Re: Types of numerals

>From: tomhchappell <tomhchappell@...>
> > 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?
Well, fairly "random-looking" numbers might be hard to put into any one
"natural" series, but then again, they're almost always expressed with two
or more numerals - that is, "9/16" will be derived from "9" and "16" rather
than "4" plus "1-1/x" and "x^2" morphemes (and that was still easy to reduce
to one natural number!)
Mathematical constants I guess will have to be an exception, especially
since e and pi are both transcendental...


> > .: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;
I may be misusing the term "lexically open"... Nevertheless you seemed to
figure out what I meant.


>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.
O_o
Are you ruling compounds out too? Because even the most number-limited
languages I've heard of (with only 1 and 2) allow "linear" number systems,
where eg 8 = 2222.
(Also interpretable as a two-level base system with base 1&1.)


> > 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.
Yes, this is true - but all languages I know have a higher number of
primitive cardinals than ordinals.


> > 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.
Yep, I can see that.


> > 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.
It does if for "need" you read "require in order to be remotely
naturalistic" rather than "logically imply".


> > In fact, all languages I know of have 2 per series tops,
>
>What does that mean, exactly?  I don't understand.
>(*)
That there are no primitive non-cardinals relating to numbers other than 1
or 2.


> > 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?
Oops, yes, I meant trial as a grammatical number. Now, if the trial marker
is unrelated to the word for "three", then we have our first example of a
primitive number morpheme relating to 3. And going on a more Sapir-Whorfy
line of thought, a language with a trial would likely have more words,
primitive or not, relating to 3, than one without. I argue that this would
also increase the likelihood of having more than one primitive one.


> > 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?
"Every halfth year" = "twice a year"
"The six-and-a-halfth" = "the one between the 6th and the 7th"
So this applies when a sequence has "whole" members and "half" members.
Eg. some areas of the world (mostly plains of some sort) have a climate with
two rainfall seasons per year; so something that's gone thru an odd number
of them definitely in its X-and-a-halfth year.

Might be also used idiomatically to something that's not exactly a half of
the whole, eg. a band releasing an extended single could also be said to
release their X-and-a-halfth album.


>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.
The reciprocal is still not ordinal in *meaning*, so just toss another
ordinal affix in. Eg 1/3th -> "thirdth". Another example in Finnish:
"kolmasosas" ("kolme" = 3, "-s" = ordinal, "osa" = part.) The final ordinal
affix tags to the last part of the expression, the word for "part" in this
case.
(Finnish does have a "more lexicalized" version of "1/3" too, employing the
explicit reciprocal affix "-nnes", but then the meaning somehow gets a
little hazier.)


> > 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.
Might be, but they're then either so obscure that I've never come across
them or the meaning of "trio" is archaic and not recognized any more.


> > .: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.
Well, yes, they're closer related to the natural ordinals than the natural
cardinals in that sense; but you still end up with cardinal real numbers. As
covered above, "halfth" is already non-trivial; what would you think of
"eth" or "negative fourth"?
(I do not know the precise set theoretical definition of "ordinal", but I
suspect it might deviate from its linguistic definition a little here.)


> > 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.
Certainly. Hence "exponents of the base number" and not "exponents of 10".


> > 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.
I mean a name unrelated to "1". See my reply to Carsten:
http://listserv.brown.edu/archives/cgi-bin/wa?A2=ind0601a&L=conlang&F=&S=&P=10083


>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.
Yes, that indeed works! Let's suppose a language that has an affix meaning
approx. "almost, but still missing a few". Add that to "none" and walla*,
there we have the name for -1.


> > 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.
*** looks dubious - I'd require n>2 there too. "Half" is so much more basic
a concept than "one and a half".
The other two I'm not going to object to.


> > .: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".
Yes, clearly compounds.
BTW, what does that apostrophe represent, elision or blurred speech or what?
I've seen similar "one letter droppings" in other archaisms / ruralisms too.


