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Abstract: In the early 1960s, as a math-loving teen-ager, I was swept up in an intoxicating binge of mathematical exploration, centered on the discovery (or invention?) of integer sequences. The discovery that launched this binge came out of an empirical exploration of how triangular numbers intermingle with squares. The strange hidden order in the sequence that reflected their intermingling was extremely unexpected and exciting to me.
This burst of joy gave rise to the desire to repeat the experience, which meant to recreate essentially the same phenomenon again, but in a new “place”, which meant to generalize outwards, and I carried out this generalization by exploring many “nearby” analogues, where the terms “place” and “nearby” suggest a metaphorical space of ideas, and in it, some kind of metaphorical metric.
Over the next few years, analogies and sequences came to me in wondrous flurries, giving rise to all sorts of discoveries, some very rich and inspirational, some less so, but in any case, these coevolving discoveries pushed the envelope of interconnected ideas outwards, revealing unsuspected new caverns in the idea-space I had stumbled upon. Some explorations gave rise to patterns I could fully understand and prove; others gave rise to mysterious, chaotic behaviors too deep for me to fathom, let alone prove. After a while, alas, I started hitting up against the limits of my own imagination, and I gradually ran out of steam, but that several-year voyage was and remains a high point of my life.
My talk will thus be all about the very human, intuition-driven, analogy-permeated nature of mathematical discovery, invention, and exploration — not at the highly abstract level of the greatest of mathematicians, to be sure — but hopefully, the essence of the mental processes mediating the modest meanderings of a middling, minor mathematician is more or less the same as that of those that mediate the marvelous and majestic masterstrokes of a major, mature mathematician.