CTK Insights

26 May

Indivisibles and Infinitesimals

If you are not a historian of mathematics and do not work in the foundation of calculus, you may easily confuse two concepts - indivisibles and infinitesimals - that are both claimed to underlie modern calculus.

For example (see Mikhail Katz and David Sherry) mention a paragraph from C. Boyer (The concepts of the calculus, p. 59):

In the seventeenth century, however, the infinitesimal and kinematic methods of Archimedes were made the basis of the differential and the fluxionary forms of the calculus.

As Katz and Sherry observe,

(That) is not quite correct. Archimedes’ kinematic method is arguably the forerunner of Newton’s fuxional calculus, but his infinitesimal methods are less arguably the forerunner of Leibniz’s differential calculus. Archimedes’ infinitesimal method employs indivisibles.

As another example, W. W. Rouse Ball (A Short Account of the History of Mathematics) writes about Johann Kepler (p. 256)

In his Stereometria, which was published in 1615, he determines the volumes of certain vessels and the areas of certain surfaces, by means of infinitesimals instead of by the long and tedious method of exhaustion.

But then on page 279 he seems to contradict himself:

The principle of indivisibles had been used by Kepler in 1604 and 1615 in a somewhat crude form.

As a matter of fact, there is a rather essential difference between indivisibles and infinitesimals. Infinitesimals are of the same dimension that the shape they combine to form. Indivisibles are one dimension less (in other words, indivisibles are of codimension 1).

For example, a square can be thought of consisting of an infinite number of thin (infinitesimal) rectangles, or of (indivisible) line segments. For another example, check two ways to determine the area of circle.

I refer to Katz and Sherry's paper for a historic exploration that led them to conclude that Leibniz's understanding of infinitesimals was very much in line with the twentieth century development of hyperreals by A. Robinson.

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