改革开放40年，搭上鲁宾逊无穷小微积分的这趟数学快车
In autumn 1960, a revolutionary, new idea was put forward（提出） by Abraham Robinson (鲁宾逊，1918–1974，享年56岁）。这种革命性的新思想是什么呢？
He realized（意识到） that recent advances in symbolic logic could lead to a new model of mathematical analysis. Using these concepts, Robinson introduced an extension of the real numbers, which he called the hyperreals（超实数）. 也就是说，可以把无穷小放在超实数系统中。这种新思想在当时具有功能性。
The hyperreals, denoted *R, contain all the real numbers and obey the familiar laws of arithmetic. But *R also contains infinitely small and infinitely large numbers. With the hyperreals, it became possible to prove the basic theorems of calculus in an intuitive and direct manner, just as Leibniz had done in the 17th century.
据此，J.Keisler，慧眼识珠，于1969年，全身心地投入实践鲁宾逊这一革命性新思想的伟大实践中去。1978年，改革开放之后，我们的故事就开始了，搭上了无穷小微积分这趟快车。
袁萌 陈启清 10月17日
附：1.7. Regained（失而复得）
In comparision with mathematicians, engineers and physicists are typically less concerned with rigor and more concerned with results. Since their studies revolve around dynamical systems and continuous phenomena, they continued to regard infinitesimals as useful heuristic aids in their calculations. A little care ensured correct answers, so they had few qualms about infinitely small quantities. Meanwhile, the formalists, led by David Hilbert (1862-1943), reigned over mathematics. No theorem was valid without a rigorous, deductive proof. Infinitesimals were scorned（受到请轻视）since they lacked sound definition.
In autumn 1960, a revolutionary, new idea was put forward by Abraham Robinson (1918–1974). He realized that recent advances in symbolic logic could lead to a new model of mathematical analysis. Using these concepts, Robinson introduced an extension of the real numbers, which he called the hyperreals. The hyperreals, denoted ∗R, contain all the real numbers and obey the familiar laws of arithmetic. But ∗R also contains infinitely small and infinitely large numbers. With the hyperreals, it became possible to prove the basic theorems of calculus in an intuitive and direct manner, just as Leibniz had done in the 17th century. A great advantage of Robinson’s system is that many properties of R still hold for ∗R and that classical methods of proof apply with little revision [6, pp. 281–287]. Robinson’s landmark book,
5Never mind the fact that their constructions were ultimately based on the natural numbers, which did not receive a satisfactory definition until Frege’s 1884 book Grundlagen der Arithmetik [14].
Non-standard Analysis was published in 1966. Finally, the mysterious infinitesimals were placed on a firm foundation [7, pp. 10–11]. In the 1970s, a second model of infinitesimal analysis appeared, based on considerations in category theory, another branch of mathematical logic. This method develops the nil-square infinitesimal, a quantity ε which is not necessarily equal to zero, yet has the property that ε2 = 0. Like hyperreals, nil-square infinitesimals may be used to develop calculus in a natural way. But this system of analysis possesses serious drawbacks. It is no longer possible to assert that either x = y or x 6= y. Points are “fuzzy”; sometimes x and y are indistinguishable even though they are not identical. This is Peirce’s continuum: a series of overlapping infinitesimal segments [2, Introduction]. Although intuitionists believe that this type of model is the proper way to view a continuum, many standard mathematical tools can no longer be used.6 For this reason, the category-theoretical approach to infinitesimals is unlikely to gain wide acceptance.