Mathematics: The Loss of Certainty (Galaxy Books) (Paperback) by Morris Kline (Author)

Editorial ReviewsBook DescriptionThis work stresses the illogical manner in which mathematics has developed, the question of applied mathematics as against 'pure' mathematics, and the challenges to the consistency of mathematics' logical structure that have occurred in the twentieth century.

About the AuthorMorris Kline is Professor Emeritus at the Courant Institute of Mathematical Sciences, New York University.

Product DetailsPaperback: 384 pages

Publisher: Oxford University Press, USA; Reprint edition (June 17, 1982) Language: English

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

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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.

30 of 55 people found the following review helpful:

Did not Convince Me, April 17, 2002Reviewer: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

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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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1 of 1 people found the following review helpful:

Excellent survey of the history of mathematics , February 9, 2007Reviewer: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

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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.

3 of 6 people found the following review helpful:

Kline's uncertainty, June 2, 2006Reviewer:Walt Peterson (Ames, Iowa) - See all my reviewsOne reviewer said, ``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.'' I won't claim that Kline doesn't understand mathematics, but it is quite clear from this book that he does not understand logic. I looked up reviews in the professional literature by logicians and found they made the same point.

Kline makes many technical errors in his account of the foundational debates in the early twentieth century. My favorite mistake, and perhaps his most blatant blooper, is Kline's statement that the Loewenheim-Skolem Theorem implies Goedel's Incompleteness Theorem; he thinks that models with different cardinalities cannot satisfy the same sentences. (For non-logicians: they can and do; Kline's alleged implication is wrong.) His account of the history of mathematics is not as bad.

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Kline was an applied mathematician, and in his last two chapters informs us in very strong terms that applied mathematics is good and true, but pure mathematics is not. He urges mathematicians to abandon the study of analysis, topology, functional analysis, etc., and devote themselves to the problems of science.

The book is lively and entertaining, if not entirely reliable.

1 of 3 people found the following review helpful:

Great book by a great author, February 23, 2005Reviewer:G. A. Meles "Sirtor" (Italy) - See all my reviews

   English: This book isn't meant to be a mathematics book, still it offers a very good qualitative view of the problems it describes - at least as long as the reader has a not-zero competence in mathematics. Don't forget what Kant wrote, in the introduction of his masterpiece "Critique of Pure Reason" i.e. "that many a book would have been much clearer if it had not made such an effort to be clear": there are topics that can't be explained in "too simple words". There are a lot of divulging books that are not clear for competent reader and seem to be clear for inadequate readers: this is not the case of Kline books, which provides a interesting reading for an interested reader. 3 of 4 people found the following review helpful:

Mathematical Uncertainty, December 23, 2004Reviewer:Jefferson D. Bronfeld "always_reading" (Binghamton, New York USA) - See all my reviews

   A delightful and important book for all math enthusiasts. A must read for budding mathematicians.

This book authoritatively chronicles the gradual realization that mathematics is not the exploration of hard edged objective reality or the discovery of universal certainties, but is more akin to music or story telling or any of a number of very human activities.

Kline is no sideline popularizer bent on de-throwning our intellectual heros - he speaks knowledgeably from within the discipline of mathematics, revealing the evolution of mathematical thought from "If this is real, why are there so many paradoxes and seeming inconsistencies?" to "If this is just something people do, why is it so damned powerful?"

Mathematics: The Loss of Certainty. by Morris Kline. Oxford. 366 pp. $19.95. Professor Kline recounts a series of ``shocks'', ``disasters'' and ``shattering'' experiences leading to a ``loss of certainty'' in mathematics. However, he doesn't mean that the astronaut should

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mistrust the computations that tell him that firing the rocket in the prescribed direction for the prescribed number of seconds will get him to the moon. The ancient Greeks were ``shocked'' to discover that the side and diagonal of a square could not be integer multiples of a common length. This spoiled their plan to found all mathematics on that of whole numbers. Nineteenth century mathematics was ``shattered'' by the discovery of non-Euclidean geometry (violating Euclid's axiom that there is exactly one parallel to a line through an external point), which showed that Euclidean geometry isn't based on self-evident axioms about physical space (as most people believed). Nor is it a necessary way of thinking about the world (as Kant had said). Once detached from physics, mathematics developed on the basis of the theory of sets, at first informal and then increasingly axiomatized, culminating in formalisms so well described that proofs can be checked by computer. However, Gottlob Frege's plausible axioms led to Bertrand Russell's surprising paradox of the the set of all sets that are not members of themselves. (Is it a member of itself?). L.E.J. Brouwer reacted with a doctrine that only constructive mathematical objects should be allowed (making for a picky and ugly mathematics), whereas David Hilbert proposed to prove mathematics consistent by showing that starting from the axioms and following the rules could never lead to contradiction. In 1931 Kurt Goedel showed that Hilbert's program cannot be carried out, and this was another surprise. However, Hilbert's program and Tarski's work led to metamathematics, which studies mathematical theories as mathematical objects. This replaced many of the disputes about the foundations of mathematics by the peaceful study of the structure of the different approaches. Professor Kline's presentation of these and other surprises as shocks that made mathematicians lose confidence in the certainty and in the future of mathematics seems overdrawn. While the consistency of even arithmetic cannot be proved, most mathematicians seem to believe (with Goedel) that mathematical truth exists and that present mathematics is true. No mathematician expects an inconsistency to be found in set theory, and our confidence in this is greater than our confidence in any part of physics.

http://www-formal.stanford.edu/jmc/reviews/kline/kline.html

Mathematics: The Loss of Certainty

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     Most intelligent people today still believe that mathematics is a body of unshakable truths about the physical world and that mathematical reasoning is exact and infallible. Mathematics: The Loss of Certainty refutes that myth. Morris Kline points, out that today there is not one universally accepted concept of mathematics - in fact, there are many conflicting ones.

