I have no recollection of how I came across Vladimir Tasic's
mathematical reconstruction of the basic ideas of postmodern philosophy, but Amazon tells me that it has been sitting on my shelf for nearly a decade, so this aporia is hardly surprising. I finally read it just now because it seemed as if it might fit in well with some of the ideas in
The Gift. There is something in the
structure of the gift economy, with its continual circulation producing a sort of de-centered proliferation, that I thought might have various mathematical analogies. And since I've been encountering this structure
all over the
place, I figured having a very abstract mathematical version of it might sharpen my understanding of it. While this book wasn't quite what I was expecting overall, it actually did advance my understanding by providing a quite unexpected new name for this structure -- the
continuum.
Tasic wrote his monograph in the shadow of the
Sokal hoax and as an indirect response to the dismissive reading of postmodernism found in
Fashionable Nonsense. Though a mathematician by profession, he wisely chooses not to take a side in the "
science wars", and instead simply attempts to read postmodernism more generously. Could it be that postmodernists had something to say that actually benefitted from the mathematical analogies they occasionally invoked? Are there at least parallels and perhaps even direct links between their questions and some of the debates that took place within twentieth-century mathematics? Though Tasic is frequently critical of the intelligibility certain 'postmodern' authors, he tries to trace the intellectual history of these ideas in a good faith effort to affirmatively answer both these questions. To sum up a long story: the roots of Derrida's concept of
différance (for Tasic the postmodern concept
par excelence) can be found in a peculiar marriage between the ideas of mathematical '
intuitionists' such as Brouwer and Poincaré and '
formalists' like Hilbert. Derrida describes a sort of intuitionism without any intuitive
subject at its center, where the free creativity and understanding formerly attributed to this (mathematical) subject have somehow become an effect of the proliferation of a network of formal signs that continually overflows itself.
Since Tasic covers a lot of philosophical ground in an interesting but frequently superficial manner, I won't try to repeat the entire train of thought that leads to this point. While his writing is fairly clear and concise, he suffers from a tendency towards tangent that constantly threatens to engulf his main argument in lengthy (though usually quite interesting) asides. In addition, he begins his intellectual history of postmodernism all the way back with the 18th century Romantic reaction to Kant. So instead, let me just quote his own summary of the first third of the book.
... it appears that romanticist thinkers managed to place two important issues relatively high on science's agenda: first, language; second, the problem of continuity—that ineffable inner flux, the sense of continuous creative action—and its relationship with language (31)
These themes of language and continuity are then developed in more detail throughout the book. The mathematical version of this 'linguistic turn' becomes Hilbert's formalism, which suggested the only thing we really know is structured strings of signs. The problem of the continuity of experience of a pre-linguistic subject becomes intuitionism -- the belief that mathematical truth and meaning necessarily go beyond mere signs.
When he finally starts discussing mathematics, Tasic's first stop is the intuitionist mathematics of
Brouwer, which he considers heir to the romantic preoccupation with subjective continuity. As a constructivist, Brouwer was not satisfied with our usual definition of the continuum as an infinite collection of points because that is not something that a finite being can
know by
construction. Typically, we see the real line as composed of infinitely many little unrelated atoms called 'numbers' that are strung infinitely closely together. Instead of treating it as a collection of
objects, Brouwer proposes to define the continuum by reference to the subjective
process of constructing those points. Tasic doesn't explain the mathematics of Brouwer's "choice-sequences" in detail, but the basic idea is roughly that the continuum is better grasped as analogous to our subjective experience of
time. For Brouwer, time is a "
falling apart" of present and past that occurs with every free act or "life-moment" of the individual. The continuum, in turn, is constructed from the realization that these various acts are never completely
finished but could be continued indefinitely, with each new imagined act inserted
between the pervious two. Since each of the acts also includes some spontaneous free choice, these constructions resist formalization in language. Thus the continuum is essentially unknowable for Brouwer; it is The Open, the subjective depths of the romantic soul.
Summing up the effects of Brouwer's construction:- For me, the "point" of the continuum is the active process of my consciousness taken together with the spontaneous choices I could make along the way.
