harnad@mind.UUCP (Stevan Harnad) (11/03/86)
Here are 8 more contributions to the A/D Distinction from (1) M. Derthick, (2) B. Garton, (3) W. Hamscher, (4) D. B. Plate, (5) R. Thau, (6) B. Kuszmaul, (7) C. Timar and (8) A. Weinstein. My comments follow the excerpts from each: ----- (1) (mad@g.cs.cmu.edu> (Mark Derthick) writes: > John Haugeland uses digital (and discrete) to mean "perfectly > definite," which is, I think, the best that can be done. Thus > representing an integer as the length of a stick in inches is digital, > but using the length of the stick in angstroms isn't. Obviously there > is a fuzzy boundary between the two. By the way, it is no problem that > sticks can be 6.5" long, as long as there can be unambiguous cases. Unfortunately, it is just this fuzzy boundary that is at issue here. ----- (2) winnie!brad (Brad Garton) writes: > A couple [of] items you mentioned in passing about the a/d issue struck > some resonant chords in my mind. You hit the nail on the head for me > when you said something about the a/d distinction possibly being a > problem of scaling (I think you were replying to the idea of quantum > effects at some level). When I consider the digitized versions of analog > signals we deal with over here <computer music>, it seems that we > approximate more and more closely the analog signal with the > digital one as we increase the sampling rate. This process reminds > me of Mandelbrot's original "How Long is the Coastline of Britain" > article dealing with fractals. Perhaps "analog" could be thought > of as the outer limit of some fractal set, with various "digital" > representations being inner cutoffs. Don't know how useful this > is, but I do like the idea of "analog" and "digital" being along > some sort of continuum. > You also posed a question about when an approximate image of something > becomes a symbol of that thing (please forgive my awful paraphrasing). > As you seemed to be hinting, this is indeed a pretty sticky and > problematic issue. It always amazes me how quickly people are able > to identify a sound as being artificial (synthesized) when the signal > was intended to simulate a 'natural' instrument, rather than when > the computer (or sunthesizer) was being used to explore some new > timbral realm. Context sensitive? (and I haven't even mentioned yet > the problems of signals in a "musical" phrase!). As you may have been noticing from the variety of the responses, the A/D distinction seems to look rather different from the standpoints of hardware concerns, signal analysis, computational theory, AI, robotics, cognitive modeling, physics, mathematics, philosophy and, evidently, music synthesis. And that's without mentioning poets and hermeneuts. My question about fractals would be similar to my question about continuity: Are they to be a LITERAL physical model? Or are they just an abstraction, as I believe the proposals based on true continuity and uncountability have so far been? Or are what I'm tentatively rejecting as "abstractions" in fact standard examples of nomological generalizations, like the ideal gas laws, perfect elasticity, frictionless planes, etc.? [I'm inclined to think they're not, because I don't think there is a valid counterpart, in the idealization of continuity in these A/D formulations, to the physical notions of friction, etc.. The latter are what account for why it is that we never observe the idealized pattern in physics (but only an approximation to it) and yet we (correctly) continue to take the idealizations to be literally true of nature.] ----- (3) hamscher@HT.AI.MIT.EDU (Walter Hamscher) replied as follows in mod.ai: > I don't read all the messages on AiList, so I may have missed > something here: but isn't ``analog vs digital'' the same thing as > ``continuous vs discrete''? Continuous vs discrete, in turn, can be > defined in terms of infinite vs finite partitionability. It's a > property of the measuring system, not a property of the thing being > measured. If you sample some of the other responses you'll see that some people think that something can be formally defined along those lines, but whether it is indeed the A/D Distinction remains to be seen. ----- (4) The next contribution, posted in sci.electronics by plate@dicome.UUCP (Douglas B. Plate) is somewhat more lyrical: > The complete workings of the universe are analog in nature, > the growth of a leaf, decay of atomic structures, passing of > electrons between atoms, etc. Analog is natural reality, > even though facts about it's properties may remain unknown, > the truth of ANALOG exists in an objective form. > DIGITAL is an invention, like mathematics. It is a representation, > and I will not make any asumptions about what it would represent > except that whatever it represents, being a part of this Universe, > would have the same properties and nature that all other things > in the Universe share. The goal of DIGITAL then would be to > represent things 100% accurately. I will not say that ANALOG is > an infinitely continuous process, because I cannot prove that > there is not a smallest possible element involved in an ANALOG > process, however taking observed phenomena into account, I would > risk to say that the smallest element of ANALOG have not been > measured yet if they do ideed exist. > Digital is finite only in the number of elements it uses to represent > and the practical problem is