FOM: Fiction and Mathematics

[Dean Buckner]

I've not had a great take-up on this theme, but I'll persevere!  The
background is the work I've been doing on explaining fiction without
invoking fictional characters, sets or numbers of such characters, (such as
e.g the statement  that Jane Austen's Mr Bennett had five daughters).  I
believe we can even do without "metaphysical" concepts like meaning and
truth.  Ordinary language can cope perfectly well with the meaning of
fiction, without being Platonistic.

Surely there is some connection between these ideas, and ideas about the
foundation of mathematics?

I know very little about maths or set theory but I looked at some of the
propositions underlying ZF.  A number of them seem to assert existence of
some sort, the most bizarre being the one about the "empty set".  Here's how
set theory would explain the brief and elegant sentence  "there are no
unicorns".

"the number corresponding to the set defined by the property of unicorn-ness
is equal to the number corresponding to the set defined by the property of
non-self identicalness"

Really?  There are two equal and opposite ways of looking at this.  Fregean
Platonists leap in excitement at the thought of proving the existence of
strange entities from minimal conceptual assumptions ("Axiom of
Specification").  Radical Ockhamists (me) will be pleased that a long
sentence containing apparently existence-invoking expressions ("the property
of non-self identicalness") can be compacted into short sentence containing
no such expressions.

(Note Frege himself abandoned this Platonism late in life.  He writes how
the use of definite article in language creates the illlusion that an object
is designated, and adds rather sadly "I myself was under this illusion when,
in attempting to provide a logical foundation for numbers, I tried to
construe numbers as sets."  (PW p.269).)

It will be argued that ordinary language though simple, may be difficult to
explain.  The language required to explain the meaning of a simple elegant
sentence may be complex and difficult and inelegant. I can't go deep into
this right now, but would just like to challenge it.

Why can't "thought" and "meaning" lie right underneath the surface of
language?  If only we could focus on the right place.  This is an intuition
that drove logic for 2,000 years until Frege.  Is the foundation of
mathematics really as difficult as it looks?

In light of the raging debate over intuition vs rigour, admit little rigour
(but can supply on demand).  The intuition is:  ordinary language  looks
simple, why shouldn't it actually be so?



Dean Buckner (research interest: semantics of fiction)
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