Re: Types of numerals

--- In conlang@yahoogroups.com, John Vertical <johnvertical@H...>
> wrote:
>>> 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.)
>>I don't get it.  Explain?  Thanks.
> Read that other reply of mine to you, with two-level base systems
> explained?
Yes, I think that's nifty.

> Base-1 is obviously 1, 11, 111, 1111, 1111, 11111... Choose "2" as
> the 2nd-level base (it's not a power of 1, but that doesn't
> matter), with still keeping with the base 1 system, so multiples of
> 2 will be 2, 22, 222, 2222, 22222...
> The gaps only have 1 number each, so there is not room for a real
> number system there - but since the lower sub-system only has one
> digit "1" it's best viewed as a base 1 system too.
OK, I get it now.  Thanks. (Pretty neat!)

>>>> 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.
>> Good. Thanks.
>> Do you know of any professional-linguist writings that
>> might corroborate or disprove that hypothesis?
> Nope.
Too bad!  Does anyone else?

>> I still don't see "why", just because
>> "languages usually have only very few non-cardinal
>> numerals which are unrelated to the corresponding
>> cardinal words", it follows that "in order to be
>> remotely naturalistic, having more than one 'root
>> series' would probably require a base of 5 or less".
> Actually, I just realized that I was wrong. See below.
OK.

>>>>> 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 think there are, though they may be rare.
>> What about English "quarter"?
> Derived from Latin.
True; Ray Brown gave us the full etymology in his recent post.

> Descendants of Latin numbers occur so often in English
> that they can be considered separate roots.
Yes; so, the simplest English word) derived from each Latin number
(other than 1 or 2) that is not obviously related to an English
number, should be considered "a primitive non-cardinal relating to a
number other than 1 or 2".  Right?

BTW: as an example of a false folk-etymology or back-formation;
"semester" comes from a Latin phrase meaning "six months";
"trimester" comes from a Latin phrase meaning "three months".
But many people these days interpret the "sem" part of "semester" to
mean "half", and decide "semester" means "half a year"
and "trimester" means "a third of a year (i.e. four months)".

> Also, this in turn is then an example of two largish sets of
> unrelated (or opaquely distantly related) number roots in one
> language!
I think Greek and Latin versions of "one, two, three, six, seven,
nine" may appear to be related to the corresponding English versions,
but Greek and Latin versions of "four, five, eight, ten" do _not_
appear to be related to the corresponding English versions IMO.

> They aren't in any trivial semantic
> relationship, but the point still stands.
> (The Greek numbers might count too.)
I think they do.

>>> "Every halfth year" = "twice a year"
>> I think it likelier that _this_ idea would be
>> expressed as in English, "every half-year", rather
>> than "every half_th_ year".
> Probably. But you still understand "every halfth year" to *also*
> have this meaning, right?
No, not until you explained it to me.

>> BTW In cities like Galveston Texas and Washington
>> D.C., where there are streets named "A Street", "B
>> Street", "C Street", and so on, there are even
>> "D-and-a-half Street" and "E-and-a-half Street", also.
> I sometimes use foreign letters for this purpose, eg. &#1041; =
> "A-and-a-half". :)
Hmm.

>>> 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.
_This_ use _does_ make sense to me, nearly immediately.

>> Because it's too "extended" to refer to as
>> merely-a-single, but too "single" to refer to as an
>> entire album?  So we "pretend" it's "half an album"?
> Yes.
That's what I thought.

>>>>> "Mathematical series"
>> Since we're talking mathematics now, I have to tell
>> you, as I didn't tell you earlier, that in
>> mathematician-speak "series", (unless modified, as in
>> "time series"; and even sometimes when modified),
>> ordinarily refers to an (infinite) _sum_; that is, to
>> the _sum_ of a sequence.
>> The things we've up-'til-this-point called "series",
>> are usually called "sequences".
> Aha. Finnish mathematics doesn't use different root words for the
> two concepts.
That explains it.  I didn't know that.

I think German uses "unendlichen Reihe" for the infinite sums, and
something else for the sequences.

