On Algebra (1950-1974)

"Whoever thinks algebra is a trick in obtaining unknowns has thought it in vain. No attention should be paid to the fact that algebra and geometry are different in appearance. Algebras are geometric facts which are proved.” (Omar Khayyam [quoted by J.J. Winter and W. Arafat, “The Algebra of ‘Umar Khayyam’”, Journal of the Royal Asiatic Society of Bengal, Volume 41, 1950)

"The designer employing Boolean algebra is in possession of a list of theorems which may be used in simplifying the expression before him; but he may not know which ones to try first, or to which terms to apply them. He is thus forced to consider a very large number of alternative procedures in all but the most trivial cases. It is clear that a method which provides more insight into the structure of each problem is to be preferred." (Maurice Karnaugh, "The map method for synthesis of combinational logic circuits", Transactions of the American Institute of Electrical Engineers Pt 1 72 (9), 1953) 

"During the last decade the methods of algebraic topology have invaded extensively the domain of pure algebra, and initiated a number of internal revolutions. 

[...] The invasion of algebra has occurred on three fronts through the construction of cohomology theories for groups, Lie algebras, and associative algebras. The three subjects have been given independent but parallel developments. ." (Henri P Cartan & Samuel Eilenberg, "Homological Algebra", 1956)

“The word ‘imaginary’ is the great algebraical calamity, but it is too well established for mathematicians to eradicate. It should never have been used. Books on elementary algebra give a simple interpretation of imaginary numbers in terms of rotations. […] Although the interpretation by means of rotations proves nothing, it may suggest that there is no occasion for anyone to muddle himself into a state of mystic wonderment over nothing about the grossly misnamed ‘imaginaries’.” (Philip E B Jourdain, “The Nature of Mathematics” in [James R Newman, “The World of Mathematics” Vol. I, 1956])

”Behind these symbols lie the boldest, purest, coolest abstractions mankind has ever made. No schoolman speculating on essences and attributes ever approached anything like the abstractness of algebra.” (Susanne K Langer, “Philosophy in a New Key”, 1957)

"A logic machine is a device, electrical or mechanical, designed specifically for solving problems in formal logic. A logic diagram is a geometrical method for doing the same thing. […] A logic diagram is a two-dimensional geometric figure with spatial relations that are isomorphic with the structure of a logical statement. These spatial relations are usually of a topological character, which is not surprising in view of the fact that logic relations are the primitive relations underlying all deductive reasoning and topological properties are, in a sense, the most fundamental properties of spatial structures. Logic diagrams stand in the same relation to logical algebras as the graphs of curves stand in relation to their algebraic formulas; they are simply other ways of symbolizing the same basic structure." (Martin Gardner, "Logic Machines and Diagrams", 1958)

"Essentially, algebraic theories are an invariant notion of which the usual formalism with operations and equations may be regarded as 'presentation'." (F William Lawvere, "Functorial Semantics of Algebraic Theories", 1963)

"Categorical algebra has developed in recent years as an effective method of organizing parts of mathematics. Typically, this sort of organization uses notions such as that of the category G of all groups. [...] This raises the problem of finding some axiomatization of set theory - or of some foundational discipline like set theory - which will be adequate and appropriate to realizing this intent. This problem may turn out to have revolutionary implications vis-`a-vis the accepted views of the role of set theory." (Saunders Mac Lane, Categorical algebra and set-theoretic foundations, 1967)

"The history of arithmetic and algebra illustrates one of the striking and curious features of the history of mathematics. Ideas that seem remarkably simple once explained were thousands of years in the making." (Morris Kline, "Mathematics for the Nonmathematician", 1967)

“[…] algebra is the intellectual instrument which has been created for rendering clear the quantitative aspects of the world.” (Simone Weil, “The Organization of Thought”, 1974)