[FOM] real numbers
Andreas Blass
ablass at umich.edu
Wed Jul 16 12:03:08 EDT 2003
For quite some time, I've owed Neil Tennant (and the rest of fom)
a reply to his message of 20 June. Neil points out that my explanation of
the "can" in statements like "the real numbers can be Dedekind cuts" gives
a de dicto interpretation of "can", and he asks for a de re
interpretation.
I suppose I could adopt the usual strategy of mathematicians when
confronted with a philosophical problem, namely to retreat into formalism.
"You want de re? What res? There's nothing here but dicta." But I feel
I should stay closer to the Platonistic attitude that I adopt when no
philosophers are watching.
Unfortunately, I don't see a good de re interpretation of "can" in
statements of the sort under consideration. Neil explained what is wanted
(by the Kripkean) as "getting a grip on some things or kinds of things in
the actual world and finding *them* again in other worlds ..." I don't
see how to do this when the things are abstract entities like the real
numbers. But my Platonism survives (at least for the time being), because
I don't know how to do what the Kripkean wants even for concrete entities
which I know quite well, for example me. I don't know how to tell what
object, if any, in some other possible world is me. And it's not just
that I don't have a decision procedure; I don't have any sort of
criterion, even an uncheckable one, that would distinguish me from other
entities in strange worlds. For example, if I exist in another world,
would I have to be human there? (Consider a possible world of the sort
depicted in "Planet of the Apes"; could I be an ape there?) Would I have
to be male? I suppose one could say that the object, in some other world,
to be identified with me is that object, if any, that has all my essential
properties. But this merely "reduces" the problem to saying which of my
properties are the essential ones, and I see no good answer to that. (I
believe Richard Heck wrote, a while back, that he could have been a
mathematician. How far can that idea be reasonably pushed? Could
Einstein have been a moron?) The upshot is that I don't know how to
handle de re modalities even for concrete objects, so I don't worry too
much about my inability to handle them for abstract objects.
By the way, when I write "I don't know ...", I don't intend the
polemical meaning that one sometimes sees on fom, "I don't know, and I
don't think you know, and if you think you know then you're probably
crazy." I just mean that I don't know. The problem seems natural enough
that many philosophers must have worked on it, and I'd like to hear about
their solutions (at least the good ones).
Neil wrote, and I agree, that a metaphysical interpretation of
"can" looks untenable in contexts like "the real numbers can be Dedekind
cuts". But then he says "That leaves the epistemic interpretation ....
Concerning the real numbers: it is possible, for all we know, that they
are Dedekind cuts ..." I'm uncomfortable with this interpretation also,
because it seems to say that we see different possibilities (Dedekind
cuts, classes of Cauchy sequences, etc.) because of our ignorance --- if I
were smarter, or if I thought about it harder, or if someone explained it
clearly to me, then I'd know whether the reals are Dedekind cuts or not.
That doesn't seem right to me. The problem isn't ignorance; it's freedom.
I can, if I wish, take the reals to be Dedekind cuts (by passing along an
interpretation to an appropriate theory). I suppose this sort of
modality, connected with free choice, must have been studied (and must
have been given a Greek name), but perhaps not in connection with abstract
entities. It's not clear to me whether such a modality could be used in a
de re sense; an attempt to do so looks rather strange to me. So I'll
stand by my original confession that I don't know a de re interpretation
of "the reals can be Dedekind cuts".
Andreas Blass
More information about the FOM
mailing list