In the constructive approach to mathematics every existence theorem must be proved by providing a construction of the object in question. As Bridger says in the preface, "existence is never established by showing that the assumption of non-existence leads to a contradiction". Thus, every constructive theorem is also true in the classical approach, and it is usually more informative.
The constructive approach was proposed by Brouwer as a response to the formalist program of Hilbert, which was to base mathematics on logic. (Gödel later showed that Hilbert's program could not fully succeed.) Brouwer's program, also known as intuitionism, was mixed with his own philosophical speculations and failed to show how to produce useful mathematics in the intuitionistic approach. This task was admirably taken up by Errett Bishop in his book Foundations of constructive analysis , which undertook a complete development of analysis within a constructive framework. (Bishop's book has been revised and extended by Douglas Bridges .) In the preface of his book, Bishop says this about Brouwer's failure: "The movement he founded has long been dead, killed […] chiefly by the failure […] to convince the mathematical public that abandonment of the idealistic viewpoint would not sterilize or cripple the development of mathematics. Brouwer and other constructivists were much more successful in their criticisms of classical mathematics than in their efforts to replace it with something better." Bishop's book has made the constructive approach worthy of serious attention.
Bridger follows the same path as Bishop, but with a different goal: to show that the constructive approach "makes sense not just to math majors, but to students from all branches of the sciences." Bridges, in his preface to , says that few undergraduates will have the mathematical maturity to fully appreciate the constructive approach and how it differs from the classical approach. Despite Bridger's efforts and nice prose, I tend to agree with Bridges. (Yes, it's confusing that their names are so similar!) However, the pace of Bridger's book is meant to allow students to learn and appreciate proof techniques.
Bridger departs from Bishop by using interval arithmetic to construct the reals from the rationals, an approach credited to Gabriel Stolzenberg. Bridger argues that this approach provides a useful metaphor for students: the output of a scientific measurement is an interval of rational numbers, due to the limited precision of instruments. Different measurements will find different intervals, but they will be consistent with each other because the "true" value of the quantity being measured must belong to all such intervals. A real number is then simply a family of rational intervals that is consistent and contains arbitrarily small intervals. (In the ideal mathematical world, measurements can be made as precise as needed.)
To give a flavor of differences between the classical approach and the constructive approach, consider the comparison of two real numbers, A and B. In the classical approach, we have the well-known trichotomy: either AB. In the constructive approach, which alternative holds must be proved constructively for each given pair of numbers. The best that can be said in general is what Bridger calls "epsilon-trichotomy": given two real numbers A and B and a rational number E, either AB, or A and B are within E of each other. This fine distinction may seem odd but is fundamental in the constructive approach. Despite first impressions, it is actually quite powerful because it leads to informative theorems, even though not all classical theorems can be proved constructively without change.
A further deviation from the classical approach is the central role played by uniformity in the development of constructive analysis: continuity and differentiablity are by definition always uniform, not pointwise. This is not as restrictive as it seems at first, because only compact intervals make sense constructively. Also, the role of the mean-value theorem (which is a hard-core existence theorem in the classical approach) is played by the Law of Bounded Change. For a summary of this development, see .
I won't spoil your discovery of constructive analysis any further. Bridger's book is a nice account and deserves to be read. Even if you do not subscribe to the constructive viewpoint, you'll learn something and find plenty of material to exploit in your classical analysis courses.
- Errett Bishop, Foundations of constructive analysis. McGraw-Hill, 1967. MR0221878 (36 #4930)
- Errett Bishop and Douglas Bridges, Constructive analysis. Grundlehren der Mathematischen Wissenschaften, 279. Springer-Verlag, 1985. MR0804042 (87d:03172)
- Mark Bridger and Gabriel Stolzenberg, "Uniform calculus and the law of bounded change". Amer. Math. Monthly 106 (1999), no. 7, 628–635. MR1720463 (2000k:26004)
Luiz Henrique de Figueiredo
is a researcher at IMPA in Rio de Janeiro, Brazil. His main interests are numerical methods in computer graphics, but he remains an algebraist at heart. He is also one of the designers of the Lua language