engaging intellectual history in the domain of mathematics, July 14, 2003
By los desaparecidos (Makati City, Philippines) - See all my reviews

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.

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30 of 58 people found the following review helpful:
Did not Convince Me, April 17, 2002
By Pedro Rosario (Río Piedras, PR USA) - See all my reviews


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.

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Chapter XII is fascinating..., June 13, 2007
By A Reader (California USA) - See all my reviews

It's entitled "Disasters." Here Godel Numbers are described (page 262):

"So to 1=1 he assigned the number 90. Note that 90 can always be decomposed uniquely to 2^1*3^2*5^1, so that we can recover the symbols 1, 2, 1."

Here's how "Disasters" ends (page 277): "The developments in this century bearing on the foundations of mathematics are best summarized in a story. On the banks of the Rhine, a beautiful castle had been standing for centuries. In the cellar of the castle, an intricate network of webbing had been constructed by industrious spiders who lived there. One day a strong wind sprang up and destroyed the web. Frantically the spiders worked to repair the damage. They thought it was their webbing that was holding up the castle."

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2 of 2 people found the following review helpful:
Excellent survey of the history of mathematics , February 9, 2007
By Michael Emmett Brady "mandmbrady" (Bellflower, California ,United States) - See all my reviews


Kline demonstates ,in a clear and detailed fashion ,that the pursuit of " pure " mathematics(the set theoretical,real analysis approach),as opposed to the applied mathematics useful to scientific discovery ( the differential and integral calculus plus ordinary and partial differential equations),leads to a dead end as far as scientific discovery is concerned.This is well illustrated in his discussion of the rise of the Nicholas Bourbaki school that has come to dominate mathematics(pp.256-257)since the mid -1930's and its impact on the social sciences.
The field of economics is an excellent example of Kline's point.Economists are notorious for trying to copy the latest technical developments that occur in mathematics,statistics,physics,biology,etc.,irrespective of whether or not such techniques will yield useful knowledge which economists can use to analyze the events/historical processes occurring in the real world so that they can explain and predict why and when these events/processes will occur/reoccur.The best examples of the non or anti-scientific approach of the economics profession are the (a) Arrow-Debreu-Hahn general equilibrium approach based on various fixed point theorems,(b)the Subjective Expected Utility approach of Ramsey-De Finetti-Savage ,and(c)the universal belief of econometricians in the applicability of multiple regression and correlation analysis based on a least squares approach which requires the assumption of normality.It is not surprising that no econometrician in the 20th century ever did a basic goodness of fit test on their time series data to check to see whether or not the assumption of normality was sound.It took a Benoit Mandelbrot to demonstrate that the assumption of normality did not stand up.
The result has been that the economists simply are incapable of dealing with phenomena in the real world.Their pursuit of the latest fad or gimmick or technique to copy leads to the type of comment made by Robert Lucas,Jr.,the main founder of the rationalist expectationist school,that his theory can't deal with uncertainty,but only risk which must be represented by the standard deviation of a normal probability distribution.It is unfortunate that Lucas never did any goodness of fit test on business cycle time series data before constructing a theory that is only applicable if business cycles can be represented by multivariate normal probability distributions.
Kline's approach to the nature of mathematical discovery is very similar to that of J M Keynes and R Carnap-"The recognition that intuition plays a fundamental role in securing mathematical truths and that proof plays only a supporting role suggests that ...mathematics has turned full circle.The subject started on an intuitive and empirical basis...the efforts to pursue rigor... have led to an impasse..."(p.319).It can easily be observed that all of the three economist approaches mentioned above have ended in an impasse also.