     Yet the effectiveness of mathematics in describing and exploring physical and social phenomena continues to expand. Indeed, mathematical activity is flourishing as never before, with the rapidly growing interest in computers and the current search for quantitative relationships in the social and biological sciences. "Are we performing miracles with imperfect tools?" Kline asks.

     This book traces the history of mathematics’s falls from its lofty pedestal and explores the reasons for its mysterious effectiveness. Kline explains in non-technical language the drastic changes that have taken place in our understanding of "pure" as well as "applied" math, and the implications for science and for human reason generally.

     Two nineteenth-century developments - non-Euclidean geometry and quaternions - forced mathematicians to realize that mathematics is not a series of self-evident truths about nature produced by infallible reasoning. They found, for example, that several different geometries fit spatial experience equally well. All could not be truths. This shocking realization impelled mathematicians to investigate the nature of their axioms and "unassailable" reasoning. To their surprise, they found that the axioms were arbitrary and inadequate and the proofs ware woefully defective.

     To rebuild the foundations of mathematics and to resolve the contradictions, four different schools of thought cropped up - each differing radically from the others in their views of what mathematics is. The pride of

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human reason and its most effective expression suffered a fall which directly and indirectly affects all employment of reason.

     Morris Kline is Professor Emeritus of. Mathematics at New York University's Courant Institute of Mathematical Sciences and associate editor of Mathematics Magazine and Archive for History of Exact Sciences. He has been a Guggenheim Fellow and a Fulbright Lecturer in Germany. His many books include Mathematics in Western Culture, Mathematical Thought from Ancient to Modern Times, Why Johnny Can't Add, and Why the Professor Can't Teach.

A quotation from the book:

     "The current predicament of mathematics is that there is not one but many mathematics and that for numerous reasons each fails to satisfy the members of the opposing schools. It is now apparent that the concept of a universally accepted, infallible body of reasoning - the majestic mathematics of 1800 and the pride of man - is a grand illusion. Uncertainty and doubt concerning the future of mathematics have replaced the certainties and complacency of the past. The disagreements about the foundations of the 'most certain' science are both surprising and, to put it mildly, disconcerting. The present state of mathematics is a mockery of the hitherto deep-rooted and widely reputed truth and logical perfection of mathematics ....

     "It behooves us therefore to learn why, despite its uncertain foundations, and despite the conflicting theories of mathematicians, mathematics has proved to be so incredibly effective."

     From Mathematics: The Loss of Certainty

     OXFORD UNIVERSITY PRESS, NEW YORK (1980)

http://www.philosophy-religion.org/handouts/mathematics.htm

Contact: Prof. Brian DaviesE.Brian.Davies@kcl.ac.ukAmerican Mathematical Society

Mathematics: The loss of certaintyPure mathematics will remain more reliable than most other forms of knowledge, but its claim to a unique status will no longer be sustainable."

So predicts Brian Davies, author of the article "Whither Mathematics?", which will appear in the December 2005 issue of Notices of the AMS.

For centuries mathematics has been seen as the one area of human endeavor in which it is possible to discover irrefutable, timeless truths. Indeed, theorems proved by Euclid are just as true today as they were when first written down

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more than 2000 years ago. That the sun will rise tomorrow is less certain than that two plus two will remain equal to four.

However, the 20th century witnessed at least three crises that shook the foundations on which the certainty of mathematics seemed to rest. The first was the work of Kurt Goedel, who proved in the 1930s that any sufficiently rich axiom system is guaranteed to possess statements that cannot be proved or disproved within the system. The second crisis concerned the Four-Color Theorem, whose statement is so simple a child could grasp it but whose proof necessitated lengthy and intensive computer calculations. A conceptual proof that could be understood by a human without such computing power has never been found. Many other theorems of a similar type are now known, and more are being discovered every year.

The third crisis seems to show how the uncertainty foreshadowed in the two earlier crises is now having a real impact in mathematics. The Classification of Finite Simple Groups is a grand scheme for organizing and understanding basic objects called finite simple groups (although the objects themselves are finite, there are infinitely many of them). Knowing exactly what finite simple groups are is less important than knowing that they are absolutely fundamental across all of mathematics. They are something like the basic elements of matter, and their classification can be thought of as analogous to the periodic table of the elements. Indeed, the classification plays as fundamental a role in mathematics as the periodic table does in chemistry and physics. Many results in mathematics, particularly in the branch known as group theory, depend on the Classification of Finite Simple Groups.

And yet, to this day, no one knows for sure whether the classification is complete and correct. Mathematicians have come up with a general scheme, which can be summarized in a few sentences, for what the classification should look like. However, it has been an enormous challenge to try to prove rigorously that this scheme really captures every possible finite simple group. Scores of mathematicians have written hundreds of research papers, totaling thousands of pages, trying to prove various parts of the classification. No one knows for certain whether this body of work constitutes a complete and correct proof. What is more, so much time has now passed that the main players who really understand the structure of the classification are dying or retiring, leaving open the possibility that there will never be a definitive answer to the question of whether the classification is true. As Davies puts it:

We have thus arrived at the following situation. A problem that can be formulated in a few sentences has a solution more than ten thousand pages long. The proof has never been written down in its entirety, may never be written down, and as presently envisaged would not be comprehensible to any single individual. The result is important and has been used in a wide variety of other problems in group theory, but it might not be correct.

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These three crises could be hinting that the currently dominant Platonic conception of mathematics is inadequate. As Davies remarks:

[These] crises may simply be the analogy of realizing that human beings will never be able to construct buildings a thousand kilometres high and that imagining what such buildings might "really" be like is simply indulging in fantasies.

