Mathematics

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Abstract

Mathematics is the most natural terrain for analytic logic; indeed, one way to think of analytic logic is that it conceives everything mathematically. Still, the other logics have an important, though generally unrecognized, presence within this discipline. The best way to see mathematics dialectically is to start from a consideration of its history: of its gradual expansion of the concept of number from the rationals to the irrationals, then to imaginary and complex numbers, then to transfinite ones. An analytic logician might appreciate all of this but would insist that the history of mathematics is external to mathematics: if there are irrational, complex, and transfinite numbers, then those numbers exist eternally; their existence has no history, though there is a history to humans coming to know them—a history of mathematicians, then, not even quite a history of mathematics. This view, though perfectly consistent, is however itself internal to the analytic point of view: what is in question here is the identity of numbers, and whether this identity is construed analytically or dialectically is an issue that cannot be decided independently from a commitment to one or the other logic. For dialectical logic, the history of mathematics (understood logically, not chronologically) simply is mathematics. The role of oceanic logic within mathematics can be best illustrated by focusing on two issues. First, recursive definitions: characterizations of what something (say, a natural number) is that indefinitely postpone the moment of fulfillment and hence never conclusively establish the distinction between what is to be defined and its contraries. Set theory allows us to prove a recursion theorem establishing that the clauses of a recursive definition identify exactly one function, but for the practicing mathematician exactly the same problem will surface in set theory itself, since the theory’s intended model (within which the practicing mathematician works) is itself defined recursively. The second issue is continuity, which analytic logic has a tendency to construe in terms of discrete entities, thus violating what from an oceanic standpoint is its very nature.

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APA

Bencivenga, E. (2017). Mathematics. In Historical-Analytical Studies on Nature, Mind and Action (Vol. 4, pp. 119–129). Springer Science and Business Media B.V. https://doi.org/10.1007/978-3-319-63396-1_9

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