(^488) 16. THE CALCULUS
16.3. Show that the point at which the tangent to the curve y = f(x) intersects
the y axis is y = f(x) - xf'(x), and verify that the area under this curve—more
precisely, the integral of f(x) - xf'(x) from ÷ = 0 to ÷ — a—is twice the area
between the curve y — f(x) and the line ay = f(a)x between the points (0,0) and
(a, /(a)). This result was used by Leibniz to illustrate the power of his infinitesimal
methods.
16.4. Recall that Eudoxus solved the problem of incommensurables by changing
the definition of proportion, or rather, making a definition to cover cases where
no definition existed before. Newton's "theorem" asserting that quantities that
approach each other continually (we would say monotonically) and become arbi-
trarily close to each other in a finite time must become equal in an infinite time
assumes that one has a definition of equality at infinity. What is the definition of
equality at infinity? Since we cannot actually reach infinity, the definition will have
to be stated as a potential infinity, that is, a statement about all possible finite
times. Formulate the definition and compare Newton's solution of this difficulty
with Eudoxus' solution of the problem of incommensurables.
16 .5. Draw a square and one of its diagonals. Then draw a very fine "staircase" by
connecting short horizontal and vertical line segments in alternation, each segment
crossing the diagonal. The total length of the horizontal segments is the same as the
side of the square, and the same is true of the vertical segments. Now in a certain
intuitive sense these segments approximate the diagonal of the square, seeming to
imply that the diagonal of a square equals twice its side, which is absurd. Does
this argument show that the method of indivisibles is wrong?
16.6. In the passage quoted from the Analyst, Berkeley asserts that the experience
of the senses provides the only foundation for our imagination. From that premise
he concludes that we can have no understanding of infinitesimals. Analyze whether
the premise is true, and if so, whether it implies the conclusion. Assuming that our
thinking processes have been shaped by the evolution of the brain, for example, is it
possible that some of our spatial and counting intuition is "hard-wired" and not the
result of any previous sense impressions? The philosopher Immanuel Kant (1724-
1804) thought so. Do we have the power to make correct judgments about spaces
and times on scales that we have not experienced? What would Berkeley have said
if he had heard Riemann's argument that space may be finite, yet unbounded?
How would he have explained the modern computer chip, on which unimaginable
amounts of data can be recorded in space far too small for the senses to perceive? Go
a step further and consider how quantum mechanics is understood and interpreted.
coco
(coco)
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