Why Mathematics Involves Both Language And Epilanguage

IF:

Halliday & Matthiessen (1999: 606):

Many socio-semiotic systems are combinations of [those realised in language] and [those parasitic on language].

Socio–Semiotic Systems Realised In Language As Registers

Halliday & Matthiessen (1999: 606):

Such higher–level systems (theories, institutions, genres), since they are realised in language, are realised as subsystems within the semantics and the grammar. These subsystems are what we have referred to as registers …

Socio–Semiotic Systems Parasitic On Language

Halliday & Matthiessen (1999: 606):

… in the sense that they depend on the fact that those who use them are articulate (‘linguate’) beings. These include the visual arts, music and dance; modes of dressing, cooking, organising living space and other forms of meaning–making behaviour; and also charts, maps, diagrams, figures and the like.

THEN:

Mathematics, as a contextual field, is realised both in language — eg the cube root of twenty-seven is three (however this is expressed) — and in semiotic systems parasitic on language ("epilanguage"), as when realised as diagrams (figures, graphs etc).

Language can be said as well as sensed:

it has wording (lexicogrammar) as well as meaning (semantics).

Epilanguage cannot be said, only sensed:

it has no wording (lexicogrammar), only meaning (semantics).
That is, a diagram cannot be read aloud like language;
though it can, of course, be described or interpreted in words.