We are witnessing a profound and irreversible change in mathematics, Davies argues, which will affect decisively its character:

[Mathematics] will be seen as the creation of finite human beings, liable to error in the same way as all other activities in which we indulge. Just as in engineering, mathematicians will have to declare their degree of confidence that certain results are reliable, rather than being able to declare flatly that the proofs are correct.

Davies's article "Whither Mathematics?" (PDF, 448KB) is available at Mathematics: The Loss of Certainty.

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Founded in 1888 to further mathematical research and scholarship, the 30,000-member American Mathematical Society fulfills its mission through programs and services that promote mathematical research and its uses, strengthen mathematical education, and foster awareness and appreciation of mathematics and its connections to other disciplines and to everyday life.

http://www.eurekalert.org/pub_releases/2005-11/ams-mtl110105.php

The Loss of CertaintyHerman von Helmholtz's Mechanism at the Dawn of Modernity. A Study on the Transition from Classical to Modern Philosophy of NatureSeries: Archimedes , Vol. 17, Schiemann, Gregor - 2007, Approx. 300 p., HardcoverISBN: 978-1-4020-5629-1, Not yet published. Available: November 3, 2007, approx.  $129.00About this book

Two seemingly contradictory tendencies have accompanied the development of the natural sciences in the past 150 years. On the one hand, the natural sciences have been instrumental in effecting a thoroughgoing transformation of social structures and have made a permanent impact on the conceptual world of human beings. This historical period has, on the other hand, also brought to light the merely hypothetical validity of scientific knowledge. As late as the middle of the 19th century the truth-pathos in the natural sciences was still unbroken. Yet in the succeeding years these claims to certain knowledge underwent a fundamental crisis. For scientists today, of course, the fact that their knowledge can possess only relative validity is a matter of self-evidence.

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The present analysis investigates the early phase of this fundamental change in the concept of science through an examination of Hermann von Helmholtz's conception of science and his mechanistic interpretation of nature. Helmholtz (1821-1894) was one of the most important natural scientists in Germany. The development of this thoughts offers an impressive but, until now, relatively little considered report from the field of the experimental sciences chronicling the erosion of certainty.

Written for: Scholars and students of  History, History of Science, History of Physics, History of Philosophy, History of the Philosophy of Science. Philosophers of Science and Technologyhttp://www.springer.com/west/home/philosophy?SGWID=4-40385-22-173701759-0

Mathematics: The Loss of CertaintyMorris KlineOxford Universiy Press, 1980You can't trust a simple axiom these days.But could we ever. After reading Kline's book, you might not think so. He's put together a sweeping history of the revolutions in mathematics. In each revolution the foundations of the previous mathematical thinking are questioned. A classic example is Euclid's fifth axiom (in two dimensions, given a line L and a point P not on L, there is only one line parallel to L that passes through P). After hundreds of years of dissatisfaction with this axiom, some mathematicians tried variations, and created non-Euclidean geometry.Kline delves much deeper into the mathematical thinking of the past two hundred years. From mathematics as a one-to-one model of the world, it became successively more divorced from reality. By the middle of this century, Kurt Gödel proved that arithmetic could be either complete (all statements are true or false) or consistent (there are no conflicts between statements), but not both. A consequence was that there are mathematical statements that are true but not provable.The disasters didn't end there. In the 1960s, Paul Cohen proved that the Continuum Hypothesis, a fundamental hypothesis about set theory (the foundations of mathematics), is not provable or disprovable given the standard axioms of set theory. The Continuum Hypothesis is about the size of infinite sets. In effect, it says that there are two sets which you either can or cannot put in a one-to-one correspondence. It's your choice whether this is allowed. It's sort of like saying you get to decide if there's another integer between 3 and 4. Mathematics would still work either way, but the decision is up to you.That's one of the reasons I didn't pursue mathematics. The axiomatic nature of the Continuum Hypothesis really shook my faith in whether mathematics has any relation to the world. This book is a must for any student interested in mathematics; it's especially good in high school. I wish I had read it many years before I found it. It's the single most important book on mathematics I've ever read.http://www.rdrop.com/~half/Personal/Hobbies/Books/RequiredReading.html

OBITUARYMorris Kline, 84, Math Professor And Critic of Math Teaching, Dies

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by Eric Pace

Morris Kline, a professor of mathematlcs who was a longtime critic of the way mathematlcs was taught, died early yesterday at Maimonides Hospital in Brooklyn. He was 84 years old and lived in Brooklyn.

He had been in declining health, and his death was caused by heart failure, said his wife, the former Helen Mann.

From 1938 to 1975, Professor Kline taught at New York University, with time out as a civilian employee in the Army in World War II. He was the author or editor of more than a dozen books, including "Mathematics in Western Culture" (1953), Mathematics: The Loss of Certainty" (Oxford University Press, 1980) and "Mathematics and the Search for Knowledge" (Oxford University Press, 1985).

In a 1986 editorial in Focus, a Journal of the Mathematical Association of America, he summarized some of his views: "On all levels primary, and secondary and undergraduate - mathematics is taught as an isolated subject with few, if any, ties to the real world. To students, mathematics appears to deal almost entirely with things whlch are of no concern at all to man".

The Key to Understandlng

The error, he contended, was that "mathematics is expected either to be immediately attractlve to students on its own merits or to be accepted by students solely on the basis of the teacher's assurance that it will be helpful in later life." And yet, he wrote,"mathematlcs is the key to understanding and mastering our physical, social and biological worlds."

He argued that teachers should stress useful applications of mathematics in various other fields: that they could have elementary schoolchildren deal wlth baseball batting averages and puzzles, get high school students work with statistics and probability, and bring college students to apply mathematics to computers and and physics.

