https://www.academia.edu/25102777/The_Technology_of_Mathematical_Persuasion

in Inscribing Science: Scientific Texts and the Materiality of Communication by Timothy Lenoir

This is an interesting paper, although I’m told that more people (philosophers) have looked at the thought experiment idea than he’s aware. He’s a mathematician, though, and talks about this great constructivist stuff in a way that works for mathematicians, which is good.

One consequence of this, radically antagonistic to prevailing Platonistic interpretation of mathematics (and to cognate realist or objectivist accounts of scientific knowledge), but familiar within Post (particularly post-structuralist) thinking about language, is that the “objects” concerned here numbers, visual scenes, values — far from belonging to prior, pre-semiotic worlds waiting to be signified, are inseparable from, and in a radically uneliminable sense owe their being to, the very codes thought to be referring, notating and describing them.”

“Another approach to mathematics, employing a discourse-based semiotic ratherthan an archelogical/structuralist one, is called for; one that would respect the fact that “The world of rigorous fantasy we call mathematics,” as Gregory Bateson put it,[5] is imagined and thought into existence.”

“…making experiments out of a process of pure ratiocination — thought experiments — is, I contend, precisely how mathematicians manufacture mathematics.”

“Each comes about by the use of an elaborately designedapparatus able to mask its rhetorical features under the guise of a neutral method for discovering pre-existing “facts” of nature or, in the case of classically conceived arithmetic, objective “truths” about the so-called natural numbers.“

“As indicated, the standard account — enshrined for example in the writing of purely internalist histories of the subject –would be that the metaCode is epiphenomenal, a matter of mere affect, psychology, heuristics and handwaving, subordinate to the “real” business of doing mathematics taking place in the Code, and in principle completely eliminable. I shall indicate how such a view of the metaCode, as a practically necessary but theoretically dispensable supplement, a tool or prop to be discarded once the passage into pure reason and formal truth has been accomplished, is entirely misconceived.“

“He makes in his imagination a sort of skeleton diagram, or outline sketch of himself, considers what modification the hypothetical state of things would require to be made in that picture, and then examines it, that is, observes what he has imagined, to see whether the same ardent desire is there to be discerned. By such a process, which is at bottom very much like mathematical reasoning, we can reach co nclusions as to what would be true of signs in all cases …[10]“

The mental representation thing again…

“How do thought-experiments persuade? They furnish the Subject with a scenario enacted by the Subject’s proxy, the Agent, of what he/she would experience.“

This is important:

“…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 organizesthe 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 theCode; they can only be told by the Person from within the metaCode.“

The gloss, the attention-directing, is what keeps the hearer along. Thinking about explicature and implicature here. Why doesn’t he talk about pragmatics?

“Is the Agent imagined to be corporeal in any way whatsoever? Since the Subject, as a sign reading/writing agency, is clearly and irremediably embodied the question is manifestly one of resemblance between imaginer and imago. If we answer yes, and maintain that our Agent has some — however idealized, vestigial or attenuated — immersion in the material world, then its action will not escape the regimes of space, time, energy use, decay and so on, that govern all physical process, and it will not, as a result, be able to go on endlessly. If we answer no, as we must if we are to cognize infinity and counting without end, then our Agent will be a disembodied ghost, and we are confronted by the question of persuasion. Why should a Subject whose embodiment is inseparable from its engagement with signs create an incorporeal phantom as its proxy? In what way can the imagined manipulation of signifiers by such a transcendental imago persuasively image the Subject’s manipulation of signs?“