> > 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?
No, it doesn't. Even with a non-trivial complex number like 0.466 ("zero dot
four six six"), one can just add the suffix -onen to the last numeral of the
expression and obtain an "ID numeral". This only works with decimal system
tho, not with fractions.

The regular forms of 8 and 9 have been supplainted by forms that, for 2-7,
mean "duo" thru "septet".


> > What other series are you aware of? I might be overlooking some
> > obvious one.
>!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".
Nice!

>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?
That's the only one I know of. But yes, it still suggests a new series.
Particularily interesting is how these relate to the group names in some
situations with regards to definiteness:
"____ pears were thrown away."
"A pair of" implies two wholly indefinite pears.
"The pair of" implies be two wholly definite pears .
"Both" implies an indefinite group of two definite pears (considered a pair
for the first time!)

(The opposite, a definite group of two indefinite objects, is probably
impossible, or at least very rare - how can you perceive a physical set
without perceiving any of its members?)


> > 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.
Also: "a few times" - for the "repeat" series.

"To divide", "multiply", "lessen" fit for the verbal series you propose
below, but they don't seem particularily random-quantitive rather than
simply qualitative. Same for "group" and "all" and their associated
numerals.


>How about verbs?
>
>English has "double" and "treble" (or "triple"), for making the object
>bigger or more numerous by a factor of two or three, respectively.
You're right! I have been completely overlooking verbs.
Someone mentioned (in a message I can't find again just now) that the
ordinals are essentially adjectival and the "repeat series" adverbial.
Cardinals seem to act as adjectives ("four tangerines on the table and three
in the bowl") and pronouns ("the four are to be eaten"). Theoretically,
these two meanings could be split apart - these new pronouns having an
interesting relation to your "all x of" words. Compare "both children got a
present" <> "the two children got a present". Some sort of part - whole
distinction. What was it called again? I remember this was discussed at some
point just last fall.

Well, the verbs you bring up are of the form "to make something into x" or
"to become x" (no tr/itr disctinction). What other verbing affixes are
relevant here? Any ideas?


>English also has the set phrase "half again as ..." for increasing some
>attribute (perhaps size or quantity) by a factor of 1.5.
Never came across it before, but "make 1+1/x" seems just as useful (or
unuseful) as the "make 1-1/x" series you're suggesting below.


>English also has the verbs "to half (smthng)" and "to quarter (smthng)"
Same series, interbred with reciprocals.


>and "to decimate (smthng)".
>
>"To decimate smthng" means "to reduce smthng by removing one-tenth of it".
Never heard this particular meaning either...


>Wouldn't it make sense for a language having verbs equivalent to "half"
>and "quarter", to have a verb for "to divide smthng into three equal
>pieces"?
>
>Wouldn't it make sense for a language having verbs equivalent to "half"
>and "quarter", to have a verb for "to reduce something to one-third as
>[attribute] as it was before"?
>
>Wouldn't it make sense for a language having verbs equivalent to "half"
>and "decimate", to have a verb for "to reduce something by removing one-Nth
>of it" for N=3, 4, 5, 6, 7, 8, or 9?
The last two are easy - just add morphemes for "1/x", or "1-1/x" and "to
make smTN into x". The first makes sense too, even if it's less trivially
derivable.
Of course derivation isn't absolutely necessary. The English words are
exactly the same as the reciprocals - and I've heard "third" used in the
first sense too: "We thirded the income of the show." "The pie was thirded."
etc.


> > ---------- Forwarded message ----------
> > From: Mark J. Reed <markjreed@...>
> > Dadgum linguists -  always messin up language.
> > Say they're just being descriptive, but at the
> > same time they're modifying whatever language
> > they're doing their describing in.
> > :)
>
>Uh-huh.
>What Heisenberg's Uncertainty Principle ("you can't observe something
>without affecting it") is to Quantum Physics, the following is to
>Linguistics; "you can't use a language without changing it".
I might say that that's why I am here in the first place - making a
personalang to minimalize the damage :)


>Tom H.C. in MI
John Vertical
*Misspelling intentional.