But, he said, many schoolteachers are simply unfamiliar with such teaching techniques, and the same is true of numerous college professors who were under "pressure to write research papers." He called on professional mathematics journals to print articles that instructed school and college teachers about ways of presenting such applications to their pupils and students.

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Doubts on its Utility.

Professor Kline himself came to doubt the utility and importance for what he called "the life of Man". He himself turned to applied mathematics since, as he once put it, "the greatest contribution mathematics has made and should continue to make was to help man understand the world about him."

He was perennially interested in the cultural significance of mathematics., and he took up that subject notably in his 1980 book "Mathematics: The Loss of Certainty," which William Barrett writing in the New York Times Book Review, called "intensely dramatic, immensely readable" in its treatment of "the decline of mathematics from its once lofty pinnacle as the supreme embodiment of human certainty to its present uncertain state rent by conflicting schools."

In his 1973 book, "Why Johnny Can't Add: The Failure of the New Math," Professor Kline was critical of the New Math, an educational trend of the 1960's. Reviewing the book in The New York Times, Harry Schwartz said "its significance goes far beyond its immediate topic. It raises the broader issue of how, in field after field in American life, there come to be sudden fixations on supposed panaceas for perceived problems. All too often however, these panaceas turn out to have unforseen consequences as bad as or worse than the original difficulties that triggered their adoption."

Morris Kline ws born in Brooklyn, the son of Bernard Kline, an accountant, and the former Sarah Spatt. He grew up in Brooklyn and in Jamaica, Queens. He graduated from Boys High School in Brooklyn and went to study mathematics at New York University, where he earned a bachelor's degree in 1930, a master's degree in 1932 and a doctorate in 1936.

In addition to his wife, to whom he was married in 1939, Professor Kline is survived by a brother, Emanuel, of Great Neck, L.I.; two daughters, Elizabeth Landers of San Francisco and Judith Karamazov of Boston; a son, Douglas, of Cambridge, Mass., and three grandchildren.

The above obituary first appeared in The New York Times, June 10, 1992.

Version: 22nd March 2001

http://www.marco-learningsystems.com/pages/kline/obituary.html

Is Arithmetic Consistent? 

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Do I contradict myself?  Very well then, I contradict myself. I am large, I contain multitudes.                                                                                                   -Walt Whitman

 The most widely accepted formal basis for arithmetic is called Peano's Axioms (PA). Giuseppe Peano based his system on a specific natural number, 1, and a successor function such that to each natural number x there corresponds a successor x' (also denoted as x+1). He then formulated the properties of the set of natural numbers in five axioms: 

(1) 1 is a natural number.(2) If x is a natural number then x' is a natural number.(3) If x is a natural number than x' is not 1.(4) If  x' = y'  then  x = y.(5) If S is a set of natural numbers including 1, and if for every x in S the successor x' is

also in S, then every natural number is in S. These axioms (together with numerous tacit rules of reasoning and implication, etc) constitute a formal basis for the subject of arithmetic, and all formal “proofs” ultimately are derived from them. The first four, at least, appear to be “clear and distinct” notions, and even the fifth would be regarded by most people as fairly unobjectionable. Nevertheless, the question sometimes arises (especially in relation to very complicated and lengthy proofs) whether theorems based on these axioms (and tacit rules of implication) are perfectly indubitable. According to Goedel’s theorem, it is impossible to formally prove the consistency of arithmetic, which is to say, we have no rigorous proof that the basic axioms of arithmetic do not lead to a contradiction at some point. For example, if we assume some proposition (perhaps the negation of a conjecture we wish to prove), and then, via some long and complicated chain of reasoning, we arrive at a contradiction, how do we know that this contradiction is essentially a consequence of the assumed proposition?  Could it not be that we have just exposed a contradiction inherent in arithmetic itself? In other words, if arithmetic itself is inconsistent, then proof by contradiction loses its persuasiveness. On one level, this kind of objection can easily be vitiated by simply prefacing every theorem with the words "If our formalization of arithmetic is consistent, then...".  Indeed, for short simple proofs by contradiction we can strengthen the theorem by reducing this antecedent condition to something like "If arithmetic is consistent over this small set of operations, then...".  We can be confident that the contradiction really is directly related to our special assumption, because it's highly implausible that our formalization of arithmetic could exhibit a contradiction over a very short chain of implication. However, with long proofs of great subtlety, extending over multiple papers by multiple authors, and involving the interaction of many different branches and facets of mathematics, how would we really distinguish between a subtle contradiction resulting from one specific false assumption vs. a subtle contradiction inherent in the fabric of arithmetic itself? Despite Goedel’s theorem, the statement that we cannot absolutely prove the axioms of arithmetic is sometimes challenged on the grounds that we can prove the consistency of PA, provided we are willing to accept the consistency of some more encompassing formal system such as the Zermelo-Frankel (ZF) axioms, perhaps augmented with the continuum hypothesis (ZFC). But this is a questionable position. Let's say a proof of the consistency of system X is

