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AIList Digest Volume 4 Issue 246
AIList Digest Wednesday, 5 Nov 1986 Volume 4 : Issue 246
Today's Topics:
Philosophy - The Analog/Digital Distinction
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Date: 29 Oct 86 17:34:08 GMT
From: rutgers!princeton!mind!harnad@lll-crg.arpa (Stevan Harnad)
Subject: Re: The Analog/Digital Distinction: Sol
Concerning the A/D distinction, goldfain@uiucuxe.CSO.UIUC.EDU replies:
> Analog devices/processes are best viewed as having a continuous possible
> range of values. (An interval of the real line, for example.)
> Digital devices/processes are best viewed as having an underlying
> granularity of discrete possible values.
> (Representable by a subset of the integers.)
> This is a pretty good definition, whether you like it or not.
> I am curious as to what kind of discussion you are hoping to get,
> when you rule out the correct distinction at the outset ...
Nothing is ruled out. If you follow the ongoing discussion, you'll see
what I meant by continuity and discreteness being "nonstarters." There
seem to be some basic problems with what these mean in the real
physical world. Where do you find formal continuity in physical
devices? And if it's only "approximate" continuity, then how is the
"exact/approximate" distinction that some are proposing for A/D going
to work? I'm not ruling out that these problems may be resolvable, and
that continuous/discrete will emerge as a coherent criterion after
all. I'm just suggesting that there are prima facie reasons for
thinking that the distinction has not yet been formulated coherently
by anyone. And I'm predicting that the discussion will be surprising,
even to those who thought they had a good, crisp, rigorous idea of
what the A/D distinction was.
Stevan Harnad
{allegra, bellcore, seismo, rutgers, packard} !princeton!mind!harnad
harnad%mind@princeton.csnet
(609)-921-7771
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Date: 29 Oct 86 16:37:55 GMT
From: rutgers!husc6!Diamond!aweinste@lll-crg.arpa (Anders Weinstein)
Subject: Re: The Analog/Digital Distinction
> From Stevan Harnad:
>
>> Analog signal -- one that is continuous both in time and amplitude.
>> ...
>> Digital signal -- one that is discrete both in time and amplitude...
>> This is obtained by quantizing a sampled signal.
>
> Question: What if the
>original "object" is discrete in the first place, both in space and
>time? Does that make a digital transformation of it "analog"? I
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.
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).
> The image of an object
>(or of the analog image of an object) under a digital transformation
>is "approximate" rather than "exact." What is the difference between
>"approximate" and "exact"? Here I would like to interject a tentative
>candidate criterion of my own: I think it may have something to do with
>invertibility. A transformation from object to image is analog if (or
>>to the degree that) it is invertible. In a digital approximation, some
>information or structure is irretrievably lost (the transformation
>is not 1:1).
> ...
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.
Anders Weinstein
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Date: 29 Oct 86 20:28:06 GMT
From: rutgers!princeton!mind!harnad@titan.arc.nasa.gov (Stevan
Harnad)
Subject: Re: The Analog/Digital Distinction
[Will someone with access post this on sci.electronics too, please?]
Anders Weinstein <princeton!cmcl2!harvard!DIAMOND.BBN.COM!aweinste>
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
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End of AIList Digest
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