The Logic and Mathematics of Charles Sanders Peirce

AbstractThis chapter explores the logic and mathematics of Charles Sanders Peirce. The chapter begins with Peirce’s sign of illation, a sign for implication that is a structural combination of the operations of or and negation. This sign is a direct notational relative of the mark of George Spencer-Brown. The work of Gottlob Frege and Spencer-Brown is compared with that of Peirce, and it is shown how to translate Peirce’s theory of existential graphs with Frege’s tree-like logical notations and how these can be superimposed on the calculus of Spencer-Brown to give a uniform method for logical quantifiers in each of these domains. Spencer-Brown’s formalisms for self-reference and re-entry are discussed in relation to Peirce’s concept of a sign for itself, and these ideas of Peirce are discussed in relation to the mathematics of infinity and infinitesimals. Peirce’s notions about infinitesimals are compared with intuitionistic logic and with the Heyting algebra of open sets in a topological space. The chapter shows the remarkable connections of Peirce’s ideas with mathematical foundations.