Philosophy Dictionary of ArgumentsHome
| |||
|
| |||
| Infinitesimal calculus: Infinitesimals are assumed infinitely small sections of a change which can be treated with the method of infinitesimal calculus. The length of the sections, e.g. represent a side of a surface to be calculated, approaches zero and can be omitted at the end. Today, a limit of this development is assumed instead. See also differential calculus, integral calculus, infinity, completeness, movement, change._____________Annotation: The above characterizations of concepts are neither definitions nor exhausting presentations of problems related to them. Instead, they are intended to give a short introduction to the contributions below. – Dictionary of Arguments. | |||
| Author | Concept | Summary/Quotes | Sources |
|---|---|---|---|
|
Christian Thiel on Infinitesimal Calculus - Dictionary of Arguments
I 171 Infinitesimal Calculation/Prehistory/Thiel: Cavallieri (died 1647, pupil of Galilei) Teaching of the Def "Indivisibles"/Cavalieri: Such "indivisibles" are the points for distances, the intersection lines for surfaces and the intersection planes for bodies, which must not be regarded as parts of the same, since they each have one dimension less, they cannot be their building blocks! The missing dimension is only added by the direction of the movement. By their movement one can imagine the corresponding shapes (e.g. section plane body) created. The "Indivisibiles" are the precursors of the "infinitely small sizes" which Leibniz, Newton and L'Hopital each introduced in different ways. I 174 Infinitesimal Calculus: for the infinitesimal view, the difference disappears when the base becomes "infinitely small" too dx. No doubt, this is a cheating. A strict justification of such area observations by correct infinitesimal observations does not assume fictitious infinitely small bases dx, but rather a division of the entire area base into very small but finite distances Dx. I 175/176 Leibniz: a route is not extended by adding dx. He saw through this as fiction, but justified it by analogies rather than eliminating it. Cantor took credit for having finally refuted the idea of "infinitely small sizes". Of course, his considerations do not quite show what they should show. Def "infinitely small": means that the product of a quantity a with this property and a basic number n, however large, cannot become larger than any given number x. In fact, this only shows that a range in which infinitely small quantities of this kind occur cannot be a "continuous" range, but not that such ranges (today referred to as "non-Archimedean") would be contradictory. I 176/177 According to some contemporary authors, such "non-Archimedean" areas have led to the subsequent justification of the "infinitely small quantities". At the end of the 19th century, it had been regarded as progress that the "arithmetization of analysis" outlined above made it possible to understand statements about "infinitely small quantities" as statements that remain valid no matter how small one chooses their size, without being forced to assume entities that were infinitely small but nevertheless were quantities, or that become infinitely small._____________Explanation of symbols: Roman numerals indicate the source, arabic numerals indicate the page number. The corresponding books are indicated on the right hand side. ((s)…): Comment by the sender of the contribution. Translations: Dictionary of Arguments The note [Concept/Author], [Author1]Vs[Author2] or [Author]Vs[term] resp. "problem:"/"solution:", "old:"/"new:" and "thesis:" is an addition from the Dictionary of Arguments. "Vs" indicates differences in content, not necessarily the chronological progression of a controversy. If a German edition is specified, the page numbers refer to this edition. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
||
> Counter arguments against Thiel
> Counter arguments in relation to Infinitesimal Calculus
