Subject: Re: [HM] A question around zero
From: Bill Everdell (Everdell@aol.com)
Date: Sat Jan 22 2000 - 23:48:28 EST
[Jean-Luc asked]
>
> So, how do you define an even number?
[Samuel S. Kutler wrote]
>
> Euclid's answer from Book VII, definition 6:
>
> An even number is that which is divisible into two parts.
>
> The first even number then is two. Zero can't be an even number, for
> it isn't a number at all. One isn't a number either. For a number is
> a multitude. One is the polar opposite of a number, since the poles
> are one and many.
[John Harper wrote]
>
> Terminology has changed since Euclid's time. I would answer the question
> "What is the number of professors who retired from your School in 1999?"
> by saying "One", not "That question is meaningless."
On the other hand, John [writes Bill Everdell, resuming here in his own
voice], Giuseppe Peano gave it as an axiom that "One is a number." There
is a fascinating discussion going on right now on the RUSSELL-L list
(russell-l-admin@informer2.cis.McMaster.CA) about whether Russell's,
Frege's or Peano's definitions of number mean and whether they still hold.
Here is a recent post:
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[From Gregory Landini to] Jan [Dejnoszka] and Torkel [Franzen],
Let me just add the following on the question as to the force of Russell's
argument that not any progression will do for counting. Russell used the
matter of counting to argue in favor of the Frege/Russell cardinals (and
natural numbers). What has often been missed is that (as Russell himself
pointed out) his concern with the matter of counting is precisely the
concern Frege pointed out (in his Grundlagen). That is, it matters to
counting what is to be the sortal under which one is counting. To take the
shopworn example, there is one deck of cards, and 52 cards in the deck. One
must first fix a property-- 'is a deck of cards', or 'is a card in this
deck', before one can properly ask (or answer) the question of how many.
Now it when Russell says that not any progression will do because one must
account for counting, he is talking about this very point. And this, he
thinks, argues in favor of the Frege/Russell cardinals. A nice passage
corroborating this is from Russell's MY PHILOSOPHICAL DEVELOPMENT, p. 53.
Crucial to this argument, however, is the distinction between the
adjectival and the substantival use of number words. In the fundamental
sense, cardinal numbers are (as Frege put it) concepts of concepts, or (as
Russell might put it) properties of properties. The concept 'is a card in
this deck' has the concept 'is exemplified by exactly 52 entities.' Frege
and Russell distinguish the predicative (adjectival) use of number words
as in "The Apostles are twelve" or "The cards in this deck are 52," from
the substantival use as in "12 is a natural number." The adjectival is
fundamental; and the substantival is part of the notion of the "extension
of a concept."
Now for Frege the object 12 is the extension of the second-level concept
'is exemplified by exactly twelve entities' (roughly) or more revealingly
it is the extension of the concept 'is similar to a concept exemplified by
exactly twelve entities.' The mutual saturation of this second-level
concept with the first-level concept 'is an apostle' is paralleled in
extension by the membership of the class of apostles (the extension of the
concept 'is an apostle') in the extension of the second-level concept 'is
similar to a concept exemplified by exactly twelve entities' (i.e., the
class of all those classes similar to a 12 membered class.)
Russell doesn't have the Fregean notion of a concept or the distinction
between concepts and objects, but his account is fundamentally the same.
The point is that for Frege and Russell it is the adjectival that comes
first, and it is in this sense that counting favors the Frege/Russell
conception of cardinals as certain second-level concepts under which first
level concepts fall (Frege) or certain properties of properties (Russell).
So counting, just as Russell argued, does provide a reason for favoring the
Frege/Russell notion of cardinal number--as certain second-level concepts
(or certain properties of properties). The trouble, of course, is that
although counting favors the Frege\Russell "adjectival" conception as the
genuine cardinals, one must get from this "adjectival" account of cardinal
numbers to an account of the objects that are to be the cardinals. Now it
won't do, as Russell originally hoped, to say that the properties of
properties that are the "adjectival" cardinals are themselves the objects
that are the cardinals. This falls to the Russell paradox of predication.
Nor, will Frege's approach to a purely logical theory of extension work. It
falls to Russell's paradox of classes.
So (pace a hitherto unknown solution of the Russell paradox) one cannot say
that the Frege/Russell account of cardinal numbers AS OBJECTS reveals the
genuine cardinal numbers. It was contradictory! Nonetheless, I would
contend that the Frege/Russell account of cardinals AS CONCEPTS is correct
and is central to the very possibility of counting (as Russell says).
The moral of the story is that if one brackets the Russell paradox, Russell
and Frege do have good grounds for preferring their cardinals to other
progressions. Their cardinals (i.e., the progression they chose) is the
extensional shadow of the fundamental notion of a cardinal as a concept (or
property)-- a fundamental notion revealed by logical analysis of the
notion of cardinal number. And their accounts of extension (classes) were
supposed to be such that the calculus of classes is but the extensional
shadow of the calculus of the pure logic of predication. Unfortunately,
their accounts [o]f extension failed.
The result is that presently NO ONE HAS ANY SATISFACTORY IDEA WHATSOEVER AS
TO THE NATURE OF ARITHMETIC AND NUMBER.
[Gregory Landini]
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Bill Everdell, Brooklyn
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