that "bits" would have to extend > into infinity or to a magnitude equalling the smallest element > of what ANALOG is made of, for digital to reach it's full potential. > The thing is, Analog has the "natural" advantage. The universe is > made of it and what is only theory to DIGITAL is reality to > ANALOG. The intrinsic goal of DIGITAL is to become like > ANALOG. Why? Because DIGITAL "represents" and until it > becomes like ANALOG in it's finity/infinity, all of it's > representions can only be approximation. > DIGITAL will forever be striving to attain what ANALOG > was "born with". In theory, DIGITAL is just as continuously > infinite as ANALOG, because an infinite number of bits could > be used to represent an infinite number of things with 100% > accuracy. In practice, ANALOG already has this "infinity" > factor built into it and DIGITAL, like a dog chasing it's own > tail, will be trying to catch up on into infinity. This personification of "the analog" and "the digital" certainly captures many peoples' intuitions, but unfortunately it remains entirely at the intuitive level. Anthony Wilden wrote a book along these lines that turned the analog and the digital into an undergraduate cult for a few years, very much the way the left-brain/right-brain has been. What I'm wondering whether this exercise can do is replace the hermeneutics by a coherent, explicit, empirical construct with predictive and explanatory power. ----- (5) On sci.math,sci.physics,sci.electronics rst@godot.think.com.UUCP (Robert Thau) replied as follows: > In article <105@mind.UUCP> harnad@mind.UUCP (Stevan Harnad) writes: >>"Preserving information under transformations" also sounds like a good >>candidate... I would think that the invertibility of analog >>transformations might be a better instance of information preservation than >>the irretrievable losses of A/D. > I'm not convinced. Common ways of transmitting analog signals all > *do* lose at least some of the signal, irretrievably. Stray > capacitance in an ordinary wire can distort a rapidly changing signal. > Even fiber optic cables lose signal amplitude enough to require > repeaters. Losses of information in processing analog signals tend to > be worse, and for an analog transformation to be exactly invertible, it > *must* preserve all the information in its input. But then wouldn't it be fairest to say that, to the degree that a signal FAILS to preserve its source's properties it is NOT an analog of it? > ...The point is that the amount > of information in the speakers' input which they lose, irretrievably, > is a consequence of the design decisions of the people who made them. > Such design decisions are as explicit as the number of bits used in a > digital representation of the signal in the CD player farther up the > pike. Either digital or analog systems can be made as "true" as you > like, given enough time, materials, and money, but in neither case is > perfection an option. But then what becomes of the often-proposed property of "approximateness" as a distinguisher of an analog representation from a digital one, if they're BOTH approximate? Thau closes by requoting me: >>And this still seems to side-step the question of WHAT information is >>preserved, and in what way, by analog and digital representations, >>respectively. to which he replies: > Agreed. I can't tell whether this is intended to be ironic or Thau is really acknowledging the residual burden of specifying what it is in the way the information is represented that makes it analog or digital, given that the approximate/exact distinction seems to fail and the continuous/discrete one seems to be either just an abstraction or fails to square with the physics. ----- (6) On sci.electronics,sci.physics,sci.math bradley@godot.think.com.UUCP (Bradley Kuszmaul) proposes the following very relativistic account: > The distinction between digital and analog is in our minds. > "digital" and "analog" are just names of design methodologies that > engineers use to build large systems. "digital" is not a property of > a signal, or a machine, but rather a property of the design of the > machine. The design of the machine may not be a part of the machine. > If I gave you a music box (which played music naturally), and > you might not be able to tell whether it was digital or analog (even > if you could open it up and look at it and probe various things with > oscilliscopes or other tools). > Suppose I gave you a set of schematics for the box in which > everything was described in terms of voltages and currents, and which > included an explanation of how the box worked using continuous > mathematical functions. The schematics might explain how various > subcomponents interpreted their inputs as real numbers (even though > the inputs might be a far cry from real numbers e.g. due to the > quantization of everything by physicists). You would probably > conclude that the music box was an analog device. > Suppose, on the other hand, that I gave you a set of schematics for > the same box in which all the subcomponents were described in terms of > discrete formulas (e.g. truth tables), and included an explanation of > how the inputs from reality are interpreted by the hardware as > discrete values (even though the inputs might be a far cry from > discrete values e.g. due to ``noise'' from the uncertainty of > everything). You would probably conclude that the music