> Another potential blind spot is equating mathematical theory with scientific
> theory. As we've seen recently with Fermat's Last Theorem, even the toughest
> propositions in Number Theory are vulnerable to proof. In contrast, even the
> simplest tenets of the Theory of Evolution cannot be proven conclusively. Let's
> not over-stretch the boundaries of analogy (though references to Star Trek are
> always welcome).
I do not deny that. Mathematical theorems can be proven conclusively
and other scientific theories cannot, and this is an important
distinction.
However, that does not contradict the point I was trying to get at:
In any field -- be it math or some other science -- we have WAYS OF
THINKING about things, not just assertions that are true or false.
Perhaps the easiest indicator here is not our list of theorems, but our
list of DEFINITIONS (which may change as time passes -- for instance, I
believe that "topological space" used to mean what "Hausdorff
topological space" means now?).
For a good analogy with Darwin's theory (which was not solely due to
Charles Darwin, I admit), we might think about Cantor's basic ideas on
cardinality (which were not solely due to Cantor, I suppose -- e.g.,
Bolzano came close to Cantor's definition). I don't mean the actual
theorems Cantor came up with; I mean the definitions. Those definitions
allowed mathematicians to think about infinity in ways that they had
never thought about infinity before -- to ask new kinds of questions and
look for new kinds of answers.
Of course, our modern way of doing mathematics has very precise
definitions. To look for paradigm shifts in earlier mathematics, I
suppose that one would have to look deeper than just the list of
definitions.
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Eric Schechter *** http://www.math.vanderbilt.edu/~schectex/
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