Name: Elements of Mathematics
ISBN: 9780691178547

Elements of Mathematics

Elements of Mathematics takes readers on a fascinating tour that begins in elementary mathematics but, as John Stillwell shows, this subject is not as elementary or straightforward as one might think. Not all topics that are part of today's elementary mathematics were always considered as such, and great mathematical advances and discoveries had to occur in order for certain subjects to become "elementary." Stillwell examines elementary mathematics from a distinctive twenty-first-century viewpoint and describes not only the beauty and scope of the discipline, but also its limits. From Gaussian integers to propositional logic, Stillwell delves into arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, and logic. He discusses how each area ties into more advanced topics to build mathematics as a whole. Through a rich collection of basic principles, vivid examples, and interesting problems, Stillwell demonstrates that elementary mathematics becomes advanced with the intervention of infinity. Infinity has been observed throughout mathematical history, but the recent development of "reverse mathematics" confirms that infinity is essential for proving well-known theorems, and helps to determine the nature, contours, and borders of elementary mathematics. Elements of Mathematics gives readers, from high school students to professional mathematicians, the highlights of elementary mathematics and glimpses of the parts of math beyond its boundaries.

What Hacker News says

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.

thread

rramadassFeb 2026

You might find the couple of books that i mention in my other comments here useful;

Concepts of Modern Mathematics by Ian Stewart - https://store.doverpublications.com/products/9780486284248

Elements of Mathematics: From Euclid to Gödel by John Stillwell - https://press.princeton.edu/books/hardcover/9780691171685/el...

Both of them give a nice tour of various domains within modern mathematics and their inter-relationships which is what i believe is most important to understand for a general reader.

thread

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.

thread

jostylrJul 2020

I remember Pi in the Sky by John Barrows very fondly. It has more of a focus on geometry and logic.

A Programmer's Introduction to Mathematics by Jeremy Kun is wide ranging and appropriate if there is also interest in programming.

Nature and Growth of Modern Mathematics by Edna Kramer is a wonderful book if history is a passion as well.

Elements of Mathematics by John Stillwell is a broad overview of subjects. It has a crisp mathematical feel to it.

Vector Calculus, Linear Algebra, and Differential Forms by John & Barbara Hubbard is a beautiful introduction to the multi-dimensional aspects, but it is a book that should happen after knowing one dimensional calculus. .

If your child hasn't been exposed to Guesstimation, then a book on that is highly recommended. The book with that title by Weinstein and Adams is a nice guide to investigating that realm.

If the child does arithmetic from right to left, as is sadly too common, the book Speed Mathematics Simplified by Edward Stoddard is a great remedy for that.

Everyday Calculus by Oscar Fernandez could also be worth a look.

thread

earthicusFeb 2019

Warning: These books are kind of all over the place as far as technical prerequisites go, but they all satisify being heavier than pop-sci and lighter than a textbook. :)

-------

Aleksandrov et al - Mathematics: Its content, Methods, ad Meaning. Overview of undergrad curriculum by a number of prominent russian mathematicians.

Levi - The mathematical Mechanic. Uses various physical intuitions and scenarios to solve mathematical problems.

Pask - Magnificent Principia. Tour of newtons principia that includes both newtons geometric language and modern vector calculus version.

Stillwell - Elements of Mathematics. Discusses some important classical problems in the history of mathematics taken from 8 subjects: arithmetic, computation, algebra, geometry, calculus, combinatorics, probability, logic. Wow!

Yandell - The Honors Class: Hilbert's problems and their solvers. Lots of mini biographies with some information describing each of Hilbert's famous problems.

Lawvere & Schanuel- Conceptual Mathematics. Some basic notions of category theory, geared towards teaching you Lawvere's clarification and generalization of the incompleteness theorems.

Cormen - Algorithms Unlocked. Pop Science version of some of the textbook.

Nahin - Inside interesting Integrals. Lots of fun integrations!

Nahin - He also wrote some books about euler and imaginary numbers which ive partially read, they seemed good.

Pierce - An introduction to information theory; symbols signals and noise. sounds and looks like a dover textbook but is a non-fake pop science book.

---------------

Boyer - The History of Calculus and its Conceptual Development. A bit old, and spends too much time on formalism in the 18th and 19th century, there might be something better out now.

Weyl - Philosophy of Mathematics and Natural Science.

Reichenbach - The philosophy of Space and time.

Klein - Elementary mathematics from the advanced viewpoint. these were written at the beginning of the 20th century so are slightly outdated, but kline was extremely forward-looking in his point of view.

Courant & Robbins - What is Mathemaics. Another classic.

Hilbert & Cohn-Vossen - Geometry and the Imagination. The gold standard of popular mathematics writing. A beautiful book.

thread

jostylrMar 2018

Recommend Guesstimation by Weinstein and Adams as the first topic to master. Getting comfortable with numbers and their sizes will make everything else easier. Also, getting in the habit of doing rough explorations is an essential skill in exploring all later material.

Technical Mathematics with Calculus by Calter is a single volume that covers stuff up through calculus in a, well, technical manner.

For a more understanding way, try Elements of Mathematics by Stillwell.

If you get past Calculus, I recommend Vector Calculus, Linear Algebra, and Differential Forms by Hubbard. It gives an amazingly clear viewpoint on the higher level analysis and algebra topics, both numerically and abstractly.

For statistics, you might try something like Think Stats by Downey which emphasizes explorations with Python, real data, and Bayesian statistics.

As a faithful companion in your journey, use something like GeoGebra or Desmos to really explore the visual side of all the topics. Computers can do the tedious computations. Your task is to learn why we are doing this and how it is being done. When you get to calculus, learn what Newton's method is doing and appreciate how amazing it is.

thread