7 Nov 2008

Continuous & Discrete Forms – and Digital & Analog Representations – of Magnitude

by Corry Shores
[Search Blog Here. Index-tags are found on the bottom of the left column.]

[Central Entry Directory]
[Mathematics, Calculus, Geometry, Entry Directory]
[Analog & Digital, Entry Directory]

[For a more extensive discussion of analog and digital, see this entry.]

Our senses seem to tell us that the world around us is too irregular to have perfectly identical magnitudes, for two reasons: 1) things are always changing, even if only a little, so two magnitudes would never sustain long enough to be considered as identically sized, and 2) it is difficult to imagine that two actual objects are the same size, if theoretically we had absolute precision of measurement; for we can always increasingly magnify their limits until we found some small difference.

On the other hand, when we imagine magnitudes abstractly by means of numerical representation, we can imagine identical and unchanging values. For long in human history such abstract values have been represented by our fingers and toes, our digits, and as well in graphic numerals. Digital representations are not continuous magnitudes but discrete ones, because they always represent determinate abstract numerical values. Maurice Trask writes:

To count, we separate each part and give it a number, or digit, in sequence. This one-to-one correspondence is digital. The digital counting of quantities is more easy to recognize, since we use numbers or number patterns to identify groups of objects in which each unit keeps its separate identity.

The word digit comes from the Latin for ‘finger.’ In counting no matter how small the step there is always a jump from one to the next. This method can be as precise as we need since we can reckon to as many places as is necessary (Trask The Story of Cybernetics 26-30).

In contrast, there is the analog representation of numbers, which is more conducive to the continuous quantities of the world we sense:

To measure, we compare an unknown quantity with one we already know, like a length marked as a ruler. An analogy is drawn between them, so this way of measuring is an ‘analog.’ The analog as a measure of size operates when you stand on a weighing machine, the pointer swings across the scale to stop at your weight. This position on the scale is an analog of your weight, since one physical variable, weight, is imitated by another, length. Such measurements are continuously variable, merging imperceptibly from one to the next. The accuracy with which we can read the measurements limits their precision (27-29).

Our senses tell us that magnitudes are continuous, but we cannot mentally compute continuous variables, hence we perform our numerical calculations with discrete and determinate digital values.

Because the Greeks were unable to solve Zeno’s paradoxes, they were unable to offer a quantitative explanation for the phenomena of motion and variability. Such experiences were treated either metaphysically as by Heraclitus, or through qualitative description, as with Aristotle’s physics. Zeno’s paradoxes, then, caused the Greeks to abandon the Democritean attempt to explain the continuous in terms of the discrete (Boyer The History of the Calculus 25-26).

Although, Plato made use of the notion of apeiron: the unbounded indeterminate.

According to Plato, the continuum, could better be regarded as generated by the flowing of the apeiron than thought of consisting of an aggregation (however large) of indivisibles.

Plato thereby fuses continuous with discrete.

The infinitely small was apparently not to be reached through a continued subdivision, but was to be regarded as analogous to the generative infinitesimal of Leibniz, or the “intensive” infinitely small magnitude which appeared in idealistic philosophy of the nineteenth century (28c).

William of Occam held, contra Aristotle, that although no part of a continuum is indivisible, “the straight line actually (not only potentially) consists of points.” For him, "points, lines, and surfaces are pure negations, having no reality in the sense that a solid is real" (67b).

Russell regarded a continuum as a “perfect set of points everywhere dense” (67d). Density is a property of analog: between any two points there is always another point. Brandwardine (and similarly Brouwer and the modern intuitionists) “conceived of the continuum as made up of an infinite number of infinitely divided continua” (67c).

Galileo, contrary to Brandwardine, held that:

continuous magnitudes are made up of indivisibles. However inasmuch as the number of parts is infinite, the aggregation of these is not one resembling a very fine powder but rather a sort of merging of parts into unity, as in the case of fluids (116b).

Trask, Maurice. The Story of Cybernetics. London: Studio Vista, 1971. ISBN 0-289-70057-4.

Boyer, Carl B. The History of the Calculus and its Conceptual Development. New York: Dover Publications, 1949.

No comments:

Post a Comment