box was a > digital device. > The idea is that a "digital" designer and "analog" designer might > very well come up with the same hardware to solve some problem, but > they would just understand the behaviour differently. > If designers could handle the complexity of thinking about > everything, they would not use any of these abstractions, but would > just build hardware that works. Real designers, on the other hand, > must control the complexity of the systems they design, and the > "digital" and "analog" design methodologies control the complexity of > the design while preserving enough of reality to allow the engineer to > make progress. If I understand correctly, Kuszmaul is suggesting that whether a representation is analog or digital may just be a matter of interpretation. (This calls to mind some Searlian issues about "intrinsic" vs. "derived" intentionality.) I have some sympathy for this view, because I myself have had reason to propose that the very same "module" that is regarded as "digital" in its autonomous, stand-alone form, might be regarded as analogue in a "dedicated" system, with all inputs and outputs causally connected to the world, and hence all "interpretations" fixed. Of course, that's just a matter of scale too, since ALL systems, whether with or without human intermediaries and interpreters, are causally connected to the world... But this does do violence to whatever is guiding some people's intuitions on this, for they would claim that THEIR notion of analogue is completely interpretation-independent. The part of me that leans toward the invertibility/information-preserving criterion sides with them. > If you buy my idea that digital and analog literally are in our > minds, rather than in the hardware, then the problem is not one of > deciding whether some particular system is digital (such questions > would be considered ill-posed). The real problem, as I view it, is to > distinguish between the digital and analog design methodologies. > We can try to understand the difference by looking at the cases > where we would use one versus the other. > We often use digital systems when the answer we want is a number. > (such as the decimal expansion of PI to 1000 digits) > We often use analog systems when the answer we want is something > physical (I don't really have good examples. Many of the things > which were traditionally analog are going digital for some of the > reasons described below. e.g. music, pictures (still and moving), > the control of an automobile engine or the laundry machine) > Digital components are nice because they have specifications which > are relatively straightforward to test. To test an analog > component seems harder. Because they are easier to test, > they can be considered more "uniform" than analog components (a > TTL "OR" gate from one mfr is about the same as a TTL "OR" gate > from another). (The same argument goes the other way too...) > Analog components are nice because sometimes they do just what you > wanted. For example, the connection from the gas peddle to the > throttle on the carburator of a car can be made by a mechanical > linkage which gives output which is a (approximately) continuous > function of the input position. To "fly by wire" (i.e. to use a > digital linkage) requires a lot more technology. > (When I say "we use a digital system", I really mean that "we design > such a system using a digital methodology", and correspondingly for > the analog case) > There are of course all sorts of places between "digital" and > "analog". A system may have digital subsystems and analog subsystems > and there may be analog subsystems inside the digital subsystems and > it goes on and on. This sort of thing makes the decision about > whether some particular design methodology is digital or analog hard. I'll leave it to the A/D absolutists to defend against this extreme relativism. I still feel agnostic. Except I do believe that the system that will ultimately pass the Total Turing Test will be deeply hybrid through-and-through -- and not just a concatenation of add-on analog and digital modules either. ----- (7) Cary Timar <watmath!watrose!cctimar> writes: > A great deal of the problem with the definitions I've seen is a > vagueness in describing what continuous and discrete sets are. > The distinction does not lie in the size of the set. It is possible to > form a discrete set of arbitrary cardinality - the set of all ordinals > in the initial segment of the cardinal. This set will start with > 0,1,2,3,... which most people agree is discrete. > I would say that a space can be considered to be "discrete" if it is not > regular, and "continuous" if it is normal. I hesitate to classify the > spaces which are regular but not normal. Luckily, we seldom deal with > models of computation using values taken from such a space. > Actually, I should have looked all of this up before I mailed it, but > I'm getting lazy. If you want to try to find mathematical definitions > of discrete and continuous spaces, I would suggest starting from texts > on Topology, especially Point-Set Topology. I wouldn't trust any one > text to give an universally agreed on definition either... Of course, if I believed it was just a matter that could be settled by textbook definitions I would not have posed it for the Net. The issue is not whether or not topologists have a coherent continuous/discrete