FOM: "The unreasonable effectiveness of mathematics"

Solomon Feferman sf at Csli.Stanford.EDU
Mon Feb 2 02:35:41 EST 1998

On Wed, 28 Jan 1998, Charles Silver wrote:

> 	I do not understand the basic points in the following exchange. 
> (If you wish to quote any of this for the list, that's ok with me, but I
> fear if I sent this to the list with all the quoted material below, I
> would receive a polite rebuke from Steve.)

Well, I also urged people not to include so many quotes.  I think your
question stands on its own, and people can look back at my response to
Davis in my posting of 26 Jan. at 11:27 if they want to see the context
again.  On to you,then:
> 	There are several questions that popped up in my mind when I read
> this exchange, but I want to stick with just the basic question here: Are
> you *agreeing* with Quine's naturalism in this context?  In other words,
> do you accept some version of the following stripped-down argument: 
> 	Objects, Ob_1, Ob_2,... are used in our best scientific theories
> to account for various facts in the world.
> _________________________________________________________________
> Therefore: Ob_1, Ob_2,... exist

NO, I don't.

> 	There are certain related arguments often used to show the
> "unreality" or "nonexistence" of certain mathematical objects, like some
> of the reals, some sets, etc.--namely those that do *not* contribute in
> the same way to our science.  So, I'm wondering whether you also accept
> this further argument (about the *non*existence of some mathematical
> objects). 

NO, I don't.

> 	(Let me tell you what's bothering me.  It looks like the argument
> could be: "We [seem to] need to talk 'about' OBJECTS_A in order to account
> for facts in the world. Therefore, OBJECTS_A exist."  And, the negative
> part: "We don't [seem to] need to talk 'about' OBJECTS_B in order to
> account for facts in the world.  Therefore, OBJECTS_B do not exist." 
> Don't both of these arguments seem weak to you?  It leads to something
> like the following view: An object exists iff we need to talk about it in
> order to account for facts in the world.  Maybe Quine does in fact hold
> such a view (with some nuances furnished), but it seems like a weak
> position to me.)
> Charlie
Yes, both arguments do seem weak to me.  And yet people may read me as
accepting the negative argument.  Rather, I am mainly using the fact that
OBJECTS_B are eliminable from our fundamental philosophy while still
accounting for their instrumental use in applications to natural science,
primarily to take issue with the Quinean view that they are on an
ontological and epistemological par with the objects of natural science.
While Quine does not say that the real numbers (or complex numbers or
sets and functions, or linear operators on Hilbert spaces, or what have
you) exist "in nature" (at least as far as I know), I think any attempted
argument to the effect that they do, as a result of their indispensability
to science, is completely vitiated.


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