Mathematical Individuality in Charles Sanders Peirce

Abstract

Peirce regards mathematics as an informative science capable of really increasing our knowledge. This means that mathematics is not limited to conceptual analysis but possesses a real object of investigation. The core of Peirce’s view of mathematics is that mathematical reasoning is not developed through general concepts alone but deals with an unavoidable element of individuality. The conclusion of a deductive inference can contain information that is not at all present in its premises and can only come into being through concrete work on the part of the mathematician. Peirce describes this work as observation and experimentation on individual diagrams. While the idea of an individual element in mathematics is already present in Kant (and can also be traced back to Aristotle), the different location Peirce assigns it attests to a marked difference in their conceptions, the basis of which lies in the difference between Kant and Peirce in categorial analysis. Peirce’s semiotic approach to mathematics involves a shift from the plane of the object denoted to that of the sign itself. This holds true for geometric as well as algebraic inferences, which Peirce can equate in this respect. In both cases, the individual element of mathematics is thus to be found within the diagram itself with no reference to the object denoted. While the diagram is in any case a token, this cannot explain its essential individuality, to which end the indexical juxtaposition of its parts should be examined.