Archive Post: Gesture, Poetry, Language, Philosophy, Math.

[This is a copy-paste of something I wrote years ago. But I'm posting this to share with a friend, and it's a line of thought that I'm still actively playing with. It's very much a draft and more of an exploration of terrain than something I am committed to right now]:

I want to explore an idea I'm only just beginning to work through, but might make for some interesting takes. Basically the idea is that if you really want to understand the nature of language, two seemingly marginal areas need to be investigated: math and gesture. My intuition is that all three terms - gesture, language, and math - all stand on a continuum of increasing abstraction, and that to understand each, we need to understand the other(s). Or to put it differently, gesture and math stand at opposite ends of a line on which language occupies the centre: they are the limit-points though which language must be understood.
 
Gesture: gesture is primarily a matter of specific movement in  space and time; in the words of Gilles Chatelet, it is a "disciplined distribution of mobility. At its base, gesture - especially pointing and ostension in general - refers to the world around it; it is, in some sense, 'of' that world too: gesture is embodied, a displacement of matter. Most importantly however, gesture finds its foundation in what linguists call 'deixis': the act or fact of its taking place. Pointing, for instance, is inseparable from the very act of pointing. Pointing does not 'symbolise': its meaning resides in its act, in its corporeality.
 
Math: Skipping to the other end of the spectrum: from a semiotic point of view, the distinguishing feature of mathematical language is the total absence of deixis. Math, considered as a formal system of signs, makes irrelevant its own 'taking-place': the act of writing of  "1+1=2" seems not to affect the fact that 1+1, in fact, equals 2. Further, unlike gestures, the basic symbols of math - numbers - are, ultimately pure 'types' (as distinct from tokens): the number "1" is not 1 of anything: it is pure form, "of which any material inscription is simply perceptual instance" (Brain Rotman). Hence the seeming incorporeal, disembodied nature of math. Thus where gesture always refers to a this (this gesture, that gesture), gestures are never - and in fact cannot be - pure forms: the material component of the gesture exhausts it entirely.

(Symbolic) Language: Now, between these two poles of gesture and math, lies symbolic language. Language, as we normally understand it, employs both elements of deixis and pure form; language can both reference things in the world, and it can 'talk about itself' in wholly abstract ways (philosophy does this alot). Somewhat more specifically, so-called 'shifters', words like "I" and "you", indicate the taking place of language between speakers (thus being the closest elements to gesture in language), while other words, like common nouns ("apples", "cities")  are types subject to the formal machinery of language. This 'mixed' character of language, both its referential capacities (able to talk about things in the world) and its formal ones (its ability to talk about things it talks about) is what lends language its inexhaustible expressive power.
 
If this is right, then this has some perhaps surprising consequences. To the degree that language exhibits both gestural and formal elements, and employs both freely and maximally (depending on pragmatics), then both gesture and math can be seen as opposite poles of language, each of which minimizes, respectively, the formal and the deietic elements of  language (the formal and the deietic can be thought of as volume knobs (i.e. as parameters), one turned all the way down, and the other turned all the way up, oppositely at each 'end' of the language spectrum).

To fill this out a bit, it might be useful to situate poetry here: on the imaginary continuum set up above, poetry probably lies somewhere in between gesture and language proper. To the degree that poetics is less subject to the constraints of pragmatics (of getting a concrete message across in order to achieve an aim), poetics can be seen to be the exhibition of gesture in language, not unlike shifters. As pointed out by Giorgio Agamben though, the limits of poetry (and the reason why it can never be pure gesture) is that it is ultimately subject to enjambment, the continuation of a sentence without a pause beyond the end of a line; as in:
 
the back wings
of the
hospital where
nothing
will grow
 
The breaks between otherwise continuous lines are enjambment. Enjambment marks the inescapable intrusion of the formal into poetry's attempts at gestural language. Agamben: "The possibility of enjambment constitutes the only criterion for distinguishing poetry from prose. For what is enjambment, if not the opposition of a metrical limit to a syntactical limit, of a prosodic pause to a semantic pause? "Poetry" will then be the name given to the discourse in which this opposition  is, at least virtually, possible; "Prose" will be the name for the discourse in which this opposition cannot take place" (Agamben The End of the Poem).
 
