Bug#210879: constitution.txt: fractured developers
Mr. A. Suffield says:
> The constitution is in prose, and because prose is different from
> math, ts norms and standards are different. Sometimes very
> different.
Which is why we discuss numbers using mathematical terminology, as is
standard for technical prose.
Is a constitution technical prose now? Maybe it should be rewritten in
symbolic logic. Seriously, describing in a constitution how to count a
quorum isn't so "technical" that it hasn't already been done well
without resorting to algebra. Also there's a democratic benefit to being
understood by the largest possible audience, as opposed to a happy
minority of experts and their translators.
> Integer has more than one meaning
Only if you're an idiot. Integer has precisely one meaning, and it is
a synonym of "whole number". The set of integers is the union of the
set of natural numbers and the set composed by subtracting every
natural number from zero.
Please try a little harder to leave my shameless idiocy out of this,
it's a bug controversy not alt.flame or whatnot. Cool your jets.
What you're doing is redefining an old word in terms of a newer more
specialized vocabulary with its own set of assumptions built in. (that
is, the vocabulary of set theory) Within the inflexible systems such
vocabularies define, any deviations from basic definitions are errors.
Then when somebody says something to an expert that disagrees with them
using a term said expert feels he owns, (or owns him), then the
expert will ignore or evade what was meant, and holler at the speaker
that they're abusing the expert's tools and are therefore low caste and
untouchable. "Shut up, he explained."
Whereas this is a question of English prose, and a word that's older
than modern set theory. Trotting out a new definition doesn't eliminate
the old definitions, it just adds one more to the heap. English and
every other true spoken language is flexible and inclusive of
incompatible and contradictory vocabularies and their underlying
worldviews. These built in contradictions are part of what makes a
language lively and interesting, they add an expressive strength that
would be impossible with any utopian one-word-one-meaning
lets-fix-all-controversy artificial substitutes.
Today, when most people say integer, (those who do say it, that is),
they think "positives and negatives" and not the set theory stanza. And
when they say "whole number" they think of natural numbers. (A Google
search of usenet shows no hits for the string "whole negative
number".) Love 'em or hate 'em, the public is any constitution's
target audience, or at least it should be.
---
More on the same topic, since I had a library trip a day later and got
to look a few things up and wound up with a decent survey. One
preliminary bias -- until yesterday, I've never in my life looked up the
term "whole number" because I'd no doubts as to its meaning. To my
mind, a whole number is a natural number plain and simple. Now the
data, then a few opinions...
Here's the 1913 Webster's unabridged 'Integer' again.
A complete entity; a whole number, in contradistinction to a
fraction or a mixed number.
Here's the same book's 'whole number':
Whole number (Math.), a number which is not a fraction or
mixed number; an integer.
The two defintions are circular, alas, and don't mention negatives.
Here's the 1934 2nd Webster's unabridged 'whole number':
an integer.
Here's the 2nd's 'integer':
A complete entity; specif., a whole number.
Still circular. Here's the 1961 3rd Webster's unabridged 'whole number':
INTEGER
Here's the 3rd's 'integer':
1. any of the natural numbers, (as 1,2,3,4,5), the negatives of
those numbers, or 0.
2. A complete entity.
governmental policy is an integer. - Dean Acheson
they become a whole, an integer, of people who have the
same aspirations and hopes - William Faulkner
So how do we know which sense, #1 or #2 the 'whole number' entry was
pointing to? I read the Explanatory Notes in the front, (note 12,
'SENSE DIVISION'), but couldn't reach any conclusion, the note says that
sometimes the sense order is historical, but not always.
Trekking on, the Random House unabridged, 1st & second editions, says of
'whole number':
1. One of the positive integers or zero, any of the numbers
(0,1,2,3...)
2. (Loosely) integer (def 1.)
1550-60
Random House's 'integer' def 1. I didn't write down, but it's pretty
much our modern definition, as in Webster's 3rd unabridged. "Loosely",
that's telling 'em.
Then there's the Random House Collegiate 'whole number':
1. INTEGER (def 1.)
2. NATURAL NUMBER
Yes, INTEGER (def 1.) is the modern version. Where's "loosely"? Note
the reversed sense order from unabridged to collegiate. That's why they
call it "Random" House...
Webster's New World American English, 1998 (Simon & Schuster), 'whole
number':
Zero or any positive or negative multiple of 1. Integer.
Webster's New Collegiate, (Merriam) didn't note its date, but it was
recent, 'whole number':
INTEGER
Doubleday Dictionary, 1970, their 'whole number' includes negatives.
Harper Collins Dictionary of Mathematics, 1991, 'whole number':
another term for NATURAL NUMBER, usually including zero.
Usage, however, varies, and the term may be used for all
INTEGERS or only positive integers.
Mathematics Dictionary, Van Nostrand Reinhold, 1968, 'whole number':
(1) One of the integers 0,1,2,3,...
(2) A positive integer; i.e. a natural number.
(3) An integer, positive, negative, or zero.
