[FOM] Realism and Platonism

[Robert Black]

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