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Re: [ontolog-forum] Historical footnote

To: "[ontolog-forum]" <ontolog-forum@xxxxxxxxxxxxxxxx>
From: Christopher Menzel <cmenzel@xxxxxxxx>
Date: Wed, 14 Feb 2007 14:30:56 -0600
Message-id: <AC435B71-A8BB-4A27-8852-0AF06F9A6304@xxxxxxxx>
> Pat and Chris,
> 1901 is hardly "early".
> CM>> Russell sent his fateful letter to Frege informing him
> >> of the paradox in the Grundgesetze der Arithmetik in 1902;
> >> Padoa published his first paper on the theory of definitions
> >> (which included a notion of non-creativity) in 1901.
> PH> Ah, I stand corrected. Thanks. I am amazed that it was
> > possible to even think of these ideas that early.
> The ancient Greeks, among others, thought about, analyzed,
> and debated paradoxes very seriously.    (01)

Yes, John, we're well aware of the history.  I believe Pat was only  
referring to the sophistication of Padoa's analysis of *definition*,  
for which you'll not find the likes in anything earlier.  Given its  
sophistication and its importance in logic and its relevance to  
paradox, it was quite reasonable for him to have thought that it was  
motivated by post-Russellian history.    (02)

> Cantor himself noticed the so-called "Russell paradox" and made
> some informal remarks to express his concerns.    (03)

This used to be the received wisdom but it has been thoroughly  
debunked.  Cantor in fact already had a firm (if informal) grasp of  
the modern set/class distinction that prevented him from seeing any  
paradoxes in his set theory.  Michael Hallett's 1984 book _Cantorian  
Set Theory and Limitation of Size_ contains the most comprehensive  
account of this, though it has been documented in a number of other  
places as well (including a 1984 paper by yours truly).  Read  
carefully, Cantor's remarks in his 1899 letter to Dedekind reveal no  
mention of, or concern about, paradoxes in set theory.    (04)

> Zermelo noted Cantor's remarks and had already developed the first  
> version of his axioms to avoid the paradox before he had heard  
> anything from Russell.    (05)

That is not so.  Zermelo's first axiomatization was in 1908 and was  
expressly in response to Russell's paradox, among others.    (06)

> Hilary Putnam remarked "Zermelo presented his axioms
> for set theory in Peirce-Schröder notation, and not, as one
> might have expected, in Russell-Whitehead notation."    (07)

Hardly a surprise given that the first volume of Principia  
Mathematica wasn't even published until 1910, which was what made  
Peano's notation famous and widely-used.    (08)

>   See
>    http://www.jfsowa.com/peirce/putnam.htm
>    Peirce the Logician
> Russell and Frege were probably unaware of Cantor's earlier remarks,    (09)

Because he didn't make any about paradoxes.    (010)

> but Russell was the loudest, if not the first.    (011)

Burali-Forti was probably the first to suggest a genuine paradox in  
1897 (which follows from the assumption that there is a set of all  
ordinal numbers).    (012)

> And note that even
> Putnam uses the phrase "Russell-Whitehead" notation despite the fact
> that every feature of their syntax (including the dots for showing
> precedence) had been published by Peano in 1895 -- and Peano gave
> full credit to Peirce and Schröder as his primary sources.  (When
> Peano corresponded with Frege, he refused to read Frege's examples
> unless Frege translated them to the algebraic notation.)    (013)

Well, perhaps understandable given how foreign the notation of the  
Begriffschift looks to someone used to a more algebraic notation.   
But Frege's has many virtues also.    (014)

-chris    (015)

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