[FOM] Mathematical explanation
4mjmu at rogers.com
Wed Nov 2 18:42:44 EST 2005
> Can you tell me what proposition the utterance "3+4=7" expresses when
> it is, say, being uttered by a mathematician and is expressing a
> necessary proposition?
Suppose I say instead that it expresses the proposition that, if there
are 3 Fs and 4 Gs, and no F is a G, then there are 7 F-or-Gs. I think
that proposition is both true and necessary. Is there an argument to the
Your argument has been to explain examples in which an utterance of
"3+4=7" is false by claiming that they express a proposition different
from the one expressed when mathematicians use it. But why not look at
it this way. Assume that any utterance of the sentence means the same
thing or expresses the same proposition (for instance as you have given
above). But then why is the utterance false in the case of the
overlapping circles? Treat whatever it takes to make the utterance
false as contextual assumptions that are not part of the proposition
expressed, but could always be written out as extra clauses appended to
the proposition expressed. Treat the utterance of "3=4=7" in the
context of mathematics (in the context of being uttered by a
mathematician) as expressing the above proposition, but sans extra
assumptions. Note that the mathematical context is a context too, and
changes the truth value
of the utterance as much as any other context.
> It is not merely a matter of making explicit unarticulated semantic
> constituents in order to transform an incomplete proposition into a
> complete one. Take the non-mathematical example again, "The ink is
> blue." Firstly, what is incomplete about the proposition it seems to
> express, that the ink is blue? But perhaps, in order to be more
> precise, we say, "The ink is blue on the page." Is this any more
> complete? I don't know, but let's assume it is, and that now we have
> a complete proposition--that the ink is blue on the page.
> But then, is the ink blue on the page under natural light, or
> ultraviolet light, or etc.?
This slip is very revealing: The ink might /look/ different colors under
different lighting conditions, but surely you do not really think that
changing the lighting conditions changes the ink's color.
I sure do. Not its physical chemical or physical makeup, obviously.
Lets say that we have an ink which is blue under natural light, but red
under ultraviolet light. Imagine that we all normally live under
Furthermore, imagine that we are in the habit of coloring things for
display under ultra-violet light. The "real" color of the ink would be
red, because ultra-violet light provides the natural conditions under
which we see the ink in this context. What the ink looks like under
non-ultra-violet light, and even if it remains the same color or rapidly
changes color under those conditions, is irrelevant.
Note also that in this situation we would still have to worry about
ink's "real color" under ultraviolet light versus how it might "appear"
under ultra-violet light conditions that are somehow unusual.
Lets' say that sentences are made true by their truth conditions. Being
blue under natural light, being blue under ultraviolet light, are two
such conditions. I think they are two wo different ones. Do you think that
under natural light is the same thing as being blue under ultra-violet
Richard concludes with some remarks on beliefs and cognitive states
which I will not respond to. Philosophers of mind or language don't have a
to what their talk of cognitive states amounts to in any terms that go
beyond the vaguely metaphoric.
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