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From: aweinste@Diamond.BBN.COM (Anders Weinstein)
Newsgroups: net.ai,net.cog-eng,sci.math,sci.physics,sci.electronics
Subject: Re: Analog/Digital Distinction: 8 more replies
Message-ID: <1747@Diamond.BBN.COM>
Date: Mon, 3-Nov-86 20:55:22 EST
Article-I.D.: Diamond.1747
Posted: Mon Nov  3 20:55:22 1986
Date-Received: Tue, 4-Nov-86 06:19:23 EST
References: <116@mind.UUCP>
Reply-To: aweinste@Diamond.BBN.COM (Anders Weinstein)
Distribution: net
Organization: BBN Labs, Cambridge, MA
Lines: 58
Keywords: continuity, density, analog, digital, Goodman
Xref: mnetor net.ai:1278 net.cog-eng:333 sci.math:117 sci.physics:105 sci.electronics:52


>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.