FOM: Maddy on method
neilt at mercutio.cohums.ohio-state.edu
Fri Feb 20 18:08:39 EST 1998
>One can think that all mathematical objects are ultimately (modelled as)
>sets without thinking that all effective mathematical methods are set
>theoretic methods. Compare: one can think that everything studied in the
>sciences is ultimately physical without thinking that the only effective
>scientific methods are those of physics. Who would expect botanists or
>psychologists to use the same methods as physicists?
The difference, however, is this: in mathematics, all methods of proof
are *reducible to* proof-in-set-theory; whereas not all botanical or
psychological theorizing is reducible to physical theorizing.
The "physicalist" can believe that ultimately everything obeys the laws
of physics, which in some sense suffice to determine, within whatever limits
are possible in principle, the trajectory in phase space of any given system;
while yet maintaining that biological explanations that appeal, say, to the principle of natural selection, and psychological explanations that appeal, say, to
the notion of rational self-interest, are irreducible to any kind of physical
explanation. The physicalist can maintain that biological and psychological
reality *supervenes* upon physical reality, in that the bio-psychological facts
are fixed by the physical facts; while denying that she would then be obliged
to provide explanations of biological evolution, or rationalizations of
human behaviour, in purely physical terms. The biological and psychological
theorizing would be sui generis, indispensable, and irreducible.
To summarize in current philosophical jargon: the physicalist can maintain
supervenience of the "higher" levels of reality on some "lower" level of
reality while yet denying the possibility in principle of reducing all
higher-level theorizing to lower-level theorizing.
But that's not how it is when we look, analogously, at sets (on the "lower"
level) and other kinds of mathematical objects (at various "higher" levels).
The set-theoretic foundationalist is making both the ontological claim
every mathematical object or structure can be modelled in the
universe of sets (i.e. treated "as if it were" a set)
and the even stronger claim of theory-reducibility:
we can prove every mathematical theorem, upon appropriate
definitional reductions, entirely within the apparatus
of set theory.
That is, the set-theoretic foundationalist is saying that we have both
set-determination and reducibility of mathematics to set theory. Indeed,
set-theoretic foundationalism would be dead in the water if the theoretical
reducibility claim could not be made good.
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