In Hans-Joachim Petsche (ed.), From Past to Future: Graßmann's Work in Context (2010)
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.
|Keywords||Epistemology Grassmann, Hermann Geometry Vector Theory|
|Categories||categorize this paper)|
References found in this work BETA
No references found.
Citations of this work BETA
No citations found.
Similar books and articles
The Unity of Logic, Pedagogy and Foundations in Grassmann's Mathematical Work.Albert C. Lewis - 2004 - History and Philosophy of Logic 25 (1):15-36.
Extension and Measurement: A Constructivist Program From Leibniz to Grassmann.Erik C. Banks - 2013 - Studies in History and Philosophy of Science Part A 44 (1):20-31.
On the Role of Constructivism in Mathematical Epistemology.A. Quale - 2012 - Constructivist Foundations 7 (2):104-111.
The Geometrical Analysis of Grassmann and its Connection with Leibniz's Characteristic.A. E. Heath - 1917 - The Monist 27 (1):36-56.
Color Perception: From Grassmann Codes to a Dual Code for Object and Illumination Colors.Rainer Mausfeld - 1998 - In W. Backhaus, R. Kliegl & J. Werner (eds.), Color Vision. Perspectives from Different Disciplines. De Gruyter.
Kant's Philosophy of Geometry--On the Road to a Final Assessment.L. Kvasz - 2011 - Philosophia Mathematica 19 (2):139-166.
Horace's Erotic Epodes Victor Grassmann: Die erotischen Epoden des Horaz: literarischer Hintergrund und sprachliche Tradition. (Zetemata, 39.) Pp. xv+180. Munich: Beck, 1966. Paper, DM. 28. [REVIEW]R. G. M. Nisbet - 1967 - The Classical Review 17 (02):163-164.
The Problem of Extension in Natural Philosophy.Erik C. Banks - 2008 - Philosophia Naturalis 45 (2):211-235.
An Enhanced Argument for Innate Elementary Geometric Knowledge and its Philosophical Implications.Helen De Cruz - 2007 - In Bart Van Kerkhove (ed.), New perspectives on mathematical practices. Essays in philosophy and history of mathematics. World Scientific.
Added to index2011-08-03
Total downloads453 ( #4,218 of 2,152,226 )
Recent downloads (6 months)22 ( #16,353 of 2,152,226 )
How can I increase my downloads?
There are no threads in this forum
Nothing in this forum yet.