On Group Theory (1950-1974)

"The notion of an abstract group arises by consideration of the formal properties of one-to-one transformations of a set onto itself. Similarly, the notion of a category is obtained from the formal properties of the class of all transformations y : X → Y of any one set into another, or of continuous transformations of one topological space into another, or of homomorphisms, of one group into another, and so on." (Saunders Mac Lane, "Duality for groups", Bulletin of the American Mathematical Society 56, 1950)

"Wherever groups disclosed themselves, or could be introduced, simplicity crystallized out of comparative chaos." (Eric T Bell, "Mathematics, Queen and Servant of Science", 1951)

"The trouble with group theory is that it leaves so much unexplained that one would like to explain. It isolates in a beautiful way those aspects of nature that can be understood in terms of abstract symmetry alone. It does not offer much hope of explaining the messier facts of life, the numerical values of particle lifetimes and interaction strengths - the great bulk of quantitative experimental data that is now waiting for explanation. The process of abstraction seems to have been too drastic, so that many essential and concrete features of the real world have been left out of consideration. Altogether group theory succeeds just because its aims are modest. It does not try to explain everything, and it does not seem likely that it will grow into a complete or comprehensive theory of the physical world." (Freeman J Dyson, "Mathematics in the Physical Sciences", Scientific American Vol. 211 (3), 1964)

"Though determinants and matrices received a great deal of attention in the nineteenth century and thousands of papers were written on these subjects, they do not constitute great innovations in mathematics. [...] Neither determinants nor matrices have influenced deeply the course of mathematics despite their utility as compact expressions and despite the suggestiveness of matrices as concrete groups for the discernment of general theorems of group theory." (Morris Kline, "Mathematical Thought From Ancient to Modern Times", 1972) 

"If indeed one tries to clarify the notion of equality, which is introduced right at the beginning of Geometry, one is led to say that two figures are equal when one can go from one to the other by a specific geometric operation, called a motion. This is only a change of words; but the axiom according to which two figures equal to a third are equal to one another, subjects those operations called motions to a certain law; that is, that an operation which is the result of two successive motions is itself a motion. It is this law that mathematicians express by saying that motions form a group. Elementary Geometry can then be defined by the study of properties of figures which do not change under the operations of the group of motions." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)

"[…] it is the whole logical structure of elementary Geometry which is contained in the group of motions and even, in a more precise manner, in the law according to which operations of that group compose with each other, independently of the nature of the objects on which these operations act. This law constitutes what we call the group structure." (Élie Cartan, "Notice sur les travaux scientifiques", 1974)