On the Shape of Mathematical Arguments

1. Front Matter

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2. Introduction

   Pages 0-6

3. A termination argument

   Pages 8-16

4. A problem on bichrome 6-graphs

   Pages 17-20

5. Proving the existence of the Euler line

   Pages 21-24

6. In adherence to symmetry

   Pages 25-28

7. On a proof by Arbib, Kfoury, and Moll

   Pages 29-33

8. Not about open and closed sets

   Pages 34-40

9. A monotonicity argument

   Pages 41-44

10. On the inverse of a function

    Pages 45-46

11. A calculational proof of Helly's theorem on convex figures

    Pages 47-56

12. The formal derivation of a proof of the invariance theorem

    Pages 57-63

13. Proving theorems with Euclid's algorithm

    Pages 64-68

14. On the deisng of an in-situ permutation algorithm

    Pages 69-79

15. Shiloach's algorithm

    Pages 80-87

16. Clarity of exposition

    Pages 88-120

17. On naming

    Pages 122-135

18. On the ue of formalism

    Pages 136-165

19. Epilogue

    Pages 166-170

20. Proof rules for guarded-command programs

    Pages 171-173

This book deals with the presentation and systematic design of mathematical proofs, including correctness proofs of algorithms. Its purpose is to show how completeness of argument, an important constraint especially for the correctness of algorithms, can be combined with brevity. The author stresses that the use of formalism is indispensible for achieving this. A second purpose of the book is to discuss matters of design. Rather than addressing psychological questions, the author deals with more technical questions like how analysis of the shape of the demonstrandum can guide the design of a proof. This technical rather than psychological view of heuristics together with the stress on exploiting formalism effectively are two key features of the book. The book consists of two independently readable parts. One part includes a number of general chapters discussing techniques for clear exposition, the use of formalism, the choice of notations, the choice of what to name and how to name it, and so on. The other part consists of a series of expositional essays, each dealing with a proof or an algorithm and illustrating the use of techniques discussed in the more general chapters.

• Algorithmenentwurf
• Lösen von Aufgaben
• algorithm
• algorithm design
• algorithms
• formalization
• mathematical methodology
• mathematische Methodik
• problem solving
• proof theory

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