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Are the theorems of mathematics discovered in nature or imposed on nature by the human mind? What is the difference between pure and applied mathematics? And just how “true” are mathematical theorems? These are but a few of the fascinating questions that the mathematician Morris Kline addresses in his book Mathematics – The Loss of Certainty. Kline provides a historical tour of man’s evolving views on the nature of mathematics – beginning with the Ancient Greeks and working his way to the discoveries made in the 20th century by mathematicians such as Kurt Gödel. This book is well worth the effort for those interested in mathematics or the philosophy of science. It is written with the lay-person in mind and is thus a highly readable book.

## Key Passages

“The disagreements concerning what correct mathematics is and the variety of differing foundations affect seriously not only mathematics proper but most vitally physical science. As we shall see, the most well-developed physical theories are entirely mathematical. . .Hence scientists, who do not personally work on foundational problems, must nevertheless be concerned about what mathematics can be confidently employed if they are not to waste years on unsound mathematics.” (Morris Kline, Mathematics – The Loss of Certainty)

“The recognition that mathematics is not a body of truths has had shattering repercussions. Let us note first the effect on science. From Galileo’s time onward scientists recognized that the fundamental principles of science, as opposed to mathematics, must come from experience, although for at least two centuries they believed that the principles they did find were imbedded in the design of nature. But by the early 19th century they realized that scientific theories are not truths. The realization that even mathematics derives its principles from experience and that they could no longer assert their truth made scientists recognize that insofar as they use axioms and theorems of mathematics their theories are all the more vulnerable. Nature’s laws are man’s creation. We, not God, are the lawgivers of the universe. A law of nature is man’s description and not God’s prescription.” (Morris Kline, Mathematics – The Loss of Certainty)

“Mathematics grows through a series of great intuitive advances, which are later established not in one step but by a series of corrections of oversights and errors until proof reaches the level of accepted proof for that time. No proof is final. New counterexamples undermined old proofs. The proofs are then revised and mistakenly considered proven for all time. But history tells us that this merely means that the time has not yet come for a critical examination of the proof. Such an examination is often willfully delayed.” (Morris Kline, Mathematics – The Loss of Certainty)