The paper aims to establish if Grassmann’s notion of an extensive form involved an epistemological change in the understanding of geometry and of mathematical knowledge. Firstly, it will examine if an ontological shift in geometry is determined by the vectorial representation of extended magnitudes. Giving up homogeneity, and considering geometry as an application of extension theory, Grassmann developed a different notion of a geometrical object, based on abstract constraints concerning the construction of forms rather than on the homogeneity (...) conditions required by the modern version of the theory of proportions. Secondly, Grassmann’s conception of mathematical knowledge will be investigated. Parting from the traditional definition of mathematics as a science of magnitudes, Grassmann considered mathematical forms as particulars rather than universals: the classification of the branches of mathematics was thus based on different operational rules, rather than on empirical criteria of abstraction or on the distinction of different species belonging to a common genus. It will be argued that a different notion of generalization is thus involved, and that the knowledge of mathematical forms relies on the understanding of the rules of generation of the forms themselves. Finally, the paper will analyse if Grassmann’s approach in the first edition of the Ausdehnungslehre should be explained in terms of the notion of purity of method, and if it clashes with Grassmann’s later conventionalism. Although in the second edition the features of the operations are chosen by convention, as it is the case for the anti-commutative property of the multiplication, the choice is oriented by our understanding of the resulting forms: a simplification in the algebraic calculus need not correspond to a simplification in the ‘dimensional’ interpretation of the result of the multiplicative operation. (shrink)

Hermann Grassmann's ideas on the nature and foundations of mathematics were published as an integral part of his mathematical treatise, the Ausdehnungslehre, in 1844. In spite of its notoriously obscure style we can better understand the work if we view it as an expression of the dialectical philosophy of his mentor, the theologian F. Schleiermacher. The relation to Schleiermacher is presented here through an analysis of the principal ideas of the Ausdehnungslehre.

Products of particlelike representations of the homogeneous Lorentz group are used to construct the degrees of spin angular momentum of a composite system of protons and neutrons. If a canonical labeling system is adopted for each state, a shell structure emerges. Furthermore the use of the Dirac ring ensures that the spin is characterized by half-angles in accord with the neutron-rotation experiment. It is possible to construct a Clebsch-Gordan decomposition to reduce a state of complex angular momentum into simpler states (...) which can be identified with α and β particles, multipole operators, etc. Finally, ground-state energy levels are calculated for all the even-even nuclei by using a differentiable manifold that is spin-graded and gauge-invariant by construction. It is shown that this manifold is Grassmann. (shrink)

Grassmann n’est pas le premier à créer un nouveau calcul :Möbius, Hamilton, Bellavitis, Cauchy, et bien d’autres l’ont précédé dans cette voie qui témoigne de toute l’importance des mutations subies par l’algèbre et de l’évolution des rapports complexes entretenus entre ce domaine et son « exacte contrepartie » la Géométrie euclidienne : à l’heure où s’élaborent les premières « structures » et les « morphismes », la géométrie euclidienne perd son statut de « critère de vérité » et d’« (...) existence » des entités algébriques, notamment dénoncé par un prétendu « retour à la rigueur ».L’introduction « philosophique », décriée par ses contemporains ne voyant là que « fausse philosophie », ne pouvait être impunément écartée : la « seconde » Ausdehnungslehre de 1862 (A2) — pourtant proposée telle une nouvelle version rédigée conformément au style « souhaité » par son époque — ne conviendra pas plus à ses trop rares lecteurs. Tout lecteur averti reconnaît que A2ne peut être parfaitement entendu sans les nécessaires éclaircissements de A1.Trois points nous paraissent aujourd’hui significatifs :– L’Ausdehnungslehre est une science formelle qui trouve sa place dans le système mathématique de Grassmann. Ce système « améliore » celui déjà classique (notamment connu du père de Grassmann) qui résulte du croisement entre les deux contrastes fluants « continu/discret » et « distinct/égal ».– L’Ausdehnungslehre admet la géométrie comme première « application » remarquable. La géométrie [Geometrie] (ou encore la théorie de l’espace [Raumlehre]) est devenue une science réelle (l’espace préexiste), elle ne peut donc légitimement figurer au sein de la mathématique pure.– Une partie précède la séparation en les quatre branches précédentes de la mathématique pure (ou théorie des formes). La théorie générale des formes présente leurs lois communes, soit cette série de vérités qui, de la même manière se rapportent à toutes les branches des mathématiques et qui ne supposent donc que les concepts généraux d égalité, de diversité, de liaison et de séparation. (shrink)

