[FOM] Mathematical explanation

[mjmurphy]

A reply mostly to R. Heck.

I quoted Searle:

"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."

Richard responded:

> What kinds of assumptions about addition are we supposed to have to
> make?

[clip]

> "3+4=7" does not say anything about nuts and circles. (That was Neil's
> point.) What it implies, and what one can prove logically, is that, if
> there are three Fs and there are four Gs, and no F is a G, then there
> are seven F-or-Gs. That has no "conditions of applicability", so far as
> I can see.

It seems to me that the latter passage here supplies the very assumption 
(condition of applicability) you
ask for: that no Fs should be Gs.  I might also include Arnon Avron's 
definition
to the effect that the four Gs and three Fs are members of disjoint sets.
Under these assumptions (perhaps not exhaustive), an utterance of "3 +4 = 7" 
will be true.  If not,
perhaps not, as in:

A: We have three nuts in this ring, and four nuts in this ring.
B:  Then we have seven nuts, since three plus four equals seven.
A: It does not in this case, as the rings overlap and etc.


Richard Heck says:

> No such example could serve to undermine the necessity of mathematical
> claims. To think that it could is to misunderstand both what
> context-dependence is and what Searle is arguing: Searle is arguing that
> the literal meaning of most (or all) sentences underdetermines the
> propositions expressed by utterances of them, which are determined only
> contextually.

As an aside, I should make it clear that Searle is re-purposing an example
from LW to make a point about literal meaning, and I am re-purposing
him.  The "stupidity and "incomprehension" that some have claimed to notice 
is most likely
mine.

But note that in the above that Richard draws a distinction between an 
utterance,  the
literal meaning of an utterance, and what gets evaluated for truth.  This is
important.  Take the non-mathematical utterance "The ink is blue."  This can 
be made
true by a number of distinct conditions of the world
(truth conditions); for example, by the ink's being blue on the page vs.
being blue in the ink bottle.  However, in both cases, the literal meaning
of the utterance is the same:  "The ink is blue." says that the ink is blue.

So when Richard claims that:

> That mathematical claims are necessary is a thesis about the
> propositions those claims express. It would be silly to think that the
> sentence "3+4=7" could not have expressed a falsehood, though that is
> sometimes said, sloppily. It could have, and it would have had "3" meant
> what "2" does.

...I think this misses the point of the example.  The point, as I read it, 
is that utterances of "3 + 4 =7" can be true or false
while its literal meaning remains the same (as your remarks above and the 
remarks of several other posters suggest).

Context is not part of meaning (or so Searle argues).


Cheers,

M.J.Murphy