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AnalogandDigital
DAVI LEWIS
PRINCETON UNIVERSiTY
The distinction between analog and digital representationof
numbersis well understoodin practice. Yet its analysis has proved
troublesome. I shall first consider the account given by Nelson
Goodman and
offer examples
to show
that some
cases of analog
representationare mis-classified,on Goodman'saccount, as digital.
Then I shall offer alternativeanalyses of analog and digital repre-
sentation.
- DIFFERENTIATEDANALOG REPRESENTATION
Accordingto Goodmanin Languages of Art,' the distinction
between digital and analog representationof numbersis as follows.
Digital representationis differentiated.
Given a number-represent-
ing "mark"-an inscription,vocal utterance, pointer position, elec-
trical pulse, or whatever-it is theoretically possible, despite our
inability to make infinitely precise measurements, to determine
exactly which other marks are copies
of the given mark and to
determineexactly which number (or numbers) the given markand
its copies represent.Analog representation,on the other hand, fails
to be differentiatedbecause it is dense. For any two marksthat are
not copies, no matter how nearly indistinguishablethey are, there
could be
a mark intermediatebetween them which is a copy
of
neither; and
for any
two marks that are not copies
and represent
differentnumbers,no matter
how close the numbersare, there is an
1
(Indianapolis
and New York: Bobbs-Merrill, 1968), sections IV.2,
IV.5, and IV.8. I have combined Goodman's syntactic and semantic differen-
tiation and combined his syntactic and semantic density;
and I have not
defined differentiation and density in full generality, but only as applied to
the representationof numbers.
321
322
NOOS
intermediate
number which would be representedby a mark that
is a copy of neither.
It is true and importantthat digital representationis
differen-
tiated, and that it differs thereby from the
many cases of analog
representationthat are undifferentiatedand dense:
those cases in
which all real
numbers in some range are represented by values
of some continuouslyvariable physical magnitude such as voltage.
But there are other cases:
representationthat is as differentiated
and non-dense as any digital representation and
yet is analog
ratherthan digital representation.Here
are two examplesof differ-
entiated analog representation.If accepted, they show
that Good-
man's distinction,interestingthough it is in its own right,
does not
coincide
with the analog-digitaldistinctionof ordinarytechnological
language.
Example 1: ordinary
electrical analog computers sometimes
receive their numerical inputs in the
form of settings of variable
resistors.A setting of 137 ohms representsthe
number 137, and so
on. There are two ways these variable resistorsmight work.
In the
first case, a contact slides smoothly along a wire with
constant
resistanceper unit length. In the second case, there is a switch
with
a very large
but finite number of positions, and at each position a
certain number of,
say, 1-ohm fixed resistorsare in the circuit and
the rest are bypassed.I do
not know which sort of variable resistor
is used in practice. In either case,
the computer
is an analog com-
puter and its representationof numbersby electrical resistances
is
analog representation.
In the sliding-contactcase the representation
is
undifferentiatedand dense; but in the multi-positionswitch case
the representation
is differentiatedand non-dense, yet analog and
not digital.
Example 2: we might
build a device which works in the fol-
lowing way to multiplytwo
numbersx and y.
There are four recep-
tacles: W, X, Y,
and Z. We put a large amountof some sort of fluid
-liquid, powder,
or little pellets-in
W. We put x grams of fluid in
X, y gramsin Y, and none in Z. At the bottom
of X is a valve,
allow-
ing
fluid to drain at a constantrate from X into a wastebasket.At
the bottom
of W is a spring-loaded
valve, allowing fluid to drain
from W
into Z at a rate proportional
to the amount of fluid in Y.
For instance, if Y contains 17 grams
of fluid,
then the rate of drain-
age
from W is 17 times the constant rate of drainage from X. We
simultaneouslyopen
the valves on W and X; as soon as X is empty,
324 NOeS
voltages vo,
. .. , V35.
(We take the magnitude to be undefined if
any vi
is 0.)
