Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP Path: utzoo!mnetor!seismo!rutgers!princeton!mind!harnad From: harnad@mind.UUCP (Stevan Harnad) Newsgroups: sci.electronics,sci.physics,sci.math Subject: A/D Distinction Message-ID: <108@mind.UUCP> Date: Sat, 1-Nov-86 11:51:23 EST Article-I.D.: mind.108 Posted: Sat Nov 1 11:51:23 1986 Date-Received: Mon, 3-Nov-86 23:02:31 EST Distribution: net Organization: Cognitive Science, Princeton University Lines: 160 Keywords: analog, digital, continuous, discrete, quantization, sampling rate Summary: Idealization, Approximation, Quantization, Continuity Xref: mnetor sci.electronics:42 sci.physics:94 sci.math:101 Anders Weinstein has offered some interesting excerpts from the philosopher Nelson Goodman's work on the A/D distinction. I suspect that some people will find Goodman's considerations a little "dense," not to say hirsute, particularly those hailing from, say, sci.electronics; I do too. One of the subthemes here is whether or not engineers, cognitive psychologists and philosophers are talking about the same thing when they talk about A/D. [Other relevant sources on A/D are Zenon Pylyshyn's book "Computation and Cognition," John Haugeland's "Artificial Intelligence" and David Lewis's 1971 article in Nous 5: 321-327, entitled "Analog and Digital."] First, some responses to Weinstein/Goodman on A/D; then some responses to Weinstein-on-Harnad-on-Jacobs: > systems like musical notation which are used to DEFINE a work of > art by dividing the instances from the non-instances I'd be reluctant to try to base a rigorous A/D distinction on the ability to make THAT anterior distinction! > "finitely differentiated," or "articulate." For every two characters > K and K' and every mark m that does not belong to both, [the] > determination that m does not belong to K or that m does not belong > to K' is theoretically possible. ... I'm skeptical that the A/D problem is perspicuously viewed as one of notation, with, roughly, (1) the "digital notation" being all-or-none and discrete and the "analog notation" failing to be, and with (2) corresponding capacity or incapacity to discriminate among the objects they stand for. > 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...? > semantic finite differentiation... for every two characters > I and K' such that their compliance classes are not identical and [for] > every object h that does not comply with both, [the] determination > that h does not comply with K or that h does not comply with K' must > be theoretically possible. I hesitantly infer that the "semantics" concerns the relation between the notational "image" (be it analog or digital) and the object it stands for. (Could a distinction that so many people feel they have a good intuitive handle on really require so much technical machinery to set up? And are the different candidate technical formulations really equivalent, and capturing the same intuitions and practices?) > A symbol _scheme_ is analog if syntactically dense; a _system_ is > analog if syntactically and semantically dense. ... A digital scheme, > in contrast, is discontinuous throughout; and in a digital system the > characters of such a scheme are one-one correlated with > compliance-classes of a similarly discontinous set. But discontinuity, > though implied by, does not imply differentiation...To be digital, a > system must be not merely discontinuous but _differentiated_ > throughout, syntactically and semantically... Does anyone who understands this know whether it conforms to, say, analog/sampled/quantized/digital distinctions offered by Steven Jacobs in a prior iteration? Or the countability criterion suggested by Mitch Sundt? > If only thoroughly dense systems are analog, and only thoroughly > differentiated ones are digital, many systems are of neither type. How many? And which ones? And where does that leave us with our distinction? Weinstein's summary: >>To summarize: when a dense language is used to represent a dense domain, the >>system is analog; when a discrete (Goodman's "discontinuous") and articulate >>language maps a discrete and articulate domain, the system is digital. What about when a discrete language is used to represent a dense domain (the more common case, I believe)? Or the problem case of a dense representation of a discrete domain? And what if there are no dense domains (in physical nature)? What if even the dense/dense criterion can never be met? Is this all just APPROXIMATELY true? Then how does that square with, say, Steve Jacobs again, on approximation? -------- What follows is a response to Weinstein-on-Harnad-on-Jacobs: > Engineers are of course free to use the words "analog" and "digital" > in their own way. However, I think that from a philosophical > standpoint, no signal should be regarded as INTRINSICALLY analog > or digital; the distinction depends crucially on how the signal in > question functions in a representational system. If a continuous signal > is used to encode digital data, the system ought to be regarded as > digital. Agreed that an isolated signal's A or D status cannot be assigned, and that it depends on its relation with other signals in the "representational system" (whatever that is) and their relations to their sources. It also depends, I should think, on what PROPERTIES of the signal are carrying the information, and what properties of the source are being preserved in the signal. If the signal is continuous, but its continuity is not doing any work (has no signal value, so to speak), then it is irrelevant. In practice this should not be a problem, since continuity depends on a signal's relation to the rest of the signal set. (If the only amplitudes transmitted are either very high or very low, with nothing in between, then the continuity in between is beside the point.) Similarly with the source: It may be continuous, but the continuity may not be preserved, even by a continuous signal (the continuities may not correlate in the right way). On the other hand, I would want to leave open the question of whether or not discrete sources can have analogs. > I believe this is the case in MOST real digital systems, where > quantum mechanics is not relevant and the physical signals in > question are best understood as continuous ones. The actual signals > are only approximated by discontinous mathematical functions (e.g. > a square wave). There seems to be a lot of ambiguity in the A/D discussion as to just what is an approximation of what. On one view, a digital representation is a discrete approximation to a continuous object (source) or to a (continuous) analog representation of a (continuous) object (source). But if all objects/sources are really discontinuous, then it's really the continuous analog representation that's approximate! Perhaps it's all a matter of scale, but then that would make the A/D distinction very relative and scale-dependent. > It's a mistake to assume that transformation from "continuous" to > "discrete" representations necessarily involves a loss of information. > Lots of continuous functions can be represented EXACTLY in digital > form, by, for example, encoded polynomials, differential equations, etc. The relation between physical implementations and (formal!) mathematical idealizations also looms large in this discussion. I do not, for example, understand how you can represent continuous functions digitally AND exactly. I always thought it had to be done by finite difference equations, hence only approximately. Nor can a digital computer do real integration, only finite summation. Now the physical question is, can even an ANALOG computer be said to be doing true integration if physical processes are really discrete, or is it only doing an approximation too? The only way I can imagine transforming continuous sources into discrete signals is if the original continuity was never true mathematical continuity in the first place. (After all, the mathematical notion of an unextended "point," which underlies the concept of formal continuity, is surely an idealization, as are many of the infinitesmal and limiting notions of analysis.) The A/D distinction seems to be dissolving in the face of all of these awkward details... Stevan Harnad {allegra, bellcore, seismo, rutgers, packard} !princeton!mind!harnad harnad%mind@princeton.csnet (609)-921-7771