«Mathematics – Form and Function» av Saunders Mac Lane

I have just finished reading «Mathematics – form and function» by Saunders Mac Lane. The main goal of the book is to present the author’s philosophy of mathematics, answering the question «what is mathematics?». In doing so, he also answers the question «is mathematics true?» and demonstrates that it is a non-question. He presents mathematics as a set of tightly intervowen formal rules, wherein deduction is only allowed following the «rules of deduction».There is no «Platonic world of ideal forms», as many philosophers of mathematics have claimed, the author says. He grounds this view on the observation that ideas in mathematics only become clear when they are formalized. One idea can have different formalizations (that is, axiomatizations), leading to different concepts. One example is the idea of geometry. The naïve approach is Euclidean geometry with the parallell axiom. However, this is not the only «geometry» (throwing away the parallell axiom gives us many other interesting geometries).

Most of the book is not philosophy, however. Before one can answer the question «what is mathematics?», one must know some mathematics. Here’s a list of the chapters:

  1. Origins of formal structure.
  2. From whole numbers to rational numbers. What is numbers really? How do we define them? In this chapter the author presents the Peano postulates, namely the usual definition of a natural number. He constructs – though rather informally – the ration numbers and discusess the concept of congruence.
  3. Geometry. What is geometry? Orientation? Is Geometry a science? Is geometry the study of invariants under groups action on the plane?
  4. Real numbers.  There are several ways to construct the real numbers. Dedekind cuts, Cauchy sequences. What matters is their formal properties, not their particular construction.
  5. Functions, transformations, and groups. What is a function? Galois theory. How do the idea of function arise? Image, composition.
  6. Concepts of calculus. Derivates, differential equations, and so on. All presented very informally – no hairy calculations, only presenting the important concepts.
  7. Linear algebra. Sources of linearity, transformations versus matrices, dual spaces, tensor products and so on.
  8. Forms of space. Curvature, manifolds, paths, sheaves. The section on sheavs is really good! It presents a difficult mathematical concept in an amazingly easy  way.
  9. Mechanics. Actually, I skipped this chapter. It was the most technical (and boring) chapter of the book.
  10. Complex analysis and topology. New viewpoint: The really good thing about complex analysis is the lack of «pathological functions», because all we demand is the derivative to exist. If it does, we have everything we want. Germs and sheaves are also treated excellently at the end.
  11. Sets, logic and categories. How is mathematics grounded? Alternative foundations via categories or sheaves.
  12. The mathematical network.  The concluding philosophical chapter.

I will definitely recommend this book for anyone interested in mathematics. It will be nice, however, to have some mathematical training to know what is being talked about – perhaps two years of university mathematics will suffice.