[FOM] Reply to Franzen, Heck, Davis

[Lucas Wiman]

Friedman writes:

 >I trust [Buckner] will get
 >to responding to my previous posting (in response to you) as well,
 >where I question whether your postings deal with any issues in the
 >foundations of mathematics. See, e.g.,
 >http://www.cs.nyu.edu/pipermail/fom/2003-April/006392.html and
 >http://www.cs.nyu.edu/pipermail/fom/2003-April/006409.html

In the former, he writes:
 >Whereas I do not doubt the possibility that there are overlooked
 >imaginative and productive connections [between natural language 
philosophy and f.o.m.], nothing in [Buckner's] postings
 >suggest any to me.

While it's not my business to defend Buckner, I do think there is 
conceivably some relevance of natural language philosophy to f.o.m.  
Michael Dummett's paper "The Philosophical Basis of Intuitionistic 
Logic" (reprinted in Benacerraf & Putnam's excellent anthology 
"Philosophy of Mathematics: Selected Readings") seems to show a strong 
relationship between the two.  Dummett argues that the strongest 
argument for intuitionistic logic is based upon the meaning of 
mathematical statements.  In his theory of meaning,  the meaning of any 
statement must be based solely upon what can be publicly manifested by 
the sayer of that statement.  If this were not the case, then statements 
could have radically different meanings for different people, but there 
would never be any means of telling that the statements have different 
meanings.  Dummett then proceeds to argue that there can be no real 
meaning given to the existential claims (say that a certain number 
exists) without constructing that number.  Similarly, this theory of 
meaning dictates that for a statement to be true or false, we must have 
a method of publicly demonstrating which, etc.  (This is a gross and 
probably inaccurate charicature of Dummett's views.  I'm not defending 
them, and I certainly don't agree with them.  They're here to prove a 
point.)

 From these sorts of arguments, which plainly have very substantial 
consequences for f.o.m., it seems clear that certainly imaginative 
connections can be found.  Dummett's arguments play a mainly negative 
role (classical mathematics cannot be given a coherent meaning), but 
their refutation is both necessary and productive for the philosophy of 
mathematics.  The extent to which the philosophy of mathematics can be 
equated with Friedman's "serious f.o.m" is another issue.

In the latter posting cited above, Friedman writes:
 >Ordinary people in ordinary activity are not doing science, which
 >requires a rather extraordinary degree of concentration and attention
 >to detail. Again, any suggestion that mathematicians have no idea
 >what they are talking about is, at least on the face of it, silly.

Noam Chomsky has addressed a related issue in his book "New horizons in 
the study of language and mind".  Chomsky addresses whether our 
intuitive notions of language and meaning have any relevance for the 
study of language.  Chomsky defends the thesis that questions about the 
intuitive usage of mental or physical terms are totally irrelevant to 
the study of language and mind.  The study of such terms are part of 
what he calls "ethnoscience."  Chomsky writes:

"It is interesting to learn how notions of language appear in the 
culture of the Navajo [...] or on the streets of New York, or even in 
the more self-consciously contrived culture of academic 
philosophy. [...] But such endeavors have to be taken seriously; they 
are not casual pursuits, and are not to be confused with naturalistic 
inquiry into the nature of what folk science addresses in its own ways, 
using possibly different faculties of the mind.  Ethnoscience is a 
branch of science that studies humans, seeking to understand their modes 
of interpretation of the world, the diversity of these systems, and 
their origins."

Similarly, one should not, as Buckner seems to do, say something along 
the following lines:  "I have this notion of number, which differs from 
what mathematicians tell me they study in the following ways ... since 
my notion of number is fine for my purposes (counting), the 
mathematicians obviously have the wrong idea of number (or set or 
whatever)."  This sort of argument (which is plainly distinguished from 
a Dummett-style argument, like the one I gave above) is a form of 
ethnomathematics, a branch of sociology.  This has absolutely nothing to 
with serious mathematics, f.o.m., or even the philosophy of language.  
Whatever Buckner's intuitive notions of number are, they are totally 
irrelevant to the serious scientific study of mathematics, and they tell 
nothing about what mathematicians actually study.  To use an example 
which Chomsky uses in several places, this tells us as much as trying to 
prove that the earth cannot revolve around the sun since the sun "rises" 
and "sets."  Chomsky writes "the correspondence [of scientific notions 
of language] to some common-sense notion would matter no more than for 
(topological) neighborhood, energy, or fish."  The same can, I think, be 
said of scientific notions of mathematics.

- Lucas Wiman