SUO: Re^n: Collections - Aggregation or Set
Chris P wrote:
> Max Black's essay (referred to in an earlier email) starts with a
> quote from a working mathematician's book on set theory - which claims
> that packs of wolves etc. are standard examples of sets.
A claim that is pretty obviously false. A pack can change its members
(if, for example, a pup is born) and remain the same pack. Not so the
set corresponding to the pack at any given time.
> If I had the time, I am sure I could come up with other examples.
There are lots of similar examples that go wrong for similar reasons --
flocks of geese, schools of fish, citizens of a country, sets of dishes,
etc. All of these collective entities can change their membership, at
least to some extent, and remain the same collective entity. Not so the
corresponding sets at any given time.
That said, there is one (and only one) clear thinking philosopher of
mathematics that I know of who has respectably defended (not to say
demonstrated) the view that sets of physical objects are themselves
physical, viz., Penelope Maddy. The locus classicus here is her 1980
paper "Perception and Mathematical Intuition," Philosophical Review 89
(1980), pp. 163-196. Maddy is no crackpot -- she's head of the Logic
and Philosophy of Science program at UC Irvine, author of two highly
regarded Oxford Univ Press books on philosophy of mathematics, and she
is a member of the American Academy of Arts and Sciences.
The most interesting thing Maddy does in defense of the physicality
thesis is to marshall neuroscientific evidence for the existence of
"set detectors" that enable us to perceive sets of physical objects.
Such detectors, she argues, if they exist, provide a sound naturalistic
basis for the possibility of set theoretic knowledge. After all, if
sets really *are* completely abstract things, then they are causally
inert and so (for the naturalist) a question remains as to how we can
know anything about them. Maddy, who is both a mathematical realist
about set theory and a scientific naturalist argues that her appraoch
fills this epistemological gap.
Chris Menzel