[.ca] Mathematics: The Loss of Certainty (ISBN 0195030850)

engaging intellectual history in the domain of mathematics:
Dedicated to Elizabeth Ruffle, M.Sci. Mathematics, on her Day of Liberation, July 15, 2003 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. P.S. Say you are Mistress, not Master, in Science.

Everyone else should be convinced by Morris Kline's book:
I want to start by saying that I agree with all of the positive reviews of Morris Kline's book (from what I can tell, all but one person gave this book high marks). Morris Kline is indeed a great mathematician as well as a great writer/expositer of his chosen field. This book -- which explores the philosophical ramifications of mathematics via its history and argues for the essentially uncertain nature of mathematics -- is definitely a great book for math fans out there. What has motivated my review is the one negative review by Sr. Barbosa of Puerto Rico (or so he claims). It is a sad but true fact that people who give their opinions on the web do not always give fair and reasonable opinions and/or are motivated by ulterior motives. Sr. Barbosa's review seems to fall into that category. First, Barbosa attacks Morris Kline (he's got some nerve doing that) for Prof. Kline's supposed lack of understanding of mathematics. This frivolous insult is so ridiculous that it isn't necessary to discuss it further. The negative review further contends that mathematics really is not uncertain. Sr. Barbosa, in that line of thought, also says that Kurt Godel didn't really believe in his own famous theorem! (Or at least that's the only way one can interpret Sr. Barbosa's statements.) Even layman that are familiar with popular works on mathematics -- *Godel, Escher, Bach*, *Godel's Proof*, etc. -- realize that mathematics as a formal, axiomatic system has been PROVEN (for all time) to be incomplete and inconsistent ... i.e., "uncertain." These ideas have been further amplified by the works of Alan Turing and Alonzo Church (the Halting Problem) as well as Gregory Chaitin (Algorithmic Information Theory -- along with Andrei Kolmogorov and Raymond Solomonoff). In fact, Chaitin has proven that the natural number system -- ie, the counting numbers (1,2,3,...) -- is itself random (i.e., uncertain). If that was not enough evidence in favor of Morris Kline (and contra Sr. Barbosa), then consider quantum physics and chaos theory. Both of those fields add further fuel to the idea that nature itself is uncertain. If nature is uncertain, then why shouldn't math (which often elegantly represents nature) be uncertain? Sr. Barbosa winds up looking foolish for arguing that Copernicus and other great thinkers of physics can be used to support Sr. Barbosa's views. On the contrary, physics seems to support Morris Kline. In short, Morris Kline's book does a valuable service by looking at how mathematics has hisorically developed in an uncertain manner in order to further highlight the uncertainity in mathematics that has been logically PROVEN by others. Shame on Sr. Barbosa and others who constantly write misleading, unfair, and irrational reviews of books that can lead customers astray and unfairly malign quality work.

What is certainty, and how is it lost?:
Clearly Morris Kline is an historical master, and his retelling of the story of mathematic is lush and rewarding. Not having had math of any sort since high school, I found the story riveting and confirming of many inchoate intuitions, especially with regard to the rather counter-intuitive status of irrational numbers, negative numbers, complex numbers, and infinitesimals, etc. However, having some training in epistemology, the book was somewhat less convincing in demonstrating its grandiose claims of the sort that (paraphrasing) "there is no truth in mathematics," or that there is no "justification for calculus" or "no factual evidence that supports the calculus," etc. His notions of truth, justification, evidence, and certainty seem entirely too dependent upon the rather limited portrait of certainty to be gotten from ancient Greek ideals of unassailable first principles and the deductions gotten from them. Epistemology itself has an evolving story that must be taken into account, and the epistemic notions (i.e. truth, evidence, etc.) require elaboration since they are central to Kline's evaluations of mathematics at every point. Since we are not ancient Greeks beholden to this limited epistemic ideal, there is only a "loss of certainty" for us to the extent that we adhere blindly to self-evident first principles and deduction as the only norms that could confer epistemic values like "certainty." Kline is persuasive in his arguments that alternate algebras and geometries are possible and useful, and one can hardly doubt that such alternatives lend themselves to a degree of modesty and potential relativity in mathematical claims to knowledge. Once one begins to admit other epistemic norms (like adequacy to empirical reality/experience or applicability to future problems, etc.) into the picture, one wonders if all the alternate algebras and geometries remain on equal footing. In any event, a degree of relativity in mathematical description and expression is not incompatible with a modest realism in mathematics. Nor are self-evidence and deduction the only norms for rationality, justification, evidence, warrant, or certainty. So long as one is cautious about the epistemic premises upon which the "loss of certainty" is predicated by Kline, the book is a great read!

