In 1943 McCulloch and Pitts, inspired by the all‑or‑none firing of biological neurons and the true‑false nature of logic circuits, proposed a mathematical model to simulate logical reasoning. Wiener deemed the model overly simplistic, yet it later became a cornerstone of modern neural networks, including large language models. Although these networks achieve remarkable empirical performance, they remain black‑box systems that struggle with reliable symbolic reasoning. Conventional approaches treat vectors as spheres of zero radius, a constraint that limits the representation of spatial concepts. By relaxing this constraint and representing concepts as spheres with non‑zero radii, additional features of biological cognition—such as region‑based representations—can be captured, enabling reasoning through model construction and inspection. Building on sphere embeddings, we introduce Sphere Neural Networks (SphNNs) that use a transition map to perform Aristotelian syllogistic reasoning at the symbolic level without any training data, taking an initial step toward the original goal of McCulloch and Pitts. We suggest that the zero‑radius constraint reflects the oversimplification highlighted by Wiener and may limit conventional networks' capacity for symbolic logical reasoning.
Review: Representing concepts as non‑zero radius spheres adds a geometric dimension to neural representations, offering a promising bridge between symbolic reasoning and deep learning.