Recently, historians have discussed the relevance of the nineteenth-century mathematical discipline of projective geometry for early modern classical logic in relation to possible solutions to semantic problems facing it. In this paper I consider Hegel’s Science of Logic as an attempt to provide a projective geometrical alternative to the implicit Euclidean underpinnings of Aristotle’s syllogistic logic. While this proceeds via Hegel’s acceptance of the role of the three means of Pythagorean music theory in Plato’s cosmology, the relevance of this can be separated from any fanciful “music of the spheres” approach by the fact that common mathematical structures underpin both music theory and projective geometry, as suggested in the name of projective geometry’s principal invariant, the “harmonic cross-ratio”. Here I demonstrate this common structure in terms of the phenomenon of “inverse foreshortening”. As with recent suggestions concerning the relevance of projective geometry for logic, Hegel’s modifications of Aristotle respond to semantic problems of his logic.

Mathematician and philosopher, Charles Howard Hinton, posited a plausible correlation between of higher-dimensional spaces, also referred to as ‘hyperspaces’, and the allegorical concept artic-ulated by the Ancient Greek philosopher Plato in his work, Republic, known as the 'Cave.' In Plato's allegory, individuals find themselves situated in an underground 'Cave,' constrained by chains on their legs and neck, perceiving shadows and sound reflections from the ‘real’ world cast on the ‘Cave’ wall as their immediate reality. Hinton extended the interpretation of these 'shadows' through the induction method, asserting that, akin to a 3D object casting a 2D shadow, the 'shadow' of a 4D hyper-object would exhibit one dimension less, manifesting as a 3D object. Building upon this conceptual framework, the authors posit a correlation between the perceived acoustic space of the chained individuals within the 'Cave' and the characteristics of a 4D acoustic space, a proposition substantiated mathematically by scientific inquiry. Furthermore, the authors introduce an interactive art application developed as a methodical approach to exploring the hypothetical 4D acoustic space within Plato's 'Cave,' as perceived by the chained individuals and someone liberated from his constraints navigating through the 'Cave.'