Grassmann n’est pas le premier à créer un nouveau calcul :Möbius, Hamilton, Bellavitis, Cauchy, et bien d’autres l’ont précédé dans cette voie qui témoigne de toute l’importance des mutations subies par l’algèbre et de l’évolution des rapports complexes entretenus entre ce domaine et son « exacte contrepartie » la Géométrie euclidienne : à l’heure où s’élaborent les premières « structures » et les « morphismes », la géométrie euclidienne perd son statut de « critère de vérité » et d’« (...) existence » des entités algébriques, notamment dénoncé par un prétendu « retour à la rigueur ».L’introduction « philosophique », décriée par ses contemporains ne voyant là que « fausse philosophie », ne pouvait être impunément écartée : la « seconde » Ausdehnungslehre de 1862 (A2) — pourtant proposée telle une nouvelle version rédigée conformément au style « souhaité » par son époque — ne conviendra pas plus à ses trop rares lecteurs. Tout lecteur averti reconnaît que A2ne peut être parfaitement entendu sans les nécessaires éclaircissements de A1.Trois points nous paraissent aujourd’hui significatifs :– L’Ausdehnungslehre est une science formelle qui trouve sa place dans le système mathématique de Grassmann. Ce système « améliore » celui déjà classique (notamment connu du père de Grassmann) qui résulte du croisement entre les deux contrastes fluants « continu/discret » et « distinct/égal ».– L’Ausdehnungslehre admet la géométrie comme première « application » remarquable. La géométrie [Geometrie] (ou encore la théorie de l’espace [Raumlehre]) est devenue une science réelle (l’espace préexiste), elle ne peut donc légitimement figurer au sein de la mathématique pure.– Une partie précède la séparation en les quatre branches précédentes de la mathématique pure (ou théorie des formes). La théorie générale des formes présente leurs lois communes, soit cette série de vérités qui, de la même manière se rapportent à toutes les branches des mathématiques et qui ne supposent donc que les concepts généraux d égalité, de diversité, de liaison et de séparation. (shrink)

Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.

Husserl’s Philosophie der Arithmetik came out in 1891. In this short essay, its place at the crossroads of two traditions in the philosophy and foundations of arithmetic is described; what preceded it and could thus have influenced Husserl. A brief look at what came later is also taken; where the choices at the crossroads led to.

Hermann Grassmann's Ausdehnungslehre of 1844 and his Lehrbuch der Arithmetik of 1861 are landmark works in mathematics; the former not only developed new mathematical fields but also both contributed to the setting of modern standards of rigor. Their very modernity, however, may obscure features of Grassmann's view of the foundations of mathematics that were not adopted since. Grassmann gave a key role to the learning of mathematics that affected his method of presentation, including his emphasis on making (...) initial assumptions explicit. In order to better understand this less well-known aspect of his work it will help to examine why some commentators have overlooked his theme of unifying logic, pedagogy and foundations, while others have recognised it. (shrink)

