“…as all mathematicians know, it is perfectly possible to agree with (fail to fault) every step of a proof without experiencing any conviction; and without such experience, a sequence of steps fails to be a proof. Presented with a new proof, mathematicians will seek the idea behind it, the principle or story that organizes the logical moves into a coherent whole: as soon as one understands that the persuasional structure of the proof can emerge. Proofs embody arguments — discursive semiotic patterns — that work over and above – before – the individual steps and which are not reducible to these steps; indeed, it is by virtue of the underlying story or idea or argument that the sequence of steps is the sort of intentional thing called a proof and not merely an inert string of formally correct inferences. The point of this in relation to thought-experiments is that such underlying stories are not available to the Subject confined to the Code; they can only be told by the Person from within the metaCode.“
From Brian Rotman (see previous post) – emphasis added.
What Rotman’s talking about could be explored in terms of explicatures and implicatures, though he’s looking at things in terms of semiotics, and that idea comes from linguistic pragmatics – from relevance theory. If the codified progression of a proof is not the thing that ‘convinces’, but if we’re avoiding the idea of inherent properties of an independently-existing mathematical reality, then what exactly is it? What’s the principle, story, organising move, persuasional structure? The explicature is the encoded meaning; the implicature is what else can be derived using contextual knowledge. The contextual knowledge part here is pretty important, then. Is this ‘organising principle’ then just something that just shows how something fits into a wider context – of other work on the same pieces of mathematics, of work on similar things, of similar approaches to other topics?
This paper is great:
“Since pictures don’t have a grammar (pace KRESS; van LEEUWEN, 2006) and a vocabulary, they would seem to be incapable of transmitting ‘coded’ information. But there are at least two reasons to think that this is too hasty a conclusion. First, as Roland Barthes pointed out in the heyday of structuralism, interpreting pictures requires cultural knowledge of the viewer, who needs to be in possession of certain ‘lexicons’ (1986, p. 35) to be successful”
These diagrams are accompanied by text, and are highly codified. They don’t refer to any other concrete thing other than themselves, if you ask me, which is ace. Other than perhaps the class of things that they are, or a general thing-of-that-ilk.