← Back to Concept Indexmathematical-elegance
The quality of a system to condense necessary information into minimal form, free from irrelevance, facilitating ease of use and error reduction.
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CHAPTER 2 consider such-and-such, suppose so-and-so, which are not commands, but invitations or directions to a way in which the implication can be clearly and wholly followed. In conceiving the calculus of indications, we begin at a point of such degeneracy as to find that the ideas of description, indication, name, and instruction can amount to the same thing. It is of some importance for the reader to realize this for himself, or he will find it difficult to understand (although he may follow) the argument (p 5) leading to the second primitive equation. In the command let the crossing be to the state indicated by the token we at once make the token doubly meaningful, first as an instruction to cross, secondly as an indicator (and thus a name) of where the crossing has taken us. It was an open question, before obeying this command, whether the token would carry an indication at all. But the command determines without ambiguity the state to which the crossing is made and thus, without ambiguity, the indication which the token will hence- forth carry. This double carry of name-with-instruction and instruction- with-name is usually referred to (in the language of mathe- matics) as a structure in which ideas or meanings degenerate. We may also refer to it (in the language of psychology) as a place where the ideas condense in one symbol. It is this condensa- tion which gives the symbol its power. For in mathematics, as in other disciplines, the power of a system resides in its elegance (literally, its capacity to pick out or elect), which is achieved by condensing as much as is needed into as little as is needed, and so making that little as free from irrelevance (or from elaboration) as is allowed by the necessity of writing it out and reading it in with ease and without error. We may now helpfully distinguish between an elegance in 81Chapter 2 explores the foundational idea that in the calculus of indications, description, indication, name, and instruction converge into a single concept, embodied in a symbol that carries both instruction and indication simultaneously. This condensation or degeneracy of meaning into one symbol is what gives the system its power and elegance, allowing mathematical operations to be both minimal and unambiguous. The chapter also reflects on the tension between mathematical rigor and ordinary interpretive habits, emphasizing the necessity of accepting instructions literally in mathematics.