[FOM] Mathematical explanation

[Paul Hollander]

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" 
can lead to two contradictory statements ("There are exactly 
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