H. G. Grassmann et l’introduction d’une nouvelle discipline mathématique : l’Ausdehnungslehre
Grassmann is not the first to create a new calculus : Möbius, Hamilton, Bellavitis, Cauchy, and many others preceded him in this way which shows all the importance of the changes undergone by the algebra and the evolution of complex connections maintained between this field and its “exact counterpar...
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| Format: | Article |
| Language: | deu |
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Éditions Kimé
2005-08-01
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| Series: | Philosophia Scientiæ |
| Online Access: | http://journals.openedition.org/philosophiascientiae/382 |
| Summary: | Grassmann is not the first to create a new calculus : Möbius, Hamilton, Bellavitis, Cauchy, and many others preceded him in this way which shows all the importance of the changes undergone by the algebra and the evolution of complex connections maintained between this field and its “exact counterpart” the Euclidean Geometry: meanwhile the first “structures” and “morphisms” are worked out, the Euclidean geometry loses its statute of “criterion of truth” and “existence” of the algebraic entities, particularly denounced by an alleged “return to rigour”.The “philosophical” introduction, denigrated by contemporaries seeing there only “false philosophy”, could not be removed with complete impunity: the “second” Ausdehnungslehre published in 1862 (A2) — howeverproposed as a new version written in accordance with the style “desired” by its time — will not be more convenient for its too few readers. Any informed reader recognizes that A2cannot be perfectly known without the necessary explanations of A1.Three observations appear significant to us :– Ausdehnungslehre is a formal science which finds its place in the mathematical system of Grassmann. This system “improves” the already traditional one (in particular known of the father of Grassmann) resulting from the crossing between two changing Contrasts, “continuous/discrete” and “distinct/equal”.– Ausdehnungslehre admits geometry as first remarkable “application”. Geometry [Geometrie] (or theory of space [Raumlehre ]), became a real science (space preexists), cannot thus legitimately appear within pure mathematics.– A part precedes the separation into the four branches of pure mathematics (or theory of forms). The general theory of forms presents their common laws, that is to say that series of truths which, in the same way refer to all branches mathematics and which thus supposes only the general concepts of equality, diversity, connection and separation. |
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| ISSN: | 1281-2463 1775-4283 |