The mathematical turns in logic

This chapter provides a short rumination around the theme of Whitehead's remark that still applies today to a considerable extent: namely, the uneasy relationship between mathematics and symbolic logics and now with computing as well. Many large topics are involved, and some are little studied, for example, the history of education in mathematics, logics, computing, and education. The term symbolic logic was introduced to characterize the kind of logic that gave prominence not only to symbols but also to mathematical theories to which they belonged. This discussion of the role of set theory in mathematical logic makes a nice entree to a survey of the differences between the two traditions of symbolic logic. They are very great, but often poorly recognized by logicians and even historians of logic; being symbolic is about the only common factor. They are best illustrated under four headings: (1) theories of collections, (2) principles and properties, (3) relationship with (some) mathematics, and (4) relationship to language.

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