In einem Brief an Christiaan Huygens aus dem Jahre 1679 schlägt Leibniz eine neue von der Cartesischen Algebra verschiedene Characteristica geometrica vor. Dieser Entwurf wurde von einigen Wissenschaftshistorikern als ein Vorläufer der Analysis situs oder Topologie betrachtet. Im Jahre 1846 erhielt Graßmann einen Preis, der ausgesetzt worden war für die Konstruktion eines geometrischen Kalküls, der dem Leibnizschen ähnlich sein sollte. Im folgenden kritisierte ich die heute gemeinhin vertretene Auffassung, die Geometnsche Analyse von Graßmann sei die legitime mathematische Weiterentwicklung (...) der geometrischen Konzeptionen von Leibniz. Zunächst breite ich einige wenig bekannte historische Fakten bezüglich Leibniz und Graßmann aus und dann gehe ich dazu über, Kritik an Graßmanns Lektüre des Leibnizschen Textes zu üben. Die Frage nach dem wissenschaftlichen Wert der Gedanken, die Leibniz Huygens gegenüber äußert, möchte ich jedoch bis zur bald zu erwartenden Publikation zahlreicher Leibniz-Texte zu diesem Thema offen lassen. (shrink)

Albert Lewis's article analysing the influence of Friedrich Schleiermacher on Hermann Grassmann, stimulated many different studies on the founder of n-dimensional outer algebra.Following a brief outline of the various, sometimes diverging, analyses of Grassmann's creative thinking, new research is presented which confirms Lewis's original contribution and widens it considerably. It will be shown that:i. Grassmann, although a self-taught mathematician, was at the centre of a hitherto understated intellectual trend, which was defining for Germany. Initiated by Pestalozzi's concept (...) of elementary mathematical education and culminating in the modern mathematics of the late 19th Century, it was reflected in the contributions of Grassmann, Riemann, Jacobi and Eisenstein.ii. Hermann Grassmann, his father Justus, and his brother Robert were all demonstrably influenced by Schleiermacher's dialectic; however the two brothers responded to it in very different ways.iii. Whilst the more philosophical parts of Hermann's 1844 Extension Theory are characterised by the influence of Schleiermacher and also by the mathematical knowledge of his father, the entire development of this work is the unfolding of a single idea based on the father's interpretation of combinatorial multiplication as a ‘chemical conjunction‘, which was developed largely dialectically by Hermann. (shrink)

Hermann Grassmann’s 1861 work [2] was probably the first attempt at an axiomatic approach to arithmetic. The historical significance of this work is enormous, even though the set of axioms has proven to be incomplete. Basing on the interpretation of Grassmann’s theory provided by Hao Wang in [4], I present its detailed discussion, define the class of models of Grassmann’s arithmetic and discuss a certain axiom system for integers, modeled on Grassmann’s theory. At the end I (...) propose to modify the set of axioms of Grassmann’s arithmetic, which consists in adding an elementary sentence and removing a non-elementary one. I prove that after this modification the only model of the theory up to isomorphism is the standard model. (shrink)

0. The ancient and honorable role of philosophy as a servant to the learning, development and use of scientific knowledge, though sadly underdeveloped since Grassmann, has been re-emerging from within the particular science of mathematics due to the latter's internal need; making this relationship more explicit (as well as further investigating the reasons for the decline) will, it is hoped, help to germinate the seeds of a brighter future for philosophy as well as help to guide the much wider (...) learning of mathematics and hence of all the sciences. 1. The unity of interacting opposites "space vs. quantity", with the accompanying "general vs. particular" and the resulting division of variable quantity into the interacting opposites "extensive vs. intensive", is susceptible, with the aid of categories, functors, and natural transformations, of a formulation which is on the one hand precise enough to admit proved theorems and considerable technical development and yet is on the other hand general enough to admit incorporation of almost any specialized hypothesis. Readers armed with the mathematical definitions of basic category theory should be able to translate the discussion in this section into symbols and diagrams for calculations. 2. The role of space as an arena for quantitative "becoming" underlies the qualitative transformation of a spatial category into a homotopy category, on which extensive and intensive quantities reappear as homology and cohomology. 3. The understanding of an object in a spatial category can be approached through definite Moore-Postnikov levels; each of these levels constitutes a mathematically precise "unity and identity of opposites", and their ensemble bears features strongly reminiscent of Hegel's Science of Logic. This resemblance suggests many mathematical and philosophical problems which now seem susceptible of exact solution. (shrink)