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"incomplete" if it's carried out within a system Y whose consistency has not been completely proven.   Theorem:  Every proof of the consistency of arithmetic is incomplete. In view of this, it isn't clear how "working in ZFC" resolves the issue.  There is no complete and absolute proof of the consistency of arithmetic, so every arithmetical proof is subject to doubt. (By focusing on arithmetic I don't mean to imply that other branches of mathematics are exempt from doubt.  Hermann Weyl, commenting on Gödel’s work, said that "God exists because mathematics is undoubtedly consistent, and the devil exists because we cannot prove the consistency".)  As Morris Kline said in his book Mathematics and the Loss of Certainty, “Gödel’s result on consistency says that we cannot prove consistency in any approach to mathematics by safe logical principles”, meaning first-order logic and finitary proof theory, which had been shown in Russell's "Principia Mathematica" to be sufficient as the basis for much of mathematics. Similarly, in John Stillwell's book Mathematics and its History we find "If S is any system that includes PA, then Con(S) [the consistency of S] cannot be proved in S, if S is consistent." On the other hand, some would suggest that the contemplation of inconsistency in our formal arithmetic is tantamount to a renunciation of reason itself, i.e., if our concept of natural numbers is inconsistent then we must be incapable of rational thought, and any further considerations are pointless. This attitude is reminiscent of the apprehensions mathematicians once felt regarding "completed infinities".  "We recoil in horror", as Hermite said, believing that the introduction of actual infinities could lead only to nonsense and sophistry.  Of course, it turned out that we are quite capable of reasoning in the presence of infinities.  Similarly, I believe reason can survive even the presence of contradiction in our formal systems. Admittedly this belief is based on a somewhat unorthodox view of formal systems, according to which such systems should be seen not as unordered ("random access") sets of  syllogisms, but as structured spaces, with each layer of implicated objects representing a region, and the implications representing connections between different regions.  The space may even possess a kind of metric, although "distances" are not necessarily commutative.  For example, the implicative distance from an integer to its prime factors is greater than the implicative distance from those primes to their product. According to this view a formal system does not degenerate into complete nonsense simply because at some point it contains a contradiction.  A system may be "locally" consistent even if it is not globally consistent.  To give a crude example, suppose we augment our normal axioms and definitions of arithmetic with the statement that a positive integer n is prime if and only if  2n 2 is divisible by n.  This axiom conflicts with our existing definition of a prime, but the first occurrence of a conflict is 341.  Thus, over a limited range of natural numbers the  axiom system possesses "local consistency". Suppose we then substitute a stronger axiom by saying n is a prime iff f(rn) = 0 (mod n) where r is any root of f(x) = x5 x3 2x2 + 1. With this system we might go quite some time without encountering a contradiction.  When we finally do bump into a contradiction (e.g., 2258745004684033) we could simply substitute an even stronger axiom.  In fact, we can easily specify an axiom of this kind for which the smallest actual exception is far beyond anyone's (present) ability to find, and for which we have no theoretical proof that any exception even exists.  Thus, there is no direct proof of inconsistency.  We might then, with enough imagination, develop a plausible (e.g., as plausible as Banach-Tarski) non-finitistic system within which I can actually prove that our arithmetic is consistent.  In fact, it might actually be

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consistent… but we would have no more justification to claim absolute certainty than with our present arithmetic. As to the basic premise that we have no absolute proof of the consistency of arithmetic, here are a few other people's thoughts on the subject: 

A meta-mathematical proof of the consistency of arithmetic is not excluded by...Goedel's analysis.  In point of fact, meta-mathematical proofs of the consistency of arithmetic have been constructed, notably by Gerhard Gentzen, a member of the Hilbert school, in 1936.  But such proofs are in a sense pointless if, as can be demonstrated, they employ rules of inference whose own consistency is as much open to doubt as is the formal consistency of arithmetic itself.  Thus, Gentzen used the so-called "principle of transfinite mathematical induction" in his proof.  But the principle in effect stipulates that a formula is derivable from an infinite class of premises.  Its use therefore requires the employment of nonfinitistic meta - mathematical notions, and so raises once more the question which Hilbert's original program was intended to resolve.                                                                                 -Ernest Nagel and James Newman Gödel showed that...if anyone finds a proof that arithmetic is consistent, then it isn't!                                                                              -Ian Stewart ...Hence one cannot, using the usual methods, be certain that the axioms of arithmetic will not lead to contradictions.                                                                              -Carl Boyer An absolute consistency proof is one that does not assume the consistency of some other system...what Gödel did was show that there must be "undecidable" statements within any [formal system]... and that consistency is one of those undecidable propositions. In other words, the consistency of an all-embracing formal system can neither be proved nor disproved within the formal system.                                                                               -Edna Kramer Gentzen's discovery is that the Goedel obstacle to proving the consistency of number theory can be overcome by using transfinite induction up to a sufficiently great ordinal... The original proposals of the formalists to make classical mathematics secure by a consistency proof did not contemplate that such a method as transfinite induction up to 0 would have to be used. To what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter for individual judgment...                                                                      -Kleene, "Introduction to Metamathematics"

 Some mathematicians assert that there is a consistency proof of PA, and it is quite elementary, using standard mathematical techniques (ie, ZF).  It consists of exhibiting a model. However, we speak of "exhibiting a model" we are referring to a relative consistency proof, not an absolute consistency proof.  Examples of relative consistency theorems are 

If Euclidean geometry is consistent then non-Euclidean geometry is consistent.If ZF is consistent then ZFC is consistent.

 Relative consistency proofs assert nothing about the absolute consistency of any system, they merely relate the consistency of one system to that of another.  Here's what the Encyclopedic Dictionary of Mathematics (2nd Ed) says on the subject: 

Hilbert proved the consistency of Euclidean geometry by assuming the consistency of the theory of real numbers.  This is an example of a relative consistency proof, which reduces the consistency proof of one

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system to that of another.  Such a proof can be meaningful only when the latter system can somehow be regarded as based on sounder ground than the former.  To carry out the consistency of logic proper and set theory, one must reduce it to that of another system with sounder ground.  For this purpose, Hilbert initiated metamathematics and the finitary standpoint...Let S be any consistent formal system containing the theory of natural numbers.  Then it is impossible to prove the consistency of S by utilizing only arguments that can be formalized in S....  In [these] consistency proofs of pure number theory..., transfinite induction up to the first e-number 0 is used, but all the other reasoning used in these proofs can be presented in pure number theory. This shows that the legitimacy of transfinite induction up to 0 cannot be proved in this latter theory.

