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No matter how correct a mathematical theorem may appear to be, one ought never to be satisfied that there was not something imperfect about it until it gives the impression of also being beautiful. (George Boole, in MacHale, 1993, p. 107)

If my teachers had begun by telling me that mathematics was pure play with presuppositions, and wholly in the air, I might have become a good mathematician, because I am happy enough in the realm of essence. But they were over-worked drudges, and I was largely inattentive, and inclined lazily to attribute to incapacity in myself or to a literary temperament that dullness which perhaps was due simply to lack of initiation. (Santayana, 1944, p. 238)

“That’s beautiful!” is the unsolicited exclamation. The response is not to a painting, a breathtaking view or a flawless musical performance, but rather to a mathematical statement or a mathematical proof. What brings such aesthetic pleasure to a mathematician or to those who wish to appreciate mathematics and engage in it?

What geometrician or arithmetician could fail to take pleasure in the symmetries, correspondences, and principles of order observed in visible things? Consider, even, the case of pictures: those seeing by the bodily sense the products of the art of painting do not see the one thing in the one only way; they are deeply stirred by recognizing in the objects depicted to the eyes the presentation of what lies in the idea, and so are called to recollection of the truth - the very experience out of which Love rises. (Plotinus, The Enneads, II.9.16; 1991, p. 129)

I begin with a story told by the French mathematician François Le Lionnais (1983) about his first experience, at age seven, of a mathematical discovery. It illustrates, perhaps more immediately than a trip to the Great Museum of ‘elegant’ mathematical proofs, how aesthetic responses, values and experiences can snugly insinuate themselves alongside logical steps and decisions throughout mathematical activity.

In the late 1970s, when the eminent anthropologist and biologist Gregory Bateson sought to codify his influential views on the ecology of mind, he chose the idea of pattern as his central heuristic device. The choice was not surprising, for Bateson, in a remarkably productive career as both scientist and educator, had by that time been using this concept to explore, identify and represent the essential features of biology and anthropology for more than a quarter of a century. In his summative , published in 1979, Bateson related one early experience in his career that illustrates particularly well the power that the notion of pattern can have in helping to organise one’s thinking in fundamental ways.

The problem is more aesthetic than ethical, philosophical, sexual, psychological, or political, though it goes without saying that such divisions are unacceptable to me because that matters is, in the long run, aesthetic. (Mario Vargas Llosa, 1999, p. 194)

The dilemma posed all scientific explanation is this: magic or geometry? (Thom, 1975, p. 5)

Studies on the foundations of mathematics and mathematical method should make substantial room for psychology, indeed even for the aesthetic. (Henri Lebesgue, 1941, p. 122)

The most imposing icon in early eighth-century Byzantium was probably the mosaic image of Christ above the Chalke, the bronze gate entrance to the palace built by Justinian to the south of Sancta Sophia. It is said to have been similar in style to a surviving fourteenth-century mosaic (Figure 1, left) in the restored church of St Saviour in Chora. [1]