Shufflebrain

To the best of my knowledge, the first formal principle of quantity remains to be found, if one really exists at all. Even Riemann's genius had to be ignited by intuition. Intuitively speaking, the basic notions of quantity imply measuring something with something else. As in our optical transform experiments in the last chapter, measuring to Riemann, "consists in superposition of the magnitudes to be compared." As in the case with the reference and object waves in the hologram, superposition of the two magnitudes or quantities occurs "only when the one is part of the other."[4] Riemann was speaking about continuity in the most exactingly analytic sense of the word.

Where can we find continuity? More important, how can we guarantee its existence in the relationship of, say, X and Y? To satisfy Riemann's requirements, we would have to show that at least one of the elements involved necessarily affects the other. Thus a frog on a lily pad won't do, especially if the animal is just sitting there enjoying the morning sunshine. We would have to find out if any change in either the animal or the plant forces a concomitant variation in the other. Thus the first requirement for establishing continuity is to get away from static situations and focus on dynamic--variable--relationships. Do they change together?

Suppose, though, that one unit of change in X procures a one-unit change in Y; that Y=X. If we graph the latter, the plot will look like a straight line. In a linear relationship, the ratio of Y to X, of course, remains constant no matter how large or small the values become. This constancy made mathematicians before Riemann shy away from points on the straight line. For an infinite number of points exist between any two points on a line; even as the values of X and Y approach zero we never close the infinite interval between two points on a straight line.

The curve is quite another story. What is a curve? My handworn 1964 edition of Encyclopedia Britannica characterizes it as "the envelop of its tangents." Remember that on a circle, the very embodiment of curvature, we can draw a tangent to a single point on the circumference. The same thing holds for tangents to a curve; and we do not draw a tangent to a straight line. Like the sine or cosine, the tangent is a function of an angle.[5] We might think of a tangent as a functional indicator of a specific direction. The points on a straight line all have the same direction; therefore, a tangent to a straight line would yield no information about changes in direction (because the directions are all the same). Neighboring points on a curve, by contrast, have different relative directions. Each point on a curve takes its own specific tangent. And the tangent to the curve will tell us something about how the directions of one point vary relative to neighboring points.[6]

Imagine that we draw a tangent to a point on the X-Y curve. The bend in the curve at that point will determine the slope of the tangent. If we could actually get down and take a look at our X-Y point, we'd find that its direction coincides with our tangent's slope. Of course, we can't reach the point. But we can continuously shrink X and Y closer and closer to our point. As we get nearer and nearer to the point, the discrepancy between the curve and the slope of the tangent becomes smaller and smaller. Eventually, we arrive at a vanishingly tiny difference between curve and tangent. We approach what Isaac Newton called--and mathematicians still call--a "limit" in the change of Y relative to X. The limit--the point-sized tangent--is much like what we obtain when we convert the value of pi from 22/7 to 3.14159...and on and on in decimal places until we have an insignificant but always persisting amount left over. The limit is very close to our point. The continuous nature of the change in Y to X permits us to approach the limit.

Finding limits is the subject of differential calculus. The principal operation, aptly called differentiation, is a search for limit-approaching ratios known as derivatives. The derivative is a guarantee of continuity between Y and X at a point. The existence of the derivative, in other words, satisfies Riemann's criterion of continuity: Y is part of X. The derivative is strictly a property of curves. For the derivative is a manifestation of changing change in the relationship of a point to its immediate neighbors. Derivatives, minuscule but measurable ratios around points, were the basis from which Riemann developed the fundamental rules of his new geometry.

Derivatives are abstractions. And, with one valuable exception, we can gain no impression of their character by representing them in perceptual space. The exception, though, will permit us to "picture" how Riemann discovered measurable relationships among points.