- I can never permanently fix this "point," precisely because the construction of the sequence involves making free choices. The "point" of the continuum is not a standard mathematical object. It is not immutable. It is an open object, a construction with indeterminate future.
- The continuum cannot be split apart. I cannot pluck a single point out of it, a point I could call "now," because this point is an open object that depends on me. It does not wait for me to discover it, because I create it, freely, spontaneously, along with the plurality of all other "points." (40)
Tasic goes on to cover other intuitionists like Weyl and Poincaré, but the point remains similar. The continuum is something that cannot be captured once and for all in language. Any definition that treats an indefinite
process as if it were a completely determined and finished
object (an
impredicative definition) will inevitably get us into trouble. Tasic compares this intuitionist continuum to Derrida's idea of
différance, to the future orientedness of Heidegger's
dasien, to Nietzsche's creative
will, and to Wittgenstein's idea that there cannot be a
private language (though there can be both language and privacy). It's frankly a bit of a tenuous connection at this point in the text, though it becomes clearer when he returns to discuss Derrida and Wittgenstein in greater depth. And of course Tasic has yet to deal with a key point in this comparison. Brouwer's intuition differs from all these other concepts for the simple reason that it is specifically the intuition of a
human subject, whereas none of these thinkers would consider themselves humanists.
Next, Tasic considers Brouwer's opponent in this mathematical debate: Hilbert's formalism. In an attempt to make math more rigorous and exorcize (almost) all subjective and intuitive elements, Hilbert tried to reduce mathematics to the mere manipulation of meaningless formal symbols. Even though this mathematical effort shipwrecked on the reef of Godel's theorem, there's little doubt that Hilbert's ideas are the more influential -- the computer has literally become the model for what we even mean by intelligence. Tasic, however, wants to illustrate the influence of formalist ideas on postmodern theory. In both cases, we see a shrinking (and at the limit a disappearance) of the subject.
Tasic sees two principle lines of direct connection. First, he traces a link from Hilbert to Foucault's idea of a "discourse" that literally
produces knowledge. He claims this influence passes by way of
Jean Cavailles a French philosopher of science I hadn't heard of before. The basic idea here is that the subject required in intuitionism is a mere "grammatical dummy" that is the product of particular operations of some formal system. In other words, the system
itself produces the appearance on meaning and subjectivity. While this connection makes sense in general -- Foucault is clearly trying to suggest that there is no single timeless definition of 'knowledge' that human subjects are gradually amassing -- I wonder whether he misreads Foucault somewhat when he reduces his contingent and historical discursive systems to a computational language.
Second, he discusses the clear analogy between formalism and structuralism. Saussure's idea that language is a system of arbitrary signs that can only carry meaning by being systematically different from one another (and not because their identity corresponds to some signified) clearly overlaps substantially with Hilbert's attempt to argue that mathematical truth lies entirely within formal symbolic demonstration (and not in our apprehension of the ideal properties of things like circles). Tasic further argues that in both Hilbert and Saussure's views, the differentiated structure in question in necessary but not sufficient to carry truth and meaning -- both men originally still thought that there must be some subject who is, at a minimum, capable of distinguishing the signs as units and of verifying that the computational system is running the correct algorithm. He contends that it is only later, when these ideas are radicalized by people like Cavailles, that we see claims that meaning is reducible to structure, or that a structured sign system is sufficient to produce what we call meaning all by itself. These later claims (which he refers to as functionalism) continue in the same anti-humanist direction one can already detect in structuralism or formalism. But they take it one step further, towards an all encompassing structure that leaves nothing out, especially not the purported subject for whom this structure was originally intended as just a means of expression or tool for justification. So the first connection he discussed is actually an extension of this second one.
