# [FOM] Mathematical explanation

Paul Hollander paul at paulhollander.com
Mon Oct 31 15:01:12 EST 2005

```I think it's important to distinguish between semantic relativity, on
one hand, and non-literality, on the other hand.  The standard view
in speech act theory (as my former professor Kent Bach taught it
about ten years ago, at least) is that literality and non-literality
are attributes of statements, while semantic relativity is an
attribute of languages and their interpretations, and hence of
sentences in those languages from one interpretation to another.

The question of whether the STATEMENT "3+4=7" is literal or
non-literal is independent of the question of just what is the
literal meaning of the SENTENCE '3+4=7'.

A speaker might state exactly the same thing in two different
utterances, but she might get there by radically different means in
each case.  She might say "3+4=7," but do so non-literally and using
a bizarre interpretation of a language of which '3+4=7' is a
sentence, so as ultimately to assert the exact same thing that would
be asserted by saying "3+4=7," except doing so literally in our own
conventional language of arithmetic.

Searle's philosophy is interesting when applied to questions of
mathematical explanation, but I don't think this discussion about
"3+4=7" gets at that more interesting debate.  One of the more
interesting aspects of Searle's philosophy for mathematical
explanation, in my opinion, consists in his combination of "external
realism" ("... that the world exists independently of our
representations of it") with "conceptual relativism," as spelled out
in his book The Construction of Social Reality.

There, Searle provides the informal apparatus for developing an
ontology of mathematics based upon functional objects.  A functional
object is one whose status is imposed upon it by us, and by the
background rules we collectively employ as a community of language
speakers (Searle's "Background").  Thus Searle provides an account of
how we can have knowledge about right and wrong ways of interpreting
and manipulating symbols, without making the additional assumption
that the expressions in question refer to independently existent
objects.  (Similar, perhaps, to how semantic assent has a
problem-solving function according to Quine.)

To my mind, this points to more interesting questions about
mathematical explanation than those about how non-literality might
work for "3+4=7."  Must mathematical objects exist in order for
mathematical explanation to work?  Can we have mathematical
explanation without always interpreting quantifiers in terms of
existence?

Searle himself does not take conceptual relativity to imply that
mathematical objects don't exist.  In The Construction of Social
Reality, pp. 160 ff., Searle borrows an example from Putnam, a
diagram showing three disjoint circles (labeled 'A', 'B', and 'C').
He asks, how many objects are there?  He answers, three.  And seven.
Searle says that if we assume Carnap's system of arithmetic, there
are three.  But if we accept Lesniewski's, there are seven objects
(A, B and C, plus objects obtained by adding A+B, A+C, etc.).  For
Searle, this example shows that two different "systems of arithmetic"
three/seven objects in this diagram.") WITHOUT forcing us to
entertain different ontologies of independently existing mathematical
objects.

Rather than mathematical realism, I think Searle's target is
conceptual fundamentalism, the view that "there is one and only one
correct conceptual scheme for describing reality."  I take Searle to
argue that if THIS were true, then the difference between Carnap and
Lesniewski WOULD be a difference between two completely distinct
ontologies, and we WOULD be forced to choose between them if we were
to make any progress.

-paul

Paul J. Hollander
Visiting Lecturer
Corning Community College
Corning, NY
```