X
-5t)
35
V
Iif vj(s't)
0-
V(st) = Z= 2t
iO
{
ifv:(st) <
= Z3=02'
Vv+
s 02] j=o L 2vis(s,t)
Analog representation,then, is representationof numbersby phys-
ical magnitudesof a special kind. Resistances,voltages, amountsof
fluid, for instance, are physical magnitudesof the proper kind for
analog representation;but V, as defined above, is a physical magni-
tude not of the properkind for analog representation.How may we
distinguish physical magnitudes that are
of the proper
kind for
analog representationfrom those that are not?
We might try saying that the magnitudessuitable for analog
representationare those that are expressed by primitive terms in
the language of physics. This will not do as it stands: a term is
primitive not relative to a language but relative to some chosen
definitionalreconstructionthereof. Any physical magnitude could
be expressed by a primitive term in a reconstruction designed ad
hoc, and there is no physical magnitudethat must be expressedby
a primitive term. We may, however,
define a primitive magnitude
as any physical magnitude
that is expressedby a primitive term in
some good reconstructionof the language of physics-good accord-
ing to our ordinarystandardsof economy, elegance, convenience,
familiarity.This definitionis scarcely precise, but furtherprecision
calls for a better general understandingof our standardsof good-
ness for definitionalreconstructions,not for more work on the topic
at hand.
Even taking the definitionof primitive magnitudes as under-
stood, however,it is not quite right to say that analogrepresentation
is representationby primitivemagnitudes.Sometimesit is represen-
tation by physical magnitudes
that are almost primitive: definable
in some simple way,
with little use of arithmetical operations,in
terms of one or a few primitivemagnitudes.(Furtherprecisionhere
awaits a better general understanding
of simplicity
of definitions.)
Products
of a currentand a voltage
between the same two points in
a circuit, or logarithmicallyscaled luminosities,seem unlikely
to be
expressedby primitive terms in any good reconstruction.But they
ANALOG AND DIGITAL
325
are almostprimitive,and representationof numbersby them
would
be analog representation.Of more interestfor our present purposes,
rounded-offprimitivemagnitudesare almost
primitive.Such are the
number-representingmagnitudesin our two examples
of differen-
tiated analog representation:resistancerounded to the nearest
ohm
and amount
of fluid rounded to the nearest gram. We use the
rounded-off
magnitudesbecause, in practice, our resistorsor pellets
are not exactly one ohm or gram
each; so if we take the resistance
of 17 resistorsor the mass
of 17 pellets to representthe number 17
(rather than some unknown
number close to 17) we are using not
primitivemagnitudesbut almost
primitivemagnitudes.
The commonplacedefinitionof analog representation
as repre-
sentation by physical magnitudesis correct, so far as I can
see, if
taken as follows: analog representationof numbersis representation
of numbers by physical magnitudes
that are either primitive or
almost primitiveaccordingto the
definitions above.
3. DIGITAL REPRESENTATION
It remains to analyze digital representation,
for not all non-
analog representationof numbers is digital. (It
may be, however,
that all practicallyuseful representationof
numbersis either analog
or digital.) If numberswere representedby the physical
magnitude
defined as follows in terms of voltages between specified pairs
of
wires
in some circuit, the representationwould be neither analog
nor digital (nor
useful).
35
P(st) = L logo( sinh(V/vi(s,t)))
Digital representationis representationby physical magnitudes
of
another special
kind, the kind exemplified
above by
V.
We may first define an n-valued unidigital magnitude
as a
physical magnitude
having as values the numbers 0, 1,.. ., n - 1
whose values depend by
a step
function on the values of some
primitive magnitude. Let U be
an n-valued unidigital
magnitude;
then there is a primitivemagnitude B which we may call the
basis
of U and there is an increasingsequence of numbersa,,. .. ., a.-
which we may call the transitionpoints of U and, for each system
s
on which
U is defined,
there is a part p of s such that U is defined
as follows on
s.
ANALOG AND DIGITAL 327
more complicated ways upon their unidigital magnitudes. Most
often, the unidigitalmagnitudesare not merely few-valuedbut two-
valued; but not so, for instance, in an odometer in which the uni-
digital
magnitudes are ten-valued step functions of angles of rota-
tion of the wheels.
I suggest, therefore,that we can define digital representation
of numbers as representationof numbers by differentiatedmulti-
digital magnitudes.
