On the Shape of Mathematical Arguments

190 Pages·1990·1.41 MB·English

by  Antonetta J. M. van Gasteren (auth.)

Lecture Notes in
Computer Science
Edited by G. Goos and J. Hartmanis
445
A.J.M. van Gasteren
On the Shape of
Mathematical Arguments
Springer-Verlag
Lecture Notes in
Computer Science
Edited by G. Goos and J. Hartmanis
445
A.J.M. van Gasteren
On the Shape of
Mathematical Arguments
Foreword by Edsger W. Dijkstra
Springer-Verlag
Berlin Heidelberg New York London
Paris Tokyo Hong Kong Barcelona
Editorial Board
D. Barstow W. Brauer P. Brinch Hansen D. Gries D. Luckham
C. Moler A. Pnueli G. Seegmiiller J. Stoer N. Wirth
Author
Antonetta J. M. van Gasteren
University of Groningen, The Netherlands
and
University of Utrecht, The Netherlands
Correspondence address:
Edelweislaan 20
5582 BWWaalre, The Netherlands
CR Subject Classification (1987): F.3.1, D.2.4
ISBN 3-540-52849-0 Springer-Verlag Berlin Heidelberg New York
ISBN 0-387-52849-0 Springer-Verlag New York Berlin Heidelberg
This work is subject to copyright. Alls rights are reserved, whether the whole or part of the material
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Foreword
While current curricula extensively teach existing mathematics, they pay
scant attention to the doing of mathematics, i.e., to the question of how
to design and to present solutions. If any attention to these issues is
paid at all, they are treated separately: design of solutions, i.e., "prob-
lem solving" or "mathematical invention", is viewed as a psychological
issue, as a matter of mathematical intuition, while presentation is viewed
as a matter of personal style or as an issue of education. Most mathe-
ran
maticians consider psychology and pedagogy as sciences too soft to be
respectable, and consequently the subject of how to do mathematics has
almost been tabooed.
The great merit of A.J.M. van Gasteren's work is to have broken
this taboo. She has done so, firstly, by recognizing that, in the case of
1
mathematical arguments, the tradition'al separation of content and form
.
is untenable, and, secondly, by showing how, via notation and formula
manipulation, d1.esign and presentation emerge as two sides of the same
C
coin. She has broken the taboo because, in this united setting, the issues
involved are purely technical: for instance, the question of whether a
cert`-t-ain notational convention -ofP,, equal relevance for the derivation as
for the presentation of a solution!- is geared to our manipulative needs
is a technical question that has nothing to do with intuition or taste.
That formalization should and can aid the mathematician rather
than add to his burden is well-known. This book reveals a wider vision,
viz., that the more formal the solutions we are heading for, the better we
r'
shall be able to teach the art and science of their 0'design. In doing so, this
work represents a crucial first step towards mathematical methodology
1-4
in the truest sense of the word, and for that reason I greatly welcome
elf
its publication in Springer's Lecture Notes Series: the more people enjoy
this book and are inspired by it, the better.
Edsger W. Dijkstra
III

Preface
For me, exploring the presentation and design of mathematical proofs
has been a very exciting activity. To see how virtually every proof a.nd
Q
program can be made much clearer and crisper, and to develop tech-
'
niques and skills for doing so, is such a rewarding experience that mDy
hope is to s0hare it with the readers of this monograph. To appreciate Cthe
m
text, the reaa.der needs some maturity in mathematics, but no specializ-ed
',
maearthematical knowledge is required to follow the example arguments in
the first part.
Of all those who made my explorations and the writing of this
text possible, I want to mention first and foremost Edsger W. Dijkstra
and W.H.J. Feijen, who educated me and who shared and guided my in-
terests. The Eindhoven Tuesday Afternoon Club, in varying formations,
has always been an indispensable forum for discussion and critique. In
their capacity as members of my dissertation committee, R.C. Back-
house, P.L. Cijsouw, and F.E.J. Kruseman Aretz carefully studied, and
Vim]
commented on, this text.
For many years, BP International Limited provided financial sup-
porRt for my research under their much appreciatedp.. Venture Research
D
Scheme; especially D.W. Braben, head of BP's Venture Research Unit,
C
has been an ever interested and stimulating ally. V.H. Backhouse and
D.B.M. Klaassen took care of the typesetting and typing of this text,
accepting and implementing all my well-considered but non-standard ty-
pographic wishes. H. Paas of the University of Groningen provided them
with technical assistance. M.C. Dijkstra-Debets and D.B.M. Klaassen,
finally, deserve mention for their moral and other support.
V

Contents
0 Introduction 0
Part 0
8
Summaries of the expositional essays
10
A termination argument
1 14
A problem on bichrome 6-graphs
2 17
3 Proving the existence of the Euler line 21
4 In adherence to symmetry 25
On a proof by Arbib, Kfoury, and Moll
5 29
6 Not about open and closed sets 34
7 A monotonicity argument d41
1
8 On the inverse of a function l45
9 A calculational proof of Helly's theorem on
convex figures 47
10 The formal derivation of a proof of the
0
invariance theorem 57
11 Proving theorems with Euclid's algorithm 64
12 On the design of an in-situ permutation algorithm 69
13 Shiloach's algorithm 80
VII
VIII
Part 1
88
14 Clarity of exposition 90
14.0 The division of labour between reader and writer 91
. . . .
14.1 On arrangement 107
f1" . . . . . . . . . . . . . . . . . . . . . . .
14.2 On disentanglement 116
. . . . . . . . . . . . . . . . . . . . .
15 On naming 122
15.0 Names in a linguistic context 122
. . . . . . . . . . . . . . . .
15.1 Names in a formal context 126
Ca' . . . . . . . . . . . . . . . . .
16 On the use of formalism 136
16.0 Manipulation without interpretation 138
. . . . . . . . . . .
16.1 On convenience of manipulation 144
. . . . . . . . . . . . . .
16.2 Avoiding formal laboriousness 155
. . . . . . . . . . . . . . .
17 Epilogue 166
18 Proof rules for guarded-command programs 171
19 Notational conventions 174
References 178
0 Introduction
This is a study about proofs; it is not about theorems or theories,
but about mathematical arguments, proofs of correctness for programs
included. The original incentive to investigate proofs in their own right
was simply a matter of necessity; later the challenge of exploring a
0
topic that has received relatively little explicit attention became a second
impetus for the investigations.
Let me briefly discuss the necessity. In the late seventies it had
become possible to derive a program and its correctness proof hand-
in-hand, and if the proof was sufficiently detailed the outcome was a
trustworthy design. That constearraint of sufficient detail, however, was
something of a problem, because what mathematicians usually consider
sufficient detail did not suffice in the context of program design, and for
all but the simplest problems the requirement, if met at all, tended to
lead to proofs that were long and verbose, or complicated and laborious
when given formally.
Our conclusion was that computing scientists would have to learn
to make their proofs more effective, if their methods were to be appli-
cable to more ambitious problems as well. At the time, however, it
was far from obvious what the characteristics of effectiveness could be
(although one thing was clear: it should be a combination of complete-
ness and brevity of argument), and even less obvious how they might be
effectuated.
0