Database-Driven Mathematical Inquiry
Steven Clontz
TL;DR
This work argues that semantic mathematical databases can accelerate mathematical inquiry by combining curated small-scale platforms with scalable, metadata-driven search. It introduces the π-Base model for small databases, organizing data into properties $P\_\{######\}$, spaces $S\_\{######\}$, and theorems that drive automated boolean deduction, all backed by a Git-based peer-review workflow. The paper demonstrates how semantic databases can illuminate known results and seed new research, recounting a case study that follows the chain $T\_2 \Rightarrow KC \Rightarrow wH \Rightarrow US \Rightarrow T_1$ and yields intermediate properties $k\_1H$, $k\_2H$ plus a novel space $X$ built as a one-point compactification, illustrating the potential for discovery within such infrastructures. It also discusses practical limitations, including the need for valued properties, handling of cardinalities, and cross-disciplinary extension, and argues for broader adoption and recognition of the scholarly labor required to build and maintain these resources.
Abstract
Recent advances in computing have changed not only the nature of mathematical computation, but mathematical proof and inquiry itself. While artificial intelligence and formalized mathematics have been the major topics of this conversation, this paper explores another class of tools for advancing mathematics research: databases of mathematical objects that enable semantic search. In addition to defining and exploring examples of these tools, we illustrate a particular line of research that was inspired and enabled by one such database.