Hofstadter on Numeric Sequences and Analogies

Source: DIMACS/Rutgers University, 2014

Douglas Hofstadter, Indiana University

Title: Analogies and Sequences: Intertwined Patterns of Integers and Patterns of Thought Processes

As a math-loving teen-ager, I got swept up into an intense and almost indescribably intoxicating binge of mathematical exploration, all of which was centered on the discovery (or invention?) of integer sequences. The very first discovery in this binge came out of an empirical exploration of the way in which triangular numbers intermingle with squares, and the discovery of the strange hidden order in the sequence that reflected their intermingling was extremely unexpected and exciting to me.

This huge burst of joy instantly 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 all sorts of nearby analogues, where the words place and nearby of course suggest (at least to mathematically inclined folk!) some kind of metaphorical, but perhaps objective, 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 to me, some of course less so, but in any case, these coevolving discoveries constantly pushed the envelope of richly-interconnected ideas outwards, revealing new and unsuspected caverns in this mysterious idea-space that I had stumbled upon. Some of my explorations gave rise to patterns that I could fully understand and prove, whereas others gave rise to mysterious, chaotic behaviors that were far too deep for me to fathom, let alone prove. After a while, I started hitting up against the limits of my own imagination, and thus, sadly, I gradually ran out of steam, but the several-year voyage had been incredible one of the greatest voyages in my life.

The talk will thus be all about the very human, intuition-driven, analogy-permeated nature of mathematical discovery, invention, and exploration not at the immensely abstract level of great 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.

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