Gilles Châtelet, graphic reason (mathematics)

"… some contemporary research … completely renews _the very notion of indexation_. No explicit intuition accompanies the ‘classical’ behaviour of calculations: in formulae of the type ∑ᵢ = , the set of indices is neutral and the indexation remains completely external to the development of these calculations, behaving like a ‘notation’ which is completely indifferent to that which it notes. These formulae remain captive to a linear successiveness, x₁ then x₂ then x₃ etc., an artificial sequence a little analogous to the chain of verbs _veni, vidi, vici_ where the temporal order of the processes of enunciation … replicate exactly the order of the processes of the statement … The contemporary point of view makes the notation _concrete_ by identifying it with a diagram already used in an _a priori_ foreign domain … This domain thus 'evokes’ gestures which are classical for it, but completely new in the domain when it is imported as ‘notation’ … This upsets the very notion of indexation which becomes _bi-dimensional_ in freeing itself from the successive: it is very much a victory of the hand that comments on itself, the indexation no longer being delivered by an external ‘set’, but by a process of deformation and modification of diagrams."