QuantX as TypeR

TypeX as QuantR

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It has become customary to represent discrete types by discrete signs, and continuously variable phenomena by other continuously variable phenomena, but this is not necessary. Iit is also possible in principle to represent quantitative difference and relationship qualitatively, by types; this is what language does in describing the continuously variable aspects of the natural world  and there are many other such mixed modes.
The mathematics of classical natural science is primarily an extension of the typological semiotics of natural language semantics to improve the description of quantitative difference, ratio, and relationship. Mathematics can be spoken, grammatically. Its semantics derives from and extends that of natural language, and natural language in turn expands through the development of the specialized linguistic register of spoken (or written) verbal mathematics. Historically natural language did not have a means of expressing complex ratios, or designating meanings that depended on precise rational (much less irrational) numbers. But from its existing resources mathematicians created in stages the notion we recognize today as proper fractions, whose delicacy and precision of meaning (differences that make a difference) grew with the growth of the number system, first to higher counting numbers, and then beyond. Arithmetic, a simple spoken algebra, grew to describe more arbitrary geometrical relationships. Without this extension, and even today for the most part with it, any verbal description of a complex shape is only very vaguely approximate. When quantitative X is represented by discrete R, information is lost to the degree that X may vary continuously and by arbitrarily small, and distinct, degrees, while R, at least in the form of nonalgebraic language may not.
As another example, consider the phoneme of linguistic analysis: the minimal distinctive sound unit by which words of different meaning can differ (e.g. good vs. goad, good vs. hood, good vs. gook). The articulatory apparatus of human speech production is capable of producing a continuous spectrum of vowels between oo and oa, and many continuous dimensions of difference in other respects, but the linguistically significant distinctions are categorial: a word belongs to our language only if it is assimilated to one of a finite number of discrete possible sequences of phonemes; a harmonically coloured, shouted, or choked vowel does not a new word make. Language reduces the continuum of the acoustic spectrum to a finite set of discrete contrasts.
Ver generally, representing quantX by typeR necessarily loses information; equivalence classes are created within which quantitative variation is filtered out and made qualitatively irrelevant.
We can also represent typeR information by quantX, but this is highly inefficient, though powerfully flexible, as the examples at top of the page show. We could in principle write our language by displaying sonograms: visual displays of the acoustic energy in each frequency range of human speech, as on an oscilloscope screen. Expert acoustic phoneticians can learn to read such displays, but they contain vast amounts of information irrelevant to determining what words are being said (though this information may be meaningful in relation to other semiotic systems, e.g. those identifying regional accents or emotional states by indexical signs). Reading sonograms teaches us that our brains learn to filter out much of this topological acoustic information, to ignore it, and also to snap a particular sound pattern to the nearest linguistic template, so that we can hear phonemes and words that are often not actually present in the sound stream, but "should" be.
Mixedmodes suggest a pathway between physics and language...