0 - Introduction: glimpses of the theory beneath Monstrous Moonshine  pp. 1-13

Introduction: glimpses of the theory beneath Monstrous Moonshine

By Terry Gannon

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When you are collecting mushrooms, you only see the mushroom itself. But if you are a mycologist, you know that the real mushroom is in the earth. There's an enormous thing down there, and you just see the fruit, the body that you eat. In mathematics, the upper part of the mushroom corresponds to theorems that you see, but you don't see the things that are below, that is: problems, conjectures, mistakes, ideas, etc.

V. I. Arnold [17]

What my experience of mathematical work has taught me again and again, is that the proof always springs from the insight, and not the other way around – and that the insight itself has its source, first and foremost, in a delicate and obstinate feeling of the relevant entities and concepts and their mutual relations. The guiding thread is the inner coherence of the image which gradually emerges from the mist, as well as its consonance with what is known or foreshadowed from other sources – and it guides all the more surely as the ‘exigence’ of coherence is stronger and more delicate.

A. Grothendieck.

Interesting events (e.g. wars) always happen whenever different realisations of the same thing confront one another. When clarity and precision are added to the mix, we call this mathematics. In particular, the most exciting and significant moments in mathematics occur when we discover that seemingly unrelated phenomena are shadows cast by the same beast. This book studies one who has been recently awakened.