Morris Kline, Professor Emeritus of Mathematics at New York University, offers us with this book a superb popular intellectual history in the domain of mathematics focusing on a single theme, the search for the perfection of truth in mathematical formalism. The outcome of this quest is described in its essence on page 257:
"The science which in 1800, despite the failings in its logical development, was hailed as the perfect science, the science which establishes its conclusions by infallible, unquestionable reasoning, the science whose conclusions are not only infallible but truths about our universe and, as some would maintain, truths in any possible universe, had not only lost its claim to truth but was now besmirched by the conflict of foundational schools and assertions about correct principles of reasoning."
Kline informs us that the current state of the science is that in which in true postmodern fashion several schools somewhat peacefully coexist--among them, Russell's logicism, Brouwer's intuitionism, Hilbert's formalism, and Bourbaki's set theory--in apparent abandonment of the nineteenth-century goal of achieving the perfection of truth in formal mathematical structures. In this coliseum of competing paradigms, the tipping point that engenders the status quo of peaceful coexistence is, of course, Kurt Godel, who in 1931 with his Incompleteness Theorem of almost cultic fame showed that any mathematical system will necessarily be incomplete because there will always exist a true statement within the system that cannot be proven within the system.
Despite this Babel, Kline believes that mathematics is gifted with the intellectual wherewithal to fruitfully pursue even the farthest and most abstruse reaches of abstraction because in this quest it is always assured the boon of the Holy Grail by virtue of the touchstone of empiricism. He concludes on the last page:
"Mathematics has been our most effective link with the world of sense perceptions and though it is discomfiting to have to grant that its foundations are not secure, it is still the most precious jewel of the human mind and must be treasured and husbanded."
In Scripture the counterpart of this outlook might be, "Test everything; retain what is good" (1 Thessalonians 5:21), while in common proverbs it would be, "The proof of the pudding is in the eating."
Although the book is written as a popular intellectual history and therefore is accessible to every educated reader, I believe that the extent to which readers would appreciate various historical portions of this book would depend on their formal mathematical preparation. From the time of Euclid's Elements to Newton's Principia Mathematica, sufficient for a deep appreciation on the reader's part is a high school background in mathematics. Beginning with Newton's fluxions and Leibniz's differentials and ending with nineteenth-century efforts to place algebra on formal footing, a finer understanding of the book requires the undergraduate-level background in mathematics that is usually obtained by scientists and engineers. Starting in the late eighteenth-century with Gauss' investigation of non-Euclidean geometry until twentieth-century disputes concerning mathematical philosophy, the discussion is probably more accessible to trained mathematicians or logicians.
Here and there I picked up interesting trivia, such as the historical fact that algebra, unlike geometry, was not initially developed as a formal system but rather as a tool of analysis, or that the intellectual enterprise to cast mathematics as a complete, consistent formal system really began in the second decade of the nineteenth century.
For lovers of mathematics, I recommend this book as engaging diversion in intellectual history. Read it on vacation.