I Dream Awake

It was from Alonzo Church that many years ago I took a course in mathematical logic for which I was unprepared — unprepared, that is, for the discipline of mathematics, unprepared for the demands of argument, and unprepared for Church’s glacial and remote style. Church was an enormously distinguished mathematician. The material was very difficult, so difficult that someone had occasion once to complain about the complexity of a proof.
Church rotated his large torso away from the blackboard and toward the ten or so of us in the lecture room. “Any idiot,” he said calmly but with immense conviction, “can learn anything in mathematics. It requires only patience.” He seemed curiously moved; a film came over his eyes. “Now to create something,” he said, “that is another matter.” In that queer moment of insight occasionally vouchsafed the very young, I understood instantly that Church was not reveling in his own accomplishments, but, with his own eyes fixed on the unattained goals to which he had aspired, was confessing obliquely to us, an audience of impossibly callow young men, that when it cam to mathematics he, too, belonged in the company of humanity’s idiots.
As do we all.

~ David Berlinski, A Tour of the Calculus p.281

As with so many other fundamental concepts, there is no saying what change is, the formula or form of words change is defined either with a knowing shrug or some verbal flourish patently the same as the concept under analysis. Change is growth. But growth is transformation. And transformations are changes. In talking about change, philosophers have made use of a vocabulary essentially no different from that engagingly presented by the ancient Greeks. There is the dusky river from which a dripping Heraclitus emerged, convinced improbably that he could never step into the same river twice. There are the paradoxes of Zeno, mad, bad, and dangerous to know. And there is not much else. But the analysis of change has been the mathematician’s stock in trade at least since the seventeenth century. It is change that is the concern of the calculus and the interpretation of change that brings a coordinate system to vibrant life; and if the mathematician cannot define change he cansort out its characteristic forms, the ways in which it appears in this, our crowded world.

We all of us live within hearing of the muted or monstrous sounds of a great clock, now ticking faster, now slower, but inevitably and inexorably ticking, and it is by reference to the clock that we measure the terrible and depressing changes in our own bodies, stomach expanding, skin sagging, arches falling, the story inconveniently reflected in the morning mirror, where a suspiciously familiar impostor apparently holds court. Such somber talk has at least the instructive effect of suggesting that change in something—change in anything—takes place against an assumed background in which time itself is changing, sagging skin sagging with respect to the time then and the time now, although how it is that time might change without some other standard of time to measure that is another mystery of the sort which mathematics is strangely replete.

p. 61-62, A Tour of the Calculus

The metaphor of a cut has meaning, because the line is ordered by the placement of its points (as a highway is ordered by the placement of its cities and towns); but the English word “cut” fails to suggest the meaty decisiveness of the German geschnitten, the energetic suggestion of action undertaken, as if the line were actually being snipped by a pair of heavy shears.

p. 43 A Tour of the Calculus

The greatly large painting of my great grandfather standing in his rose garden with his snipping shears comes vibrantly to mind…