I Dream Awake

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