FOM: constructive mathematics

[Charles Stewart]

Matthew Frank mentioned Dummett's views on mathematics, but decided,
understandably enough not to expand upon them.  There are a few
subscribers to FOM who are interested in Dummett's views, so it is
worth elucidating them.  I hope I do a passable job of it: it is quite
easy to make a mess of.

I think it is not too difficult to describe *where* Dummett situates
himself, though it is rather harder to explain *why* he situates
himself there.  Roughly speaking Dummett is an idealist who believes
that what there is is tightly dependent upon what sentences involving
their descriptions mean.

Dummett embraces the idea that meaning is constrained by our
linguistic competence (a widely held view, disputed mostly by
adherents of truth-conditional theories of meaning), but rejects the
idea that meaning is subjective (for the reason that you and I can
both mean the same thing).

There are good reasons to believe both of these things, rather harder
to explain why meaning should be supervenient on ontology.  Dummett's
real reason is that he believes that the only way we can give a
correct account of the nature of abstract entities, such as numbers
and equators, is by making use of Frege's context principle, namely
that the Bedeutung (reference) of an individual is picked out by the
role it plays the class of sentences in which it occurs, and that in
the case of abstract entities there is no more to them than this
role.

Dummett's commitment to idealism varies.  It is strong in the case of
mathematics and ethics, rather weaker when it comes to the nature of
past events and mundane physical entities.

Charles Stewart
Boston University
http://achilles.bu.edu/cas