[FOM] Mathematical explanation

Richard Heck rgheck at brown.edu
Fri Oct 28 16:48:35 EDT 2005

mjmurphy wrote:

>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. [snip] 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."
What kinds of assumptions about addition are we supposed to have to
make? The example mentioned is one in which there are three nuts in one
circle and four in another, but the circles overlap. So what Searle is
suggesting is that you can't `apply' "3+4+7" in such a situation. That's
obviously true, if what that is supposed to mean is that you shouldn't
conclude that there are seven nuts in the union of the circles. But
"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. And if you're tempted to say it's useless if there are Fs
that are G, don't forget you can reason by modus tollens: If you find
there aren't seven F-or-Gs, and you know there are three Fs and four Gs,
then you can conclude that some F is G. (Frege was rather fond of this
sort of point.)

>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 counter[example] to its application under any particular set of assumptions, and so mathematics is not necessary.
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. What Murphy says is that "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". That comment is
ambiguous, and I'll guess that the ambiguity is part of the trouble.
What Searle is saying is not that the *assessment* is relative to
context (Searle isn't a relativist) but that what proposition is
expressed is relative to context. If so, however, then the literal
meaning of a sentence is not, in general, truth-evaluable, so of course
its literal meaning isn't necessary. But that's trivial.

Compare: The proposition expressed by the sentence "I am Richard Heck"
is necessarily true if I utter it, but necessarily false if anyone else
does. The *sentence* is neither necessary nor contingent, since it isn't
truth-evaluable, and the same goes for its literal meaning. Searle is
claiming that all sentences are kind of like "I am Richard Heck", but he
really thinks they are more like "That is Richard Heck", about which
similar things could be said.

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. Similarly, if there were certain contexts in which
utterances of "3+4=7", that would not show that what it expressed in
certain other contexts was not necessary. So to argue is seriously confused.

Richard Heck

Richard G Heck, Jr
Professor of Philosophy
Brown University
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