Summary: Methodology and metaphysics in the
development of Dedekind's theory of ideals
April 13, 2005
Philosophical concerns rarely force their way into the average mathematician's
workday. But, in extreme circumstances, fundamental questions can arise as to
the legitimacy of a certain manner of proceeding, say, as to whether a particular
object should be granted ontological status, or whether a certain conclusion is
epistemologically warranted. There are then two distinct views as to the role
that philosophy should play in such a situation.
On the first view, the mathematician is called upon to turn to the counsel of
philosophers, in much the same way as a nation considering an action of dubious
international legality is called upon to turn to the United Nations for guidance.
After due consideration of appropriate regulations and guidelines (and, possibly,
debate between representatives of different philosophical factions), the philoso-
phers render a decision, by which the dutiful mathematician abides.
Quine was famously critical of such dreams of a "first philosophy." At the
opposite extreme, our hypothetical mathematician answers only to the subject's
internal concerns, blithely or brashly indifferent to philosophical approval. What