[FOM] Realism and Platonism
Robert Black
Robert.Black at Nottingham.ac.uk
Wed Oct 29 05:07:33 EST 2003
Joe Shipman's distinction between realism and platonism is one I have
argued for before on this list; let me say a bit more about it.
Platonism is the view that mathematical objects (abstract structures
and their inhabitants) genuinely exist, as real as sticks and stones,
just not so chunky.
Realism is the view that mathematical sentences about one or more of
these structures can have objective but undiscoverable truth values.
This means that the realist thinks we can *specify* a structure in
more detail than is given by what we can *prove* about that structure
(prove in principle: I'm not concerned here with problems about
unfeasably long proofs). The most obvious tool for doing this is
second-order logic. Typically. the realist will think that the phrase
'*every* (possible) subset' (of a given infinite set) has a
determinate (maximal) reference, and thus for example that the
structures of the natural numbers, or the reals, or the cumulative
hierarchy below the first inaccessible are determinately given by
their categorical second-order specifications (and thus that every
sentence of arithmetic has a truth value, CH has a truth value, and
so on). But you might be a more restricted realist, e.g. if you
weren't confident that we understood second-order logic in a
determinate way but thought that we did understand the quantifier
'there are only finitely many ..'. That way you'd get a categorical
specification of the natural numbers and thus a truth-value for every
arithmetical sentence but not get a truth value for CH.
The easiest way to be a realist without being a platonist is to
modalize: there aren't (or need not be) any numbers, but if there
*were* such-and-such a structure then ... *would have to be* true in
it. Call these structures fictions if you like, but as Martin Davis
has pointed out they have to be possible fictions and once you've set
up the fiction determinately enough there's no free choice any more
about what goes on in it. You're a realist if you think you can
specify the fiction beyond what you can prove about it.
The choice between modalist realism and platonist realism is what
Quine many years ago called a choice between ontology and ideology.
With a big enough ontology (famous example: David Lewis's ontology of
possible worlds) you can get a reductive analysis of modality. The
platonist can be a realist while dispensing with modal notions
(though all the mathematical platonist needs is really existing
abstract structures rather than Lewis's really existing concrete
worlds).
Joe says that it's clear that platonism entails realism.
Unfortunately, it's *not* clear. Suppose platonistically that all the
models of first-order ZFC are sitting out there grinning at us in
their platonic heaven. For CH to have a determinate truth value we
have to be able to specify which of these models we really intend.
And it's not clear that we can do this (that's basically the argument
of Putnam's 'Models and Reality', JSL 1980).
Robert
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