On Algebra (1900-1924)

"A knowledge of statistics is like a knowledge of foreign languages or of algebra; it may prove of use at any time under any circumstances." (Sir Arthur L Bowley, "Elements of Statistics", 1901)

"All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions." (Mary E Boole, "Lectures on the Logic of Arithmetic", 1903)

"But in the mathematical or pure sciences, - geometry, arithmetic, algebra, trigonometry, the calculus of variations or of curves, - we know at least that there is not, and cannot be, error in our first principles, and we may therefore fasten our whole attention upon the processes. As mere exercises in logic, therefore, these sciences, based as they all are on primary truths relating to space and number, have always been supposed to furnish the most exact discipline." (Joshua Fitch,"Lectures on Teaching", 1906)

"Today it is no longer questioned that the principles of the analysts are the more far-reaching. Indeed, the synthesists lack two things in order to engage in a general theory of algebraic configurations: these are on the one hand a definition of imaginary elements, on the other an interpretation of general algebraic concepts. Both of these have subsequently been developed in synthetic form, but to do this the essential principle of synthetic geometry had to be set aside. This principle which manifests itself so brilliantly in the theory of linear forms and the forms of the second degree, is the possibility of immediate proof by means of visualized constructions." (Felix Klein, "Riemannsche Flächen", 1906)

"In the beginning of algebra, even the most intelligent child finds, as a rule, very great difficulty. The use of letters is a mystery, which seems to have no purpose except mystification. It is almost impossible, at first, not to think that every letter stands for some particular number, if only the teacher would reveal what number it stands for." (Bertrand Russell, "Mysticism and Logic: And Other Essays", 1910)

"Geometry formerly was the chief borrower from arithmetic and algebra, but it has since repaid its obligation with abundant usury; and if I were asked to name, in one word, the pole-star round which the mathematical firmament revolves, the central idea which pervades as a hidden spirit the whole corpus of mathematical doctrine, I should point to Continuity as contained in our notions of space, and say, it is this, it is this!" (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"Time was when all the parts of the subject were dissevered, when algebra, geometry, and arithmetic either lived apart or kept up cold relations of acquaintance confined to occasional calls upon one another; but that is now at an end; they are drawn together and are constantly becoming more and more intimately related and connected by a thousand fresh ties, and we may confidently look forward to a time when they shall form but one body with one soul." (James J. Sylvester, Presidential Address to the British Association, [The Collected Mathematical Papers of James Joseph Sylvester Vol. 2, cca. 1904–1912])

"The ends to be attained [in mathematical teaching] are the knowledge of a body of geometrical truths to be used. In the discovery of new truths, the power to draw correct inferences from given premises, the power to use algebraic processes as a means of finding results in practical problems, and the awakening of interest In the science of mathematics." (J Craig, "A Course of Study for the Preparation of Rural School Teachers", 1912)

"This diagrammatic method has, however, serious inconveniences as a method for solving logical problems. It does not show how the data are exhibited by cancelling certain constituents, nor does it show how to combine the remaining constituents so as to obtain the consequences sought. In short, it serves only to exhibit one single step in the argument, namely the equation of the problem; it dispenses neither with the previous steps, i.e., 'throwing of the problem into an equation' and the transformation of the premises, nor with the subsequent steps, i.e., the combinations that lead to the various consequences. Hence it is of very little use, inasmuch as the constituents can be represented by algebraic symbols quite as well as by plane regions, and are much easier to deal with in this form." (Louis Couturat, "The Algebra of Logic", 1914)