LL-L "Lexicon" 2004.01.21 (12) [E]

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From: Sandy Fleming [sandy at scotstext.org]
Subject: "Lexicon"

> From: Floor en Lyanne van Lamoen <fvanlamoen at planet.nl>
> Subject: LL-L "Lexicon" 2004.01.21 (04) [E]
>
> I don't expect you to be a mathematician!! The thing is that
> mathematicians
> sometimes use language that differs from the everyday use. The
> discussion of
> historia matematica was funny, because it was agreed that "Anzahl" means
> "cardinality", a concept of which I doubt that "normal people"
> have any idea

Speaking as a mathematician, but _not_ as a linguist (in fact
I don't know much German) while "Anzhal" might be taken to mean
"cardinality", this is not a good way, mathematically speaking,
to distinguish it from "Zahl".

> Die Anzahl der auf diese Weise gebildeten Zahlen ist unendlich.
> The number (Anzahl) of numbers (Zahlen) formed in this way are infinite.

In this sentence _all_ numbers mentioned, both the number of
numbers involved and each number in the set, are cardinal numbers.

To explain cardinality with everyday language, you might
say that the following are cardinal numbers:

"One, two, three, four,..."

and these are distinguished from the ordinal numbers:

"First, second, third, fourth,..."

Not terribly complicated, even for non-mathematicians!

I think the reason "Anzahl" was taken to mean "cardinality"
is because of the mathematical function "Card" which returns
the number of elements in a set, eg:

Card {4, 7, 2} = 3

In English you might read this as "the number of numbers in
the set {4, 7, 3} is 3". However, by the definition of numbers
in set theory, each of the numbers _in_ the set are also "Anzahl":

    4 is defined as the number of numbers in
    some unspecified set containing 4 numbers

    7 is defined as the number of numbers in
    some unspecified set containing 7 numbers

and so on (actually the sets can be specified, but it wouldn't
anything to the linguistic discussion!).

In other words the distinction between "number" and "number of"
isn't important in mathematics and even as used by Cantor is
merely grammatical. "Cardinality" is contrasted with "ordinality"
in mathematics, but there are no ordinal numbers in Cantor's
sentence, only cardinals.

> Another remarkable thing is the English plural in this translation. The
> original by Cantor stated "ist unendlich" and also Dutch would use here a
> singular form.

In English I would say "The number of numbers formed in
this way is infinite", just as I'd say "The number of
stars in the sky seems infinite", _not_ *"The number of
stars in the sky seem infinite". The translation actually
seems to contain an error which I find to be quite common
amongst English-speaking technical writers, that of making
a verb agree with the previous noun instead of the noun it
applies to.

Sandy
http://scotstext.org/

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From: Kenneth Rohde Christiansen <kenneth at gnu.org>
Subject: LL-L "Lexicon" 2004.01.21 (04) [E]

This is how it is in Danish (and probably swedish, norwegian and related
dialects):

antal = amount
tal   = number

Kenneth

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