5. Generalization

• The process of generalization involves starting with a situation you know and love and seeing what you can do that’s a bit similar but different. Eugenia starts with a cake recipe that contains no gluten, no dairy, and no sugar but still seems like a cake. She calls it a generalization of a cake. She explains the idea of how five axioms allowed Euclid to deduce all of the facts of geometry. She also explains proof by contradiction where you prove that the opposite of something can’t be possible. Further examples of generalization include different kinds of quadrilaterals and the world of topology where all things with one hole like a donut and a coffee cup are the same in a general way.

6. Internal vs. External

• The internal and external here refer to motivation also known as intrinsic and extrinsic. The recipe analogy refers to making something out of leftovers you happen to have on hand (internal) and rounding up the exact ingredients you need to make a specific recipe (external). In school, the external part happens most of the time as students follow teacher instructions to get good grades. In math, almost everything is external as you take on problems set before you. Mathematicians often just play with math for internal reasons. The implication here is that schools should try to let students do this too.

7. Axiomatization

• In math the basic ingredients are called axioms and the process of breaking something down into its basic ingredients is called axiomatization. Axioms are the basic truths that we accept in a particular situation. Axioms are like Lego blocks that are the basic shape you build things with. In the kitchen, basic ingredients are like axioms. Chocolate for some may be an ingredient, while for others it is something made of three more basic ingredients. With the basics, however, you can come up with great complexity. The rules or axioms of chess are simple, yet the game is very complex. The chapter ends with explanations of some axioms from the world of math.

8. What Mathematics Is

• Mathematics is the study of anything that obeys the rules of logic, using the rules of logic. It provides a language for making precise statements about concepts, and a system for making arguments about them. It also idealizes concepts so that a diverse range of notions can be compared and studied at the same time. Believe it or not, math is there to make difficult things easier. It allows us to construct arguments that are too difficult for ordinary intuition. It allows us to eliminate ambiguity so we can know something precisely. It also cuts corners by answering many questions at the same time by showing that they are the same question. It throws out ambiguity and ignores details that are irrelevant. The things that can’t be subsumed by math can be beautiful things like language, communication, poetry, art, and fun.

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