Elementary mathematics from an advanced standpoint

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1. rramadassFeb 2026
   Seems similar to John Stillwell's classic Elements of Mathematics: From Euclid to Gödel - https://press.princeton.edu/books/hardcover/9780691171685/el...

   It has one chapter each for Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.

   He positioned it as a sort of a modern update to Felix Klein's Elementary Mathematics from an Advanced Standpoint series of books.

   From the preface;

   This book grew from an article I wrote in 2008 for the centenary of Felix Klein’s Elementary Mathematics from an Advanced Standpoint. The article reflected on Klein’s view of elementary mathematics, which I found to be surprisingly modern, and made some comments on how his view might change in the light of today’s mathematics. With further reflection I realized that a discussion of elementary mathematics today should include not only some topics that are elementary from the twenty-first-century viewpoint, but also a more precise explanation of the term “elementary” than was possible in Klein’s day.

   So, the first goal of the book is to give a bird’s eye view of elementary mathematics and its treasures. This view will sometimes be “from an advanced standpoint,” but nevertheless as elementary as possible. Readers with a good high school training in mathematics should be able to understand most of the book, though no doubt everyone will experience some difficulties, due to the wide range of topics...

   The second goal of the book is to explain what “elementary” means, or at least to explain why certain pieces of mathematics seem to be “more elementary” than others. It might be thought that the concept of “elementary” changes continually as mathematics advances. Indeed, some topics now considered part of elementary mathematics are there because some great advance made them elementary...

   Note: "Elementary" here does not mean Easy.

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2. rramadassDec 2025
   > I kept adding new material, and a text consisting of math 'selections'

   That is what makes it so interesting to a "mathematically inclined" layman; a smorgasbord of Mathematics! More value for the time/money ;-)

   If MAA does not understand that there is huge market for Mathematics targeted towards Computer Programmers, they are just dumb. Programmers are the ones with the money and the interest in learning Mathematics presented in a manner more to their understanding.

   Please do find some other low-cost publisher to publish this; "Dover Publications" might be a good one since they publish a lot of classics particularly if you position this as a modern update to the Engel book.

   For example; John Stillwell positioned his Elements of Mathematics: From Euclid to Gödel as a sort of modern update to Felix Klein's Elementary Mathematics from an Advanced Standpoint. From the preface;

   This book grew from an article I wrote in 2008 for the centenary of Felix Klein’s Elementary Mathematics from an Advanced Standpoint. The article reflected on Klein’s view of elementary mathematics, which I found to be surprisingly modern, and made some comments on how his view might change in the light of today’s mathematics. With further reflection I realized that a discussion of elementary mathematics today should include not only some topics that are elementary from the twenty-first-century viewpoint, but also a more precise explanation of the term “elementary” than was possible in Klein’s day.

   So, the first goal of the book is to give a bird’s eye view of elementary mathematics and its treasures. This view will sometimes be “from an advanced standpoint,” but nevertheless as elementary as possible. Readers with a good high school training in mathematics should be able to understand most of the book, though no doubt everyone will experience some difficulties, due to the wide range of topics...

   The second goal of the book is to explain what “elementary” means, or at least to explain why certain pieces of mathematics seem to be “more elementary” than others. It might be thought that the concept of “elementary” changes continually as mathematics advances. Indeed, some topics now considered part of elementary mathematics are there because some great advance made them elementary...

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3. rramadassOct 2025
   There is no one book which can give you the overall sweep of Mathematics. However, you might find the following (though not textbooks per se) useful.

   1) The Princeton Companion to Mathematics by Timothy Gowers et al. and The Princeton Companion to Applied Mathematics by Nicholas Higham et al. - The closest you have to a Modern Encyclopedia of Mathematics. You get unmatched breadth after which you can move on to dedicated books as needed. Well worth the money.

   2) Mathematics: Its Content, Methods and Meaning by Aleksandrov, Kolmogorov et al. - Absolutely brilliant overview of Basic Mathematics. Published by Dover and hence very affordable.

   3) Elements of Mathematics: From Euclid to Gödel by John Stillwell - Written as sort of an update to the great Felix Klein's Elementary Mathematics from an Advanced Standpoint books. The Topics are particularly well chosen given modern advances; they include Arithmetic, Computation, Algebra, Geometry, Calculus, Combinatorics, Probability, Logic.

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4. jasomillMay 2024
   Nice. After five minutes: 95.6% with a mouse; 96.2% with a Wacom tablet; trackpad (Apple Magic 2), hopeless.

   As for opportunistically relevant posting: see the first chapter of Felix Klein's Elementary Mathematics from an Advanced Standpoint: Geometry[1] for a bit on the geometric theory behind the planimeter.

   [1] https://archive.org/details/elementary_mathematics_geometry/

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