Preface to the Second Edition. Preface. 1. The Irrational. 2. The Imaginary. 3 .The Horizon. 4. The Infinitesimal. 5. Curved Space. 6. The Fourth Dimension. 7. The Ideal. 8. Periodic Space. 9. The Infinite. Epilogue.
John Stillwell was born in Melbourne, Australia. From 1970 to 2001 he taught at Monash University in Melbourne, and since 2002 he has been Professor of Mathematics at the University of San Francisco. He has been an invited speaker at several international conferences, including the International Congress of Mathematicians in Zurich, 1994. His works cover a wide spectrum of mathematics, from translations of classics by Dirichlet, Dedekind, Poincare, and Dehn to books on algebra, geometry, topology, number theory, and their history. For his expository writing, he was awarded the Chauvenet Prize of the Mathematical Association of America in 2005, and the AJCU National Book Award in 2009.
John Stillwell has found a thread that runs through all of
mathematical creativity – an openness to dangerous mathematical
ideas that once seemed irrational, imaginary, or paradoxical, and a
yearning to bring them into the fold. With clear writing and
marvelously original insights, Stillwell conveys the soul of
mathematics as well as anyone ever has.—Steven Strogatz, Cornell
University, and author of The Joy of xStillwell unifies nine major
branches of mathematics by one common theme: yearning for the
impossible. Yearning for the impossible, for instance in romantic
love or in utopian politics, is a central theme of the human
predicament. It is familiar in Shakespeare or in Dostoyevsky. But
in mathematics, as Stillwell recounts, we ACHIEVE the
impossible!For example, in the "rational" number line, the square
root of 2 is missing. Where it ought to be, we just find a gap. So
the German mathematician Dedekind proposed simply decreeing that
this very gap IS the square root that we are looking for! Creating
the impossible.The history of the calculus is inseparable from the
concept of the infinitesimal. Mathematicians used it successfully
for centuries, even while well aware that its definition is
self-contradictory.For thousands of years, mathematicians agreed
that Euclid's fifth postulate is true, although they all found it
impossible to prove that "fact." Then a mathematician in Germany,
and another one in Russia, and another one in Hungary within a few
years all discovered that the unsuccessful attempts to "prove"
Euclid's postulate were actually revealing a new "non-Euclidean"
geometry where it is falsified! Creating the impossible.Stillwell
goes beyond retelling these familiar examples of "yearning for
impossibility". In algebra, the German mathematician Kummer,
determined somehow to force algebraic numbers to behave as he knew
they OUGHT TO behave, identified a new substitute for a number, a
set which he called "the ideal." It does the job that for the known
numbers was "impossible."A chapter on periodic functions and
periodicity uses higher dimensional pseudo-cylinders, to make
seemingly discontinuous functions continuous and smooth.An easy,
readable chapter on projective geometry, creates another kind of
impossible. The point at infinity is dragged into a finite
position, and the horizon, that infinitely distant circle, is
merely a line at infinity.A charming, daring and irresistible
elementary textbook.—Reuben Hersh, University of New MexicoPraise
for the previous edition:Like the White Queen in Lewis Carroll's
Through the Looking-Glass, mathematicians are called upon to
believe in things that, at first glance, defy common sense and
appear impossible. . . . As Stillwell puts it, 'Mathematics is a
story of close encounters with the impossible because all its great
discoveries are close to the impossible.'
—Ivars Peterson, Science News , July 2006Stillwell weaves
historical details into his writing seamlessly, helping to give the
reader the true feeling that mathematics is more than just a bunch
of people playing games with symbols, but rather a rich and
rewarding intellectual endeavor important to the human
enterprise.
—Marcus E. Barnes, MAA Reviews, August 2006Yearning for the
Impossible is as much of a celebration of the greater understanding
mathematics has brought to the world as it is a history and
discussion of innovative concepts. and is highly recommended for
library and personal reading shelves.
—Wisconsin Bookwatch, August 2006Rises nobly to the challenge of
describing these topics to a genuine novice...There is much to
admire in Stillwell's attempt... he's accomplishing something very
important and difficult here in demonstrating that there's some
real struggle present in the process of mathematical discovery.
—Daniel Biss, Notices, June 2007Yearning for the Impossible offers
a fascinating, historical look at some popular mathematical
concepts used in music, art and philosophy. . . This book is an
interesting find and provides a readable approach to some
higher-level mathematics. The chapters can be read independently,
and the reader can dig deeper into textbooks and history books for
additional problems and details. I give a high recommendation for
this book!
—Lynn Godshall, Convergence Magazine (MAA), June 2007A wonderful
journey through mathematical discoveries... this book is an
excellent vehicle for giving mathematics students new research
ideas and, most important, for planting the seed in their minds to
'yearn for the impossible' as they investigate new truths.
—Mathematics Teacher, August 2007Stillwell has achieved what many
might well have come to believe to be nearly impossible in
mathematical exposition for the masses...[he] succeeds, in every
topic treated, in bringing a fresh eye to questions even
mathematicians might think have been mined in the past to boring
exhaustion [and] shows there is still a lot of gold to be found, if
one only thinks about things in a new way. Stillwell brings new,
unorthodox insights to his writing that will stimulate readers
(from high schoolers to emeritus professors) to think about old
topics in new, nonstale ways... Yearning for the Impossible will be
a treat for teachers, too, who are looking for new ways to bring
stimulating, fresh examples into their courses.
—SIAM Review, May 2007Stillwell does an excellent job laying the
historical foundations for these discoveries; he is to be commended
for his historical accuracy.
—Recreational Math, March 2007Mathematics may be described as a
story of close encounters with the impossible because all great
discoveries are close to the impossible. The aim of this book is to
tell this story, briefly and with few prerequisites, by presenting
some representative encounters across the breadth of
mathematics.
—CMS Notes, May 2007This book explores history through a lens
focused on the creative tension between common sense and the
'impossible' . . . Drawing connections to art, literature,
philosophy, and physics, this book examines the place of
mathematics in our intellectual landscape.
—L'Enseignement Mathématique, January 20062009 Alpha Sigma Nu Book
Award in the Discipline of Mathematics/Computer Science
—The Association of Jesuit Colleges & Universities, November 2009
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