Going back to the way we learn math, it strikes me that while the general way we are introduced to this type of cognition depends on gestures, as means of spatial designation and delimitation, as well as on language to slide us toward abstraction, neither of those domains recovers the essence of mathematical reasoning. The understanding of 1+1=2 depends on an initial suspension of actuality on the operational side, followed by a recovery of this actuality through a projection of this operation on an abstracted world which we pretend is a proper translation of ours.
I would suggest that if gesture is primordial here, in this context, its because it is a form of informational mapping. The educational gestures behind 1+1=2 serves the purpose of establishing boundaries and then lifting them, putting emphasis on the sequential aspect of the event so as to give the impression of operationality, while the language serves the purpose of obfuscating the fact that none of this is actually happening in reality. — Akanthinos
Hmm, but I'm not sure that the
spatial aspect of mathematical reasoning is quite as suppressed as I
think you're stating (I'm not sure if I'm reading you right on this so bear with me if I'm not). I mean, just to set the stage, 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]". (
source).
I'm sure there is a better way for me to try and articulate these issues, but I'm very much groping here. Also, I'm not sure if I'm agreeing or disagreeing with your post here, but this is what its prompted outta me!