Piece Of Mathematics Research Articles

Sophie Germain: or Was Gauss a feminist?

This article is focussed on a letter that perhaps the greatest mathematician ever, Carl Frederich Gauss, sent to a female mathematician, Sophie Germain. The letter is interesting because it is the first that Gauss wrote to Germain in the knowledge that she was a woman, and it contains a piece of mathematics that reveals better than the text of the letter what Gauss really thought of Germain. To put the letter in context I shall give some of Germain’s biography and also briefly describe female mathematicians that pre-date Germain.

Gertrude Stein, the Cone Sisters, and the Puzzle of Female Friendship

For ten years, between 1903 and 1913, Gertrude Stein saw human relationships as painful mathematical puzzles in need of solutions. Again and again, she converted the predicaments of her personal life into literary material, the better to solve and to exorcise them. The revelation that relationships had a structural quality came to her during the composition of Q.E.D. (1903), when she grasped the almost mathematical nature of her characters' emotional impasse. Stein's persona in the novel comments on their triangular affair, Why it's like a piece of mathematics. Suddenly it does itself and you begin to see.' The theory encouraged her to examine such situations as if they were case histories: she continued to study the same piece of mathematics from different angles in Fernhurst (1904), Three Lives (1905-6), and The Making of Americans (1906-11). But whatever the sexual arrangements in these triangles, the powerful generally managed to impose their wills upon the less powerful, and the triangles resolved themselves into oppositional structures, pitting two against one. Gradually, when the couple began to replace the triangle as her structural model, Stein composed numerous verbal portraits of couples and their relationships. In two of these, Ada and Two Women, Stein applied her general theory of relationships to the particular puzzle of female friendships because, I think, she felt that women's characters were most intensely molded in same-sex involvements. Although she attempted to prove these theories in distanced, de-

Die Theorie des Regenbogens als Beispiel für beziehungshaltige Analysis im Oberstufenunterricht

Every student should come to know the force of mathematical methods in tackling problems belonging to quite different fields of knowledge. Here, I propose the study of the theory of the rainbow as such a piece of mathematics fraught with relations. The student tests the power of the hitherto developed mathematical methods and simultaneously practises the transfer from the problem to the mathematical modell and the translation back to the starting problem. It seems important that students who do not specialice in mathematics learn this aspect of mathematical thinking.

Notes on the civil applications of mathematics

Mathematics derives its meaning from its civil applications. It enters civil-life as a response to requests for conceptual control in the domination of nature. It is incorporated in a technical cognitive interest. An interesting question arises as to whether mathematics can be related to knowledge-constitutive interests concerned with emancipation. Some of the dangers of a technical-context mathematics entering emancipatory language-games are pointed out. Atkin's contributions to mathematical sociology are examined critically. He offers a piece of mathematics suitable for modelling deprivations as objects of reflection in the social life-world. The way this can enter emancipatory language-games presupposes a consensus theory of truth and a theory of communicative competence.

Economic life of incandescent lamps for domestic lighting

The length of life of incandescent lamps has been fixed by manufacturers at too low a level as a result of a widely accepted piece of mathematics which, this paper argues, is erroneous. On amending the mathematics it is shown that the length of life should be fixed at about 50% above present values; it is also suggested that, if other elements in the calculation were amended, a further lengthening would be required.

Function boxes: a model for differentiation

The paper begins with a description of a diagrammatical model which I used when lecturing on elementary calculus to a class of engineers. The second part suggests how the use of the model might be extended to provide a framework on which one could build up the theory of differentiation of functions of one or more real variables. For the purposes of this article it is assumed that the reader is familiar with this theory—the paper, therefore, merely gives the outline of a possible approach. The keynote of the lecture course to the engineers was ‘plausibility’ rather than ‘rigour’ and my use of ‘function boxes’ was sparked off by the fact that the class were using flow diagrams and ‘black boxes’ in a number of other courses—most noticeably in that given by the person who lectured immediately before me and who, in his progress up the technological scale to the electronic computer, had clearly not encountered that mundane educational aid, the blackboard duster. The grand finale of the one-term course was to be an introduction to partial differentiation and, in particular, the chain rule—a piece of mathematics which students often find difficult and which I hoped to make easier by means of an approach using a simple type of ‘flow’ diagram.

An Application of Base-Three Numeration

From a pedagogical point of view, it is desirable to be able to show a subtle relationship between two pieces of mat hematics that are seemingly unrelated. When such a relationship can be shown at a rather elementary level and when the “marriage” helps to analyze still another interesting question, the potential benefits to the teacher are great. Such a situation will be described in this paper. It involves the use of the base-three positional numeration system in analyzing some topics from function theory.

Abstract Algebra From Axiomatic Geometry

This article is entirely expository. It is devoted to the derivation of an abstract algebraic structure from a certain projective geometry. Though this piece of mathematics is already well known to the experts, it is so remarkable and so beautiful that it deserves to he much better known than it is to schoolteachers and to those whose business it is to modernize school mathematics.

Teaching engineering students how to use mathematics

Mathematics teaching for engineers must include specific instruction on applications, although this must not obscure the logical presentation of the mathematics. Examples are given of simple physical apparatus to illustrate various pieces of mathematics.

Experimental projects and research: The Greater Cleveland Mathematics Program

The G.C.M.P. attempts to lead its students to t he challenging and exciting phases of mathematics-in-the-making by having them rediscover the fundamental patterns and by having them re-create the basic symbolism, thus making the learning of mathematics to a large extent selfmotivating. Problem situations are presented to students as if they had not been explored already by the great geniuses of the past and present. These situations are presented in such a manner that pattern discovery has a good chance of taking place almost spontaneously; then students are guided to a point where the established symbolism for the rediscovered pattern seems almost a necesity. It is hoped that students will come away with the feeling that they, for themselves, have seen the relations, invented the symbols, put them together in meaningful sentences, and, in general, have had a genuine part in making this or that piece of mathematics.

The combinatory foundations of mathematical logic

In investigations of the foundations of mathematics we can distinguish two separate tendencies. On the one hand, one may seek to define his subject with greatest possible explicitness: to obtain a formulation which satisfies the most exacting demands for precision, and which is at the same time free from paradoxes and adequate for the purpose. On the other hand, besides the problem of formulation, there is that of simplification; one can seek to find systems based upon processes of greater and greater primitiveness. The reduction of a piece of mathematics to a formal system, and still further to a completely formalized system (as explained, for example, in my New York address a year ago), is a step toward the first of these objectives. But it is evident that one can formalize in various ways, and that some of these ways constitute a more profound analysis than others. Although from some points of view one way of formalization is as good as any other, yet a certain interest attaches to the problem of simplification, as is shown by the attention which some of the greatest mathematicians have devoted to it.The researches about which I am reporting today are directed toward the second of these objectives. In fact we are concerned with constructing systems of an extremely rudimentary character, which analyze processes ordinarily taken for granted. This is properly part of the business of mathematical logic. Of course there are those, even among logicians, who doubt the utility of this sort of thing—who profess to have no interest in improvements which do not lead to increases in deductive power or what not. However that may be, this second objective certainly has some interest in its own right; and it is this interest which has formed the primary motivation for these researches.

Quaternion Note on a Geometrical Problem

The problem referred to is that of inscribing in a sphere a closed n-sided polygon, whose sides shall pass respectively through n given points which are not on the surface. Hamilton evidently regarded his solution of this question as a very tough piece of mathematics (see his Life, vol. iii. pp. 88, 426). In preparing a new edition of my Quaternions, I was led to a mode of treating this question which enables us to dispense with the brilliant feats of analysis which seem to be required in Hamilton's method.