莱布尼兹懂得超实数吗?
古希腊阿基米德教导人们,世界上不存在无穷小量,后人称其为“阿基米德原理”。
到了十七世纪,牛顿与莱布尼兹抛弃阿基米德原理,利用无穷小概念创立; 微积分学,后来,欧拉与高斯继承无穷小微积分的理论传统。
但是,无穷小概念受到诸多批评。柯西与魏尔斯特拉斯利用极限的(ε, δ)
定义给牛顿、莱布尼兹的无穷小理论打“补丁”,保无穷小微积分“过关”。
袁萌
附件:
From Leibniz to Robinson
When Newton and (more explicitly) Leibniz introduced differentials, they used infinitesimals and these were still regarded as useful by later mathematicians such as Euler and Cauchy. Nonetheless these concepts were from the beginning seen as suspect, notably by George Berkeley. Berkeley's criticism centered on a perceived shift in hypothesis in the definition of the derivative in terms of infinitesimals (or fluxions), where dx is assumed to be nonzero at the beginning of the calculation, and to vanish at its conclusion (see Ghosts of departed quantities for details). When in the 1800s calculus was put on a firm footing through the development of the (ε, δ)-definition of limit by Bolzano, Cauchy, Weierstrass, and others, infinitesimals were
largely abandoned, though research in non-Archimedean fields continued (Ehrlich 2006).
However, in the 1960s Abraham Robinson showed how infinitely large and infinitesimal numbers can be rigorously defined and used to develop the field of non-standard analysis.[6] Robinson developed his theory nonconstructively, using model theory; however it is possible to proceed using only algebra and topology, and proving the transfer principle as a consequence of the definitions. In other words hyperreal numbers per se, aside from their use in nonstandard analysis, have no necessary relationship to model theory or first order logic, although they were discovered by the application of model theoretic techniques from logic.