 All known consistency proofs of arithmetic rely on something like transfinite induction (or possibly primitive recursive functionals of finite type), the consistency of which is no more self-evident than that of arithmetic itself. Oddly enough, some people (even some mathematicians) are under the impression that Goedel’s results apply only to very limited formal systems. One mathematician wrote to me that “there is no proof in first-order logic that arithmetic is consistent, but that has more to do with the limitations of first-order logic than anything else, and there are other more general types of logic in which proofs of the consistency of arithmetic are available.” Of course, contrary to this individual’s claim, Goedel's results actually apply to any formal system that is sufficiently complex to encompass and be modeled by arithmetic. Granted, if we postulate a system that cannot be modeled (encoded) by arithmetic then other things are possible, but the consistency of such a system would be at least as doubtful as the consistency of the system we were trying to prove.  For example, Gentzen's proof of the consistency of PA uses transfinite induction, but surely it is pointless to try to resolve doubts about arithmetic by working with transfinite induction, since the latter is even more dubious.  The inability of even many mathematicians to absorb the actual content and significance of Goedel’s theorems is interesting in itself, as are the various misconstruals of those theorems, which tend to reflect what the person thinks must be true. For example, we can see how congenial is the idea that Goedel’s results apply only to a limited class of formal systems. The fact that mathematicians at universities actually believe this is rather remarkable. It seems to be a case of sophomoric backlash, in reaction to what can often seem like sensationalistic popular accounts of Goedel’s theorems. Apparently it becomes a point of pride among math graduate students to “see through the hype”, and condescendingly advise the less well-educated as to the vacuity of Goedel’s theorems. (This would be fine if Goedel’s theorems actually were vacuous, but since they aren’t, it isn’t.) Another apparent source of misunderstanding is the sheer inability to believe that any rational person could doubt the consistency of, say, arithmetic. Imagine the reaction of the typical mathematician to the even more radical suggestion that every (sufficiently complex) formal system contains a contradiction at some point. When I mentioned this to an email correspondent, he expressed utter incredulity, saying "You can't possibly believe that simple arithmetic could contain an inconsistency!  How would you balance your check book?" This is an interesting question.  I actually balance my checkbook using a formal system called Quicken. Do I have a formal proof of the absolute consistency and correctness of Quicken? No. Is it conceivable that Quicken might contain an imperfection that could lead, in some circumstances, to an inconsistency?  Certainly. But for many mathematicians this situation must be a real paradox, so it’s worth examining in some detail.

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 Suppose I balance my checkbook with a program called Kash (so as not to sully the good name of Quicken), and suppose this program implements arithmetic perfectly - with one exception.  The result of subtracting 5.555555 from 7.777777 is 2.222223.  Now if B is my true balance then I should have the formal theorems 

 for every value of q.  Thus, in the formal system of Kash I can prove that 1 = 2 = 3 = 4.23 = 89.23 = anything.  Clearly the Kash system has a consistency problem, because I can compute my balance to be anything I want just by manipulating the error produced by that one particular operation.  ("So oft it chances in particular men...")  But here is the fact that must seem paradoxical to many mathematicians: thousands of people have used Kash for years, and not a single error has appeared in the results.  How can this be, given that Kash is formally inconsistent? The answer is that although Kash is globally inconsistent, it possesses a high degree of local consistency.  Traveling from any given premise (such as 5+7) directly to the evaluation (e.g., 12), we are very unlikely to encounter the inconsistency.  Of course, in a perfectly consistent system we could take any of the infinitely many paths from 5+7 to the evaluation and we would always arrive at the same result, which is clearly not true within Kash (in which we could, by round-about formal manipulations, evaluate 5+7 to be -3511.1093, or any other value we wanted).  Nevertheless, for almost all paths leading from a given premise to its evaluation, the result is the same. Now consider our formal system of arithmetic.  Many people seem agog at the suggestion that our formalization of arithmetic might possibly be inconsistent at some point.  Clearly our arithmetic must possess a very high degree of local consistency, because otherwise we would have observed anomalies long before now.  However, are we really justified in asserting that every one of the infinitely many paths from every premise to its evaluation gives the same result?  As with the system Kash, this question can't be answered simply by observing that our checkbooks usually seem to balance.  Moreover, the question cannot even be answered within any formal system that can be modeled by the natural numbers.  It is evidently necessary to assume the validity of something like transfinite induction to prove the consistency of arithmetic.  But how sure are we that a formal system that includes transfinite induction is totally consistent? (If, under the assumption of transfinite induction, we had found that arithmetic was not consistent, would we have abandoned arithmetic... or transfinite induction?)  The only way we know how to prove this is by assuming still less self-evidently consistent procedures, and so on. The points I'm trying to make are 

(1) We have no meaningful proof of the consistency of arithmetic.(2) If arithmetic is inconsistent, it does not follow that our checkbooks must all be out of

balance.  It is entirely possible that we could adjust our formalization of arithmetic

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to patch up the inconsistency, and almost all elementary results would remain unchanged.

(3) However, highly elaborate and lengthy chains of deduction in the far reaches of advanced number theory might need to be re-evaluated in the light of our patched-up formalization.