2002 Encyclopedia Britannica, under 'Number... something or other', but
it was vol 8, p. 827, col. 1b:
Numbers can be classified in many ways. The simplest class
comprises the counting, or natural numbers (1,2,3,...), which
with the addition of zero (not widely used until the 13th
century) are known as the whole numbers. The use of the
negatives of these numbers (-1,-2,-3) and negative numbers in
general was not widely accepted until the 17th century. The
natural numbers, their negative equivalents and the zero
constitute the 'integers'.
E. Brit. 1771, (the first), v2. p.884 'integer':
In arithmetick, a whole number, in contradistinction to a
fraction.
(No entry in 1771 for 'negative' or 'whole'.)
And so forth. A 1972 E. Brit. article 'Numbers, Theory of' begins:
The theory of numbers is concerned, in its elementary parts,
with properties of the integers, or whole numbers 0, +-1, +-2, +-3 . . .
It seems that even the E. Brit. is not at peace with this.
Popularisations like Asimov's 'Realm of Numbers' use 'whole number' as a
synonym for modern integers, but Irving Adler uses it as a synonym for
'natural'.
Under it's long article on 'whole', vol, W, p.91, the Oxford English
Dictionary says:
Math, Of a number: Denoting a complete and undivided thing, or
a set of such things (not a part of a thing), integral, not
fractional.
1557 RECORDE Whetst. Aij, Some are whole nombers... others are
broken nombers, and are commonly called fractions.
1608 R. NORTON Stevin's Disme A 3b, A whole number is either a
vnitie, or a compounding multitude of vnities.
For 'integer', vol I, p.366, the word derives from the French 'integre',
and the Latin 'tangere' (to touch), and 'in', meaning 'untouched'.
Math, Denoting a whole thing or number of whole things; denoted
by a whole number; 'whole', not fractional.
Under 'negative' the oldest usage that I noted was:
1727-38 CHAMBERS ... Negative or privative quantities are those
less than nothing.
1798 HUTTUN Course Math. ... Negative integer...
Is this the furthest back we can go? No. Here's a section of a helpful
web page:
Earliest Known Uses of Some of the Words of Mathematics (I)
http://members.aol.com/jeff570/i.html
-------quote begins-----------
INTEGER and WHOLE NUMBER. Writing in Latin, Fibonacci used numerus sanus.
According to Heinz Lueneburg, the term numero sano "was used extensively
by Luca Pacioli in his Summa. Before Pacioli, it was already used by
Piero della Francesca in his Trattato d'abaco. I also find it in the
second edition of Pietro Cataneo's Le pratiche delle due prime
matematiche of 1567. I haven't seen the first edition. Counting also
Fibonacci's Latin numerus sanus, the word sano was used for at least 350
years to denote an integral (untouched, virginal) number. Besides the
words sanus, sano, the words integer, intero, intiero were also used
during that time."
The first citation for whole number in the OED2 is from about 1430 in
Art of Nombryng ix. EETS 1922:
Of nombres one is lyneal, ano(th)er superficialle, ano(th)er
quadrat, ano(th)cubike or hoole.
In the above quotation (th) represents a thorn. In this use, whole
number has the obsolete definition of "a number composed of three prime
factors," according to the OED2.
Whole number is found in its modern sense in the title of one of the
earliest and most popular arithmetics in the English language, which
appeared in 1537 at St. Albans. The work is anonymous, and its long
title runs as follows: "An Introduction for to lerne to reken with the
Pen and with the Counters, after the true cast of arismetyke or awgrym
in hole numbers, and also in broken" (Julio González Cabillón).
Oresme used intégral.
Integer was used as a noun in English in 1571 by Thomas Digges
(1546?-1595) in A geometrical practise named Pantometria: "The
containing circles Semidimetient being very nighe 11 19/21 for exactly
nether by integer nor fraction it can be expressed" (OED2).
Integral number appears in 1658 in Phillips: "In Arithmetick integral
numbers are opposed to fraction[s]" (OED2).
Whole number is most frequently defined as Z+, although it is sometimes
defined as Z. In Elements of the Integral Calculus (1839) by J. R.
Young, the author refers to "a whole number or 0" but later refers to "a
positive whole number."
-------quote ends-------------
The same website, under 'N' has good entries for 'Natural' and
'Negative', with an alarming quote that some mathmeticians, (like John
Conway, inventor of the cellular automata 'Game of Life'), have taken to
using 'Natural' for integer as well!
That's enough historical data. It starts out clear enough, a 'numerus
sanus', a "virgin number", not a broken number... and ends up with some
being positive of one opinion and others quite negative.
Is there a right answer?
Mathematically, I think, no. With math, authors can be as arbitrary as
they like, and the word histories above demonstrate that.
Mathematicians are like computer programmers choosing memorable names
for their functions and variables. The important thing is that the
names are used consistently. The McDonalds Restaurant menu would do as
well, so Cheeseburger Numbers are positive, and Hamburger Numbers are
negative, and Big Mac numbers are both, and so on. There's no necessary
right answer.
