Database-Driven Mathematical Inquiry

The paper provides a survey of semantic methods for solution of fundamental tasks in mathematical knowledge management. Ontological models and formalisms are discussed. We propose an ontology of mathematical knowledge, covering a wide range of fields of mathematics. We demonstrate applications of this representation in mathematical formula search, and learning.

We present a short survey of the literature on indexing and retrieval of mathematical knowledge, with pointers to 72 papers and tentative taxonomies of both retrieval problems and recurring techniques.

In the words of the esteemed mathematician Paul Erd\"os, the mathematician's task is to \emph{prove and conjecture}. These two processes form the bedrock of all mathematical endeavours, and in the recent years, the mathematical community has increasingly sought the assistance of computers to bolster these tasks. This paper is a testament to that pursuit; it presents a robust framework enabling a computer to automatically generate conjectures - particularly those conjectures t...

In this paper, we present an ontology of mathematical knowledge concepts that covers a wide range of the fields of mathematics and introduces a balanced representation between comprehensive and sensible models. We demonstrate the applications of this representation in information extraction, semantic search, and education. We argue that the ontology can be a core of future integration of math-aware data sets in the Web of Data and, therefore, provide mappings onto relevant da...

Over the last decades, a class of important mathematical results have required an ever increasing amount of human effort to carry out. For some, the help of computers is now indispensable. We analyze the implications of this trend towards "big mathematics", its relation to human cognition, and how machine support for big math can be organized. The central contribution of this position paper is an information model for "doing mathematics", which posits that humans very efficie...

One of the most promising applications of mathematical knowledge management is search: Even if we restrict attention to the tiny fragment of mathematics that has been formalized, the amount exceeds the comprehension of an individual human. Based on the generic representation language MMT, we introduce the mathematical query language QMT: It combines simplicity, expressivity, and scalability while avoiding a commitment to a particular logical formalism. QMT can integrate var...

In mathematics, LaTeX is the de facto standard to prepare documents, e.g., scientific publications. While some formulae are still developed using pen and paper, more complicated mathematical expressions used more and more often with computer algebra systems. Mathematical expressions are often manually transcribed to computer algebra systems. The goal of my doctoral thesis is to improve the efficiency of this workflow. My envisioned method will automatically semantically enric...

We introduce T2Ku, an open source project that aims at building a semantic wiki of mathematics featuring automated reasoning(AR) techniques. We want to utilize AR techniques in a way that truly helps mathematical researchers solve problems in the real world, instead of building another ambitious yet useless system. By setting this as our objective, we exploit pragmatic design decisions that have proven feasible in other projects, while still employs a loosely coupled architec...

We discuss the advantages of searchable, collaborative, language-independent databases of mathematical results, indexed by "fingerprints" of small and canonical data. Our motivating example is Neil Sloane's massively influential On-Line Encyclopedia of Integer Sequences. We hope to encourage the greater mathematical community to search for the appropriate fingerprints within each discipline, and to compile fingerprint databases of results wherever possible. The benefits of th...

Since the early twentieth century, it has been understood that mathematical definitions and proofs can be represented in formal systems systems with precise grammars and rules of use. Building on such foundations, computational proof assistants now make it possible to encode mathematical knowledge in digital form. This article enumerates some of the ways that these and related technologies can help us do mathematics.