In order to better understand the discoveries of mathematics and aid in the assimilation of their ramifications to my conceptual vocabulary I have a need for narratives that give the big picture of each subfield in a typical mathematics curriculum. What I mean by this is that for each of the courses say in a mathematics undergraduate program, all the axioms, theorems, corollaries, lemmas, and vocabulary are brought together to create a story of what these things mean and how they relate to each other and what they portend for other areas of mathematics. What i envsion is someone (probably myself) taking a look at something like Calculus or Linear Algebra or Abstract Algebra and collecting together the key ideas captured by the vocabulary, theorems, and other matter that have been discovered and established. Reading such a narrative I can then assimilate the ramifications of the material easily. By understanding the larger picture during the actual detailed study of the material for each subfield in mathematics I can look at each of the actual theorems, corollaries, lemmas, and appreciate how they contribute to the development of the subfield. It also clearly answers questions of what is known and what is not known in this field as well as understanding any cross connections if any.
I think then a narrative of the sort that I envision would have to be arranged by the questions which have motivated research in that area of mathematics. There is a book on prime numbers titled "The Little Book of Big Prime Numbers". The author does a wonderful job of organizing the research and discoveries pertinent to Prime Numbers by questions such as "What are Prime Numbers", "How are Prime Numbers Distributed?". What really struck me and remained with me after looking through the book and looking at the last chapter was that the primary questions regarding prime numbers is in some deep sense still a mystery. Work has been done by many great minds at coming to grips with the Prime Numbers but so much still remains unanswered about them.
In my attempts to unearth my understanding of what I consider mathematics, I have discovered tht narrative is of prime importance to me. This narrative is driven by my questions such as "What is the point of Abstract Algebra? What does it aim to do? What are its goals? What does it seek to establish? What does it establish? What are its discoveries?" The answers to these questions can in part be derived from looking at the established knowledge: vocabulary, axioms, theorems, corollaries, lemmas. I suppose by diegesting and assimilating this knowledge I can accomplish the task of creating some sort of narrative.
I envision this narrative as being far more deep than vaguely pointing to the existence of structures that link senses and mathematics or more prosaically algebra and topology as I do in my earlier posts. I hope to be able to show by doing this that there are some recurrent themes, for lack of a better alternative word to "structure", that lends itself to a mathematical organization of reality. The relevance of such work to me resides in the promise of deep practical applications to my life.