With general languages like English, I think there is a right answer,
but that answer is not so convenient as merely exerting a top down
authority. Words are already "out there" in the minds, utterances and
writings of their speakers. We can observe and record meanings, but for
individuals to go dictating general meaning itself is unreliable at
best, and at worst futile or opressive. The approach must be scientific
and observational rather than a priori, like cataloging animals -- the
ostrich is out there, we can note what he does, or we can make up
morally instructive mythology like "the ostrich sticks its head in the
sand when frightened" in hopes of encouraging people to not "act like
ostriches", or "make pigs of themselves" and so on.
The morally instructive dictionary writer or censor is always up to that
stuff. Some may recall a Stan Freburg number called "Elderly Man
River", in which Freburg attempts to sing Steven Foster's "Old Man
River" before a sensitive network censor who makes him change "Old" to
"Elderly", "ain't" to "isn't", and so forth "to protect the tiny tots."
Radical egalitarians attemping to abolish discrimination have coined
such doomed euphemisms like "womyn" for "women" (having "men" in there
would be patriarchal), "pern" for "person" ("son" is too masculine), or
"herstory" for "history"; they call actresses "actors", waitresses
"waiters", and proactively employ "she" as an indefinite personal
pronoun to balance out the outnumbering usage of "he". To induce virtue
the medieval Church promoted naming babies after saints. The Puritans
named their kids after abstract virtues. Did it work?
What is the secret passion of the morally instructive mathmetician,
should he be installed as a dictionary or encyclopedia editor? Order!
Forget the word history, to hell with common usage, it's high time to
set down some rules. Math is like the word of God which they will
interpret, to them a common usage is like a fart in church. Redefining
messy words in an orderly way helps everyone think more clearly, they
believe; but people are still dumb after decades of this, and dumping
history just makes the definitions even duller and dryer than they were
to start.
Anyway, on to the right answer, or my best attempt at it, with metaphors
computer buffs might better appreciate.
Word definitions have a kind of mass and weight, which might be
calculated by how many people use them in a given way. It's impossible
to really count that, but we can sample and guess with some accuracy.
A word with many definitions has connotiations that come from the
interactions of its various meanings and distinctions, its etomology,
how it sounds, and even how it looks written down on paper. "Night"
looks good in old typefaces, but 'Nite' looks cheap. The connotations
might be considered as a vectored force, floating in n-space. As with
long range ballistics, it's easier to calculate a shell's trajectory if
we ignore wind, the earth's rotation, and other factors, but if those
are ignored, the shell is more likely to miss the mark. Ignoring
connotations of words, by simplifying them to one-meaning ideal usages,
makes for prose that misses the mark. The shell lands miles away, the
enemy is unhurt, but the gunner's friends and family applaud to help him
save face, or maybe they're nearsighted.
Adding to the difficulty is that groups of words seem to share and
intermingle some of their connotations, as if those vectors were
combined to some degree. Two seperately innocent words can together be
very naughty. Some words are almost always used in the company of other
words, and are rarely seen elsewhere. Foreign speakers often don't know
what the usual companion words are, and odd their choices of synonyms in
place of compainion words may even pinpoint their native language.
So, we have mass, vectors, and combinations. It's a veritable four-body
problem. Scrutinize those three things, and you can define and rank the
usages of a common word. Or don't scrutinize and go make up terms like
"womyn", but that's like trying to stop a locomotive with a pea shooter.
An old word has considerable intertia, while a custom made coinage is
usually weak.
On 'whole number' vs. 'integer', it's the connotations that distinguish
them. 'integer' is Latin, and has few English meanings, 'whole' is
English and has pages of them. Sometimes 'whole' means "plenty", as in
"a whole lot", "a whole passel", "a whole bunch", "I can't believe I ate
the whole thing", or "a whole lotta shakin' going on". "A passel" isn't
the same thing as "an entire passel" or "a complete passel". 'Whole'
connotes a surplus of a positive quantity. That connotation is so
strong that it makes applying 'whole number' to a debt, lack, or
negative quantity seem nerdy and pedantic. 'Integer' on the other hand,
being Latin, is of 'low mass', and was light enough for scholars to drag
away wherever they wanted.
For technical matters, do not trust wordnet. It's not particularly
accurate.
You're right, and I hadn't noticed the 'negative naturals' bit, though
it would probably gratify John Horton Conway.
a) you think "integer" is appropriate
In the sense of "whole number" as synonymous with "integer", no. I'd go
with the common usage, where they're not usually synonyms. Or
'natural' is fine too. Let's not loop around this one again though.
You've got a fixed set theory definition, while I'd hold and am ready to
defend that such fixity is useful in its realm, but contrary to the goal
of communicating to a broader public.
b) you are wasting our time
The kettle is black? Mr. Winsleydale a cheese purveyor shot
senselessly. Some might say debate is invariantly 50% waste, and maybe
they're right, but others think that unopposed authority, however
benign, wastes so much more than 50%, it makes a mere 50% look like
heaven on Earth.
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