The New York Times published a short but good article about Emmy Noether yesterday.
This month’s Communications of the ACM (Association for Computing Machinery) has an interesting article which explains how SQL databases and noSQL databases are duals of each other in the Category Theoretical sense.
Did you know that there is a museum of mathematics? I didn’t know either, but now that I regularly get junk from the AMS (American Mathematical Society), I recently learned that there is a museum dedicated to math!
The new Museum of Mathematics will open in Manhattan in early 2012; see http://www.momath.org. MoMath has the enthusiastic support of notables such as actor, director, and science buff Alan Alda, among others. MoMath is showing the fun of math at the upcoming U.S. Science & Engineering Festival and in its traveling Math Midway http://www.mathmidway.org. In the coming months, the Math Midway will travel to Texas, California, Ohio and Maryland.
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.
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.
Steven Strogatz, a mathematician at Cornell University and the author of a really cool book on chaotic dynamical systems was asked to guest blog at the New York Times today. He has written a short but interesting piece about power laws originating in biological systems and even in cities.