Chapter 13: The Rise of Symmetry: How not to solve an equation
“But by 1900 formulas and transformations were viewed as things, not processes, and the objects of algebra were much more abstract and far more general.”
Group Theory: Henri Poincaré (1900), Évariste Galois (algebraic equations, quintic, system of permutations as a group. Is one of the most tragic figures in the history of mathematics.), Camille Jordan (1870, Traité de Substitutions et des Équations Algébriques, closed groups)
“If mathematicians had taken the easy route, and assumed the solution to be impossible, mathematics and science would have been a pale shadow of what they are today. That is why mathematicians insist on proofs.”
Group Theory: Henri Poincaré (1900), Évariste Galois (algebraic equations, quintic, system of permutations as a group. Is one of the most tragic figures in the history of mathematics.), Camille Jordan (1870, Traité de Substitutions et des Équations Algébriques, closed groups)
“If mathematicians had taken the easy route, and assumed the solution to be impossible, mathematics and science would have been a pale shadow of what they are today. That is why mathematicians insist on proofs.”
Chapter 14: Algebra Comes of Age: Numbers give way to structures
“By 1860 the theory of permutation groups was well developed. The theory of invariants – algebraic expressions that do not change when certain changes of variables are performed – had drawn attention to various infinite sets of transformations… in 1868 Camille Jordan had studied groups of motions in three-dimensional space, and the two strands began to merge.”
This chapter is about more sophisticated concepts such as permutations, transformations, matrices, rings and fields, Lie groups and Lie algebras.
Important Figures: Sophus Lie, Felix Klein, Wilhelm Killing.
An interesting thing about this chapter is how the development of all these concepts (rings and fields) allowed Andrew Wiles to solve Fermat’s Last Theorem, also with the contributions of Yutaka Taniyama (Taniyama-Weil conjecture), Gerhard Frey (link between the theorem and elliptic curves).
“But the issue is no longer whether abstraction is useful or necessary: abstract methods have proved their worth by making it possible to solve numerous long-standing problems, such as Fermat’s Last Theorem. And what seemed little more than formal game-playing yesterday may turn out to be a vital scientific or commercial tool tomorrow.” (page 264)
This chapter is about more sophisticated concepts such as permutations, transformations, matrices, rings and fields, Lie groups and Lie algebras.
Important Figures: Sophus Lie, Felix Klein, Wilhelm Killing.
An interesting thing about this chapter is how the development of all these concepts (rings and fields) allowed Andrew Wiles to solve Fermat’s Last Theorem, also with the contributions of Yutaka Taniyama (Taniyama-Weil conjecture), Gerhard Frey (link between the theorem and elliptic curves).
“But the issue is no longer whether abstraction is useful or necessary: abstract methods have proved their worth by making it possible to solve numerous long-standing problems, such as Fermat’s Last Theorem. And what seemed little more than formal game-playing yesterday may turn out to be a vital scientific or commercial tool tomorrow.” (page 264)