218 - Österreichische Mathematische Gesellschaft

Nonnegative Matrices by S. M. Fallat and C. R. Johnson (2011), appeared just a
couple of months after the book under review, thus giving evidence of the current
lively interest in TP matrices.]
A. R. Kräuter (Leoben)
J. Stillwell: Roads to Infinity. The Mathematics of Truth and Proof. A. K. Peters,
Natick, Massachusetts, 2010, xi+203 S. ISBN 978-1-56881-466-7 H/b $ 39,–.
This is a most amazing book about the interaction of set theory with logic and
their impact on mainstream mathematics. The author explores the surprising consequences
of infinity and presents results that show that all levels of infinity, even
those that defy belief, have effects on the level of finite objects. (For an example
consider the insert “From infinite Ramsey to finite Ramsey.”) He muses that
infinity may be more down to earth than theoretical physics.
Of course Cantor and Gödel play a central role, but the author also makes a point
in telling the story of two neglected logicians, namely Emil Post and Gerhard
Gentzen. Post had discovered incompleteness before Gödel, although he published
later. His proof makes the origin of incompleteness in Cantor’s set theory
more transparent than that of Gödel, and also the connections with the theory of
computation. Gentzen, on the other hand, extended the theorem of Gödel that
the consistency of number theory depends on an assumption from outside number
theory, by providing the minimum such assumption. Thus he paved the way into
new insights into unprovability in number theory and combinatorics.
The book is astonishingly easy reading. It assumes little background, every chapter
beginning with a natural mathematical question, following a sequence of historical
responses. Every response leads to new questions and a fortiori to new concepts
and theorems. Finally, every chapter ends with a “Historical Background”.
One can read the theorems and skip the background, or one can profitably read the
background and then fill in the details, depending on one’s interests and previous
familiarity with the topic.
W. Imrich (Leoben)
G. G. Szpiro: Die verflixte Mathematik der Demokratie. Aus dem Englischen
von M. Junker. Springer, Berlin, Heidelberg, 2011, vii+212 S. ISBN 978-3-642-
12890-5 H/b 29,95.
Der Autor beschreibt in 13 Kapiteln einige Fragestellungen und Ergebnisse der
mathematischen Wahltheorie. Es werden u.a. Platon, Plinius der Jüngere, Ramon
Llull, Nikolaus Cusanus, Jean Charles de Borda, der Marquis de Condorcet, Pierre
Simon de Laplace, Charles Lutwidge Dodgson, Walter Willcox und Kenneth
Arrow erwähnt.
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