 Of course, the consistency or inconsistency of arithmetic can only be appraised in the context of a completely formalized system, but the very act of formalizing is problematic, because it invariably presupposes prior knowledge on the part of the reader. Thus it can never be completely clear that our formalization corresponds perfectly with what we call `arithmetic'. Our efforts to project (exert) our formalizations past any undefined prior knowledge tend to lead to the possibility of contradictions and inconsistencies. But when such an inconsistency comes to light, we don’t say the ideal of “arithmetic” is faulty. Rather we say "Ooppss, our formalization didn't quite correspond to true ideal arithmetic.  Now, here's the final ultimate and absolutely true formalization... (I know we said this last time, but this time our formalization really is perfect.)" As a matter of fact, this very thing has occurred historically. In this sense we are tacitly positing the existence of a Platonic ideal "ARITHMETIC" that is eternal, perfect, and true, while acknowledging that any given formalization of this Platonic ideal may be flawed.  The problem is that our formal proofs are based on a specific formalization of arithmetic, not on the ideal Platonic ARITHMETIC, so we are not justified in transferring our sublime confidence in the Platonic ideal onto our formal proofs. Claims that arithmetic is indubitable, while acknowledging that our formalization may not be perfect, are essentially equivalent to saying that we are always right, because even if we are found to have said something wrong, that's not what we meant. Any given theorem can be regarded as a theorem about the ideal of ARITHMETIC, prior to any particular formalization, but then the first step in attempting to prove it is to select a formal system within which to work.  Of course, it's trivial to devise a formal system labeled "arithmetic" and then prove X within that system. For example, take PA+X.  But the question is whether that system really represents ARITHMETIC, one requirement of which is consistency. We don't know what, if any, parts of our present mathematics would be rendered uninteresting by the discovery of an inconsistency in our present formalization of arithmetic, because it would depend on the nature of the inconsistency and the steps taken to resolve it.  Once the patched-up formalization was in place, we would re-evaluate all of our mathematics to see which, if any, proofs no longer work in the new improved "arithmetic".  One would expect that almost all present theorems would survive.  The theorems most likely to be in jeopardy would be the most elaborate, far-reaching, and "deep" results, because their proofs tax the resources of our present system the most. Some mathematicians respond to the assertion that we have no meaningful proof of the consistency of arithmetic by claiming that “the usual ZFC proof is quite meaningful." But this seems to hinge on different understandings of the meaning of “meaningful”. Consider the two well known theorems           (1)    con(ZF)  implies  con(ZFC)

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          (2)    con(ZF)  implies  con(PA) From a foundational standpoint, these two theorems act in opposite directions.  In case (1), if the result had been {con(ZF) implies NOTcon(ZFC)} then it would have undermined our confidence in the "C" part of ZFC.  However, if the result of case (2) had been {con(ZF) implies NOTcon(PA)}, it would presumably would have undermined our confidence in ZF, not in PA (because the principles of PA are considered to be more self-evidently consistent that those of ZF). The only kind of proof that would enhance our confidence in PA would be of the form                       con(X) implies con(PA) where X is a system whose consistency is MORE self-evident than that of PA. (This is the key point.)  For example, Hilbert hoped that with X = 1st Order Logic it would be possible to prove this theorem, thereby enhancing our confidence in the consistency of PA.  That would have been a meaningful proof of the consistency of PA.  However, it's now known that such a proof is impossible (unless you believe in the existence of a formal system that is more self-evidently consistent than PA but that cannot be modeled within the system of natural numbers). Others argue that it may be possible to regard the theory of primitive recursive functionals as more evidently consistent than PA. It's well known that both transfinite induction and the theory of primitive recursive functionals cannot be modeled within the system of natural numbers, but we do not need to claim that it would be impossible to regard such principles as more evidently consistent than PA. We simply observe that no one does – and for good reason.  Each represents a non-finitistic extension of formal principles, which is precisely the source of uncertainty in the consistency of PA. Again, there is a little thought experiment that sometimes helps people sort out their own hierarchy of faith:  If, assuming the consistency of the theory of primitive recursive functionals, one could prove that PA is NOT consistent, would we be more inclined to abandon PA or the theory of primitive recursive functionals? Some mathematicians assert that doubts about whether PA is consistent, and whether it can be proven to be consistent, are trivial and pointless, partly because this places all of mathematics in doubt. However, as to the triviality, much of the most interesting and profound mathematics of this century has been concerned with just such doubts. As to the number of proofs that are cast into doubt by the possibility of inconsistency in PA, the "perfect consistency or total gibberish" approach to formal systems evidently favored by many mathematicians is not really justified. It just so happened that Russell and Whitehead's FOL was a convenient finitistic vehicle to use as an example, although subsequent developments showed that this maps to computability, from which the idea of a universal Turing machine yields a large segment (if not all) of what can be called cognition. Of course, people sometimes raise the possibility of a finitistic system that cannot be modeled within the theory of natural numbers but, as Ernst Nagel remarked, "no one today appears to have a clear idea of what a finitistic proof would be like that is NOT capable of formulation within arithmetic". PA can be modeled within ZF.  It follows that con(ZF) implies con(PA). This was simply

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presented as an illustration of how the formal meaning of a theorem of this form depends on our "a priori" perceptions of the relative soundness of the two systems. Some mathematicians have alluded to the "usual" proof of con(PA) but have not specified the formal system within which this "usual" proof resides.  Since there are infinitely many possibilities, it's not possible to guess which specific one they have in mind. In general terms, if there exists a proof of con(PA) within a formal system X, then we have the meta-theorem { con(X) implies con(PA) }, so we can replace X with whatever formal system we consider to be the "usual" one for proofs of PA.  The meaningfulness of such a theorem depends on our perception of the relative soundness of the systems X and PA.  I assert that no system X within which con(PA) can be proved is evidently more consistent that con(PA) itself. Here's a related quote from "Mathematical Logic", 2nd Ed, by H. D. Ebbinghaus, J. Flum, and W. Thomas, Springer-Verlag, 1994: 

The above argument [Goedel's 2nd Thm] can be transferred to other systems where there is a substitute for the natural numbers and where R-decidable relations and R-computable functions are representable.  In particular, it applies to systems of axioms for set theory such as ZFC...  Since contemporary mathematics can be based on the ZFC axioms, and since...the consistency of ZFC cannot be proved using only means available within ZFC, we can formulate [this theorem] as follows:  If mathematics is consistent, we cannot prove its consistency by mathematical means.

