The Use of Symbols in Mathematics and Logic

It is commonly believed that the use of arbitrary symbols and the process of symbolisation have made possible the discourse of modern mathematics as well as modern, symbolic logic. This paper discusses the role of symbols in logic and mathematics, and in particular analyses whether symbols remain arbitrary in the process of symbolisation. It begins with a brief summary of the relation between sign and logic as exemplified in Indian logic in order to illustrate a logical system where the notion of ‘natural’ sign-signified relation is privileged. Mathematics uses symbols in creative ways. Two such methods, one dealing with the process of ‘alphabetisation’ and the other based on the notion of ‘formal similarity’, are described. Through these processes, originally meaningless symbols get embodied and coded with meaning through mathematical writing and praxis. It is also argued that mathematics and logic differ in the way they use symbols. As a consequence, logicism becomes untenable even at the discursive level, in the ways in which symbols are created, used and gather meaning. The role of symbols in the formation of the disciplines of logic (particularly modern and symbolic logic) and mathematics is often acknowledged to be of fundamental importance. However, symbols have become so essential that their function in these disciplines is rarely queried. In the epoch of any discipline it is always worthwhile to periodically reconsider the foundational elements. It is in this spirit that I approach the reconsideration of the role of symbols in logic and mathematics. Signs, in the most fundamental sense of the word, can refer to anything which stands for something else (the signified). Thus, a word is a sign; for example, the word ‘cow’ stands for the object cow. There are many ways by which a sign can come to stand for something else. There could be a natural relation which immediately suggests the relation between a sign and the signified. Or, the