Did not Convince Me:
I wish to point out first the positive aspects of the book. First of all, it should be noted that Morris Kline is one of the greatest mathematicians and now discusses a very important philosophical issue that is pertinent today. Kline shows a great insight concerning the history of the development of mathematics, a recount of the problems that different mathematicians had throughout history, the way they pretended to solve the problem, their logical and illogical reasons for doing so. He at least defends himself very well looking to history to prove how uncertain mathematics is. However, his book lives up according to a fallacy. Let's say that somebody thinks that certainty depends on a property "F" characteristic of some "a" mathematical system. Then the fact that up to that point it was believed by many people that F(a), then mathematics was certain, while when they discovered that it was not the case that F(a) then certainty of mathematics can no longer be established. An analogy with science will make clear the fallacy. Galileo insisted that the certainty of science on the universe depended greatly on the fact that the planets and stars moved in perfect circular orbits; Kepler on the other hand proved that the planets move in eliptical orbits. It would be an exaggeration to think, that the certainty of science is lost just because planets move in eliptical orbits. Another problem is that he states that mathematics is also uncertain because the irrational reasons to admit certain mathematical entities or axioms. However, the *validity* of the axioms is what is at stake in mathematics, not the subjective reasons that somebody had to admit them. An analogy again with science can show this second fallacy. Some of the reasons Copernicus admited that the Sun was the center and not the Earth, was because the Sun was the noblest star, and because it would restore the perfection of the circles in which planets revolve, because it had been lost in the Ptolemaic geocentric view of the universe. Do these reason should really dismiss the validity of Copernicus' theory? No. The same happens with mathematics. The illogical reasons that somebody might have to discover something, is irrelevant concerning the validity and certainty of mathematics. Also, there is the fallacy that because that there is a development of mathematics in one area that seems to be unorthodox at some moment, might compromise the certainty of mathematics. For example, he uses the development of "strange" algebras or "strange" geometries as examples of this. Non-Euclidean geometry doesn't invalidate Euclidean geometry, as Morris seems to suggest, nor does imply the loss of certainty of Euclidean geometry. It only means that Euclidean geometry is one of infinite possible mathematical spaces. Certainty is guaranteed in each one of them. Also, he seems to use the word "disaster" concerning Godel's theorems. But it was a "disaster" only to *some* philosophical schools. Godel's theorems doesn't seem at all to imply the uncertainty of mathematics, since Godel himself believed in its certainty during his entire life. In fact, Platonist propoposals such as Husserl's, though Edmund Husserl posited the completeness of mathematics, his main philosophy of mathematics is supported *even after* Godel's discovery. The only thing refuted in his philosophy is the completeness of mathematics, but not his mathematical realism, nor his account of mathemathical certainty. Interestingly, Husserl is never mentioned in the book (just as many philosophers of mathematics ignore his philosophy). Though the book is certainly instructive and Morris shows his knowledge of history of mathematics, due to these fallacies, he never proves his case.

A great book on the nature of mathematics!:
I wouldn't normally write a review of any book, but this book is really good (read the other reviews if you don't believe me), and I felt I had to write something. I highly recommend it for anyone who has ever wondered about the nature of mathematics. I have always been fascinated by mathematics, but doubts started creeping into my mind about it when I was taught about the calculus, and all of a sudden, I began to question whether this was reality I was being taught, or just some convenient invention. After all, zero divided by zero doesn't make sense, and the idea of the "ultimate limit" seemed to be a trick, or dangerously close the Infinite, which is isn't much easier to swallow either.... Many years of engineering didn't make me feel any more comfortable, although clearly, it worked! On reading this book, to my surprise (and somewhat to my consolation), I realized that even the great Newton and Leibniz did not justify their thoughts on this in a totally logical way, even though they helped to invent it. Which makes you wonder...why does the physical world seem to follow mathematical patterns (or does it really...)? And did the thinkers justify their "laws" of mathematics and establish them beyond any doubt? Did "constructive intuition", whatever that might be, play the most important role in the creation of mathematics? You may not get all the answers to these questions in this book (you won't get it in any other book this side of the universe), but you will certainly get a very thorough, deep and entertaining discussion these and many other questions you may not even have thought of. It is almost like being in a room with all these historical figures and listening to them arguing it out! Best part is, the book is quite cheap! You'll like it!