Math Forum / Mathematics / Mathematical Logic / October 2008

translogi wrote:

Instead of earning their keep supporting axiomatic systems,
these slacker axioms just keep sneaking off to lie around
the park, feed the pigeons, panhandle, etc.

Basically, they're a public nuisance.

--
hz

I suppose a 'slack' axiom might be a redundant one.  Well-known examples
are Whitehead and Russell's (p v (q v r)) -> (q v (p v r)) shown to be
redundant by Lukasiewicz and Bernays, and Frege's (p -> (q -> r)) -> (q
-> (p -> r)) shown to be redundant by Lukasiewicz.

David Ullrich asked:

Oops, silly me, I answered the question "What's nominalism?" instead of
the question 'What's "nominalism"?'.

Ha ha, well done.

I'm not sure that Quine is quite the nominalist poster boy.

At http://en.wikipedia.org/wiki/Willard_Van_Orman_Quine#Set_theory
it says:

 "He flirted with Nelson Goodman's nominalism for a while, but
  backed away when he failed to find a nominalist grounding of
  mathematics."

At http://en.wikipedia.org/wiki/Nelson_Goodman#Nominalism_and_mereology
it says:

 "Goodman, along with Stanislaw Lesniewski, is the founder of the
  contemporary variant of nominalism, which argues that philosophy,
  logic, and mathematics should dispense with set theory. Goodman's
  nominalism was driven purely by ontological considerations. After
  a long and difficult 1947 paper coauthored with W. V. O. Quine,
  Goodman ceased to trouble himself with finding a way to reconstruct
  mathematics while dispensing with set theory -"

In Quine's "Mathematical Logic" revised edition of 1951, Quine says in
section 22:

"Once classes are freed thus of any deceptive hint of tangibility,
 there is little reason to distinguish them from properties ...

 Discourse in general, mathematical and otherwise, involves
 continual reference to to abstract entities of this sort --
 classes or properties.  One may prefer to regard abstractions
 as fictions or manners of speaking; one may hope to find a
 method whereby all ostensible reference to abstract entities
 can be explained as mere shorthand for a more basic idiom
 involving reference to concrete objects (in some sense or
 other).  Such a nominalistic program presents extreme
 difficulty, if much of standard mathematics is to be really
 analyzed and reduced rather than merely repudiated; however
 it is not known to be impossible.  If a nominalistic theory
 of this sort should be achieved, we may gladly accept it as
 the theoretical underpinning of our present ostensible reference
 to so-called abstract entities; meanwhile, however, we have no
 choice but to admit those abstract entities as part of our
 ultimate subject matter ...

 Our working ontology is thus pretty liberal.  But in mitigation,
 it may now be said that this is the end; no abstract objects
 other than classes are needed -- no relations, functions,
 numbers, etc., except insofar as these are construed simply
 as classes.  In addition to concrete objects we need recognize
 only classes having such objects as members, then classes whose
 members are drown from the thus supplemented totality, and so on.
 This is presumably all the ontology that is needed for discourse
 in general; certainly it is all that is needed for mathematics ...

 It would even be possible, compatibly with the projected formal
 developments and indeed with the whole of mathematics, to repudiate
 concrete objects altogether -- to recognize just classes, each of
 which has classes in turn as members or else no members whatever ...
 This exclusively abstract ontology has little naturalness to
 recommend it, but there is no need here to reject or accept it."

I don't know whether the above paragraphs were written in the 1940
edition (pre-Goodman paper) or for the revised 1951 (post-Goodman)
edition.

--
hz