It was Paul Erdös (“You trivial beings!”) who said that “a mathematician is a device for turning coffee into theorems.”
This may be something deeper than it at first seems to be.
A mathematician is a human being, that is, an evolved meat bag, a snorting, grubbing, hungering, randy machine of self-propagation and gene transmission. Humans and other animals, as a general rule, eat, excrete, breed, and die, all without any rhyme or reason to their brief lives. For most of all history of life, all species have impacted the world only in similarly blind and unintentional ways: catastrophes unplanned, revolutions unforeseen, mathematical laws of biology obeyed without comprehension or awareness.
Then came the first mathematician; not the traditional Thales of Greek legend, but someone much earlier. And what was she or he, and what do mathematicians do? They take the dross and brutishness of life, and see there are patterns beneath and within: they draw triangles in the sand, and discover Pythagoras’s Theorem; they delineate the fields of Nile, and discover geometry. They distill the essential out of the crude whole, and abandoning that crudity to its own devices and gyrations build on their pure suppositions. Biologians and physicists are concerned with what is; philosophers and leaders often consider what should be; artists, closest to the people of number, consider what they would like to be, the imaginations that most tickle their various fancies; but mathematicians are intent on invading the land of imagination and seeking not their wishes or needs, but simply all that what must be.
And what emerges from their work is a structure of pure intellect divorced from the distractions of corporeal existence, an unfallen Babel-tower of axiom and result. Their work is imagination, clouds piled on clouds, not anchored to anything solid, but it is consistent against itself. In that stark sculpture-glory it is something immeasurably different from all pursuits before, and most pursuits since.
Mathematicians are crude creatures, and live in the crude world of flesh and matter: but they are also gates and conduits into the high worlds of logic and suppositional truth. And being that, they are the most precious alchemists, for they transform the raw crudeness of their existence and sustenance into something universal, eternal, beautiful and — if one wishes to use the word — divine.
Coffee into theorems it is, but that is something of an understatement.
Or then what I said just now was an overstatement; I don’t know. Why plumbers never get these lyrical fits, or are they just better in self-control?