Discreteness, Continuity and Infinity

Abstract: 
This paper proposes a discrete-continuous model of cognition to resolve enduring 
philosophical difficulties surrounding time, motion, and infinity. The argument begins 
by establishing the transcendental necessity of discreteness: any cognitive act must 
already contain a distinction between consciousness and its object, making non
continuous recognition a structural condition of cognition itself. Within this framework, 
a critical distinction is drawn between two types of discrete operation. Signifier
discretization is a legitimate operation within the sign system that establishes 
differential relations among terms in coexisting continuity (space). Signified
discretization, conversely, is an illegitimate transgression that covertly attributes a 
discrete structure to the substrate itself. The paper demonstrates that this transgression 
occurs through a two-stage process: first, “retention” (Husserl) converts the non
juxtaposable flow of successive continuity (time) into a juxtaposed surrogate in 
consciousness; second, the legitimate operation of signifier-discretization is applied to 
this surrogate, thereby concealing the original structural conversion. This model is then 
applied to critically re-examine several philosophical problems. Zeno’s paradoxes are 
diagnosed as an illicit conversion of successive motion into static, coexisting positions. 
Bergson’s intuitionism is shown to be self-destructive, as its proposed method of 
“intuition” inevitably presupposes the very discreteness it seeks to overcome. Russell’s 
logical atomism is criticized for its foundational commitment to signified-discretization, 
which is revealed as the source of its insoluble difficulties with relations, change, and 
self-reference. Finally, the paper extends this analysis to the foundations of mathematics. 
The Cantorian actual infinite (ἐνεργείᾳ ἄπειρον) is critiqued as an illegitimate product 
of converting the potential infinite’s (δυνάμει ἄπειρον) endless process in successive 
continuity into a completed object in coexisting continuity. Consequently, the Axiom 
of Infinity and the Axiom of Foundation in Zermelo-Fraenkel set theory are revealed to 
be non-logical, and in the latter case counter-logical, stipulations introduced to stabilize 
this structural distortion. The paper concludes that discreteness, while the necessary 
mode of cognitive operation, can never exhaust the continuous substrate, leaving a 
remainder that perpetually escapes its grasp. 
Keywords: discreteness, continuity, infinity, transcendental philosophy, retention, 
Zeno's paradoxes, Bergson, logical atomism, Russell, Zermelo-Fraenkel set theory