[FOM] Mathematical explanation

[mjmurphy]

On Mon, 24 Oct 2005, I wrote:

> ... there is an example of Searle's (borrowed from Wittgenstein, but I

> don't quite know where).  The example involves the mathematical
proposition
> 3 + 4.

Neil said:

That's not a proposition, but let's not bother ...
-----

Actually, lets, as it may make Searle's point clearer.  His basic
contention (in the paper "Literal Meaning", where the example is
presented, and elsewhere) is that, in general, the literal utterance of
a statement does not supply us with a proposition which can be used to 
assess its
truth value without reference to context.  For example (not Searle here,
but Travis, who has made a similar argument on a number of occasions)
one might say "The ink is blue", literally asserted, expresses the
proposition that the ink is blue, and still not be clear how to assess
the sentence for its truth value.  For instance, the ink might be blue
on the page but not in the bottle, or blue in the bottle but not on the
page. The proposition expressed may be made true by one condition, or
the other, depending on context.

Searle then asks: are there any types of sentence which are immune from
this kind of context dependence, and considers mathematical statements
in this light:

"Perhaps one might show, for example, that an arithmetical sentence such
as "3+4=7" is not dependent on any contextual assumptions for the
applicability of its literal meaning.  Even here, however, it appears
that certain assumptions about the nature of mathematical operations
such as addition must be made in order to apply the literal meaning of
the sentence."

It is at this point that he refers to the example I origonally mentioned:.

So, to the question "Is mathematics necessary?"


It seems to me that if an arithmetical sentence with its literal meaning
can be applied under differing assumptions about the nature of
mathematical operations, than we have a counterfactual to its
application under any particular set of assumptions, and so mathematics
is not necessary.


I wrote:


> Imagine two overlapping circles, A and B.  A contains three dots, B
> four.  However, two of the dots fall into the area where A and B
overlap.
> Here, Searle and LW contend, A + B = 5.

Neil responded:

...If you mean, instead (on
behalf of Searle and LW), that the sum of the number of dots in circle A
and the number of dots in circle B is 5, then of course that's a howler.
The sum of the number of dots in circle A and the number of dots in
circle
B is 7.

Me:

If the dots were nuts and you were stranded on a desert island, you
would probably starve to death adding like this.  We have three nuts
within circle A, four nuts within B.  Do we therefore have seven nuts in
total? Well, if we count each nut once, regardless of how many circles
they appear in, they only add up to five, and since you can't eat a nut
twice, there is some justification for counting this way in this context.
For other purposes, your method might be fine.  The point is that we
have more than one method, and hence a counterfactual with respect to
any particular method.

Neil wrote:

If you could supply a reference for this howler in Wittgenstein's
writings, it would be most useful. It would reveal a remarkable, not to
say incomprehensible, degree of conceptual/mathematical illiteracy....


As I say, the Searle piece is "Literal Meaning" in Expression and
Meaning.  I would prefer to reference that alone since you, like many in
this field, seem to go batty at the mention of LW.

Cheers,

M.J.Murphy