I will not elaborate too much here on the complementary roles of difference of degree and difference of kind in semiosis. It should be clear that the integrated view of dynamics and semiosis presented above requires that we should be able to describe how semiosis can transcode (quasi-) continuous variability in X’s into discrete variants of R’s (and I’s) and vice versa.

The usual case in semiotics is that a discrete R is taken to stand for a discrete X, so there are discrete words (and numeral symbols) for discrete concrete and abstract classes of phenomena. There are no intermediate words between the discrete items of a dictionary, just as there are no intermediate chemical elements between the discrete types on the periodic table. One discrete element, one discrete chemical symbol.

But we also represent continuous difference by continuous difference, as when temperatures are represented by the length of a line or a color in the visible spectrum. There are in fact all three main Peircean types of such signs: iconic (a scale model or map represents continuously variable spatial conformations by similarly variable drawn shapes), indexical (electrical current represented in an ammeter by the angle of deflection of a needle), symbolic (continuously variable movement represented by a Cartesian distance-time plot). Computerized scientific visualization renders all sorts of continuously variable data by continuous variables such as color, brightness, position in space, etc. One could also do this using representation by acoustical frequencies.

To complete the picture, we need to consider the mixed-mode or transcoding semiosis, where discrete differences are represented by continuous variation and vice versa. When we represent a continuous function, or its graph, by an algebraic expression, we are using discrete algebraic symbols and combinations (syntagms) to represent continuous variation. Likewise when we represent the continuous acoustical variation of human speech by writing with discrete alphabetical symbols. In both cases information is lost, except insofar as there is a larger-scale context of conventions regarding (a) what matters, and (b) how to reverse the conversion.

The last case is one in which we take a discrete phenomenon, such as a word of a language, and represent it by means of continuously variable signs, as we might do if we wrote our language with the acoustical sonograms that show energy-frequency distributions in time. We might do this as well by representing a discrete numeral by a closely approximate Fourier transform spectrum that describes its shape, or representing any object by its Schroedinger wave-packet. In these cases generally a great deal more information is provided by the continuous representamen than by the discrete object, but only again insofar as some larger system embodies conventions about what information is relevant and what is not.

For our present purposes it is only important to see that semiosis can and does operate equally for both discrete and continuous modes of sign-relations among (X, R, I). One can actually see in the history of mathematics and mathematical notation (Cajori 1928), that mathematical symbolism began as an extension of natural language into a domain where language is semiotically weak: representing continuous variation. Before the elaboration of a mathematical vocabulary, it was, and in ordinary natural language remains, quite difficult to describe an irregular shape or a non-simple ratio or a complex motion in 3-dimensions in words. It was always easier to represent such phenomena in gestures or drawings, where the continuous degrees of freedom of 3-space allowed easier mapping between X’s and R’s. Many of the great achievements of mathematics and physics, however, have come when ways were found to integrate discrete signs (words for concepts, algebraic symbols, numeral number systems) with continuously variable phenomena (motion, energy, flux, density-dependence) and continuous forms of representation (Cartesian graphs, readouts of analogue measuring devices, scientific visualization systems). See also Lemke (in press).

In earlier work I have used the terms 'typological' for the case where the representamina were discrete elements, and 'topological' for the case where they were values in a quasi-continuum (i.e. indexing the topology of the real numbers). For further details and discussion see Lemke 1999, 2000a.