Circular Logic

Abstract

My objective for this chapter is to draw your attention to a few peculiarities inherent in the logic of periodic functions. I find a visual approach the most fruitful for thinking about such matters. As the pictures involved consist mainly of mappings between circles and products of circles, I must first say a few words about the notions of topological spaces and mappings. This chapter thus has four sections:

• A. Spaces, with emphasis on rings (i.e., closed loops. To avoid the more exact connotations of the word circle I use ring, trusting the reader do not confuse my meaning with algebraic rings.)

• B. Mappings, with emphasis on the winding number of mappings to a ring

• C. Phase singularities of maps (Parts I and II), with emphasis on the consequences of a nonzero winding number

• D. Technical details on the application of circular logic to biological rhythms

Philosophy is written in this great book—by which I mean the universe—which stands always open to our view, but it cannot be understood unless one first learns how to comprehend the language and interpret the symbols in which it is written, and its symbols are triangles, circles, and other geometric figures, without which it is not humanly possible to comprehend even one word of it; without these one wanders in a dark labyrinth.

Galileo Galilei, 1623