On Algebra (1925-1949)

"What had already been done for music by the end of the eighteenth century has at last been begun for the pictorial arts. Mathematics and physics furnished the means in the form of rules to be followed and to be broken. In the beginning it is wholesome to be concerned with the functions and to disregard the finished form. Studies in algebra, in geometry, in mechanics characterize teaching directed towards the essential and the functional, in contrast to apparent. One learns to look behind the façade, to grasp the root of things. One learns to recognize the undercurrents, the antecedents of the visible. One learns to dig down, to uncover, to find the cause, to analyze." (Paul Klee, "Bauhaus prospectus", 1929)

"The word is of Arabic origin. ‘Al’ is the Arabic article the, and ‘gebar’ is the verb to set, to restitute." (Tobias Dantzig & Joseph Mazur, “Number: The Language of Science”, 1930)

 “By the help of God and with His precious assistance I say that algebra is a scientific art. The objects with which it deals are absolute numbers and (geometrical) magnitudes which, though themselves unknown, are related to things which are known, whereby the determination of the unknown quantities is possible. Such a thing is either a quantity or a unique relation, which is only determined by careful examination. […] What one searches for in the algebraic art are the relations which lead from the known to the unknown, to discover which is the object of algebra as stated above.” (Omar Khayyam [quoted by Daoud Suleiman Kasir in “The Algebra of Omar Khayyam”, 1931)

"The steady progress of physics requires for its theoretical formulation a mathematics which get continually more advanced. […] it was expected that mathematics would get more and more complicated, but would rest on a permanent basis of axioms and definitions, while actually the modern physical developments have required a mathematics that continually shifts its foundation and gets more abstract. Non-Euclidean geometry and noncommutative algebra, which were at one time were considered to be purely fictions of the mind and pastimes of logical thinkers, have now been found to be very necessary for the description of general facts of the physical world. It seems likely that this process of increasing abstraction will continue in the future and the advance in physics is to be associated with continual modification and generalisation of the axioms at the base of mathematics rather than with a logical development of any one mathematical scheme on a fixed foundation." (Paul A M Dirac, "Quantities singularities in the electromagnetic field", Proceedings of the Royal Society of London, 1931)

"The theory of probability as a mathematical discipline can and should be developed from axioms in exactly the same way as geometry and algebra." (Andrey Kolmogorov, "Foundations of the Theory of Probability", 1933)

“Every moment of time dictated and determined the following moment, and was itself dictated and determined by the moment that came before it. Everything was calculable: everything happened because it must; the commandments were erased from the tables of the law; and in their place came the cosmic algebra: the equations of the mathematicians.” (George Bernard Shaw, “Too True to Be Good”, 1934)

“Algebra is applied to the clouds, the irradiation of the planet benefits the rose, and no thinker would dare to say that the perfume of the hawthorn is useless to the constellation.” (Victor Hugo, “Les Miserables”, 1938)

"Algebra tends to the study of the explicit structure of postulationally defined systems closed with respect to one or more rational operations." (George D Birkhoff, "Some Recent Advances in Algebra", The American Mathematical Monthly Vol. 46, 1939)

"The invariant character of a mathematical discipline can be formulated in these terms. Thus, in group theory all the basic constructions can be regarded as the definitions of co- or contravariant functors, so we may formulate the dictum: The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

"The subject of group theory is essentially the study of those constructions of groups which behave in a covariant or contravariant manner under induced homomorphisms. More precisely, group theory studies functors defined on well specified categories of groups, with values in another such category." (Samuel Eilenberg & Saunders Mac Lane, "A general theory of natural equivalences", Transactions of the American Mathematical Society 58, 1945)

“Algebra reverses the relative importance of the factors in ordinary language. It is essentially a written language, and it endeavors to exemplify in its written structures the patterns which it is its purpose to convey. The pattern of the marks on paper is a particular instance of the pattern to be conveyed to thought. The algebraic method is our best approach to the expression of necessity, by reason of its reduction of accident to the ghost-like character of the real variable.” (Alfred N Whitehead, “Essays in Science and Philosophy”, 1948