>>> but you still end up with cardinal real numbers.
>> What do you mean, "cardinal real numbers"?
>> Do you mean "real numbers are more cardinal than
>> ordinal"?
>> Because I think a real number applies to mass nouns,
>> not to count nouns; and I'm not sure a "real" number
>> is either cardinal _or_ ordinal, though I suppose both
>> a cardinal-like number and an ordinal-like number can
>> be made out of a "real" number.
> I think mass-noun-numerals are still cardinals;
> like count-noun-numerals, they describe an _amount_
> rather than a rank or order.
To me, that means they aren't ordinals;
it _doesn't_ make them cardinals, IMO.

> I don't mean to imply anything about real numbers as a mathematical
> concept.  I used the term simply because that seems to be the only
> commonly named set where it's possible to accommodate all the
> possible sequences derived from the natural numbers
> (eg e^1, e^2, e^3..)
Yes, I see.  Too me, the "real" numbers come in when you want a
_continuous_ set of measurements; that is, all the irrational and
transcendental numbers as well as the rational numbers (fractions).

>>> As covered above, "halfth" is already
>>> non-trivial; what would you think of "eth"
>>> or "negative fourth"?
>>
>> See my previous paragraph.
>> The "eth" value of some function (say "f") would be
>> f(e), that is, f(2.718281828459045...).
>> The "-4th" value of "f" would be f(-4).
>
> Yes, they can certainly be interpreted in *some* contexts as
> meaningful, but I doubt you're going to think of any *useful* ones.
Since I consider mathematics, even pure mathematics, "useful", I'll
have to ask you to change "any" to "many".

>>>>> AFAIK, only reciprocals (half, third,
>>>> quarter...) and exponents of the base number
>>>> (ten, hundred, thousand...) are lexical anywhere.
>> What about English pair, dozen, score, gross?
> They don't exactly suggest any further series of numbers to me - I
> wasn't talking about the lexicality of _numbers_, but the
> lexicality of _series_.
Aha.  You're right, that makes a difference.
None of my examples are part of a _sequence_.

> And "pair" isn't particularily mathematical anyway.
Oh yes it is! [Details withheld in the interests of space.]

>> In particular, you don't object to,
>> "For each n>2, many languages which have their own
>> "words for 1+(1/2) and 2/3, and which have a
>> "special word for 1/n, will also have special words
>> "for 1-(1/n) and 1+(1/n)."
> Correct, assuming that regularly derived words aren't "special".
I _think_ that might have been part of what I meant by "special",
or "own".

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>> In this particular case, the "'" in "se'ennight"
>> represents that the "v" is left out, but the two "e"s
>> are both pronounced.  So the "se'en" part of the word
>> is pronounced as, in length, stress, and tone, two
>> syllables.  The second "e", because it is unstressed,
>> should technically be pronounced as a schwa, but since
>> it directly follows a stressed short "e" (IPA symbel
>> [e]), it sounds very much like an unstressed and
>> lower-toned [e].
> So _all_ apostrophes are pronounced as hiatus (or schwa?)
> No risk of running into dialectal glottal stops?
Well, in Standard English, both Standard American English and British
Received Pronunciation, an apostrophe always represents "sounds left
out".  It won't usually be pronounced "schwa", and in fact usually
won't be pronounced at all.

In Standard English or Received Pronunciation, glottal stops are very
rare in legato, allegro speech, except epenthetically before
utterance-initial vowels.

Situations in which glottal stops are likely, from most likely to
least likely;
1. Epenthetically before an utterance-initial vowel.
2. Epenthetically before a sentence-initial vowel after a
sentence-final vowel.
3. Epenthetically before a sentence-initial vowel after a
sentence-final semi-vowel.
4. Epenthetically before a word-initial vowel after a word-final
vowel.
5. Epenthetically before a word-initial vowel after a word-final
semi-vowel.
6. Epenthetically before a morpheme-initial vowel after a
morpheme-final vowel.
7. Epenthetically before a morpheme-initial vowel after a
morpheme-final semi-vowel.

Situation 1 is the only one which is obligatory in all registers.
Situations 2 through 5 are increasingly likely as speech becomes
either more emphatic, or more careful, or more exaggerated for
whatever reason.
Situations 6 and 7 may be optional even in the most careful and/or
emphatic speech.

Situation 6 -- a word-internal epenthetic glottal stop between a
morpheme-final vowel and a morpheme-initial vowel -- may alternate
freely with an epenthetic semi-vowel in the same position.

