id: 05279601
dt: b
an: 2015f.00533
au: Krantz, Steven G.
ti: The proof is in the pudding. The changing nature of mathematical proof.
so: New York, NY: Springer (ISBN 978-0-387-48908-7/hbk;
978-0-387-48744-1/ebook). xvi, 264~p. (2011).
py: 2011
pu: New York, NY: Springer
la: EN
cc: E50 A30
ut: nature of proof; proof methodology; mathematical proof; history of proof
ci:
li: doi:10.1007/978-0-387-48744-1
ab: The somewhat cryptic title of the book is a modification of the bonmot
“The proof of the pudding is in the eating” which in turn is an
analogue to Cervantes’ citation “Al freir de los huevos lo verá”
from {\it Don Quijote} (1615). Both citations mean, loosely speaking,
that you can say that something is a success only when you have tried
it. “Successful” is a mathematician if he has found an interesting
result (or counterexample), and “trying” it means being able to
give a proof which is accepted by a sufficiently large part of the
community. The primary concern of this book is to describe the essence,
nature, and methodology of mathematical proof, with a strong emphasis
on the change of these concepts in time. In fact, the book covers the
full history and evolution of the proof concept. The notion of rigorous
thinking has evolved over time and changed more than once, and the
author documents that development. He gives examples both of decisive
developments in the technique of proof and also of magnificent blunders
that taught us about how to think rigorously. In contrast to the
“sloppy” style some centuries ago, in our times strict rules for
generating and recording proofs have been established (which, however,
are not always observed). At the same time, many new external factors
have now an influence over the way mathematics is practiced; for
example, the computer plays an important role in many (not only
application-oriented) mathematical investigations. Enumerating the
headings of the 13 chapters gives a rough idea of its contents. 1. What
is a proof and why? 2. The ancients. 3. The middle ages and an emphasis
on calculation. 4. The dawn of the modern age. 5. Hilbert and the
twentieth century. 6. The tantalizing four-color theorem. 7.
Computer-generated proofs. 8. The computer as an aid to teaching and a
substitute for proof. 9. Aspects of modern mathematical life. 10.
Beyond computers: the sociology of mathematical proof. 11. A legacy of
elusive proofs. 12. John Horgan and “the death of proof?” 13.
Closing thoughts. This enumeration illustrates the large variety of
topics treated by the author in this wonderful book. It is written in a
very clear and suggestive manner that makes the reading pleasant and
rewarding; just consider Chapter 5 (on distinguished mathematicians of
the last century) or Chapter 11 (on some famous open problems) which
are really worth reading. The reviewer’s only criticism concerns the
author’s typical restricted American view of the “rest of the
world”: for example, what he writes in Section 5.2 about European
mathematics is simply not true. Moreover, the “he/she ‒ his/her”
style used by many authors, especially in the US, who mix up, for the
alleged sake of political correctness, grammatical and biological
gender, is annoying and ridiculous. However, these minor flaws did not
spoil the reviewer’s pleasure in reading the whole book. As the
author himself writes in the Preface, the purpose of the book is to
collect the ideas connected with all aspects of proof, [\dots] and to
acquaint the reader with the culture of mathematics: who mathematicians
are, what they care about, how they think, and what they do. Any reader
will notice that the author has reached this goal in very convincing
way, and the outcome is a brilliant work which should be found in every
math library and department office.
rv: Jürgen Appell (Würzburg)