It was observed in an earlier thread that G.H. Hardy made the following

observation in his 194o book A Mathematician's Apology:

"The mathematician's patterns, like the painter's or the poet's, must be beautiful, the ideas, like the colors or the words, must fit together in a harmonious way. Beauty is the first test; there is no permanent place in the world for ugly mathematics...It may be very hard to define mathematical beauty, but that is just as true of beauty of any kind-we may not know quite what we mean by a beautiful poem, but that does not prevent us from recognizing one when we read it."

The main issue for this thread is whether beauty or structure comes first in doing mathematics. In terms of structure, consider

A mathematical structure is a set (or sometimes several sets) with various associated mathematical objects such as subsets, sets of subsets, operations and relations, all of which must satisfy various requirements (axioms). The collection of associated mathematical objects is called the structure and the set is called the underlying set. http://www.abstractmath.org/MM/MMMathStructure.htm

For example, a set endowed with a proximity (nearness) relation acquires a structure, namely, collections of near sets (each collections contains all sets that are near a given set in a proximity space). Any set endowed with a relation has a structure. Then the question here is Do we first look for structures when we do mathematics?

The question here is not whether mathematics is beautiful but rather whether beautiful patterns are the starting point of mathematics. For example, is it the painting Mona Lisa or the golden ratios in the painting that first attracts us? See the attached image.

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