In defense of anti-essentialism, pragmatist Richard Rorty holds that we may think all objects as if they were numbers. I find Rorty’s metaphysics hinges on two rather weak arguments against the essences contrast, Plato’s offers a plausible definition essentiality by which numbers do have essential properties. Further, argue argumentative mistake is mischaracterizing definition. conclude “essent...

We give a simple proof based on symmetries that there are no geodesics from vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.

let $r$ be a commutative ring with identity. let $g(r)$ denote the maximal graph associated to $r$, i.e., $g(r)$ is a graph with vertices as the elements of $r$, where two distinct vertices $a$ and $b$ are adjacent if and only if there is a maximal ideal of $r$ containing both. let $gamma(r)$ denote the restriction of $g(r)$ to non-unit elements of $r$. in this paper we study the various graphi...

In each of Plato's major cosmological works, the Timaeus , the Statesman myth and the Philebus , he asserts that the body of the whole universe is alive and possesses a single WorldSoul which extends throughout it. I wish to offer a new interpretation of the role of the World-Soul which gives the World-Soul a special function in the economy of the Platonic cosmology and which explains why Plato...

There exist precisely 914, 58 and 46 equivariant types of tile-transitive tilings of 3-dimensional euclidean space by topological cubes, oc-tahedra and tetrahedra, that fall into 11, 3, and 9 topological families, respectively. Representatives are described for all topological families. A general method for obtaining results of this kind is introduced.

We construct globally regular gravitating solutions, which possess only discrete symmetries. These solutions of Yang-Mills-dilaton theory may be viewed as exact (numerical) solutions of scalar gravity, by considering the dilaton as a kind of scalar graviton, or as approximate solutions of Einstein-Yang-Mills theory. We focus on platonic solutions with cubic symmetry, related to a rational map o...