[FOM] Mathematical explanation

Richard Heck rgheck at brown.edu
Sun Oct 30 11:10:14 EST 2005

Sorry for the extensive quotation. It seems necessary in this case.

mjmurphy wrote:

> 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.  

This misses both my points. (i) That no Fs should be Gs is not a
condition for the applicability of "3+4=7", since "3+4=7" neither says
nor implies that if there are three Fs and four Gs, then there are seven
F-or-Gs. What it says is that three plus four is seven, and the most it
implies is: (*) If there are three Fs and four Gs, and no F is G, then
there are seven F-or-Gs. is what I wrote above. (ii) If that doesn't
seem convincing, then consider (*): I claim that it has no "conditions
of applicability". It is certainly not a condition for its applicability
that no F is G, since it can be used in /modus tollendo/ reasoning.

>> 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.
> 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 [the proposition expressed--RH]. [snip]
> 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).

Obviously that is the point of the example. I've argued above that the
example fails to make that point. But let's assume I'm wrong. Then the
question at issue is what, even if Searle were correct, would follow
about the necessity of *the //proposition expressed* by "3+4=7" on
various occasions of utterance. I am conceding, for the moment, that
this sentence may express different propositions on different occasions.
In that case, it is correct that the literal meaning of the sentence is
not necessarily true, but that is because the literal meaning of the
sentence is not a complete proposition. The literal meaning of "3+4=7"
is neither true nor false, period: It makes no more sense to ask whether
(the literal meaning of) "3+4=7" is true or false than it does to ask
whether (the literal meaning of) "I am Richard Heck" is true or false. A
fortiori, the literal meaning of "3+4=7" is neither necessarily true nor
necessarily false, but trivially so. The only thing that can be
necessarily true or necessarily false is a thing that might be true or
false, namely, *the proposition expressed* on some occasion of utterance.

But once that point has been made, it becomes obvious that Searle's
example simply doesn't bear upon the question whether *the proposition
expressed* by "3+4=7", on any occasion of use, is necessarily true. The
example purports to show that "3+4=7" sometimes expresses a proposition
that is false, but for all Searle says that proposition is *necessarily*
false, and there is a different proposition that "3+4=7" expresses on
other occasions---say, when mathematicians utter it---that is
necessarily true. That's how it is with "I am Richard Heck". When I
utter it, it expresses a proposition that is both true and necessary;
when someone else does, it expresses a proposition that is both false
and impossible. To argue that it is not necessary that I am Richard
Heck, on the ground that "I am Richard Heck" would have expressed a
different proposition had someone else uttered it, is to make an
elementary mistake.

There is nothing original in anything I've said here. It is absolutely
standard, run-of-the-mill, Philosophy of Language 101 stuff, though it
took great effort on the part of David Kaplan, Saul Kripke, and many
others to sort all of this out in the 1960s and 1970s. It is, sadly,
true that there are still people who haven't grasped these simple
points, but I don't think Searle is one of them. His point is supposed
only to be that "3+4=7" can express different propositions on different
occasions of use. As I've said, I think he's wrong, but that's at least

Richard Heck

Richard G Heck, Jr
Professor of Philosophy
Brown University
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