[FOM] Freeman Dyson on Inexhaustibility

[Don Fallis]

Hi,

In his review of *The Fabric of the Cosmos* (
http://www.nybooks.com/articles/17094 ), Freeman Dyson offers two main
criticisms of Brian Greene's views on string theory and science. 
First, Dyson argues that developing a theory of quantum gravity is
pointless since it is likely to be "impossible in principle to observe
the existence of individual gravitons."  Second, Dyson argues that
"science is inexhaustible" since mathematics is part of science and
Go"del has established that mathematics is inexhaustible.  However,
this second argument is very strange in light of Dyson's insistence
that theories make empirically testable predictions about the physical
world.

Go"del established that mathematics is inexhaustible by giving a recipe
for constructing, for any proposed axiomatization of mathematics, a
statement that is undecidable.  However, this particular statement
(which essentially says "This statement is not provable from these
axioms") is not of much interest to physicists or mathematicians. There
are a number of other undecidable statements, such as the Continuum
Hypothesis, that are more interesting (cf. Maddy, P. 1997. Naturalism
in Mathematics. New York: Oxford University Press, pp. 63-70).  But
while some statements of mathematics (e.g., 2 + 2 = 4) certainly make
empirically testable predictions about the physical world, it is not
clear how physicists would go about empirically testing these
undecidable statements about extremely large and complicated sets of
real numbers.  In fact, it is not even clear that *any* use of the real
numbers in physics is literally true of the physical world.  In
particular, the assumption that space-time is continuous may simply be
a useful idealization (cf. Maddy, pp. 143-154).  As a result, it is
likely that these undecidable statements will be decided on purely
mathematical grounds rather than on the basis of empirical observation
(cf. Maddy, pp. 206-215).  Thus, I would suggest that Go"del's result
does not provide an effective argument for the inexhaustibility of
science.

take care,
don

Don Fallis
School of Information Resources
University of Arizona
http://www.sir.arizona.edu/faculty/fallis.html