Infinitesimal #2

What are infinitesimals? That term was coined around 1670. A related term is indivisibles.  The author of Infinitesimal says: "To understand why the struggle over indivisibles became so critical, we need to take a close look at the concept itself, which appears deceptively simple but is in fact deeply problematic. In its simplest form the doctrine states that every line is composed of a string of points, or "indivisibles," which are the line's building blocks, and which cannot themselves be divided" (p. 8-9).

Some mathematicians, especially Cavalieri, held that a line is composed of an "infinite" number of points, a plane is composed of an "infinite" number of lines laid parallel and a solid is composed of an "infinite" number of planes laid parallel. Of course, for any polygon, sets of parallel lines can be drawn in different directions, e.g. vertical or horizontal or diagonal, which gives a different number of lines.

Infinitesimal mentions that the math symbol used for infinity, ∞, was invented by John Wallis (who has a prominent role in the book). Wikipedia confirms this. Wallis used 1/∞ for an infinitesimal.

One conundrum is how many indivisibles (or points) there are in a given line and how small they are. Assume a given line has a huge, yet finite, number, we can call a gazillion. That suggests each indivisible is 1/gazillion in size. Problems start arising when considering other lines of different length. A "deep" problem -- discovered by the Greeks -- is that some lines can not be subdivided into an integer number of uniform-sized units, no matter how small they are. For example, the square root of two length units (e.g. inches) cannot be converted to an integer number of anything, because the square root of two is an irrational number. Ditto for pi, the circumference of a circle with a diameter of 1 length unit, and Euler's number e, the base base of natural logarithms.

A related topic is the continuum. The usual meaning of continuous is “unbroken” or “uninterrupted.” Thus a continuous entity—a continuum—has no “gaps.” Aristotle addressed it in book 6 of his Physics. He concluded that the concept of infinitesimals was erroneous, and that continuous magnitudes can be divided ad infinitum (p. 10). The Jesuit Benito Pereia proposed the thesis that a line is composed of separate points and presented all the arguments in its favor by others. He then demolished them one by one, and concluded, like Aristotle, that the continuum is infinitely divisible, and not composed of  indivisibles (p. 121). The Jesuits thought Cavalieri's were untenable.

The Epilogue of Infinitesimal describes how the use of "infinitesimals" later grew to have a prominent part of mathematics. I believe it would have been improved if it explained the concept of a limit as used in math, such as in calculus. Said concept was developed in the 19th century. It involves, but is not identical to, "infinitesimals" as described in the book. The similarity is stronger in integral calculus.

A much longer history of infinitesimals and related topics is here. The historical period the book is mainly about is a very small part of it.