EAI OnQuotienting

Quotienting (other possible word: "absorption"?) is forgetting details judged irrelevant or unobservable, and considering only the resulting structure of relevant and observable facts in a substratum. It's making the irrelevant implicit.

Examples: relative integers as quotient of pairs of natural integers of "same difference"; rational numbers as quotient of pairs of relative integers (the second of which is non-zero) of "same ratio"; real numbers as quotient of Cauchy sequences of rational numbers of "same limit"; etc. The notions of "difference", "ratio", "limit" are actually defined a posteriori as emerging from the structural operation of quotienting.

Considering the world up to isomorphism. The simplest non-trivial class of isomorphism is renaming. Peter Lax reportedly reported this nice story about An English mathematician whose name was lost being was asked by his very religious colleague: "Do you believe in one God?", and replying "Yes, up to isomorphism!".

Next, beware that isomorphisms introduce a (constant) cost in terms of complexity; thus, everyone will choose whichever way of seeing things suits his habits best; the cost of the isomorphism will thus show during communication; people who engage in rational discussion must understand this cost, and work towards reduce it by accepting common conventions.

An extreme case to which to apply quotienting. See the way we translate among languages: words don't intrinsically refer to a same absolute idea; instead, it is by examining the relationships of words each with the other, and each with the external world, that we may match words each with the other; the actual "ideas" are not attached to particular words of any particular language, but to that structure of relationships between words, up to a renaming of words, up to a change in the grammar of the language, up to a translation to any another language in which to express them; that is, the structure of ideas is a quotient structure. This holds even when humans talk to each other for the internal "language" in which human brains encode ideas varies from individual to individual (and even from an individual to a further self, since "he" is not the same as "he" will be a few seconds or a few years later). [See below about the timeliness of information]

This page is linked from: EAI OnFormalReasoning   Ethics and Information