Mathematics: The Loss of Certainty is a book by Morris Kline on the developing perspectives within mathematical cultures throughout the centuries.
This book traces the history of how new results in mathematics have provided surprises to mathematicians through the ages. Examples include how 19th century mathematicians were surprised by the discovery of non-Euclidean geometry and how Godel's incompleteness theorem disappointed many logicians.
Kline furthermore discusses the close relation of some of the most prominent mathematicians such as Newton and Leibniz to God. He believes that Newton's religious interests were the true motivation of his mathematical and scientific work. He quotes Newton from a letter to Reverend Richard Bentley of December 10, 1692:When I wrote my treatise about our system The Mathematical Principles of Natural Philosophy, I had an eye on such principles as might work with considering men for the belief in a Deity; and nothing can rejoice me more than to find it useful for that purpose.He also believes Leibniz regarded science as a religious mission which scientists were duty bound to undertake. Kline quotes Leibniz from an undated letter of 1699 or 1700:It seems to me that the principal goal of the whole of mankind must be the knowledge and development of the wonders of God, and that this is the reason that God gave him the empire of the globe.Kline also argues that the attempt to establish a universally acceptable, logically sound body of mathematics has failed. He believes that most mathematicians today do not work on applications. Instead they continue to produce new results in pure mathematics at an ever-increasing pace.
Criticism
In the reviews of this book, a number of specialists, paying tribute to the author's outlook, accuse him of biased emotionality, dishonesty and incompetence.
In particular, Raymond G. Ayoub in The American Mathematical Monthly writes:
For centuries, Euclidean geometry seemed to be a good model of space. The results were and still are used effectively in astronomy and in navigation. When it was subjected to the close scrutiny of formalism, it was found to have weaknesses and it is interesting to observe that, this time, it was the close scrutiny of the formalism that led to the discovery (some would say invention) of non-Euclidean geometry. (It was several years later that a satisfactory Euclidean model was devised.)
This writer fails to see why this discovery was, in the words of Kline, a "debacle". Is it not, on the contrary, a great triumph?...
Professor Kline does not deal honestly with his readers. He is a learned man and knows perfectly well that many mathematical ideas created in abstracto have found significant application in the real world. He chooses to ignore this fact, acknowledged by even the most fanatic opponents of mathematics. He does this to support an untenable dogma. One is reminded of the story of the court jester to Louis XIV: the latter had written a poem and asked the jester his opinion. "Your majesty is capable of anything. Your majesty has set out to write doggerel and your majesty has succeeded." On balance, such, alas, must be said of this book.
John Corcoran in Mathematical Reviews:
The overall purpose of the book is to advance as a philosophy of mathematics a mentalistic pragmatism which exalts "applied mathematics" and denigrates both "pure mathematics" and foundational studies. Although its thesis is predicated in part on the deep foundational achievements of twentieth century logicians, the basic philosophy is a close cousin of various philosophies which were influential in the nineteenth century. Moreover, as can be seen from the above-listed ideas, the author's grasp of twentieth century logic is not reliable. Accordingly he finds it surprising (p. 322, 323) that Hilbert, Gödel, Church, members of the Bourbaki school, and other "leaders in the work on foundations affirm that the mathematical concepts and properties exist in some objective sense and that they can be apprehended by human minds". His only argument against the Platonistic realism of the mathematicians just mentioned is based on his own failure to make the distinction between (human) error and (mathematical) falsehood (p. 324)...