Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!seismo!think!husc6!Diamond!aweinste From: aweinste@Diamond.BBN.COM (Anders Weinstein) Newsgroups: net.ai,net.cog-eng Subject: Re: The Analog/Digital Distinction Message-ID: <1701@Diamond.BBN.COM> Date: Thu, 30-Oct-86 21:45:56 EST Article-I.D.: Diamond.1701 Posted: Thu Oct 30 21:45:56 1986 Date-Received: Fri, 31-Oct-86 03:39:00 EST References: <15@mind.UUCP> <1670@Diamond.BBN.COM> <20@mind.UUCP> Reply-To: aweinste@Diamond.BBN.COM (Anders Weinstein) Organization: BBN Labs, Cambridge, MA Lines: 75 Keywords: analog, digital, continuous, discrete, quantization, density, Goodman Summary: Further explanation of Goodman on A/D Xref: mnetor net.ai:1262 net.cog-eng:317 In article <20@mind.UUCP> harnad@mind.UUCP (Stevan Harnad) writes: > I suspect that some people will find Goodman's >considerations a little "dense," not to say hirsute, ... Well you asked for a "precise" definition! Although Goodman's rigor may seem daunting, there are really only two main concepts to grasp: "density", which is familiar to many from mathematics, and "differentiation". >> A scheme is syntactically dense if it provides for infinitely many >> characters so ordered that between each two there is a third. > >I'm no mathematician, but it seems to me that this is not strong >enough for the continuity of the real number line. The rational >numbers are "syntactically dense" according to this definition. But >maybe you don't want real continuity...? Quite right. Goodman mentions that the difference between continuity and density is immaterial for his purposes, since density is always sufficient to destroy differentiation (and hence "notationality" and "digitality" as well). "Differentiation" pertains to our ability to make the necessary distinctions between elements. There are two sides to the requirement: "syntactic differentiation" requires that tokens belonging to distinct characters be at least theoretically discriminable; "semantic differentiation" requires that objects denoted by non-coextensive characters be theoretically discriminable as well. Objects fail to be even theoretically discriminable if they can be arbitrarily similar and still count as different. For example, consider a language consisting of straight marks such that marks differing in length by even the smallest fraction of an inch are stipulated to belong to different characters. This language is not finitely differentiated in Goodman's sense. If, however, we decree that all marks between 1 and 2 inches long belong to one character, all marks between 3 and 4 inches long belong to another, all marks between 5 and 6 inches long belong to another, and so on, then the language WILL qualify as differentiated. The upshot of Goodman's requirement is that if a symbol system is to count as "digital" (or as "notational"), there must be some finite sized "gaps", however minute, between the distinct elements that need to be distinguished. Some examples: A score in musical notation can, if certain conventions are adopted, be regarded as a digital representation, with the score denoting any performance that complies with it. Note that although musical pitches, say, may take on a continuous range of values, once we adopt some conventions about how much variation in pitch is to be tolerated among the compliants of each note, the set of note extensions can become finitely differentiated. A scale drawing of a building, on the other hand, usually functions as an analog representation: any difference in a line's length, however fine, is regarded as denoting a corresponding difference in the building's size. If we decide to interpret the drawing in some "quantized" way, however, then it can be a digital representation. To quote Goodman: Consider an ordinary watch without a second hand. The hour-hand is normally used to pick out one of twelve divisions of the half-day. It speaks notationally [and digitally -- AW]. So does the minute hand if used only to pick out one of sixty divisions of the hour; but if the absolute distance of the minute hand beyond the preceding mark is taken as indicating the absolute time elapsed since that mark was passed, the symbol system is non-notational. Of course, if we set some limit -- whether of a half minute or one second or less -- upon the fineness of judgment so to be made, the scheme here too may become notational. I'm still thinking about your question of how Goodman's distinction relates to the intuitive notion as employed by engineers or cognitivists and will reply later. Anders Weinstein