[FOM] On Platonism and Formalism

Dmytro Taranovsky dmytro at mit.edu
Tue Sep 30 20:38:24 EDT 2003

Of the many mathematical papers that I have read, every one of them
treats the mathematical objects as though they exist.  In all papers
deriving the consequences of ZFC that I have read, ZFC is a theory about
sets.  It is difficult to even think about how to derive the
consequences of ZFC without invoking the notion of a set.  The thoughts
and discussions on sets appear to refer to something, but one cannot
refer to something unless it exists (or existed or will exist).  

Thus, sets appear to exist.  On the other hand, sets are not directly
observable, and there is no scientific experiment that will directly
prove the existence of an inaccessible cardinal, so formalists claim
that the notion of the mathematical universe is superfluous, and that,
by Ockham's razor, there is no such thing as a real number.

However, physical reality is not directly observable either:  All we
observe are our feelings.  No experiment directly disproves the belief
that the only objects that exist are human souls but their feelings are
subject to the same patterns as they would if the physical universe
existed.  The belief is non-empirical and hence not subject to
experimental refutation.  Yet, we believe that the physical world
exists.  It exists because it feels like it exists and there is no
evidence to the contrary.   Inaccessible cardinals metaphysically exist
because once one has spent a lot of time studying them, it appears that
they exist and there is no evidence to the contrary.

We believe that electrons exist because the explanation of the patterns
of our experience becomes much more natural if we assume their
existence.  In other words, any natural explanation of say, chemistry,
invokes the notion of an electron.  However, any such natural
explanation also invokes the notion of a real number:  Physical theories
are supposed to be quantitive--and hence mathematical.  Mathematics is
extremely effective at making physical predictions.  Such effectiveness
would certainly be unreasonable if mathematics is the theory about
nothing, that is if sets do not exist.  Mathematical objects are as
central to our understanding of physics as are physical objects.  To
claim that mathematical sets do not exist is as reasonable as claiming
that the physical world does not exist.

It appears as though most mathematicians believe that mathematical
objects semi-exist.  They would deny that the empty set exists in the
same sense that pens, pencils, and buildings exist, yet they would
accept that ZFC is not merely a string of symbols but a theory about
sets.  However, what are sets if they do not exist? The mere reference
to sets implies that they are something, and hence exist.  An object
either exists or it does not; there is no such thing as semi-existing
object, and for the reasons stated above, mathematical sets do exist.

Dmytro Taranovsky

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