematical series" are technically still cardinal series, formed
> by filtering the natural numbers {0, 1, 2, 3...} thru some random
> function.
I think it may make more sense to think of them as functions whose
domain-of-variation for their argument is the finite _ordinals_,
rather than the finite _cardinals_.  "X sub n" represents the nth
member of the series X; it does not represent n of anything.

> AFAIK, only reciprocals (half, third, quarter...) and exponents of
> the base number (ten, hundred, thousand...) are lexical anywhere.
Not forgetting that bases of four, five, twelve, and twenty, are also
pretty common compared to base-ten, according to Greenberg's book.

> Unusually geeky loglangs might have more, but even then, I doubt
> whether expressing eg. -6 as something along the lines of "unsix"
> would be useful.
> ...And speaking of negative numbers, why doesn't -1 have a name on
> its own, but i does?
But it does; "-1" is its name.

Actually many languages have a way to express missing-a-few or
lacking-a-few.  King James's version of one of St. Paul's letters
translates his expression as "forty lashes save one", i.e., a certain
number of times he was sentenced to receive 39 strokes with a scourge.

> There are also often a handful of numbers which have an original
> name in addition to a derived one. Most of the ones I know have
> been used as units of measure (eg. Finnish "tiu" is a unit of 20
> eggs), but are there others?
You must mean like "dozen"=12 and "score"=20.
English has several compounds
with "score"; "twoscore"=40, "threescore"=60, "fourscore"=80, etc.
I would be willing to bet that most small numbers have such a word in
some natlang or other; and, that most products of two such numbers
(especially most squares of one of them) also have such words in
natlangs.  An example is English "gross" for a dozen dozens.

> Eg. is the Latin prefix sesqui- really a *root* morpheme?
In English it is monomorphemic (which is what I think you meant),
although not really a _root_;
but in Latin it was a contraction of "semis que", where "que" is the
enclitic form of "and"; "semis que" means "half and".
Among English prefixes: "sesquidi-" means "two and a
half"; "sesquitri-" means "three and a half"; etc.

> If yes, I could imagine lexicalizing other simple fractions too,
> like 2/3 and 3/4.
"Two-thirds" was indeed "lexicalized" in Ancient Egyptian, so I've
read; and I wager "three-fourths" is lexicalized in some natlangs, as
well.

In fact, I think the following to be likely "statistical
implicational 'universals'";
*** For each n, many language which have a special word for 1/n, will
also have special words for 1-(1/n) and 1+(1/n).
**** Languages which have their own words for 1+(1/2) and 2/3 are
probably much more consistent about satisfying *** than other
languages.
***** A language which satisfies *** for a natural number n>2,
probably also satisfies *** for most or all natural numbers m where
1<m<n.

The above predict that:
:Most languages (such as English) which have a special word ("half")
for 1/2, should have a special word for 1+(1/2).
::Most languages which have special words for 1+(1/2) and 1/3, should
have a special word for 1+(1/3).
:::Most languages which have special words for 2/3 and 1/4, should
have a special word for 3/4.

> Also I might add the golden and silver ratios
> (the latter is sqrt(2)) to uwjge...
The golden ratio solves t = 1/(t-1); it is (1+sqrt(5))/2.
Sqrt(2) is about 1.41421356...
I do not know of any natlang which has words for either of these two
numbers, nor for any irrational number; including both algebraic
irrationals like these, and transcendental like "pi" and/or the base
of the natural logarithms (about 2.18281828459045...)

> Have any of you lexicalized any unusual numbers in your conlangs?
> I'd be interested to hear.
I haven't gotten beyond the cardinal natural numbers yet.
I don't think I intend to lexicalize any irrational numbers.
As for counting numbers, I haven't lexicalized them yet, but I've
worked on it some.

I intend to use a base of twelve, and allow a system to produce words
for exact counts up to anything less than:
twelve to the power of (twelve to the power of twelve).

Basically, for large numbers, I will have a system whose base is
(twelve to the twelfth power); its "coefficients" will be drawn from
the smaller-number system of:
"numbers with base twelve, up to
anything less than 'twelve-to-the-twelfth'".

> .:OTHER NUMERALS:.
> So what other numerals are there?
> English has at least the
> "group numerals" (single, duo, trio...), the
> "repeat numerals" (once, twice, thrice...)
Well-spotted.  You're ahead of me.

> Polygons, time-period names ("biweekly") etc. are probably best
> considered compound words.
As in "fortnight" (fourteen nights) and "twelvemonth"?
Also I have seen "se'ennight" to mean "week".

> In Finnish, the simplest polygon names are derived instead
> (with the generic agentative affix -iO),
> and we also have a series which are used as
> the names of the number symbols,
In my "Math for Elementary Education Majors" courses, these were
called "numerals".  (They were made up of "digits".)

> as well as sort of pronouns for things with ID numbers...
Interesting!  The closest I've seen in any natlang is the ordinals.
The idea of "ID numbers" is too recent to have been incorporated into
any natlang developed before the 19th century, I thought; I'm
surprised Finnish has something (besides ordinals) for this.

Does an "ID number" have to be a natural number in Finnish?

> What other series are you aware of? I might be overlooking some
> obvious one.
If you're overlooking something, it isn't obvious to me yet (assuming
you have not overlooked negatives and reciprocals and powers-of-the-
base, which you did in fact mention before).

If I think of something else, I'll let you know.

!I just thought of one!
"Both" is to "two" as "???" is to "three", and so on.
In some English writing from the 19th and early 20th
centuries, "both" is used to mean "all three of", as well as "all two
of".
Might a series meaning
"all two of" (both), "all three of", ..., "all n of", ...
be useful?  How much of this series is attested in natlangs?

> There's also the possibility of adding "generic" numerals to each
> series. "Number" is essentially a generic cardinal, and "nth" might
> count for a generic ordinal... but it's a little iffy beyond those.
"Numeral" is a "generic name-of-number", as well as meaning "name of
number".
"Digit" is a "generic numeral-less-than-the-base", as well as other
meanings.
"Part" or "fraction" might be a "generic reciprocal"; it might also
be a "generic positive-number-less-than-one".  There is no reason
against polysemy, having it mean both.

> Would you think others were likely to exist?
Yes.  (I know how helpful that one-word answer is; sorry.)

> ---
> That's all I can think of now; more maybe later, if the topic
> gathers any discussion...
It did.

Thanks, John.

Tom H.C. in MI

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