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Most intelligent people today still believe that mathematics is a body of unshakable truths about the physical world and that mathematical reasoning is exact and infallible. Mathematics: The Loss of Certainty refutes that myth. Morris Kline points, out that today there is not one universally accepted concept of mathematics - in fact, there are many conflicting ones.
Yet the effectiveness of mathematics in describing and exploring physical and social phenomena continues to expand. Indeed, mathematical activity is flourishing as never before, with the rapidly growing interest in computers and the current search for quantitative relationships in the social and biological sciences. "Are we performing miracles with imperfect tools?" Kline asks.
This book traces the history of mathematics’s falls from its lofty pedestal and explores the reasons for its mysterious effectiveness. Kline explains in non-technical language the drastic changes that have taken place in our understanding of "pure" as well as "applied" math, and the implications for science and for human reason generally.
Two nineteenth-century developments - non-Euclidean geometry and quaternions - forced mathematicians to realize that mathematics is not a series of self-evident truths about nature produced by infallible reasoning. They found, for example, that several different geometries fit spatial experience equally well. All could not be truths. This shocking realization impelled mathematicians to investigate the nature of their axioms and "unassailable" reasoning. To their surprise, they found that the axioms were arbitrary and inadequate and the proofs ware woefully defective.
To rebuild the foundations of mathematics and to resolve the contradictions, four different schools of thought cropped up - each differing radically from the others in their views of what mathematics is. The pride of human reason and its most effective expression suffered a fall which directly and indirectly affects all employment of reason.
Morris Kline is Professor Emeritus of. Mathematics at New York University's Courant Institute of Mathematical Sciences and associate editor of Mathematics Magazine and Archive for History of Exact Sciences. He has been a Guggenheim Fellow and a Fulbright Lecturer in Germany. His many books include Mathematics in Western Culture, Mathematical Thought from Ancient to Modern Times, Why Johnny Can't Add, and Why the Professor Can't Teach.
A quotation from the book:
"The current predicament of mathematics is that there is not one but many mathematics and that for numerous reasons each fails to satisfy the members of the opposing schools. It is now apparent that the concept of a universally accepted, infallible body of reasoning - the majestic mathematics of 1800 and the pride of man - is a grand illusion. Uncertainty and doubt concerning the future of mathematics have replaced the certainties and complacency of the past. The disagreements about the foundations of the 'most certain' science are both surprising and, to put it mildly, disconcerting. The present state of mathematics is a mockery of the hitherto deep-rooted and widely reputed truth and logical perfection of mathematics ....
"It behooves us therefore to learn why, despite its uncertain foundations, and despite the conflicting theories of mathematicians, mathematics has proved to be so incredibly effective."
From Mathematics: The Loss of Certainty
OXFORD UNIVERSITY PRESS, NEW YORK (1980)