The visible and the invisible 1 The visible and the invisible: mathematics as revelation

Mathematics educators have shown increasing interest in theorizing knowing and learning as something alive or as something that comes alive through the involvement of the body. Almost all current efforts attempt doing so by focusing on the body in which the otherwise invisible living exhibits itself, thereby failing to consider everything that is available only in pure immanence. One advance, however, exists in the taking up of the ideas of mathematical philosopher G. Châtelet, who focuses attention on the virtual. In this study, we contribute to the further development of this line of work, linking it to the phenomenological studies of the role of the invisible in painting. The invisible and whatever precedes mathematical insight inherently cannot be the object of intention: doing mathematics becomes a revelation and mathematics is the revealed. We address two major shortcomings in previous work: (a) the integration of forces as immanent rather than external; and (b) the passive-affective dimension, which allows the new to be received and which therefore is generative, transcending the limitations of agency, which only repeats forms that already exist. We develop and thereby contextualize our advance by means of an in-depth analysis of an episode where a second-grade student finds herself having made a square from tangram shapes