FOM: Feferman's first thesis

[steel@math.berkeley.edu]

   Some comments on Feferman's first thesis:

1. Mathematics consists in reasoning about more or less clearly and
coherently [conceived?] groups of objects which exist only in our
imagination.


   If you drop "which exist only in our imagination", there isn't much to
disagree with there. What is the meaning of "exist only in our
imagination"? The only examples of meaningful uses of this phrase that
occur to me are along the lines of "Sherlock Holmes exists only in our
imagination". Is this the meaning Feferman intends? If so, how does
such a "fictionalist" account of mathematics deal with its applicability?
We don't go to 222 Baker street looking for Mr. Sherlock Holmes. We do
use facts about real numbers to build bridges and send men to the moon.

   The Realist has deep suspicions that anything interesting can be made
of "qualified" existence for mathematical objects, such as existence "only
in our imagination" ( or only in our minds, or only in our social
conventions). Different kinds of existence amount to no more than the
existence of different kinds of things. ( The existence of Sherlock Holmes
in our imaginations amounts to the existence of a certain story.) An
operational version of this last slogan might be: the logic of "there is"
is the same in mathematics as in the rest of science. Sets are certainly
different in kind from electrons; both exist, but then so does
everything else. 

   One of the main differences is of course that one cannot attribute
spacetime locations to sets in any useful way. Similarly, one cannot
attribute causal relations to pure sets and physical objects (such as
ourselves) in any useful way.  The interesting things that CAN be said
about sets are said in set theory, and the sciences which apply it. There
are lots of really useful things to be said in this domain--that's why
society supports mathematicians. Virtually everything said in this domain
logically implies that there are sets. None of it is about how
these sets are related to our imaginations or social conventions.

   If one shifts to the question of how we know about sets, then perhaps
imagination, in some sense, plays a role which it does not play in other
areas. The Realist objects, however, to confusing this question with the
question of whether there are sets, as seems to be done when one says that
they "exist only in our imagination".

John Steel

Position: Prof. of Math., UC Berkeley
Interests: set theory, descriptive set theory, phil. of math.