I am sorry that I seem to have irritated some people with my
exposition of 9 November on various forms of platonism. I certainly
do not want to attack A-platonists. Is is a basic rule of our
society to let people believe what they want, as long as they do
not start any attacks themselves.
For non-A-platonists it seems to be impossible to grasp what
goes on in the minds of the A-platonists. But it does not matter:
one can still talk mathematics together. And from what the
A-platonists reveal I think we have to conclude that their working
methods are not different from those of others.
My message of 9 November should have been written a bit more
carefully, of course. One thing is, that I did not mention the role
of equality in mathematics. Equality seems to be strongly related
to A-platonism. I remember an old text-book where the author gave
as a Definition: "two complex numbers are called equal if and only
if they are exactly the same". It gave me the feeling that "the
same" was an absolut notion in his platonic reality, otherwise he
would have given a definition of "the same".
I was sloppy about metalanguage, saying that it is reasonable
to be platonistic in metalinguistic discussion where mathematical texts
are the objects. Randall Holmes pointed out that this is not always
reasonable, and I agree. We can get platonism all over again. Admittedly,
it can be restricted to the world of finite strings of characters,
but this world is still immense.
We create a language in order to talk about a few things we
know, and later we notice that our language can also talk about a
lot of thing we did not think of in the first place. In particular,
the language may be able to deal with fiction. The language can
introduce imaginary objects and talk about them as in exactly the
same way as about the real objects.
In mathematics one might think of rows of slashes, like /, //,
///, ////, and a few more, as real things. We can represent them
physically. But we can imagine very long sequences that are too
long to represent, due to the limited size of the universe. Still, we
can talk about such rows. We might call them IMAGINABLE. But we can
also use our "row talk" in different situations, which might be
called IMAGINARY. Here are some examples of them.
a. "Let n be a positive integer, and consider a row of n slashes".
b. "Assuming that a contradiction can be derived from the
Zermelo-Fraenkel axioms, we define n as the length of the shortest
possible derivation. Now take a row of n slashes".
c. "The number n is defined as 15 if the four color theorem is true,
and 18 if it is false. Nowake a row of n slashes".
d. "Let the number n be equal to 15 and moreove assume that n=18.
Now take a row of n slashes."
The distinction between imaginable and imaginary situations can
also be made for a metalanguage that talks about another language.
The actual language texts are physically representable, but the
metalanguage can talk about imaginable and imaginary situations as
well.
I do not think it has any use to try to convince A-platonists
that they are wrong. The words "right" and "wrong" simply do not
apply here.
I have no idea what goes on in the minds of those who take
A-platonism seriously, but nevertheless I have some feeling for the
psychology. Talking about real things works so well, and is so
easily extended to imaginable and even to imaginary situations, that
one is easily inclined to extend the notion "real things" in order to
get a much wider scope. It might be even felt as helpful to think
that one is talking about a big universe that actually contains all
the things one talks about.
By the way, I suspect that there are people who will claim
that the example d above is incorrect mathematical language. I am
afraid that such people will never be able to understand a mathematical
verification system.
In my message of 9 November I wrote
>An essential difference is that the mathematician may even be
>cheating: he starts with some assumptions that he wants to disprove,
>in the course of the argument he constructs objects, treats them
>psychologically as existing things, until at the end a contradiction
>is reached and the whole edifice falls into pieces.
Roger Jones interpreted this as if I had said that Platonists are cheating.
I did not write "Platonists", but "mathematicians". And with such
cheating, the deceived party is not mathematics, nor the
mathematician, but only the Platonic reality. What I actually
intended to say is that the psychological phenomenon that one feels
to be talking about real things should not be taken as a proof for
the actual existence.
I wonder whether the reason that students find it hard to
understand the existence quantifier is connected with a kind of
A-platonistic attitude where "existence" has a meaning already.
And I wonder whether there is a trace of A-platonism in
teachers who raise their voices when they have to say that
something EXISTS (in cases where explicit example cannot be given).
But again, it is hard to find out what goes on in other people's minds.
What we should be concerened with instead, is whether and how one
should run QED. That discussion may go on for some time.
N.G. de Bruijn
Eindhoven University of Technology
Department of Mathematics and Computing Science
PO Box 513, 5600 MB EINDHOVEN, The Netherlands
FAX 31-40-436685, TELEX 51163