So visually, the continuum runs:
 
Language: {Gesture --- Poetry --- Language --- Philosophy --- Math}

Another extension of the schema here: what is counting? I want to say that counting is not exactly math. Rather, math would be (among other things), a formalization of counting (just like logic is a formalization of reasoning). One way to see this is to note that while its true that some animals can count, their treatment of numbers - even among the smartest of them - is not in terms of tokens and types, as in a formalized mathematical system.

For instance, even monkeys can't process numbers as numbers (qua form). They can only really process numbers as coupled with material things: 1 of X, 2 of Y and so on, and not as sheer 'types', never as 'just' 1 or 'just' 2. The formalization of counting (qua math) can be seen as an attempt to get around this cognitive limit, which even we are subject to. By formalizing counting, we can treat entire sets of numbers (and even types of numbers) as single things to be manipulated, thus bypassing the need to deal with a whole range of sheer particulars. Math involves the stratification and proliferation of type and token, which is present only to a single degree at the level of counting.

Math and Space

Some more words on math and space: one of the things that math does is to erase or rather compress time into space: relations between mathematical objects - mappings, translations, computations - can all ultimately be seen as spatial relations between entities distributed among imaginary space(s). I mean, the whole 'structure' of math - I'm not sure how else to put it - things like the number line, invariants which define groups and their corresponding abstract topologies, higher-dimensional numbers (imaginaries, quaternions, octonions, etc), the very idea of ordinals: all these things can be understood (and perhaps ought to be understood) in spatial terms. 

It's not a coincidence that math in many ways can be understood as the study of various broken-symmetries (another spatial notion!). I would only add that the invariants which characterize the different mathematical asymmetries belong strictly to the level of form (that is, are invariants relating to types, and never tokens): math is the study of how pure types can be mapped and related to each other depending on the invariants in question (we just happen to call these pure types 'numbers'). 

 

This is obviously a super, super abstract definition of math (could it be otherwise?), but if one can accept this, then the major point is that the spatial characteristics that define math are not different in kind from the spatial characteristics that are found anywhere else in the 'real' world: the 'only' difference is that mathematical objects are not bound by so-called material constraints (or energetic constraints), whereas 'real things' are; in fact 'real things', are bound by both material/energetic constraints and formal ones. Mathematical objects simply have an extra 'degree of freedom'. 

 

Yet the point would be that in both cases, what constitutes 'space' for both mathematical and extra-mathematical objects is exactly the same. And if this is the case, then we can understand why gesture (as a "disciplined distribution of mobility") is foundational to math qua math,and not merely a contingent tack-on that helps humans learn it: mathematical manipulations are gestures in mathematical space (where the idea of spatiality is irreducible and fundamental). Both these gestures and these spaces must in turn be seen as extrapolations from a more originary space without which these corresponding abstract spaces could not exist and could not be thought. 

 

Here is Rotman: "Contemporary mathematics, though habitually understood in terms of static disembodied object-concepts, is constructed in/by a language whose basic conceptual vocabulary is rooted in gestural movement–schemata of the body. Chief among these are: the gestures of pairing two things together; combining two things to make a third; replacing one thing by another; pointing at a thing; showing, exhibiting or manifesting a thing; displacing or extending the body in its space; making/altering a mark; and the meta-gesture of repetition, of doing the gesture again. So that, for example, the object-concept ‘number’ can be seen as rooted in the gesture of making a stroke, an undifferentiated mark, and then repeating it; likewise the object-concepts ‘equation’, and ‘relation’, are different conceptualizations of the gesture of pairing... [etc]".

 

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