Though it may surprise most people who know me, I have actually read all of Brian Rotman’s book. Nevertheless, for several posts now, I have been dwelling in the first chapter, “Toward a Semiotics of Mathematics.” I just haven’t gotten through all I want to get through yet. In particular, I haven’t gotten to the semiotic account of mathematical objects he moves toward in the first chapter. I hope that this post will get there.
I think I’ve taken a while to get there because, as I said in an earlier post, Rotman essentially skips over the part of semiotics that deals with the referent.
Early in the chapter, Rotman explains the perils of approaching math by each of the three axes of semiotic investigation: beginning with signifiers leads to a roughly formal approach (see Hilbert); beginning with signifieds leads to a roughly psychological approach (see Brouwer); and beginning with referents leads to a realist approach (see Frege) (4-5). Though he says each approach is problematic, he seems to begin with signifiers. But he doesn’t fall into the trap of old-fashioned pure formalism because he deals with signifiers only as material things used by and existing in relation to embodied semiotic subject-agents. He then proceeds from the signifiers used, to the signifieds imagined by these subject-agents.
Rotman doesn’t get “more specific about the semiotic answer to the fundamental question of what (in terms of sign activity) the whole numbers are or might be” until about pg. 38. Like Ruben Hersh, Brouwer, Kronecker, and most others who pursue foundational projects, Rotman approaches a description of mathematical objects, in general, via the natural numbers, in particular. And, like Hersh, who says that they are “social institutions,” Rotman suggests that mathematical objects are “social, cultural, and historical artifacts” (38). But I find Rotman’s account much more exciting. I want to move through his description a bit at a time:
- “numbers do not arise, nor can they be characterized, as single entities in isolation from one another: they form an ordered sequence, a progression” (38 emphasis in original)
- Doesn’t this conception of whole numbers being an ordered sequence presume that the whole numbers are already there…? Doesn’t there have to be whole numbers for anything to be “ordered”?
- “It seems impossible to imagine what it means for ‘things’ to be the elements of this progression except in terms of their production through a process of counting. And since counting rests on the repetition of an identical act, any semiotic explanation of the numbers has to start by invoking the familiar patter of figures [...] created by iterating the operation of writing down some fixed but arbitrarily agreed upon symbol type” (38 italics original, bold mine).
- For all Rotman says about Deleuze later in the book, I don’t think he’s thinking repetition here in a Deleuzean way. Repetition, in Deleuze’s Difference and Repetition, isn’t about identity…. Rotman goes somewhere else with Deleuze. I’m sure I’ll get back to that.
- “Such a pattern achieves mathematical meaning as soon as the type ‘1′ is interpreted as the signifier of a mathematical sign and the ‘etc.’ symbol as a command, an imperative addressed to the mathematician, which instructs him to enact the rule: copy the previous inscription then add to it another type. Numbers, then, appear as soon as there is a subject who counts. [...]. With the semiotic model I have proposed, the subject to whom the imperative is addressed is the Subject, while the one who enacts the instruction, the one who is capable of this unlimited written repetition, is his Agent” (38-9 bold mine).
- Here it comes: “Seen in this way, numbers are things in potentia, theoretical availabilities of sign production, the elementary and irreducible signifying acts that the Subject, the one-who-counts, can imagine his Agent to perform via a sequence of iterated ideal marks whose paradigm is the pattern 1, 11, 111, etc. [...]. Thus, the numbers are objects that result-that is, are capable of resulting-from an amalgam of two activities, thinking (imagining actions) and scribbling (making ideal marks), which are inseparable: mathematicians think about marks they themselves have imagined into potential existence. In no sense can numbers be understood to precede the signifiers that bear them; nor can the signifiers occur in advance of the signs (the numbers) whose signifiers they are” (Rotman 39 italics original; bold mine).
- So numbers are signs or (if this is different and I think it is) the signified parts of signs…. More precisely, numbers are potential signifying acts, which are signified by numerals, which are signifiers. This begins to come into line with Felman’s performative referent, which is always an act, “a dynamic movement and modification of reality” (77).
- The objects of math–”the things which it countenances as existing and can be said to be ‘about’–are unactualized possibilities, the potential sign productions of a counting subject who operates in the presence of a notational system of signifiers. [...]. And moreover, what is true of numbers is in fact true of the entire totality of mathematical objects: they are all signs-thoughts/scribbles-which arise as the potential activity of a mathematical subject” (Rotman 40-1 bold mine).
- One the one hand, this makes the objects of math “mappings” in the Deleuzean sense. Mathematical things must always be made.
- At the same time that they must be made, however, they are also never actually made; they are always going to be made, in the future that never comes to pass. This is a second connection to Felman, whose performative “referentiality [...] can be reached and defined only through the dimension of failure: on the basis of the act of failing” (82 emphasis hers).
- “Thus, mathematics, characterized here as a discourse whose assertions are predictions about the future activities of its participants, is ‘about’–insofar as this locution makes sense–itself. The entire discourse refers to, is ‘true’ about, nothing other than its own signs. And since mathematics is entirely a human artifact, the truths it establishes–if such is what they are–are attributes of the mathematical agency of Agent/Subject/Person who reads and writes mathematical signs and suffers its persuasions” (41 bold and italics mine).
- The italicized phrase reiterates the connection between Rotman and Hersh. And the sentence it appear in reiterates the rhetorical nature of mathematical languge.
- More importainingly, for me at least, the bold-face phrases make another connection with Felman’s performative referent: “the performative has the property of subverting the alternative, the opposition, between referentiality and self-referentiality” (79-80). This connection between Rotman and Felman is less direct and it is a connection that runs through Rotman to the beyond. That is, Rotman, as I’ve said, is not exactly, explicitly talking about reference and referents. (PS–It’s no mistake that the endings of these two words–the former naming an act and the latter multiple objects–have similar pronunciations.) He skips over reference/ents early on and is now talking about mathematical objects. His point here is that, on a semiotic account, math is about math, mathematical language is about mathematical language. (Can we say that the latter formulation is purely formalist? Rotman insists that language is material….) So, for Rotman, math is self-referential, but it is not necessarily referential. It may only be by considering math on the model of the performative, the performative as worked (out) by Felman, that math can be referential too.
- Rotman goes on from here to conclude that “truth” is basically useless in mathematics. This is also due to the fact that mathematical objects are always deferred objects. Mathematical assertions are predictions, so it makes no sense to ask if the are true. “A prediction-about some determinate world for which true and false make sense-might in the future be seen to be true, but only after what is foretold has come to pass; for only then can what was predicted be dicted” (41). Thus, mathematics is about belief… but belief based on persuasion. So it’s rhetorical… but it’s more.
