# Fearless Symmetry

Number theorists are like lotus-eaters. Having tasted this food they can never give it up.
Kronecker

There are current expository texts of wide scope that can guide one to the briefest glimpse of the true way of mathematics. B&K gifted me such a big picture math book, Fearless Symmetry, an expository text (math book largely without proofs) that attempts to explain Barry’s research field, Algebraic Number Theory (ANT).

‘Fearless Symmetry, Exposing the Hidden Patterns of Numbers’ is a book by Avner Ash and Robert Gross. The book began as an expository article to explain the proof of Fermat’s last theorem to professional mathematicians. Because of the positive reviews the article generated, it was turned into a book for a more general audience (a challenge that is hard to overestimate, even assuming any with the interest will also have enough rudimentary mathematics exposure to gain something from the exposition). My congratulations to the Fearless authors, although I am disappointed with their editors for allowing incorrect language, ‘associated to’. [Oops, pedantry alert.]

With Fearless, I hope to gain some hints regarding the meaning of ANT, and so to get in touch with what our fearless number theorist is up to. It’s a tall order, even for a casual observer with a math degree. But regurgitating (writing stuff down) aids learning. Also, by a ruthless pruning, the essence is distilled, facilitating future review. Just the facts, ma’am.

Fearless does a fast fly-by of mathematics topics. To gain a closer view, I have added, for my enlightenment, occasional supplemental material not contained in Fearless. A visitor here may well want to avoid these musings, by skipping all the paragraphs beginning with “Supplement:” to the end of that chapter.

Basic concepts are referenced without elaboration, e.g. set, cardinality, function, bijection, …; these can be looked up on Wikipedia as necessary.

I occasionally tackle a chapter in this book, and then report on what I learned, adding some background material of my own to help fill in the gaps in my working knowledge. This is the level of exposition that can jump-start exploration, an exposition that shows the inter-relatedness of the various important results.

In some chapters, Fearless includes an aside, or digression, conveying some meta-mathematical information intended to demonstrate, to the uninitiated, the way of mathematics. So as not to interrupt the flow, I have factored these out and added them to appropriate segments of the menu item About Mathematics, which provides background information to prepare the reader for communing with mathematics. These nuggets are placed without attribution – if you find ideas therein that seem enlightening, they probably came from someone else.