Mathematical Milestones

by Clement Falbo © 2020

This is an expository book emphasizing the progress of mathematics and its uses.  Concentrating on the last four centuries and acknowledging worldwide historical beginnings, I want the reader to experience number theory problems of 400 BCE right on up to Artificial Intelligence problems of today.  Come with me and learn about the practical problems solved over the centuries by matrices. Marvel at the invention of non-Euclidean geometries and abstract algebras.  Sense the emotions on a rollercoaster ride from Hamilton’s jubilance when he found the missing piece for his quaternions to Frege’s agony on the receipt of Russell’s letter.

I believe the beauty and the power of mathematics shines through from the stories and applications I have written about here. I rejoice in equations, because I can, and do, explain them, and because they are the most eloquent way to express ideas in mathematics.  I establish, early, the fact that equations are sentences.  


In this book, I treat mathematics as one of the liberal arts, a branch of the humanities.  I assert that mathematics need not be obscure and unfathomable.  I use plain English to touch on some of the fascinating current mathematical mountain tops, while not forgetting to fill in the valleys.

Throughout the book, I introduce well-chosen examples and problems that the readers can work on in order to catch the fine points. It cannot be denied that modern mathematics plays an enormous role in our lives today. By using equations in a book for a broad audience, I am communicating these ideas in an unusual, if not unique way. Readers may be surprised to learn of the significant contributions to mathematics from women such as Emmy Noether and Sonya Kovalevsky.  And let us not forget the Middle East, Persia and Egypt.  In this book, I am mindful of the mathematical ideas that have come from the ancient and modern Eastern world including China and Japan.

I am addressing intelligent, curious, and, perhaps, adventurous readers who are willing to take chances at understanding difficult subjects that are interestingly written about.  For example, I am betting that such readers will dive into the discussion of additive prime number theory, where I introduce a little history and some new insights concerning the famous Goldbach Conjecture.


Clement Falbo

Clement Falbo
July, 2020

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