Dedekind, structural reasoning, and mathematical understanding
| Abstract | The last few decades have witnessed a broadening of the philosophy of mathematics, beyond narrowly foundational and metaphysical issues, and towards the inclusion of more general questions concerning "mathematical methodology" and "mathematical practice" (a development parallel to an earlier broadening of the philosophy of science). There is now widespread, and growing, interest in topics such as: concept formation and conceptual change in mathematics, the use of heuristics in mathematical research, the applicability of mathematics, and even sociological or anthropological questions concerning the mathematical community. Part of this broadening, although a part that remains relatively close to foundational and metaphysical issues, is the turn towards a "new epistemology" for mathematics. The latter includes the study of topics such as: the role of visualization in mathematics, the use of computers in proving mathematical theorems, and the notion of explanation as applied to mathematics.1 The present paper is a contribution to this new epistemology. More particularly, it is an attempt to bring into sharper focus, and to argue for the relevance of, two related themes: "structural reasoning" and "mathematical understanding". As the notion of understanding is vague and slippery in general, as well as very loaded in philosophical discussions of the sciences, the latter label has to be handled with care, though. It will have to be clarified what, if anything (or anything reasonably precise), is to be meant by "understanding" in connection with mathematics. Similarly, while talking about "structural" reasoning in mathematics may be suggestive, that term too requires further elaboration. My clarifications and elaborations will be tied to a specific historical figure and period: Richard Dedekind and his contributions to algebraic number theory in the nineteenth century. This is not an incidental choice; Dedekind's case is particularly pertinent in this context, as I also hope to establish in this paper. I will proceed as follows: In the first section, I will provide a brief summary of Dedekind's work on the foundations of mathematics, as well as of its usual perception in.... | |||||||||
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