Examples:
"coordinate" may be pronounced as if it were either
"co(w)ordinate" or "co(?)ordinate".
"microorganism" may be pronounced as if it were either
"micro(w)organism" or "micro(?)organism".
"milliohm" may be pronounced as if it were either
"milli(y)ohm" or "milli(?)ohnm".
"megaohm" may be pronounced as "mega(?)ohm".

But there is no glottal stop in "chaos", nor in "vacuum".

-----

Certain English dialects do have more glottal stops than SAE or RP.
In many cases these glottal stops result from "dropping an h".
In such cases it is natural to write the word with an apostrophe
taking the place of the h.  Note that in English the [h] sound occurs
almost exclusively prevocalically (just before a vowel).

In non-careful speech of some dialects, the glottal stop can fill in
for just about any stop which is part of an unaccented syllable.
E.g. "Sum?m" for "something".

-----

Some English dialects "drop a g"; what this means is, if the "ng"
velar nasal is word-final in an unstressed syllable, it is often
replaced with an alveolar nasal "n".
E.g. "everything" --> "ever'thin'"
"anything" --> "anythin'"
"nothing" --> "nothin'"
"something" --> "somethin'"
"seeing" --> "seein'"
"believing" --> "b'lievin'"
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>> BTW I understand English once had a labio-dental
>> semivowel (approximant);
> /v\/ in English? Really?
So I've read somewhere -- I don't know where.

> So where did it come from and whence it went?
(You mean, "whence did it come and whither did it go?" ;-) )

I don't know.  Does anyone else?

>> Modern English L1-speakers have no problem telling
>> that Spanish-accented L2-English-speakers are
>> substituting a bilabial voiced fricative for the
>> labio-dental "v" in English words.  Perhaps it wasn't
>> so difficult for Old English speakers to tell the
>> difference between a bilabial approximant ("w") and a
>> labiodental approximant.
> Myself having /v\/ in my L1, I can confirm this.
Interesting!  Thanks.

> It's much more difficult to hear the difference of homorganic
> fricative and approximant.
Still interesting -- still "Thanks".

> The velarization of /w/ of course also eases things up.
Yeah, I guess it would, if /v\/ is _not_ velarized.

>>>>> In Finnish, the simplest polygon names are
>>>>> derived instead (with the generic agentative
>>>>> affix -iO),
>>
>> So a pentagon is a "fiver" and a decagon is a
>> "tenner"?
>
> No, only "triangle" and "square" are derived this way.
Ah.  A triangle is a "three-er" and a square is a "four-er"?

>>>> Does an "ID number" have to be a natural number
>>>> in Finnish?
>>> No, it doesn't. Even with a non-trivial complex
>>
>> "complex", here, is used in the linguistic sense; you
>> are not talking about a "number with a non-zero
>> imaginary part".
>
> Good job managing my mishmash of linguistic and mathematical
> terminology.
Thanks.  It wasn't hard, I guess; I just felt I ought to ask you to
confirm it, rather than simply assume it "on my own hook".

>>> This only works with decimal system tho, not with
>>> fractions.
>>
>> That is, not with "common fractions" like one-third or
>> two-sevenths or four-ninths?  Clearly it works OK with
>> decimal fractions like 0.466.
>
> "Four-hundred-sixty-six thousandths" or "four tenths six hundreths
> six thousandths" certainly is a fraction,
Yes;

> but "zero dot four six six" isn't.
I think it is; I think "zero period four six six" is synonymous
with "four tenths six hundredths six thousandths".

> Actually, now that I think of this, technically fractions, if
> expressed in the form "466 per 1000", could also be used, that
> construction just doesn't appear very often.
Right.  "Percent" occurs frequently, but "parts per thousand",
"parts per million", and "parts per billion" occur less frequently.

>>> "____ pears were thrown away."
>>> "A pair of" implies two wholly indefinite pears.
>>> "The pair of" implies two wholly definite pears.
>>> "Both" implies an indefinite group of two
>>> definite pears (considered a pair for the first
>>> time!)
>>
>> Interesting analysis; definitely at least _some_ uses
>> of these phrases, _do_ mean what you've proposed they
>> mean.  You _may_ be _entirely_ correct.
>>
>>> (The opposite, a definite group of two indefinite
>>> objects, is probably impossible, or at least very
>>> rare
>>
>> Maybe so.
>>
>>> - how can you perceive a physical set without
>>> perceiving any of its members?)
>>
>> I'm not sure perceivability is related to definiteness
>> in the way you seem to be implying.
>
> Sorry, my mistake.
> And I thought of a counterexample to my point nevertheless:
> "The trio of fish comprised of a salmon, a perch, and a trout."
Oh.

