Mathematicians generally like to present their work in the wraps of extreme rigor and pure logic. This professional posture is in some ways very admirable. However, where do their ideas really come from? Might mathematicians be cut from a different cloth than ordinary humans, or could that be the case at least temporarily, whenever they do their professional work? Could it be that mathematicians, just before sitting down to do their work, don blur-suppression helmets or anti-intuition caps, in somewhat the same way passengers in cars fasten their seatbelts? Do they then strictly follow the straight-and-narrow pathways of pure, rigorous, logical axiomatic deduction in order to reach their often astonishing conclusions?

No.

This talk will be about how deeply and universally mathematical thought at all levels of sophistication is riddled with impure, nonrigorous, illogical intuitions originating in analogies, often highly unconscious ones. Some of these analogies are good and some of them are bad, but good or bad, it is they that lurk behind the scenes of all mathematical thought. And far from being a disappointment, it is a source of joy to recognize and savor the swarm of analogies that lurk behind the scenes.

What is curious, to my mind, is that so few mathematicians seem to take pleasure in examining and exploring this crucial and wonderful aspect of their minds, their thoughts, and their deep discoveries. Perhaps, however, they can be stimulated to examine their own hidden thinking processes if the ubiquity of analogies can be made sufficiently vivid as to grab their interest. So in this talk, I will do my best to provoke mathematicians. At the same time, I will try equally hard to convey to non-mathematicians the sheer joy of mathematical thinking, of mathematical invention, of mathematical discovery, of mathematical revelation.

To convey this intense type of joy, I will conclude the talk with some highly personal tales of analogical invention/discovery in mathematics, because I still recall the profound and heady exhilaration they gave me over 50 years ago so vividly that it all might as well have happened just yesterday.