Machiavelli spoke about the return to fundamentals as a key step in keeping strength of an ideology. He was referring to religions with examples chosen from Ancient Greece and Rome. However, he is a great genius and it is worthwhile generalizing his advice to a large class of religion-like ideologies with the interesting case being Algebra. Now within geek circles religion-like ideologies grow up naturally. Within physics and mathematics, there have been geek prophets and geek seers and geek Christs and Buddhas all within the microcosm of mathematicians. This is a sort of self-similarity of the social networks that form in any ideology. While Algebra is named after an Arab for good reasons, it was seventeenth through nineteenth century Europe that has shaped the actual dimensions of the modern mathematical discipline of algebra, and today no one wants to produce divisions between algebra and topology as the merge had been so fruitful that the basic objects in geometry are principal bundles with fiber the most beautiful union of algebra and analysis: Lie groups. All this does not remove the fact that there are religions beneath all intellectual traditions, something that Greeks wrote consciously with their long list of ‘isms’ and devotees of specific philosophical schools. Algebra today begins with group theory. Groups were first defined abstractly in the mid-nineteenth century by Arthur Cayley by explicitly looking for underlying structure of a disparate set of problems in mathematics — Galois theory where permutations come about and some others. Now if Machiavelli is right, the ‘fundamentals’ of the religion of algebra say begat by Cayley are the examples from which he derived his notion of group; Sylvester defined ‘matrix’ for the first time during the same time period, although contrary to how our children learn about these things, determinants for solving linear systems was known before matrices were defined. In my youth, Paul Halmos’ book treating matrices as equivalent to linear transformation of a vector space already seemed a bit bizarre at first, having been beaten with calculus and analysis before. My first real math book, given as a gift to me by my chemistry teacher and my Savior in the sense of ensuring my entry to Princeton by enormous hopes and efforts, Mrs. Elizabeth Waltien, was Raoul Bott and Tu’s Differential Forms in Algebraic Topology which for a young man seemed too bizarre to be mathematics. Of course these childish reactions to standard machinery in modern mathematics today makes people smile.

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