Brian Rotman and Shoshana Felman: mathematical objects and referents

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.

Brian Rotman’s (rhetorical) approach to math

Rotman’s project is interesting because it directly engages not only the question of persuasion, but the question of the construction of mathematical objects through mathematical language. This is interesting to me because, as I argue in my Master’s Report, “Unlike other sciences and public discourses, whose arguments proceed from shared assumptions or agreed upon perceptions of the natural world, mathematics begins with writing that performatively posits abstract fictions” (46). For Rotman, as for Reuben Hersh and Phillip Davis and countless others, mathematical knowledge is rhetorical: rough, approximate consensus arrived at through specialized, yet inexact, language, and argument among those familiar with its specialized terms. But Rotman also recognizes the aspect of mathematical language that differentiates it from natural language, in general, and from the special, altered subsets of it used in other sciences. Although neither Rotman nor Mitchell Reyes, who brought Rotman to the pages of Quarterly Journal of Speech, seems especially attuned to the uniqueness of this quality, they both appreciate the constitutive, constructive nature of the mathematical signifier.

In my Master’s Report, I was looking to Brouwer for an account of math that would bring it into line with the theories of performative language articulated by J.L. Austin, Derrida, Shoshana Felman, and/or Mikkel Borch-Jacobson. I was only partially successful. Brouwer left me hangin’. Rotman comes even closer than Brouwer, but also misses the mark. His account of math makes its signs constructive and creative but not performative in the post-Austin sense.

But Rotman’s account of math is definitely rhetorical. Here’s how it works:

1. Rotman’s starting point: “Mathematics is an activity, a practice” (7).
2. Three interrelated agencies, Person-Subject-Agent are at work in mathematical activity; the activity is primarily communication; and the agencies are, therefore semiotic (to Rotman) or rhetorical (to “us”).
3. Mathematical assertion and proof are two essential mathematical activities: the assertion is a prediction and the proof is a process of persuasion.
4. When the Person asserts and proves, he (sic) “scribbles/thinks,” thereby becoming the “scribbling/thinking” Subject.
5. The Person-as-Subject makes a prediction about his (sic) own future, about what would be true if the (Person-as-)Subject manipulated signs infinitely, if he (sic) repeated certain scribblings again and again.
6. The Person-as-Subject writes a proof to persuade someone/s (the Subject for Rotman…, maybe the “they” or the “we” for me) that the prediction is valid (true?). The proof is a thought experiment and the writing that it is inextricably linked to. Each step of the written proof, i.e. each discreet act of writing performed by the Subject, corresponds to some imagined action performed by the Agent in the thought experiment. *(We’ll eturn to this briefly.)
7. The Subject is persuaded to accept the prediction by writing the proof and, thereby, executing the thought experiment (as the Agent-imagined-by).
8. Rotman adds that, along with this mechanical logic to persuade the Subject, there must be what C.S. Pierce calls a “leading principle.” The leading principle, which is related but not reducible to the proof, persuades the Person. An important feature of the leading principle is that it explains the relationship between the Subject and the Agent. “Precisely in the articulation of this relation lies the semiotic source of a proof’s persuasion. [...]. It is the business of the underlying narrative of a proof to articulate the nature of this resemblance” (19).When an assertion is validated, “it is the Person who, by being able to articulate the relation between Subject and Agent within a thought experiement, is persuaded that a prediction about the Subject’s future encounter with signs is to be accepted” (24).

*Brief return: The writing/thinking Subject imagines the Agent to act. The Subject “must act indirectly and set up an imagined experience-a thought experiement-in which not he but his Agent, the skeleton diagram of himself, is required to perform the appropriate infinity of actions” (17). So what are the actions of the Agent? It seems that the Agent manipulates (virtual) signs. If this is the case, then th Agent must write and think… right? But is that problematic? Does it imply the existence of a Virtual Agent? Does he (sic) do something else? Something more or less?

Leaving aside the potentially problematic details of the Agent’s activities, the preceding is Rotman’s account of mathematical language in action, of communication by transactions with mathematical sign. It is his rhetoric of math, which explains how people with bodies and imaginations interact with meaningful material signs in order to produce knowledge, which then influences their perception of and future interactions with material reality.

Brian Rotman: signifier, signified, and referent

For Rotman, the signified meaning of mathematical signifiers “exists” in the mind of the scribbling/thinking mathematician-in-action. If we adopt this position and rigorously pursue it, I think we end up in a solipsistic position much more sophisticated than, but ultimately rather similar to, the one associated with Brouwer’s intuitionism. I don’t have time to make such a pursuit at the moment, nor do I actually know how that road goes… I have yet to map it. However:
The difference is that Rotman doesn’t deny the importance of mathematical language, the sign, the inseparable signifier-signified pair. Brouwer ignores the signifier to focus exclusively on, and give a very satisfying account of, the signified meaning. Rotman, on the other hand, acknowledges that the signifier is just as crucial since, as everyday experience proves, it is the means by which mathematics is a public, intersubjective experience.

This is wonderful. Thank you Brian Rotman. For how brilliant Brouwer seems to have been, I’m shocked by the stupidity of his claim that math has nothing to do with language.

