A general outline of the genesis of vector space theory (original) (raw)

The role of formalism in the teaching of the theory of vector spaces

Linear Algebra and its Applications, 1998

In the French tradition of Bourbaki, the theory of vector spaces is usually presented in a very formal setting, which causes severe difficulties to many students. The aim of this paper is to analyze the underlying reasons of these difficulties and to suggest some ways to make the first teaching of the theory of vector spaces less ineffective for many students. We do not reject the necessity for formalism. On the contrary, on the basis of a historical analysis we can explain the specific meaning it has in the theory. From this mathematical analysis with a historical perspective, we analyze the teaching and the apprehension of vector space theory in a new approach. For instance, we will show that mistakes made by many students can be interpreted as a result of a lack of connection between the new formal concepts and their conceptions previously acquired in more restricted, but more intuitively based areas. Our conclusions will not plead for avoiding formalism but for a better positioning of the formal concepts with regard to previous knowledge of the students as well as special care to be given in making the role and the meaning of formalism in linear algebra explicit to the students. 0 1998 Elsevier Science Inc. All rights reserved.

Towards a history of the geometric foundations of mathematics

Revue de Synthèse, 2003

Enriques, et autres-ont joué un rôle important dans la discussion sur les fondements des mathématiques. Mais, contrairement aux idées d'Euclide, ils n'ont pas identifié « l'espace physique » avec « l'espace de nos sens ». Partant de notre expérience dans l'espace, ils ont cherché à identifier les propriétés les plus importantes de l'espace et les ont posées à la base de la géométrie. C'est sur la connaissance active de l'espace que les axiomes de la géométrie ont été élaborés ; ils ne pouvaient donc pas être a priori comme ils le sont dans la philosophie kantienne. En outre, pendant la dernière décade du siècle, certains mathématiciens italiens-De Paolis, Gino Fano, Pieri, et autres-ont fondé le concept de nombre sur la géométrie, en employant des résultats de la géométrie projective. Ainsi, on fondait l'arithmétique sur la géométrie et non l'inverse, comme David Hilbert a cherché à faire quelques années après, sans succès.

A Study on the Linear Algebra & Matrix in Mathematics

isara solutions, 2019

In this we are presenting a study on the linear algebra and matrix in mathematics. Linear algebra is the branch of mathematics concerned with the study of vectors, vector spaces (also called linear spaces), linear maps (also called linear transformations), and systems of linear equations. Vector spaces are a central theme in modern mathematics; thus, linear algebra is widely used in both abstract algebra and functional analysis. Linear algebra also has a concrete representation in analytic geometry and it is generalized in operator theory. It has extensive applications in the natural sciences and the social sciences, since nonlinear models can often be approximated by linear ones.

Notes on Linear Algebra

Some basic ideas from linear algebra play an essential role in our subject, so I'm o¤ering this summary as a reference. These notes won't substitute for a proper textbook. In particular, I won't give proofs of anything here. Nevertheless, I hope they will be of some use. I may add to these notes later the semester. (I won't add to them earlier).

On the Mathematical Representation of Spacetime

This essay is a contribution to the historical phenomenology of science, taking as its point of departure husserl's later philosophy of science and Jacob klein's seminal work on the emergence of the symbolic conception of number in european mathematics during the late sixteenth and seventeenth centuries. since neither husserl nor klein applied their ideas to actual theories of modern mathematical physics, this essay attempts to do so through a case study of the concept of "spacetime." in §1, i sketch klein's account of the emergence of the symbolic conception of number, beginning with Vieta in the late sixteenth century. in §2, through a series of historical illustrations, i show how the principal impediment to assimilating the new symbolic algebra to mathematical physics, namely, the dimensionless character of symbolic number, is overcome via the translation of the traditional language of ratio and proportion into the symbolic language of equations. in § §3-4, i critically examine the concept of "minkowski spacetime," specifically, the purported analogy between the Pythagorean distance formula and the minkowski "spacetime interval." finally, in §5, i address the question of whether the concept of minkowski spacetime is, as generally assumed, indispensable to einstein's general theory of relativity.