The line between the objects that exist and the objects which do not
is easy to draw; everything is on one side of it.
When I show (by contradiction) that "the largest prime number" does
not exist, I am not at any point talking about an object called "the
largest prime number" in a way which cannot be explained logically
without admitting the existence (however shadowy) of such an object,
any more than a general worrying about a large enemy force which
_might_ be on his left flank where he has poor intelligence and takes
quite sensible precautions against it is suffering from delusions if
there happens to be no such force. The way I would handle it myself
is to discuss a number N which is the largest prime number (if there
is a largest prime number) and 4 otherwise; I then proceed to prove
that N is 4. The existence of N is not in doubt at any point in the
argument. Notice that "being the largest prime number" is a perfectly
well-defined predicate whose formal definition does not involve any
object "the largest prime number"; another way to approach the problem
is to reason about this predicate (show that nothing satisfies it)
instead.
This approach is implemented in HOL, for example.
This philosophical problem was solved by Russell with his theory of
descriptions a long time ago (there are alternative reasonable
solutions), If you admit it has substance as an objection to
mathematical platonism, it is equally dangerous to common-sense
realism (think about the force that the general fears might be
attacking him from his (objectively secure) left flank). But I don't
admit that it has any substance except as a word game.
DeBruijn did suggest that Platonism would lead to sloppiness in proof
standards in a paragraph quite separate from his discussion of proof
by contradiction. I, at least, did not take offence, merely
disagreed.
The opinions expressed | --Sincerely,
above are not the "official" | M. Randall Holmes
opinions of any person | Math. Dept., Boise State Univ.
or institution. | holmes@math.idbsu.edu