[FOM] Founders of set theory and natural language

Dean Buckner Dean.Buckner at btopenworld.com
Wed Oct 22 14:14:25 EDT 2003

On Friedman's claim that Cantor had no "natural language concerns", well
yes, but Cantor was not the only founder of set theory.  My rough
understanding of events is: Frege, Cantor,  Peano and Dedekind are all
contenders for being founding fathers.  Russell, Zermelo & others were

Frege is universally revered as the founder of natural language philosophy,
as we know.  Peano originated the "epsilon" sign for set membership in 1888,
which Russell discussed at length in P of M, and also seems to have spotted
the difference between "x is a man" and "if x is a man, x is mortal", which
I always thought of as a Fregean innovation.  However, Russell writes

"Frege's work abounds in subtle distinctions, and avoids all the usual
fallacies which beset writers on Logic.  His symbolism ... [though cumbrous
& difficult] ... is based on an an analysis of logical notions much more
profound than Peano's, and is philosophically very superior to its more
convenient [i.e. uncumbrous] rival." ( Pof M § 475)

The matter is complicated by Zermelo's contribution, of a nearly unflawed
set theory in 1908.  The first part of the paper is devoted to addressing
"Russell's antinomy", however he claims in an earlier paper that he
discovered the antinomy earlier than Russell (and there is some evidence
from Hilbert & Husserl to support this).

Zermelo was the first teacher of "mathematical logic" in Germany.  However
he leaves the logic of his paper implicit.   He uses a mixture of
Peano's symbols, Frege's notion of assertion and "truth-values", plus
(according to Volker Peckhaus) a technique of first order quantification
derived from Schroder.  He uses the vague adjective "definite" to
characterise a propositional function in Axiom III - Russell noticed this
defect at once (in a letter to Jourdain 8 March 1908).  Later Weyl (Uber die
Definitionen der mathematiscen Grundbegriffe, Math-naturw. Blatter 7, 93-5,
1910) made it clear that "definite" means the function is constructed of a
finite number of logical connectives, quantifiers and set-theoretic

So the question is, whether Zermelo's system owes what rigour it has to the
work of philosophers of language such as Frege and Russell?  Or not?  I
don't know.  Finally, on the relevance of the philosophy of language to
logic itself:

"The study of grammar, in my opinion, is capable of throwing far more light
on philosophical questions than is commonly supposed by philosophers.
Although a grammatical distinction cannot be uncritically assumed to
correspond to a genuine philosophical difference, yet the one is prima facia
evidence of the other, and may often be most usefully employed as a source
of discovery.  On the whole, grammar seems to me to bring us much nearer to
a correct logic than the current opinions of philosophers: and in what
follows, grammar, though not our master, will yet be taken as our guide."
(Pof M § 46).


Dean Buckner,    : If one person can see it as a paradise
London,              : of mathematicians, why should not
ENGLAND        : another see it as a joke?

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