distinction but (among other things) whether that distinction (1) corresponds to the A/D Distinction, (2) captures the intuitions, usage and practice of the several disciplines puporting to use the distinction and (3) conforms with physical nature. ----- (8) aweinste@Diamond.BBN.COM (Anders Weinstein) replies on net.ai,net.cog-eng to an earlier iteration about the philosopher Nelson Goodman's formulation: > 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". > 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). There seems to be some difference of opinion on this matter from the continuity enthusiasts, although they all advocate precision and rigor... > "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. Do you mean cases like 2 vs. 1.9999999..., or cases like 2 vs. 2 minus epsilon? They both seem as if they could be either "theoretically discriminable" or "theoretically indiscriminable," depending on the theory. > 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:... musical notation [vs]... [an unquantized] scale > drawing of a building > 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." So apparently it does not matter whether the watch is in fact an "analog" or "digital" watch (according to someone else's definition); according to Goodman's the critical factor is how it's used. > 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. Please be sure to take into consideration the heterogenous sample of replies and rival intuitions this challenge has elicited from these various disciplines. -- Stevan Harnad {allegra, bellcore, seismo, rutgers, packard} !princeton!mind!harnad harnad%mind@princeton.csnet (609)-921-7771
aweinste@Diamond.BBN.COM (Anders Weinstein) (11/04/86)
>Stevan Harnad: > >> 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). > >There seems to be some difference of opinion on this matter from the >continuity enthusiasts, although they all advocate precision and rigor... I don't believe there's any major difference here. The respondants who require "continuity" are thinking only in terms of physics, where you don't encounter magnitudes with dense but non-continuous ranges. Goodman deals with other, artificially constructed symbol systems as well. In these we can, by fiat, obtain a scheme that is dense but non-continuous. I think that representation in such a scheme would fit most people's intuitive sense of "analog-icity" if they thought about it. >> Objects fail to be even theoretically discriminable if they can be >> arbitrarily similar and still count as different. > >Do you mean cases like 2 vs. 1.9999999..., or cases like 2 vs. 2 minus epsilon? >They both seem as if they could be either "theoretically >discriminable" or "theoretically indiscriminable," depending on the >theory. I'm not sure what you mean here. I don't see how a length of 2 inches would count as "theoretically discriminable" from a length of 1.999... inches; nor is a length of 2 inches theoretically discriminable from a length of 2 minus epision inches if epsilon is allowed to be arbitrarily small. On the other hand, a length of 2 inches IS theoretically discriminable from a length of 1.9 inches. In his examples, Goodman rules out cases where no measurement of any finite degree of precision would be sufficient to make the requisite distinctions. >> "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." > >So apparently it does not matter whether the watch is in fact an >"analog" or "digital" watch (according to someone else's definition); >according to Goodman's the critical factor is how it's used. Right. Remember, Goodman is not talking about whether this is what an engineer would class as an analog or digital WATCH (ie. in its internal workings); he's ONLY talking about the symbol system used to represent the time to the viewer. And he's totally relativistic here -- whether the representation is analog or digital depends entirely on how it is to be read.
myers@hpfcph.HP.COM ( Bob Myers) (11/06/86)
After following the "analog vs. digital" discussion for some time,
I have finally felt moved to post my very first (golly!....a drum roll,
please!) response here.
The difference between "analog" and "digital" is nothing more than the
difference between a table of numbers and the corresponding graph; in a
digital representation, we assign a finite-precision number to indicate the
value of something (usually a signal) at various points in time (or frequency,
or space, or whatever). An "analog" representation is just that - we choose
to view some value (voltage, current, water pressure, anything) as hopefully
being a faithful copy of something else. An excellent example is a
microphone, which converts a varying pressure into an "analogous" signal -
a varying voltage. This distinction has nothing to do with the accuracy of
the representation obtained, the technology used to obtain, or any of a host
of other items that come to mind when we think of the terms "analog" and
"digital".
This whole discussion brings to mind a quote that, I believe, Carl Sagan used
in his recent novel, "Contact". It had to do with a conference of SETI
researchers trying to define "a living organism". After a lengthy and rambling
discussion, one member of the British delegation said, "Gentlemen, everyone
here knows the difference between a dead horse and a live one. Pray, let us
cease flogging the latter."
Bob Myers
Somewhere Out of My Mind,
HP Fort Collins