Methodology and Metaphysics in the Development of Dedekind's Theory of Ideals

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 philosophers 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 is at stake to our mathematician friend is whether the questionable practice provides a proper mathematical solution to the problem at hand, or an appropriate mathematical understanding; or, in pragmatic terms, whether it will make it past a journal referee. In short, mathematics is as mathematics does, and the philosopher’s task is simply to make sense of the subject as it evolves and certify practices that are already in place. In his textbook on the philosophy of mathematics (Shapiro, 2000), Stewart Shapiro characterizes this attitude as “philosophy last, if at all.” The issue boils down to whether fundamental questions as to proper mathematical practice should be adjudicated with respect to general, and potentially extra-mathematical, considerations, or with respect to inherently “mathematical” standards, values, and goals. Of course, what typically happens lies somewhere in between. Mathematics is not a matter of “anything goes,” and every mathematician is guided by explicit or unspoken assumptions as to what counts as legitimate — whether we choose to view these assumptions as the product of birth, experience, indoctrination, tradition, or philosophy. At the same time,