John,
The distinction between nominal thinking and real thinking is distinct from
the distinction between extensional thinking and intensional thinking, as one
can see from the fact that extreme nominalists do not admit sets as entities.
Peirce admitted both extensions and intensions of concepts, as integrated in
his theory of information. This is just another one of the ways that Peirce
was able to bypass the whole aporia, boondoggle, debacle, gridlock, whatever
you want call it that had bedeviled the issue of universals up to that time.
Regards,
Jon
JFS: For a nominalist, a function or relation *is* a set of n-tuples.
For a realist, the _intension_ of a function or relation is a rule,
law, principle, or axiom. The _extension_ is the set of tuples
determined by that rule, law, principle, or axiom.
On 1/29/2017 12:39 PM, John F Sowa wrote:
Eric and list,
EC
My initial inclination is to say that everything you pointed to does
seem important, but doesn't seem obviously to hinge on anything I can
easily understand as a difference between nominalists and realists
The simplest explanation I have ever read was by Alonzo Church --
in a lecture to Quine's logic group at Harvard:
http://www.jfsowa.com/ontology/church.htm
The Ontological Status of Women and Abstract Entities
This excerpt from Church’s 1958 lecture was preserved by Tyler Burge.
Cathy Legg posted it to her web site, from which I downloaded it.
(I really wish we had a YouTube of that lecture and the debates
between Church and Quine.)
In my web page, I added URLs for a 1947 paper by Goodman and Quine
and a response by Church in 1951.
For anyone who wants to see an important *practical* difference
between nominalism and realism, see the following excerpt from
Church's book, _The Calculi of Lambda Conversion_:
http://www.jfsowa.com/logic/alonzo.htm
Nominalists like Quine deny the distinction between essence and
accident in philosophy. In mathematics and computer science, they
extend their ideology to deny the distinction between intensions
and extensions.
For a nominalist, a function or relation *is* a set of n-tuples.
For a realist, the _intension_ of a function or relation is a rule,
law, principle, or axiom. The _extension_ is the set of tuples
determined by that rule, law, principle, or axiom.
Peirce would add *habit* to that list. A habit is an informal law
that could be made formal -- but only at the expense of losing its
flexibility (AKA vagueness). Peirce said that vagueness is essential
for mathematical discovery. George Polya did not cite Peirce in
his books, but he made that point very clear.
Carnap was a nominalist who denied the reality of all value
judgments, including Truth. After talking with Tarski, he accepted
the notion of truth because it could be defined in terms of sets.
That led Carnap (1947) to define modal logic in terms of a set of
undefined things called possible worlds.
Other nominalists, such as Kripke and Montague adopted Carnap's
method, but I believe that Michael Dunn's definition in terms
of laws (related to methods by Aristotle, Peirce, and Hintikka)
is more fundamental: http://www.jfsowa.com/pubs/5qelogic.pdf
Quine & Co. also deny the existence of propositions. They insist
on talking only about sentences. For a definition of proposition
that was inspired by Peirce, but stated in a way that a nominalist
could accept, see http://www.jfsowa.com/logic/proposit.pdf
This article is a 5-page excerpt from a longer article that discusses
the philosophical issues: http://www.jfsowa.com/pubs/worlds.pdf
John
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