​    ​2016年元旦，美國R Vinsonhaler教授發表最新研究論文，題為“Teaching Calculus with Infinitesimals”，詳細闡述了非數學專業低年級大學生學習無窮小微積分的有益性與必要性。

    ​    ​實際情況是，國內此類研究是一項空白。對此，雖然無窮小微積分網站呼籲多年，但是沒有實際效果。

    ​    ​回到國內，Vinsonhaler研究成果值得我們關注。

        ​袁萌  陳啟清   10月13日

附：Teaching Calculus with Infinitesimals

    By R Vinsonhaler（2016.01.01）  

    ​This article argues that first semester calculus courses for non-mathematics majors should be taught using infinitesimals. This applies to both high school and undergraduate calculus courses. The use of infinitesimals in calculus, though more intuitive than the approach developed in the 19th century, has been controversial for over two millennia. However, in the 20th century their use was shown to be equa-consistent with the approach developed in the 19th century. Here I first provide a brief history of infinitesimals, why they were controversial, and how they were finally put on a firm footing. Next I illustrate the intuitive nature of the use of infinitesimals. Thus I conclude that at least students not continuing on to more advanced analysis courses would be better served by learning calculus via infinitesimals.

    ​Keywords: calculus reform; history of mathematics; non-standard analysis

1. Introduction

     ​Infinitesimals not only could serve as an important intuitive aid to learning some key concepts in calculus, they also have a fascinating history. Infinitesimals were used as early as the Ancient Greeks by mathematicians such as Archimedes, were still being used by Leibniz in 1600 C.E. [14] and informally by physicists and mathematicians until at least the end of the 19th century. However, because it would not be until the 20th century that infinitesimals were shown to be realizable without contradiction, mathematicians usually sought alternative methods of proof when presenting their results to the “public” [10]. Thus, for example, Archimedes replaced his arguments

    ​Journal of Humanistic Mathematics Vol 6, No 1, January 2016

250 Teaching Calculus with Infinitesimals

    ​using “indivisibles” (found, for example, in his The Method of Mechanical Theorems1) with extensions of Eudoxus’s method of exhaustion. Although Leibniz and Newton invented calculus by using infinitesimals, the latter were not rigorously established until the 1960s, forcing the development of a formal calculus along a different route. This formalized calculus, referred to as modern calculus, or standard analysis, was made possible by the work of such mathematicians as Cauchy and Weierstrass with the introduction of the “epsilon-delta” definition of limit [5].

There are deep ironies in this history. For example, standard analysis was certainly facilitated by the acceptance of a rigorous definition of real numbers. And this in turn was facilitated by the invention of set theory by Cantor. One twist here is that Cantor’s thesis advisor, Kronecker, rejected the concept of an infinite set, something critical to both set theory and the definition of reals that evolved. Another unexpected fact is that if historical weight counted for anything, then the moniker “standard analysis” would have to go to the infinitesimal approach, since it was used by some of the most prominent mathematicians and physicists working in the field, from Archimedes to those in the 18th century. Instead, the use of infinitesimals in calculus is dubbed “non-standard analysis”. Once standard analysis was developed, infinitesimals were exiled.

2. Standard Analysis vs. Non-Standard Analysis

    ​Standard analysis uses the “epsilon-delta” definition of limit which in turn is used to define such key concepts as convergence, continuity, derivative and integral. Standard analysis is currently used as the rigorous foundation for a majority of traditional calculus courses. “Non-standard” analysis, on the other hand, uses infinitesimals and what are called “hyperreals”, and most importantly, does not depend on a notion of “limit” to define the fundamental concepts of elementary calculus. To speak loosely, the limit concept formalizes a concept of converging to small differences, whereas with the concept of the infinitesimal, one is “already there”.
    ​1This work was only rediscovered early in the 20th century in what has come to be called Archimedes Palimpsest — a 10th century Byzantine copy had been overwritten with Christian religious text by 13th century monks.

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    ​Let us look at the notion of the continuity of a function. Intuitively a function with domain and range that are subsets of the real numbers is continuous at a real number r if: 1) the number r is in the domain of the function; and 2) for real numbers s in the domain of the function that are close to r, the value of the function at s is close to the value of the function at r. Thus logically one must check a “for all” statement: “for all real numbers s in the domain of the function close to r [statement]”. And if one uses infinitesimals then this is exactly the logical complexity of the statement, since “s is close to r” is translated formally into “the arithmetic difference of s and r is an infinitesimal”; denoted by r ≈ s. However, if infinitesimals are not available, one must search for a different translation of “s is close to r”. The solution introduced by Weierstrass required a definition whose logical complexity is “for all   > 0 there exists a δ > 0 such that for all s [statement]” (where the missing statement is “if the absolute value of the difference of s and r is less than δ, then the absolute value of the difference of the function values at s and r is less than  ”). This is the famous or infamous (depending on your experience in calculus) “epsilon-delta argument”.

