Thoughtprovoking remarks taken from the notes to chapter 2 of ‘Laws of Form‘ by G. Spencer Brown:

“It may be helpful at this stage to realize that the primary form of mathematical communication is not description, but injunc­tion. In this respect it is comparable with practical art forms like cookery, in which the taste of a cake, although literally indescribable, can be conveyed to a reader in the form of a set of injunctions called a recipe. Music is a similar art form, the composer does not even attempt to describe the set of sounds he has in mind, much less the set of feelings occasioned through them, but writes down a set of commands which, if they are obeyed by the reader, can result in a reproduction, to the reader, of the composer’s original experience.

Where Wittgenstein says [4, proposition 7]
‘whereof one cannot speak,
thereof one must be silent’

he seems to be considering descriptive speech only. He notes elsewhere that the mathematician, descriptively speaking, says nothing. The same may be said of the composer, who, if he were to attempt a description (i.e. a limitation) of the set of ecstasies apparent through (i.e. unlimited by) his composition, would fail miserably and necessarily. But neither the composer nor the mathematician must, for this reason, be silent.

In his introduction to the Tractatus, Russell expresses what thus seems to be a justifiable doubt in respect of the rightness of Wittgenstein’s last proposition when he says [p 22]

‘what causes hesitation is the fact that, after all, Mr Witt­genstein manages to say a good deal about what cannot be said, thus suggesting to the sceptical reader that possibly there may be some loophole through a hierarchy of languages, or by some other exit.’

The exit, as we have seen it here, is evident in the injunctive faculty of language.

Even natural science appears to be more dependent upon injunction than we are usually prepared to admit. The pro­fessional initiation of the man of science consists not so much in reading the proper textbooks, as in obeying injunctions such as ‘look down that microscope’. But it is not out of order for men of science, having looked down the microscope, now to describe to each other, and to discuss amongst themselves, what they have seen, and to write papers and textbooks describ­ing it. Similarly, it is not out of order for mathematicians, each having obeyed a given set of injunctions, to describe to each other, and to discuss amongst themselves, what they have seen, and to write papers and textbooks describing it. But in each case, the description is dependent upon, and secondary to, the set of injunctions having been obeyed first.

When we attempt to realize a piece of music composed by another person, we do so by illustrating, to ourselves, with a musical instrument of some kind, the composer’s commands. Similarly, if we are to realize a piece of mathematics, we must find a way of illustrating, to ourselves, the commands of the mathematician. The normal way to do this is with some kind of scorer and a flat scorable surface, for example a finger and a tide-flattened stretch of sand, or a pencil and a piece of paper.”