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%3Mathematics: The Loss of Certainty%1. by Morris Kline. Oxford.
366 pp. α$19.95.

%3John von Neumann and Norbert Wiener: From Mathematics to the
Technologies of Life and Death%1. by Steve J. Heims. M.I.T. Press.
547 pp. α$19.95.

.turn on "→"
→by John McCarthy

	Professor Kline recounts
a series of "shocks", "disasters" and "shattering" experiences
leading to a "loss of certainty" in mathematics.  However, he
doesn't mean that the astronaut should mistrust the computations
that tell him that
firing
the rocket in the prescribed direction for the prescribed
number of seconds will get him to the moon.

	The ancient Greeks were "shocked" to discover that
the side and diagonal of a square could not be integer multiples
of a common length.  This spoiled their plan to found all
mathematics on that of whole numbers.  Nineteenth century
mathematics was "shattered" by the
discovery of non-Euclidean
geometry (violating Euclid's axiom that there is exactly one
parallel to a line through an external point), which
 showed that Euclidean geometry
isn't based on self-evident axioms about physical space (as most
people believed).  Nor is it
a necessary way of thinking about the world (as Kant had said).

	Once detached from physics, mathematics developed
on the basis of the theory of sets, at first informal and then increasingly
axiomatized, culminating in formalisms so well described that proofs
can be checked by computer.  However, Gottlob Frege's plausible axioms
led to Bertrand Russell's surprising
paradox of the the set of all sets that are not members of themselves.
(Is it a member of itself?).  L.E.J. Brouwer reacted
with a 
doctrine that only constructive mathematical objects should be allowed
(making for a picky and ugly mathematics), whereas David Hilbert proposed to
prove mathematics consistent by showing that starting from
the axioms and following the rules could never lead to
contradiction.  In 1931 Kurt Goedel showed
that Hilbert's program cannot be carried out, and this was another
surprise.

	However, Hilbert's program and Tarski's work
led to metamathematics, which
studies mathematical theories as mathematical objects.  This
replaced many of the disputes about the foundations of mathematics by
the peaceful study of the structure of the different approaches.

	Professor Kline's presentation of
 these and other surprises as shocks that
made mathematicians lose confidence in the certainty and in the future
of mathematics seems overdrawn.
While the consistency of even arithmetic cannot be proved,
most mathematicians
seem to believe (with Goedel) that mathematical truth exists and that present
mathematics is true.  No mathematician expects an inconsistency
to be found in set theory, and our confidence in this is greater
than our confidence in any part of physics.

	Heims has a thesis and presents the lives of two mathematicians
as an illustration.

	John von Neumann (1903-1957) was perhaps the brightest
of a remarkable group of Hungarian born mathematicians and physicists.
Heims describes his contributions to mathematical logic, to the
mathematical foundations of quantum mechanics, to the theory of games,
to the development of computers, to the development of atomic bombs
and peaceful nuclear energy, and to the relation of brain and
computer.

	von Neumann's enormous popularity and reputation
also came from his willingness to listen to other scientists and
his ability to clarify their ideas and often solve the problems
they were posing.  No-one else then or since had anything like his
reputation for this, but he might have made greater original
contributions had he been less helpful to others.

	Norbert Wiener (1894-1964) was child prodigy intensely educated by
his father, a professor of languages at Harvard.  He received his PhD at
18 and immediately began a career of contributions to many branches of
mathematics.  After World War II, he proposed a science of "cybernetics,
the theory of feedback and control in animal and machine".

	Wiener stories are often about his constant solicitation of assurance
that his contributions to mathematics were outstanding - which they were.
Although he was inclined to pontificate and had a higher opinion
of the importance of some of his contributions than many others did, his
two volumes of autobiography are usually very objective, especially
about his  earlier life.

	In the late forties, Wiener and von Neumann shared an interest
in the relation of computers to the brain, met often, and jointly
organized meetings.  Wiener's approach was through the notion of
feedback, wherein the output of a process was compared with a goal
and the difference used to control the process.  He coined the term
cybernetics for the whole field.
  von Neumann began
to construct a "general logical theory of automata" and produced
some fragments including a way of constructing reliable computers
from unreliable components and a theory of self-reproducing machines.

	While Heims doesn't attempt to evaluate the subsequent influence
of the work of either Wiener or von Neumann, neither cybernetics nor the
general theory of automata has been as successful as the approach first
proposed about 1950 by the British logician and computer scientist Alan
Turing.  Turing proposed that mental processes be studied by programming a
computer to carry them out rather than by building machines that imitate
the brain at the physiological level.  Programming concrete mental
processes such as learning and heuristic search in connection
with problem solving programs has proved more
fruitful in psychology, computer science and the philosophy of mind.

	When the first conference on artificial intelligence was organized
for the summer of 1956, everyone had great hopes for a contribution
from von Neumann, but he was already too sick.

	Both men were interested in human affairs.
von Neumann developed a mathematical theory of an expanding economy
in the 1920s
and a theory of games in the 1940s for studying competition and conflict.
Both theories are still being applied and extended.
He was alarmed by Soviet expansionism after World War II and
advocated a strong U.S. military position including the development
of the hydrogen bomb to which he also made technological
contributions.

	Wiener proposed that there be a cybernetic theory of human
biology and sociology emphasizing both random processes and stabilization
by feedback.  These attempts
achieved considerable acclaim, but (I think) few results of lasting
value, because significant problems require more than just the ideas of
feedback and filtering.  Perhaps because he didn't see the problems
of pattern matching and heuristics, he expected automatic factories
to replace most manual labor before 1970.  He worried about the
expected unemployment but had few concrete proposals.
His attitude to defense was the opposite of
von Neumann's; he opposed work on defense problems after the end of
World War II, sometimes holding that a scientist should keep secret work that
he thought could be used for military purposes.

	Heims's thesis is that Wiener was moral and von Neumann was
immoral in their attitudes toward the uses of science, especially
military applications, but also industrial.  Aspects of their
family backgrounds, early work, and personal lives are interpreted
as precursors of their postwar positions.  The "critical science"
style he adopts involves loaded adjectives and other unfairness
and often assumes what he has undertaken to prove.  Thus Truman's decision
to use the atomic bomb is ascribed solely to a desire to intimidate
the Soviet Union, and Eisenhower's 1955 atoms-for-peace
proposals are described as a "benign veneer".  Both propositions are
unsupported by argument.
  The series of photographs
ends with two of deformed Japanese babies.
Like the curate's egg, parts of the book are good.