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Published in: Neurocomputing. 2001 Jan;42(2):331.
Casti JL, DePauli W.
Gödel. A Life of Logic.
Cambridge, MA: Perseus Publishing. 2000.
ISBN 0-7382-0274-6, 210 pages.
This book is a brief biography of arguably the greatest
mathematician of the twentieth century.
The work originally appeared as a tribute to Gödel
on German television,
and was subsequently expanded and translated into English.
Gödel's seminal work was published in 1931, but
only mathematicians and philosophers had ever heard of him until
a popular book covering his work appeared two decades ago.
Most middle-class Americans have a computer
sitting on their desktop, but few realize
that the computer's basic design of pointing
instructions to numbers was Gödel's.
Gödel grew up in Brno, Czech Republic, in the early 1900s.
This was a time and place of intellectual ferment and ethnic diversity.
Gödel was fluent in four languages,
with an amateur interest in several others.
He was influenced by existential writers such as Kafka;
by Jewish Kabbalists, who associated letters and words in the Hebrew Bible
with numerical values and relationships;
and by the emerging doctrine of non-Euclidean geometry.
Gödel came of age at a time of crisis in physics.
The unintuitive result that the speed of light
in a vacuum is everywhere constant
conflicted with Newton's Laws at high speeds,
and the geometry of space did not satisfy the classical Euclidean model.
Each new physics experiment seemed to shake another foundation.
David Hilbert, the leading German mathematician of his day,
called upon his colleagues to design a consistent set of axioms
from which all true statements of mathematics could be proved.
Such an achievement would stake out an irreducible core of true mathematics,
which would not totter at each new advance in physics.
The fundamental tool of mathematical reasoning is Aristotle's SYLLOGISM.
For example: (1) All men are mortal; (2) Socrates is a man;
(3) Therefore, Socrates is mortal. If one knows that assertions
(1) and (2) are true, then one is entitled to INFER that assertion (3)
is true. This stepwise derivation of additional true statements
from known true statements is a MATHEMATICAL PROOF. Aristotle also
proposed the paradox of Epimenides the Cretan, who asserted that
all Cretans are liars. This so-called PARADOX OF SELF-REFERENCE
has no truth-value, for if the assertion is true, then it is false;
if the assertion is false, then it is true.
There are many forms of this paradox, including: "this statement is false";
"the barber shaves everyone who doesn't shave himself"; and
"the set of all sets" (Frege-Russell paradox).
In a few years Gödel had demolished Hilbert's fondest dream,
by proving that EVERY system of mathematics at least as rich as
as arithmetic, geometry, or set theory, must necessarily
contain true but unprovable statements.
The method by which Gödel achieved this result
was as stunning as the result itself.
Gödel assigned a unique number to every
grammatically well-formed statement in mathematics.
He then constructed this true statement in his enumeration model:
"this statement is unprovable."
The book concludes by discussing the impact of Gödel's work
during the remainder of the twentieth century. Gödel's ideas
have influenced computer science, artificial intelligence, neural nets,
and possible limits on human sentience and creativity, all discussed
in this book. John von Neumann, an early supporter of Gödel's work,
clearly had Gödel's enumeration model in mind when von Neumann
designed the first modern computer in the 1940s. The initial
philosophical pessimism over the impossibility of establishing
a complete and consistent mathematical system has weakened:
the reverse of the argument is that there will always
be future work for creative mathematicians.
Toward the end of his career, Gödel
speculated that biological and human cultural diversity
could serve as an inexhaustible wellspring for mathematical creativity.
As an undergraduate, I had a mathematics professor who would
have forbidden me from reading this book. He felt that a young student
should focus more on the technical content of mathematics, and
should not be distracted by some popular account of "Gödel's Dreams".
I only earned a B in his course; perhaps he was right. On the other hand,
there is a a romance and mystery in mathematics, which is appreciated
even by some mathematicians only late in their training, and by the
general educated public almost not at all. Here is a book that begins
in Eastern Europe in the early 1900s, with echoes from Aristotle
and the mediaeval Kabbalists, and concludes with computers,
artificial intelligence, neural nets, and the limits on human creativity.
Anyone who cares about the great intellectual achievements
of the twentieth century should read this book.
Last updated: 1/6/2006, by G. William Moore, MD, PhD.