     ​In some sense the techniques introduced by Weierstrass for standard analysis reflected the method of exhaustion introduced by Eudoxus more than 2400 years earlier. Once it was understood by the ancient Greeks that there were geometric lengths whose ratios were not rational (such as the ratio of the hypotenuse to either of the other sides in an isosceles right triangle), a decision had to be made: tackle the question of the existence of numbers that were not rational, or find a way around the issue. Eudoxus did the latter. For he said that two lengths were equal if, using modern terminology, the sets of rational lengths less than both were equal, and the sets of rational lengths greater than both were equal. This comes very close to the definition of the real numbers introduced by Dedekind in the 19th century. Eudoxus’s method allowed the development of two and three dimensional versions used to prove such things as the relationship of the area and circumference of a circle or the area and volume of a sphere (all by Archimedes). Recall that Archimedes convinced himself of the truth of these relationships by using infinitesimals (and in the case of the sphere also using the Law of the Lever − see below), but then resorted to a version of the method of exhaustion probably because he recognized that an argument using infinitesimals would not be considered a proof.

252 Teaching Calculus with In?nitesimals

    ​We will see below in more detail how standard analysis avoids the question of the possible existence of a useful extension of the reals to what are now called the hyperreals by introducing an alternation of quantifiers that dynamically bounds the behavior of a function in a way similar to how Euclid, Archimedes, and others of that era statically captured certain lengths, areas, and volumes by variations on Eudoxus’s method of exhaustion. The invention of the reals, though strongly resisted by constructivists such as Kronecker, facilitated the rigorous development of analysis and many other fields. The invention of the reals also simplified the number of cases necessary in some of the arguments by such luminaries as Euclid and Archimedes. It is the thesis of this paper that the invention of a sound system that incorporates infinitesimals, though passively resisted by most mathematicians of this and the previous century, could facilitate a more nuanced understanding of calculus by a larger part of humanity.

3. Historical Perspective - the development of both approaches and the reason infinitesimals are not popular

    ​Today most mathematicians learn standard analysis first, few encounter non-standard analysis, and an even smaller number actually learn it. This is not surprising, since it took until the 1930s and 1960s respectively, for the work of logicians Kurt Go¨del (Compactness Theorem) and Abraham Robinson (Non-standard analysis, North-Holland Publishing Co., Amsterdam 1966) to put the use of infinitesimals on a firm logical foundation. With Robinson’s formalization non-standard analysis was proven sound, but the mathematical community did not readily warm to the concept. Although this formalization was not quickly accepted, it did eventually lead to attempts to teach calculus using non-standard analysis. Logicians such as Robinson and Jerome Keisler wrote calculus textbooks using infinitesimals and hyperreals, which are the basis of non-standard analysis. These texts more closely modeled the way Leibniz and others had considered the subject, and reduced the formal logical complexity of central notions in calculus, such as that of continuity, derivative, and integral (see below).

    ​Although there is evidence (see Section 12 below) that students found the approach more intuitively accessible, the pedagogical approach did not take hold. A probable cause was that practicing mathematicians were unable to embrace the approach (Keisler, personal communication, April 17, 2014).

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    ​Standard analysis is currently used to teach almost all calculus courses at both the high school and college level. In this paper I argue that infinitesimal calculus should be taught in place of a standard first-semester calculus course for non-mathematics majors. Not only is there evidence that infinitesimals accord with the intuitions of students, but they also have a long and interesting history. Highlighting aspects of the intellectual history of mathematics can furthermore strengthen the idea that mathematics is not static and that it is not apolitical.

    ​In what follows I will briefly explain the history behind infinitesimals, explain what they are and why they are intuitive, and reflect on some attempts of their integration into classrooms. The history of infinitesimals is an interesting one, and can be used by teachers to motivate lessons as well as to illustrate the usefulness of mathematics. Story telling can be a valuable tool when teaching mathematics [7]. Regardless of whether the stories are true, they capture our attention, and add to any lesson. The story that Newton’s study of gravity was prompted by an apple falling on his head is fictitious. The story of Archimedes running naked from the baths in Sicily after realizing how to decide if the king’s crown was made of pure gold may also be fictitious [14]. However, these stories help illustrate how and why mathematics was invented. They are entertaining and grab our attention.（容量有限，以下省略。請查閱原文）