> Anyway, there appears to be also several other words similar
> to "both" - principally "neither" and "either".
How are "either" and "neither" similar to "both"?

> These seem to be related to the logical
> connectivities AND, NOR and OR.
Oh, I get it.
both --- and
either --- or
neither --- nor

> NAND and XOR would be easy additions to this
> family, were they not tricky to generalize for
> more than 2 arguments.
XOR is commutative and associative -- it should present no problems
generalizing to any finite positive number of arguments.

NAND is more of a problem.

>>> "To divide", "multiply", "lessen" fit for the
>>> verbal sequences you propose below,
>>
>> You mean, as generic members?
>
> *nod*
>
>>> but they don't seem particularily random-
>>> quantitive rather than simply qualitative.
This was the part I was asking "What do you mean, exactly?" about.
What (exactly) is "random-quantitative"?
How (exactly) is it different from "simply qualitative"?

>>> Same for "group" and "all" and their associated
>>> numerals.
>>
>> What do you mean, exactly?
>
> *thinking*
> ...Well, "number" refers either, somewhat pronominally, to all
> numbers in  general ("the number of CDs"), OR to an unknown amount
> ("a number of CDs").
> "To divide" for example works well in the 1st meaning,
> but not the 2nd.  It doesn't really
> draw attraction
You mean, "draw attention"?

> to the fact that we don't know how many parts
> we're going to end up with.
> I guess this is really more of a semantic hue
> than a real distinction.
Still, could be interesting.

>>>> English also has the verbs "to halve (smthng)"
>>>> and "to quarter (smthng)"
>>> Same sequence, interbred with reciprocals.
>>
>> Not strictly; these two verbs have two meanings.
>> One meaning is "to make half" or "to make 1/4";
>> (*) but "halve" also means "two divide into two
>> (nearly) equal pieces", and "quarter" also means "to
>> divide into four (nearly) equal pieces."
>
> Hmm, true. I think this 2nd meaning is not even really directly
> reciprocal; you did write "to divide into 2 parts" and not "to
> divide into halves".
"To halve" can mean, "to divide into two equal parts", which will
then be halves, or "to divide into two nearly equal parts", which
will then be "near-halves".

Similarly, "to quarter" can mean, "to divide into four equal parts",
which will then be quarters, or "to divide into four nearly equal
parts" (as when executing a regicide),
which will then be "near-quarters".

> I mean, if we assume it IS reciprocal, what would be the natural
> number equvalent? "To divide into 1/2 parts?" Would that be "to put
> two similar things together" or what?
Interesting question.  I suppose it could mean "join pairs of similar
things together so that you end up with half as many separate pieces
as you started out with, but without discarding any of them."

I don't know; does "marry" or "mate" cover this idea?
Especially in the Inca empire where all of the village's single women
and all of their single men would line up, and the imperial official
would go down the line marrying one from column A to one from column
B until he ran out of one sex or the other.
Or, one of those Moonie mass-weddings Sung Myung Moon used to perform.
Or, a "punaluan" -- a Polynesian multiple-wedding in which more-than-
one siblings married an equal number of other siblings (e.g. the
brides are each other's sisters and the grooms are each other's
brothers; or, the bride in each couple is sister to the groom in the
other couple and the groom in each couple is brother to the bride in
the other couple).

>>> and I've heard "third" used in the first sense too: "We thirded
>>> the income of the show." "The pie was thirded." etc.
>>
>> I have never heard either of these expressions in English.
>
>> Are you sure these aren't translationisms?
>
> No I'm not, actually. Do these make sense to a native speaker?
A native speaker of my idiolect would never use these; but would
probably guess what they meant right away.

I can't speak for all speakers of English.

>> Thanks for writing.
>
> Thank you too! Your previous reply was so far the most helpful
> on this subject.
Thanks, John.  That's good to know.

Tom H.C. in MI

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