The crucial insight for making infinitesimals feasible mathematical entities was that they could still retain certain properties such as angle or slope, even if these entities were infinitely small. Nevertheless, it is still necessary to have command of it. Consequently, present-day students are not fully in command of this language. Nowadays, when teaching analysis, it is not very popular to talk about infinitesimal quantities. Following this, mathematicians developed surreal numbers, a related formalization of infinite and infinitesimal numbers that include both hyperreal cardinal and ordinal numbers, which is the largest ordered field. Infinitesimals regained popularity in the 20th century with Abraham Robinson's development of nonstandard analysis and the hyperreal numbers, which, after centuries of controversy, showed that a formal treatment of infinitesimal calculus was possible. As calculus developed further, infinitesimals were replaced by limits, which can be calculated using the standard real numbers. This definition was not rigorously formalized. Infinitesimal numbers were introduced in the development of calculus, in which the derivative was first conceived as a ratio of two infinitesimal quantities. Infinitesimals do not exist in the standard real number system, but they do exist in other number systems, such as the surreal number system and the hyperreal number system, which can be thought of as the real numbers augmented with both infinitesimal and infinite quantities the augmentations are the reciprocals of one another. The word infinitesimal comes from a 17th-century Modern Latin coinage infinitesimus, which originally referred to the " infinity- th" item in a sequence. In mathematics, an infinitesimal or infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero.
All rights reserved.Infinitesimals (ε) and infinities (ω) on the hyperreal number line (ε = 1/ω)
Moreover, the paper shows that Cohen never supported, but instead explicitly opposed, the doctrine of the centrality of the 'concept of function', with which Marburg Neo-Kantianism is usually associated.Įrnst Cassirer Hermann Cohen Inifnitesimal calculus Marburg school Neo-Kantianism Paul Natorp.Ĭopyright © 2016 Elsevier Ltd. The "puzzle of Cohen's Infinitesimalmethode," as we will call it, can be solved by looking beyond the scholarly results of the book, and instead focusing on the style of philosophy it exemplified.
By dissecting the ambiguous attitudes of the best-known representatives of the school (Paul Natorp and Ernst Cassirer), as well as those of several minor figures (August Stadler, Kurd Lasswitz, Dimitry Gawronsky, etc.), this paper shows that Das Princip der Infinitesimal-Methode is a unicum in the history of philosophy: it represents a strange case of an unsuccessful book's enduring influence. This paper offers an introduction to Hermann Cohen's Das Princip der Infinitesimal-Methode (1883), and recounts the history of its controversial reception by Cohen's early sympathizers, who would become the so-called 'Marburg school' of Neo-Kantianism, as well as the reactions it provoked outside this group.