structures in a cohesive (∞,1)-topos
infinitesimal cohesion?
In Georg Hegel’s Encyclopedia of the Philosophical Sciences there is discussion of the cohesion of some substance. William Lawvere argued that the “objective logic” of this discussion is to be formalized via categorical logic by the axiomatics of cohesive toposes, i.e. by modal type theory equipped with shape modality and flat modality.
Entries discussing aspects of cohesion include the following
Hegel goes on to speak of cohesion being refined to elasticity:
PN§297Zusatz Elasticity is the whole of cohesion.
Moreover, according to PN§298 this elasticity is related to the unity of opposites that consistute Zeno's paradox of motion, hence to the modern concept of differentiation via a limit of a sequence. In terms of categorical logic this is precisely what is encoded in the infinitesimal shape modality and infinitesimal flat modality of
(shape modality $\dashv$ flat modality $\dashv$ sharp modality)
$(ʃ \dashv \flat \dashv \sharp )$
dR-shape modality $\dashv$ dR-flat modality
$ʃ_{dR}\dashv \flat_{dR}$
(reduction modality $\dashv$ infinitesimal shape modality $\dashv$ infinitesimal flat modality)
$(\Re \dashv \& \dashv \Im)$
Examples
Applications