holmes at catseye.idbsu.edu
Thu Mar 25 16:18:38 EST 1999
What does it mean "well-defined"? Say, is any property given by
an arithmetic formula involving unbounded quantifiers (if you do
not like my first example) well-defined? Are you sure that such
a property is either true or false for each natural number? How
can you check that? Is this an act of belief? Sorry, I am not
religious, especially in science (and since some time also in
arithmetic). I can adopt such a belief, but only _temporary_,
just as a game.
I doubt my answer will give you much satisfaction, but here it is:
properties aren't defined or undefined; strings of symbols are (in a
given language). The question of definedness applies to purported
descriptions of properties, not to the properties themselves. I would
say that any arithmetic formula involving unbounded quantifiers
succeeds in referring to a property, which property holds or does not
hold of each natural number.
By the way, what is then "true", standard, unique up to
isomorphism model for arithmetic if second order quantification
is so doubtful?
I don't regard second-order quantification as doubtful. I have doubts
as to the existence of models of the natural numbers only insofar as I
have doubts as to whether there are really infinitely many objects.
Since I'm quite certain that it is logically possible for there to be
infinitely many objects, I don't regard this as a real concern for
mathematics. (I also think that there really probably _are_ infinitely
many objects, but that's another kind of question altogether).
When we are working with a first-order theory we are normally
"wearing Platonist's hat" (as Simpson once said). Is not this
weak, innocuous Platonism quite enough to our comfortable
working with any such theory? Why do we need anything more
in mathematics and its foundations?
It is the beauty of the formal character of mathematics that we can
play the game and get uniform results regardless of our individual
opinions on such questions. I think that mathematical language either
has reference or it does not. If it has reference, this suggests
other (possibly embarrassing) questions; if it does not have reference
this suggests other (possibly embarrassing) questions. One can
practice mathematics without answering these questions, but the
questions don't go away because they are being ignored. (Which is not
to say that I think you are ignoring them; you obviously do think
about them). The reasons that I think the questions matter are not
internal to mathematics (but obviously do bear on the foundations of
mathematics); on the other hand, I think that it is very telling that
a mathematician when he is actually working will "put on his Platonist
hat" -- this suggests that other views are hard to reconcile with the
actual nature of the subject (which is not a knockdown argument that
a realist view is correct!).
And God posted an angel with a flaming sword at | Sincerely, M. Randall Holmes
the gates of Cantor's paradise, that the | Boise State U. (disavows all)
slow-witted and the deliberately obtuse might | holmes at math.idbsu.edu
not glimpse the wonders therein. | http://math.idbsu.edu/~holmes
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