The mathematicians — or shall we say, the skilled  ones — often proclaim beauty as their criterion for truth. It is, in the view of the angels, and of the genius or saint, a world where order prevails, and thus serenity and peace. It is a place where we are reminded of the simplicity and purity that could govern our affairs, if men did not reject them. There is elegance, without all the slovenly trappings that we have associated with elegance.

I refer to things which are “just so”; and somehow inevitable, though inevitably unpredictable, in art and in science.

Conversely, the mediocre mathematician produces results that are ugly, trivial, squalid, and a mess. He is by nature clumsy, but also lazy: looking for proofs that can be easily found (because they are clichés.) Like all the higher intellectual disciplines, math reserves painful punishments for the lethargic and incurious, and awards brutal treatment to those intent on ignoring its “aesthetic” dimension; the element of form.

The other arts — one thinks of music, and sculpture — differ from mathematics in their selection of blindness. Math is sensually blind, or vacated, and must become musical or sculptural to assume “shape.” (A subtly different term from “form.”) But it is not sensual, and resists the transformation into something alien to itself. No mathematical model can be exact.

And yet, mathematics has beauty, or should have, for it is an art.

I knew once a collector of mathematics — a Polish gentleman, of course. He had, it was true, many mathematical books in his retinue, but these were the shadow not the substance of his subject, as print may offer a shadow of literature, poetry, song. These items have yet to be animated by their singers, and indeed, the visual arts also do not really exist until they are animated or “sung.” They must be found, or brought back, into life.

My Polish friend collected from among the things he had animated. He would eagerly show his most valuable possessions. He was remarkably patient with the slow-witted, such as me. But he realized his task was to give his possessions away; for only then could he keep them. He had the instinct of a teacher, which is to say, a kind of collector.

Those who collect material goods, sometimes exclusively, tend to operate on one of two radically opposite principles. Either they collect things that “might be useful,” though few things will ever be. Most do this, and make arguments for doing it. They are the common collectors of junk, as opposed to normal people, who use what is meant to be used, so that it is constantly disappearing.

The other type collects things which are beautiful, being indifferent instead to whether they will ever be needed. These hardly restrict themselves to lumpish, awkward, physical goods. However, note, they make shocking exceptions.