In his 1896 lecture course on logic–reportedly a blueprint for the Prolegomena to Pure Logic –Husserl develops an explicit account of logic as an independent and purely theoretical discipline. According to Husserl, such a theory is needed for the foundations of logic (in a more general sense) to avoid psychologism in logic. The present paper shows that Husserl’s conception of logic (in a strict sense) belongs to the algebra of logic tradition. Husserl’s conception is modeled after arithmetic, and respectively logical (...) inferences are viewed as analogical to arithmetical calculation. The paper ends with an examination of Husserl’s involvement with the key characters of the algebra of logic tradition. It is concluded that Ernst Schröder, but presumably also Hermann and Robert Grassmann influenced Husserl most in his turn away from psychologism. (shrink)

This paper tackles the question of whether the order of concepts was still a relevant aspect of scientific rigour in the 19th and 20th centuries, especially in the case of authors who were deeply influenced by the Leibnizian project of a universal characteristic. Three case studies will be taken into account: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The main claim will be that the choice of primitive concepts was not only a question of convenience in modern hypothetico-deductive investigations, (...) but sometimes also the result of philosophical investigations onto the foundation of scienti­fic disciplines. The question of the "right" order of concepts is an ideal to be followed rather than a task that can be fulfilled, but remains nonetheless an essential part of the axiomatic enterprise. This paper aims to question whether there is in fact such a stark contrast, as there is often claimed to be in the literature, between the debates relating to the right order of concepts and the foundational questions concerning modern axiomatics. The scientific rupture determined by the appearance of hypothetico-deductive systems in mathema­tics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements. (shrink)

This paper tackles the question of whether the order of concepts was still a relevant aspect of scientific rigour in the 19th and 20th centuries, especially in the case of authors who were deeply influenced by the Leibnizian project of a universal characteristic. Three case studies will be taken into account: Hermann Graßmann, Giuseppe Peano and Kurt Gödel. The main claim will be that the choice of primitive concepts was not only a question of convenience in modern hypothetico-deductive investigations, (...) but sometimes also the result of philosophical investigations onto the foundation of scienti­fic disciplines. The question of the "right" order of concepts is an ideal to be followed rather than a task that can be fulfilled, but remains nonetheless an essential part of the axiomatic enterprise. This paper aims to question whether there is in fact such a stark contrast, as there is often claimed to be in the literature, between the debates relating to the right order of concepts and the foundational questions concerning modern axiomatics. The scientific rupture determined by the appearance of hypothetico-deductive systems in mathema­tics and logic should thus not be dissociated from some relevant continuities concerning the ideal of knowledge as the search for a general theory of concepts deriving from some fundamental elements. (shrink)

Hermann Grassmann is known to be the founder of modern vector and tensor calculus. Having as a theologian no formal education in mathematics at a university he got his basic ideas for this mathematical innovation at least to some extent from listening to Schleiermacher’s lectures on Dialectics and, together with his brother Robert, reading its publication in 1839. The paper shows how the idea of unity and various levels of reality first formulated in Schleiermacher’s talks about religion in 1799 (...) were transformed by him into a philosophical system in his dialectics and then were picked up by Grassmann and operationalized in his philosophical-mathematical treatise on the extension theory in 1844. (shrink)

In 1887 Helmholtz discussed the foundations of measurement in science as a last contribution to his philosophy of knowledge. This essay borrowed from earlier debates on the foundations of mathematics, on the possibility of quantitative psychology, and on the meaning of temperature measurement. Late nineteenth-century scrutinisers of the foundations of mathematics made little of Helmholtz’s essay. Yet it inspired two mathematicians with an eye on physics, and a few philosopher-physicists. The aim of the present paper is to situate Helmholtz’s contribution (...) in this complex array of nineteenth-century philosophies of number, quantity, and measurement.Author Keywords: Helmholtz; Measurement; Arithmetic; P. Du Bois-Reymond; H. and R. Grassmann; J. von Kries. (shrink)