无穷小最古老的祖先

    众所周知，鲁宾逊的最大历史功绩是为无穷小奠定了严格的数学理论基础。

    但是，无穷小最古老的祖先是什么？一般人不甚了解。

根据Joel A.Tropp的最新研究“无穷小：历史与应用”，我们发现无穷小最古老的祖先是毕德哥拉斯学派提出的“单子”（Monad）概念。单子没有长度，但是又不退缩为一个几何点，是世界上最古老的无穷小。

    有兴趣的读者可参阅本文附件。

袁萌  陈启清 10月3日

附件：1.2. Origins

The rst deductive mathematician, Pythagoras (569?–500? b.c.), taught that all is Number. E.T. Bell describes his fervor:

He ...preached like an inspired prophet that all nature, the entire universe in fact, physical, metaphysical, mental, moral, mathematical—everything—is built on the discrete pattern of the integers, 1,2,3,... [1, p. 21].

Unfortunately, this grand philosophy collapsed when one of his students discovered that the length of the diagonal of a square cannot be written as the ratio of two whole numbers. The argument was simple. If a square has sides of unit length, then its diagonal has a length of √2, according to the theorem which bears Pythagoras’ name. Assume then that √2 = p/q, where p and q are integers which do not share a factor greater than one. This is a reasonable assumption, since any common factor could be canceled immediately from the equation. An equivalent form of this equation is

p2 = 2q2.

We know immediately that p cannot be odd, since 2q2 is even. We must accept the altrnative that p is even, so we write p = 2r for some whole number r. In this case, 4r2 = 2q2, or 2r2 = q2. So we see that q is also even. But we assumed that p and q have no common factors, which yields a contradiction. Therefore, we reject our assumption and conclude that √2 cannot be written as a ratio of integers; it is an irrational number [1, p. 21]. According to some, this proof Pythagoras so much that he hanged its precocious young auth stories or. Equally apocryphal reports indicate that the student perished in a shipwreck. These tales should demonstrate how badly this concept unsettled the Greeks [3, p. 20]. Of course, the Pythagoreans could not undiscover the proof. They had to decide how to cope with these inconvenient, non-rational numbers.

The solution they proposed was a crazy concept called a monad. To explain the genesis of this idea, Carl Boyer presents the question:

If there is no nite line segment so small that the diagonal and the side may both be expressed in terms of it, may there not be a monad or unit of such a nature that an indenite number of them will be required for the diagonal and for the side of the square [3, p. 21]?

The details were sketchy, but the concept had a certain appeal, since it enabled the Pythagoreans to construct the rational and irrational numbers from a single unit. The monad was the rst innitesimal. Zeno of Elea (495–435 b.c.) was widely renowned for his ability to topple the most well-laid arguments. The monad was an easy target. He presented the obvious objections: if the monad had any length, then an innite number should have innite length, whereas if the monad had no length, no number would have any length. He is also credited with the following slander against innitesimals:

That which, being added to another does not make it greater, and being taken away from another does not make it less, is nothing [3, p. 23].

The Greeks were unable to measure the validity of Zeno’s arguments. In truth, ancient uncertainty about innitesimals stemmed from a greater confusion about the nature of a continuum, a closely related question which still engages debate [1, pp. 22–24].