Deduction, perception, and modeling: The two peirces on the essence of mathematics

Abstract

Charles Sanders Peirce, the celebrated philosopher of pragmatics and semiotics, viewed mathematics as the basic science. But, according to him-what is it? In providing an answer, he gave reference to his father Benjamin Peirce, a leading Harvard mathematician. Charles quoted him with: Mathematics is the science which draws necessary conclusions. However, he went further than his father's position by asking what is necessary reasoning. His analysis led him from the clean world of pure reasoning to the more down-to-earth circumstances of perception and experimentation. Even deductive reasoning proceeds by using signs and their iconic qualities and is based on the perception and experimental manipulation of diagrams. Moreover, Peirce accomplished a pragmatic shift that was oriented toward mathematical practice and especially included the process of modeling as a mathematical key activity. This Peircean standpoint will be explored in more detail, and it will be shown (so I hope) that it offers a perspective for a genetic philosophy with an impact on the didactics of mathematics. © 2005 Springer Science+Business Media, Inc.