So it’s great that Rotman points out that mathematical writing, signifiers, are important. Nevertheless, his account is problematic to me. This is, in part, because it doesn’t explain the relationship of signifier and signified. Nor does it explain the stability of the signifier over time, between semiotic agents, across various significations. It assumes but says nothing about the way in which mathematical signifiers carry their signified meaning along with them. It assumes but says nothing about the way in which signifiers “create” or “give rise to” the signified meaning “in” the mind or, to remain within Rotman’s terms, “in” the imagination of the person-doing-math (the Person? the Subject? both?).

Where/what is meaning?

This is a bit surprising to me since Rotman seems to know somethings about somethings: in particular, in this case, Derrida (and what he wrote about potential and not-coming-to-pass) and theories of situated, distributed cognition. What about all the potential meanings overflowing from detached/detachable and re-attachable, iterating and iterable mathematical signifiers? What about the flimsiness, pointed out by Thompson, Varela, Hutchins, Brooks, et. al., of an account of “knowledge in the mind”?

I have to return to these questions.

The other problem I have with Rotman’s approach—and this one is less surprising—is the way he (mis)treats and (dis)regards the referent. He accepts Umberto Eco’s (deconstructive?–don’t know; haven’t read it) critique of the sense/reference distinction, claiming that there is no such thing as a prelinguistic referent (31). If every referent is socio-historically constructed and contingent, there’s not much sense in talking about anything but sense, i.e. signification, i.e. the signified.

Why throw away the referent? Of course, it would be laughable to claim it’s prelinguistic. But what about a postlinguistic referent? What about the kind of referent Shoshana Felman posits into existence through her trinity of Don Juan-Austin-Lacan?

Another thing I have to return to…

P.S. I wanted to name this post B. Rotman & L.E.J. Brouwer: initial thoughts .

(A) thought on/in/for Brian Rotman

I am making my way through Brian Rotman’s Mathematics as Sign, slowly, non-linearly, and on-and-off-ly. Although I haven’t read it all, something/s need to be said at the moment:

Rotman criticizes the Fregean-contemporary (is it still popular/dominant?) version of Platonist philosophy of mathematics for many reasons. Here’s one point Rotman makes: Fregean Platonism posits “thought” as something eternal and unchanging that can be “apprehended” by thinking thinkers (people, more or less) but it “is incapable of giving a coherent account of knowing” (33). Rotman quotes Frege: “‘The apprehension of a thought presupposes someone who apprehends it, who thinks it. He is the bearer of the thinking but not of the thought. Althought the thought does not belong to the thinkers consciousness yet something in his consciousness must be aimed at that thought. But this should not be conufsed with the thought itself’ (Frege 1967, 35)” (33 emphasis mine). Rotman finds this problemaic because it leaves knowing, “the means by which we manage to apprehend [thoughts] in total mystery” (33).

Rotman then attempts to salvage this conception of math by translating it into semiotic terms. Luckily, for Rotman, this attempt at reform fails and, better yet, points up a more *constructive* way to re-theorize mathematics. I hope to return to Rotman’s constructive re-construction at a later date. For now, I’d like to wonder, briefly, about another response to Frege’s frigg-up.

Can Nietzsche come to Frege’s rescue? (After all, he is Superman, right?) Maybe Frege’s mistake is not that he leaves out the process by which the thinker and his (sic) thinking gets at the thought. Maybe the problem is that, for Frege, “The apprehension… presupposes someone who apprehends it, who thinks it (emphasis mine),” that, for Frege, the thinking must be born of a (masculine) thinker.

What if we follow Nietzsche’s line (one that runs to me through Judith Butler): ~”there is no doer behind the deed; there is no thinker behind the thought”~? If we approach the thinker as only (an) after-thought, maybe we can save something else or something more from Fregean Platonism than Rotman does…?

But what? And/or why?

I want to come back to this, but for now, I will just suggest that the answer to “why” has something to do with Brouwer’s conception of the solipsistic subject, expressivist composition pedagogy, and/or the religiosity/faith that some have identified (re)surfacing in/around/through poststructuralist/postmodern theory (the body of work that Rotman calls something like contemporary Post thought).

Probably Probabble

One thing I’m doing while I’m not quite in school is studying probability. To be more specific, I am learning “the mathematics of probability [and] the many possible applications of this subject” by reading and exercising with Sheldon Ross’s A First Course in Probability, 6th ed. By “exercising with,” I mean attempting the examples, as well as the problems, exercises, and self-tests listed at the end of each chapter. At least… so far.

Chapter 1 is about “Combinatorial Analysis,” a subject I took a class or two on in college. It was mostly review, but for a number of reasons, it took me quite some time to finish. So, after spending several months, off-and-on, reading and exercising with the first chapter, I’ve just started reading chapter 2, on the “Axioms of Probability.” I’ve just started reading it; haven’t even really warmed up yet. But despite my lack of direct exposure to this material, I’ve been having a thought… a premature, unsubstantiated thought:

Probability theory is about experiments, possible outcomes, and events that have and have not occurred. This is obviously why it’s rhetorical. This is why so many rhetoricians have done said so many times that rhetoric is so probabilistic. But probability is math, dam-nit. And, more importantly… maybe… probability is… maybe… Deluzean.