Category Archives: Books

Books: “A Mathematician’s Lament” by Paul Lockhart

A Mathematican's Lament book cover
Anybody who teaches math should read this book!

This book perfectly summarizes my disgust with the state of “education” today. The author’s main thesis is that math is an art that should be taught in a way to cultivate appreciation and understanding. It is not merely a set of formulas and definitions that students should commit to memory. Here is a quote from page 29:

By concentrating on what, and leaving out why, mathematics is reduced to an empty shell. The art is not in the “truth” but in the explanation, the argument. It is the argument itself which gives the truth its context, and determines what is really being said and meant. Mathematics is the art of explanation. If you deny students the opportunity to engage in this activity— to pose their own problems, make their own conjectures and discoveries, to be wrong, to be creatively frustrated, to have an inspiration, and to cobble together their own explanations and proofs—you deny them mathematics itself. So no, I’m not complaining about the presence of facts and formulas in our mathematics classes, I’m complaining about the lack of mathematics in our mathematics classes.

And another choice quote about the evils of High School Geometry from page 67:

…never was a wolf in sheep’s clothing as insidious, nor a false friend as treacherous, as High School Geometry. It is precisely because it is school’s attempt to introduce students to the art of argument that makes it so very dangerous.

Posing as the arena in which students will finally get to engage in true mathematical reasoning, this virus attacks mathematics at its heart, destroying the very essence of creative rational argument, poisoning the students’ enjoyment of this fascinating and beautiful subject, and permanently disabling them from thinking about math in a natural and intuitive way.

The mechanism behind this is subtle and devious. The student-victim is first stunned and paralyzed by an onslaught of pointless definitions, propositions, and notations, and is then slowly and painstakingly weaned away from any natural curiosity or intuition about shapes and their patterns by a systematic indoctrination into the stilted language and artificial format of so-called “formal geometric proof.”

This book is available as a freely downloadable PDF.

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Books: “Euler’s Gem The Polyhedron Formula and the Birth of Topology” by David S. Richeson

Euler's Gem book cover
This book is a beautifully illustrated gem! It examines Euler’s famous formula: V-E+F=2 which holds for all polyhedra or surfaces which are topologically equivalent to the sphere. The formula is an example of a topological invariant, something which can be computed for any surface, and thus allows one to categorize surfaces. The book also covers the famous classification theorem which categorizes all surfaces as either homeomorphic (topologically equivalent) to a sphere, n-handled torus, or sphere with n cross-caps (projective plane). Next it dives into knot theory and Seifert surfaces, before moving on to the interplay between topology and geometry, and ends by mentioning homology and how topology is done in higher dimensions.

Books: “Particle Physics A Very Short Introduction” by Frank Close

Particle Physics book cover
I found this book in the library the other day and found it to be a very good introduction to the basics of particle physics. I liked the fact that it explains the basics of the experimental side of particle physics as well as introducing the three families of leptons and quarks. Too often, popular science books explain the various subatomic particles with out giving you any idea of how we actually know of their existence. Because of this, I think disbelief tends to creep in. But Close does an excellent job of explaining the basics of how accelerators operate and also the basics of how detectors work. And he does all this in only 129 pages! Very short indeed, and highly recommended.

Algebraic Geometry: “Ideals, Varieties, and Algorithms” C++ Code


Ideals, Varieties, and Algorithms book cover
I did an REU (Research Experience for Undergraduates) this summer and we used the textbook “Ideals, Varieties, and Algorithms” (isbn 978-0-387-35650-1) by David Cox, John Little, and Donal O’Shea. As the title mentions, algorithms are an important tool in the study of varieties via their corresponding algebraic ideals. One of the main tools used in algebraic geometry is what is known as a Groebner basis, which is somewhat akin to a basis set in Linear Algebra. Anyway, I wrote some C++ code which implements a few of the algorithms in the book, but stops short of implementing Buchberger’s algorithm for computing a Groebner basis. The code should be useful as a starting point because it implements a Monomial class and a Polynomial class both with lots of useful methods thus eliminating the drudgery of implementing the book’s algorithms in C++.

The code can be found at: my web site as a gzipped tarball. Please let me know if you find it useful. I developed the code on an Ubuntu Linux machine, but it is generic enough that it should compile on any platform. You will need to read the README to understand the simple input, output format.

ADS Digital Library

While googling for information on the Virial Theorem, I ran across a really cool page hosted at the Harvard ADS system. ADS stands for Astrophysics Data System, and it is primarily an abstract search engine that helps you locate journal articles mostly in the realm of Astronomy and Astrophysics, but they also have a small virtual library of books that you can freely download.

Here are the titles I found interesting:

If you want to concatenate all the separate PDF chapters, I recommend using the texexec method oulined in this web page by Matthew Skala.

Book Review: The Trouble with Physics

The Trouble with Physics book coverThis book is Lee Smolin’s attempt to diagnose the physics community. Smolin entered graduate school in 1976, at the end of arguably the most fruitful era ever known to physics. However, since the 1980s no new fundamental challenges to the canon of physical theory have succeeded in reshaping how we view our universe. Although this fact troubles and frustrates Smolin, it is not this lack of substantial progress that he cites as the problem with physics. The problem as he sees it is that the majority of theoretical physicists are barking up the wrong tree, namely string theory. He argues very convincingly that string theory fails as a unifying theory. In the introduction he cites Nobel laureate Gerard ‘t Hooft as saying,

“I would not even be prepared to call string theory a ‘theory,’ rather a ‘model,’ or not even that: just a hunch. After all, a theory should come with instructions on how to deal with it to identify the things one wishes to describe, in our case the elementary particles, and one should, at least in principle, be able to formulate the rules for calculating the properties of these particles, and how to make new predictions from them. Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrest will perhaps be delivered soon. Whatever I did give you, can I still call it a chair?”

Smolin sees the widespread adoption of string theory as a continuation of the “shut up and calculate” mentality of the particle physics era that was so successful. He also argues that this follow the leader attitude is hurting diversity, especially the young physicists with bold new ideas. Not becoming part of an established research group and striking out on one’s own research path is tantamount to career suicide in theoretical physics, at least for those without established reputations. The notable exception of course is Einstein, but people like him are rare.

He divides his argument into four parts. In the first section, he sets the stage by giving a brief overview of the five great problems in theoretical physics, and the early attempts at unification. I especially liked how he described his set of heuristics for judging the promise of a new theory.

In the second section he gives a whirlwind history of string theory up to current times. This part of the book will be difficult for someone not familiar with the jargon of string theory. I recommend reading Brian Green’s “The Elegant Universe” before tackling this book, as it is a good introduction to string theory.

The third section examines some alternatives such as Smolin’s own research in loop quantum gravity as well as other programs such as Alain Conne’s non-commutative geometry research.

The final section looks at the physics community from a sociological perspective, attempting to understand how its internal power structure strongly encourages certain behaviors and punishes others. As a soon-to-be graduate student, I found this section to be very informative.

I strongly recommend this book. I however admit, that with only a bachelor of science degree in physics, I’m not yet qualified to judge string theory let alone even the Standard Model, but I can say that the book is well-written and I found it very interesting.

Book Review: Letters to a Young Mathematician

Letters to a Young Mathematician book coverI found this book in my local library and read it over the weekend. It is a quick read, but has some excellent advice. One thing I liked is the advice on how to read a math text or other technical material. The main idea is that if you get stuck, don’t go backwards assuming you missed something, instead keep going because often you’ll find the new or unfamiliar term defined in a short while. I only learned this myself about a year or two ago. It definitely seems counter-intuitive to keep going, but in the long run it usually works.