[FOM] The deductive paradigm for mathematics
Tom Dunion
tom.dunion at gmail.com
Sat Aug 7 13:42:09 EDT 2010
On August 6, Marc Alcobe wrote:
[snip]
>I think that the deductive paradigm is too restrictive, assigning the
mathematician a unique role as a "theorem prover", and so explicitly
excluding the role as a "theory framer" every other kind of scientist
must play. Why shouldn't mathematicians explore different theories
through their consequences and choose those that best meet their
needs, just like any other kind of scientists do?>
In agreement with this sentiment, I would just add a couple of brief
comments/observations.
Some time ago, working in an academic "Division" of Science and Math,
I argued (successfully) for a change in bulletin wording from "our two
disciplines" to "our several disciplines" as I was distressed at the idea
of mathematics being perceived as somehow radically different in kind
than the natural sciences.
If those who take the real number continuum as something with objective
independent existence, given that we know there are questions not settled
within ZFC, how might one proceed, if one is to act scientifically?
Possibilities include thought experiments (e.g. Freiling's darts); plausibility
arguments (e.g. those of Goedel); conjectures by people with long experience
manipulating sets of reals (Borel); and drawing out logical consequences of
appealing ad hoc "axioms" or principles, until they clash with other results.
If and when such a clash occurs, refine the above procedures and have at it
again. That's acting scientifically.
Tom Dunion
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