Reflections on Structure
Edward Beckenstein and Lawrence Narici, Department of Mathematics and Computer Science, St. John’s College of Liberal Arts and Sciences
Abstract
An idea gestating throughout 19th century mathematics was that certain concretely different things were the same somehow. By the mid-twentieth century, the idea had been formalized and adopted. A “mathematical structure” came to mean: Start with a set of objectives that interact with each other according to certain rules. Results about this general structure will then apply to anything that you later discover to obey the rules. Stripped of the accountrement of the concrete situation, some things became much clearer. What are the objects? What can they be? This is the rare case where ignorance is mandatory. The great German mathematician David Hilbert summarized it in a 1941 letter to the philosopher/mathematician F. L. G. Frege:
"If among my ‘points’ [i.e., ‘objects’] I consider some systems of things (e.g., the system of love, law, chimney sweeps……) and then accept only my complete axioms as the relationships between these things, my theorems (e.g., the Pythagorean) are valid for these things also."
Our recent research is about a novel way to determine that two abstract structures–Banach spaces with bases–are the same.