Third, he sharpens the idea of a split between a still humanist structuralism and an anti-humanist functionalism by devoting a chapter to the way this ambiguity shows up
within Wittgenstein's conception of a language game. On the one hand, Tasic discusses an interesting example that makes it sound like Wittgenstein is rejecting formalism. Consider the question of the correct way to extend the sequence 2, 4, 6 ... We would all answer 8, but in fact there are endless rules for constructing sequences that would correctly provide a different response. The data we have are simply not enough to distinguish between these and assess which one the questioner had in mind. What's worse is that we cannot even be sure we
ourselves even know which algorithm we are
really using. I have only ever computed a finite number of iterations of what I call "multiplications by two". Perhaps the rule I was using secretly was a completely different one that just
happened to coincide in its results in these cases. How could I ever distinguish these. And if I don't even know what rule I'm using, how can I be trusted to
drive a formal system? On the other hand, Wittgenstein's proposed solution to this problem appears to itself be a version of formalism. He proposes that all meaning is constructed as simple intersubjective agreement in playing a language game. And what are these games if not little formal systems we agree to (or are coerced to into or habitually) abide by? The space that seemed to open up for the unjustifiable intuition that I am using the "multiply by two" algorithm has suddenly snapped shut. So it seems we can take Wittgenstein's arguments in either direction.
One could say, based on what the argumentation demonstrates, that the community, the collective, is involved in motivating my interpretive acts but that it does not necessarily supersede my conviction. The community of players of a particular language game guides the interpretation of rules. This is natural and in some sense obvious. It would be strange to say that culture, education, tradition, community, or my experiences of the physical world have no bearing whatsoever on my interpretive practices. It is also fair to admit that I am indeed "trained" and indoctrinated in various ways. But it also follows from the above argument that even upon extensive training individuals can always challenge the grounds of justification of any given rule, as we saw in the case of Sue above.
Nonetheless, it is possible to steer the conclusions from the "private language argument" in a completely different direction. If I believe that meaning anything by anything involves my being able to justify it—or if I happen to be one of the people who believe that meaning resides solely "in" justification—then it seems to follow that I cannot have any semantics of my own. To have any semantics whatsoever, I must follow a cultural convention. These conventions are drilled into me daily by my culture, a tradition into which I enter upon birth.
Putting it somewhat crudely, the community programs me and debugs me during the language game that is my life. Conversely, I use the community just like the functionalist shrink suggested Sue should use a PC: I identify my meaning with what it does. It therefore appears that in this case we have a kind of functionalism on our hands. Words perform a certain function ("use") in the system of cultural conventions. I am trained to observe these conventions; the only way in which I can escape them is by making a mistake, by unwittingly causing some "infraction" of the rules. These infractions are what I mistakenly attribute to my own "creativity."(129)
Tasic concludes by bringing all these threads together in a discussion of Derrida (who is basically presumed to stand in for whatever substance there might be in postmodern ideas). He essentially argues that unlike ze waffling Wittgenstein, Derrida is actually both an intuitionist and a formalist at the same time. On the one hand, Derrida is famously skeptical that we can ever know the the true meaning of a text. All we have is writing about writing about ... Each supposed subjective meaning defers to earlier ones ad infinitum. This seems to be the same radicalization of formalist ideas we saw before. On the other hand, this same process of open-ended writing could be seen as a reduction to the absurd of formalism. Derrida's logic here is similar to Wittgenstein's argument that we can't even know what rule we are falling and Poincaré's critique of impredicative definitions. If a formal system has a generative grammar capable of producing new syntactical structures, then how can we guarantee that the functionalist definition of its units as those things we must use in this way will remain forever constant? In short, new writing changes the meaning of old writing as the "text-in-general" continues to grow. Formalism fails because things keep escaping whatever language we use to describe them. This conclusion resembles intuitionism in that there seems to be some sort of free and creative principle at work beyond language. But of course, for Derrida, this principle does not coincide with the individual subject but with the capacity of "writing-in-general" to produce différance -- simultaneously the distinction between signs necessary for a formal language, and the endless deferral of meaning that overwhelms fixity of this language. Hence, différance is analogous to the continuum. QED
P.S. There are several passages that Tasic quotes in his final chapter that make Derrida's ideas appear very close to Deleuze's work in Difference and Repetition. Does this mean I have to read Derrida!?
