An original book using recipes to explain sophisticated math concepts to students and even the math-phobic.
In a chapter on generalization, Cheng (Mathematics/Univ. of Sheffield and Univ. of Chicago) begins with a recipe she adapted to produce a cake that was vegan as well as gluten-, sugar-, and dairy-free, thus extending the recipe’s usefulness to serve more people. A chapter on axiomatization describes the difference between basic ingredients and things you can make with basic ingredients (e.g., marmalade). Math uses basic ingredients—axioms—that are assumed to be true and proofs that use hard logic to derive new truths. That’s what math is all about, writes the author; it is different from science, which gathers evidence to draw conclusions. By this time, Cheng has introduced readers to number systems, groups and sets, algebra, and topology. She also discusses internal vs. external motivation. In cooking, this is the difference between looking at what is on the shelves and figuring out how to use it in a recipe you invent (internal motivation) versus having a recipe in mind and gathering all the ingredients you need to make it (external). The author laments the way math is often taught, with the teacher providing a problem to solve and students finding the correct answer. She is strongly internally motivated in the pursuit of her specialty, category theory. She calls it the mathematics of mathematics, a field that seeks the most abstract generalizable concepts in relation to the worlds of mathematical objects. Cheng explains how category theory works by emphasizing contexts, relationships, structure, and universal properties, giving examples. The reading is tougher going here, probably because readers are in a state she describes as believing what she is teaching but not fully understanding it. However, Cheng is such a gifted teacher, readers will want to dive in again.
A sharp, witty book to press on students and even the teachers of math teachers.