莱布尼兹是微积分奠基人吗？

莱布尼兹是微积分奠基人吗？

  本文附件是“History of calculus”中的一段文字，极其清楚地表明：莱布尼兹是微积分奠基人。当之无愧。但是，这个断语与我国现行高等数学教学大纲是相互冲突的。为什么？

  简而言之，因为，莱布尼兹创立微积分的奠基石是非阿基米德无穷小概念，而《高等数学大纲》规定微积分基础却是阿基米德有序域，完全拒绝无穷小。换句话说，莱布尼兹不是微积分奠基人。

  《数学大纲》错了，培养大批数学“小糊涂”有何用？

注：本文附件最后的一句话是：  Abraham Robinson （鲁宾逊）showed（证明） that using infinitesimal quantities in calculus could be given a solid （坚固的基础）foundation.

  妙也！

袁萌   陈启清 9月7日

附件：

Leibniz

Leibniz: Nova methodus

 pro maximis et minimis, Acta Eruditorum, Leipzig, October 1684. First page of Leibniz' publication of the differential calculus.

 

 

Graphs referenced in Leibniz' article of 1684.

While Newton began development of his fluxional calculus in 1665–1666 his findings did not become widely circulated until later. In the intervening years Leibniz also strove to create his calculus. In comparison to Newton who came to math at an early age, Leibniz began his rigorous math studies with a mature intellect. He was a polymath, and his intellectual interests and achievements involved metaphysics, law, economics, politics, logic, and mathematics.   

In order to understand Leibniz’s reasoning in calculus his background should be kept in mind. Particularly, his metaphysics which described the universe as a Monadology, and his plans of creating a precise formal logic whereby, "a general method in which all truths of the reason would be reduced to a kind of calculation."

In 1672 Leibniz met the mathematician Huygens who convinced Leibniz to dedicate significant time to the study of mathematics. By 1673 he had progressed to reading Pascal’s Traité des Sinus du Quarte Cercle and it was during his largely autodidactic research that Leibniz said "a light turned on". Like Newton, Leibniz, saw the tangent as a ratio but declared it as simply the ratio between ordinates and abscissas. He continued this reasoning to argue that the integral was in fact the sum of the ordinates for infinitesimal intervals in the abscissa; in effect, the sum of an infinite number of rectangles. From these definitions the inverse relationship or differential became clear and Leibniz quickly realized the potential to form a whole new system of mathematics. Where Newton over the course of his career used several approaches in addition to an approach using infinitesimals, Leibniz made this the cornerstone of his notation and calculus.

In the manuscripts of 25 October to 11 November 1675, Leibniz recorded his discoveries and experiments with various forms of notation. He was acutely aware of the notational terms used and his earlier plans to form a precise logical symbolism became evident. Eventually, Leibniz denoted the infinitesimal increments of abscissas and ordinates dx and dy, and the summation of infinitely many infinitesimally thin rectangles as a long s (∫ ), which became the present integral symbol

∫

While Leibniz's notation is used by modern mathematics, his logical base was different from our current one. Leibniz embraced infinitesimals and wrote extensively so as, “not to make of the infinitely small a mystery, as had Pascal.”[26] According to Gilles Deleuze, Leibniz's zeroes "are nothings, but they are not absolute nothings, they are nothings respectively" (quoting Leibniz' text "Justification of the calculus of infinitesimals by the calculus of ordinary algebra").[27] Alternatively, he defines them as, “less than any given quantity.” For Leibniz, the world was an aggregate of infinitesimal points and the lack of scientific proof for their existence did not trouble him. Infinitesimals to Leibniz were ideal quantities of a different type from appreciable numbers. The truth of continuity was proven by existence itself. For Leibniz the principle of continuity and thus the validity of his calculus was assured. Three hundred years after Leibniz's work, Abraham Robinson showed that using infinitesimal quantities in calculus could be given a solid foundation.