It became clear by the 1930s that steps of formal computation are also steps of formal deduction as defined by recursion equations and other principles of arithmetic. This development began with attempts at giving a foundation to the laws of commutativity and associativity of sum: Followers of Kant's doctrine of the synthetic a priori in arithmetic, a certain Johann Schultz as foremost, missed ...

Let R be a ring with center Z, Jacobson radical J , and set N of all nilpotent elements. Call R semiperiodic if for each x ∈ R\ (J ∪Z), there exist positive integers m, n of opposite parity such that x − x ∈ N . We investigate commutativity of semiperiodic rings, and we provide noncommutative examples. Mathematics Subject Classification (2000). 16U80.