Re: Types of numerals

Hi all,

On Fri, 13 Jan 2006 John Vertical wrote:

[snip]
> > > .: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 don't think "halfth" makes sense, even in the
senses :-) you explained it!


> (I do not know the precise set theoretical definition of "ordinal", but I
> suspect it might deviate from its linguistic definition a little here.)
Set theory says nothing about order.  You have
to go to algebra (specifically, the theory of
partial orders and lattice theory) to discuss
a "complete order" such as the ordinal series
first , second, third, ... corresponding to the
integer cardinals 1, 2, 3.


> > AFAIK, only reciprocals (half, third, quarter...) and exponents of
> > the base number (ten, hundred, thousand...) are lexical anywhere.
Which form an "initial segment" (the first few)
of a cardinal (but not an ordinal) series: half, third,
quarter, fifth, ... being the integer-reciprocal
cardinals 1/2, 1/3, 1/4, 1/5, ...



[snip]

> >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?)
How about "Both electrons of the Helium atom
normally occupy and fill the same valence shell."
You can't see the electrons, but you can observe
that Helium is chemically inert under normal
conditions, and even that it requires a certain
energy (visible as a wavelength of emitted light)
for one electron to escape this level.


> > > 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.
English (both everyday and mathematical) has
many "generic" terms for calsses of numbers.
Some examples follow:

"Even" and "odd" are "generic integers [not]
divisible by two".

"Primes" are (generic) integers exactly divisible
only by themselves and units.

"Superparticular" and "epimoric" are adjectives
for "generic reciprocals whose numerators exceed
their denominators by one".   I think the noun is
"epimore".

"Subparticular" is an obvious extension.

"Units" are integers that ... hmmm,  I forget
the exact definition, but the units of a field
of integers form a "basis" for that field - in
that all integers in that field can be expressed
as a sum of the units.

Example 1:  In the field of natural numbers N,
the only unit is 1.  Every natural number is a
sum of 1s.  eg 3 = 1+1+1.

Example 2:  In the field of integers Z, the
units are 1 and -1.  Every integer is a sum of
1s and -1s.  (The decomposition is not unique!)
eg -2 = -1+-1+-1+1.

Example 3:  In the field of complex integers CZ,
the units are 1, -1, i and -i.  Every complex
integer is a sum of 1s, -1s, is and -is.  eg 2+i =
1+1+i.

Example 4:  In the field of quaternion integers
QZ, the units are 1, -1, i, -i, j, -j, k and -k.  Every
quaternion integer is a sum of these 8 units.

Note: this usage of "unit" is not canonical maths!
Many authors restrict unit to be positive, so that
every member of the field is either a sum of units,
or the additive inverse of such a sum of units.
using either definition, the product of two units
is always a unit. eg i * i = -i * -i = -1, i * -i = 1 and
-1 * i = -i.

"Negatives" are (generic) "numbers less than zero"
or "additive inverses of positive numbers".

"Inverses" are ...



[snip]

> >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)"
That's "to halve".


> Same series, interbred with reciprocals.
>
> >and "to decimate (smthng)".
> >
> >"To decimate smthng" means "to reduce smthng by removing one-tenth of
it".

That's what it originally meant. Story is that
the punishment meted out to a Roman legion
for failure in battle was to execute one-tenth
of its members, chosen by lot.  This was before
the concept of individual responsibility took
hold in a big way ...!

Nowadays,in English at least, "decimate" is
usually taken to mean "to reduce to one-tenth",
ie "to almost annihilate".  Some reactionaries
rail against this use, which they regard as an
"enormity" (a monstrous crime, NOT "something
enormous").


> 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?
Very logical!  Makes sense in a loglang.  But do
you know of any natlang, or even any "praclang"
(constructed practical language), that has more
than a handful of such constructions?


> 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.

[snip]

Regards,
Yahya

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