Re: Types of numerals

> > 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?
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.


> >> 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.


>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.


> >>> 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. Descendants of Latin numbers occur so often in English
that they can be considered separate roots. Also, this in turn is then an
example of two largish sets of unrelated (or opaquely distantly related)
number roots in one language! They aren't in any trivial semantic
relationship, but the point still stands.

(The Greek numbers might count too.)


> > "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 too have *this*
meaning, right?


>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. Б =
"A-and-a-half". :)


> > 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.
>
>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.

> >>> "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.


> > 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.
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..)


> > 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.


> >>> 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_.
And "pair" isn't particularily mathematical anyway.


(...)
>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 regularily derived words aren't "special".


>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?


>BTW I understand English once had a labio-dental
>semivowel (approximant);
/v\/ in English? Really? So where did it come from and whence it went?


>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. It's much more difficult to
hear the difference of homorganic fricative and approximant.
The velarization of /w/ of course also eases things up.


> >>> 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.


> >> 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.


> > 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, but "zero dot four six six" isn't.
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 offen.


> > "____ 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."

Anyway, there appears to be also several other words similar to "both" -
principally "neither" and "either". These seem to be related to the logical
connectivities AND, NOR and OR. NAND and XOR would be easy additions to this
family, were they not tricky to generalize for more than 2 arguments.


> > "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.
> > 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 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.


> >> 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". 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?


> > 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?


>Thanks for writing.
>
>Tom H.C. in MI
Thank you too! Your previous reply was so far the most helpful on this
subject.

John Vertical

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