              I wanted to include this quote because every time I say there is no meaningful formal proof of the consistency of arithmetic (PA), someone always says "Oh yes there is, just work in ZFC, or ZF, or PA + transfinite induction, or PA + primitive recursive functionals, or PA + con(PA), or just use the "usual" proof, or (as one mathematician advised me) just stop and think about it!"  But none of these proposed "proofs" really adds anything to the indubitability of con(PA).  Of course, it's perfectly acceptable to say "I'm simply not interested in the foundational issues of mathematics or in the meaning of consistency for formal systems".  However, disinterest should not be presented as a substitute for proof. In response to the rhetorical question “Are we really justified in asserting that every one of the infinitely many paths from every premise to its evaluation gives the  same result?”, some mathematicians will say that if we allow associativity of modus ponens, then there is essentially only one proof of any classical formula. However, this misses a crucial point. If we define the "essential path" between any two points in space as a function only of the beginning and ending points, then there is essentially only one path from New York to Los Angeles.  This definition of a "path", like the corresponding definition of a "proof" is highly reductionist. We are certainly free to adopt that definition if we wish, but in so doing we ignore whole dimensions of non-trivial structure, as well as making it impossible to reason meaningfully about "imperfectly consistent" systems which, for all we know, may include every formal system that exists. The unmeasured application of the modus ponens (rule of detachment) is precisely what misleads people into thinking that a formal system must be either absolutely consistent or total gibberish.  Then, when they consider systems such as naive set theory, PA, or ZFC, which are clearly not total gibberish, they conclude that they must be absolutely consistent. Some mathematicians claim that PA is a collection of simple statements, all of which are

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manifestly true about the positive integers. However, although the explicit axioms look simple, they are meaningful only in the context of a vast mathematical apparatus that includes conventions regarding names of variables, rules about substitutions, and more generally rules of inference and implication. For example, if we propose a sequence of variable names x, x', x'' and so on, we need to know that all these names are different. Of course, most of us would accept this as intuitively obvious, but if the formal system is to be absolutely watertight, we must prove that they are different. Thus we end up needing to apply the Peano axioms to the language used to talk about them. As Jan Nienhuys observed, one of the basic subtleties related to Peano’s axioms is that they implicitly assert that any set of positive integers has a smallest element. In view of Ramsey numbers, it’s clear that even in simple cases this is more a metaphysical assertion than a matter of "manifestly true". How can we have confidence that any set of positive integers, no matter how unwieldy its definition, has a smallest element?  (The unwieldiness of definitions is made possible by having an unlimited amount of positive integers and variable names at our disposal.) The Peano axioms plus a formal system in which they are embedded must be free of contradictions, i.e. for no statement  X is both X and not-X provable. (Also, note that we allow variables such as X to denote statements, so the formal system should have watertight descriptions - no hand waving or illustration by example allowed - of how to go about making variables represent statements.) Only such a system as a whole can conceivably contain a contradiction. Without the formal framework a contradiction doesn't make much sense. Here’s a sampling of comments on this topic received from various mathematicians: 

Personally I think that mathematicians are sometimes too hung up on the details of formalizations and lose track of the actual math they are talking about. As the old saying goes, mathematicians are Platonists on weekdays and formalists on Sunday. I personally think that this tendency is a good and healthy one! But Goedel's Theorem can be derived even with a substantially weakened Axiom of Induction, and I believe that such a weakened axiom would then also lead to a contradiction, so it seems that we would have to throw away all induction. If the consistency of FOL is somehow a hypothetical matter, why should I assume that you or I make any sense whatever in our babbling? If arithmetic was inconsistent, wouldn't all the bridges fall down? If so, it's an awfully trivial point, and hardly worth making.  She might as well say "Everything we think might have an error in it because some demon somewhere is messing with our brains."  Quite true.  So what? A proof from ZF brings with it the supreme confidence that a century of working with ZF and beyond has given us.

 One mathematician argued that (apparently with a straight face) “The consistency of ZFC is provable in ZFC+Con(ZFC), the consistency of ZFC+Con(ZFC) is provable in FC+Con(ZFC)+Con(ZFC+Con(ZFC)), etc., so the infinite hierarchy of such systems provides a complete proof of the consistency of ZFC.  Or, just to press the point, there is actually a system which proves the consistency of any system... This system even proves its own consistency.” This proposed hierarchy of systems possesses an interesting structure.  For example, each system has

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a successor which is assumed to be a system, and distinct from any of the previous systems (i.e., no loops).  And of course it's necessary to justify speaking of the completed infinite hierarchy, implying induction.  The whole concept might almost be formalized as a set of axioms along the lines of: (i)   ZFC is a System.(ii)  For any System X, the successor X+con(X) is also a System.(iii) ZFC is not the successor of a System.(iv)  If the successors of Systems X and Y are the same, then X and       Y are the same.(v)   If ZFC is in a set M, and if for every System X in M the       successor X+con(X) is in M, then M contains every System. Now we're getting somewhere!  If we can just prove this is consistent... By the way, here's an interesting quote from Ian Stewart's book "The Problems of Mathematics": 

Mathematical logicians study the relative consistency of different axiomatic theories in terms of consistency strength.  One theory has greater consistency strength than another if its consistency implies the consistency of the other (and, in particular, if it can model the other).  The central problem in mathematical logic is to determine the consistency strength in this ordering of any given piece of mathematics. One of the weakest systems is ordinary arithmetic, as formalized axiomatically by Giuseppe Peano...  Analysis finds its place in a stronger theory, called second-order arithmetic.  Still stronger theories arise when we axiomatize set theory itself.  The standard version, Zermelo-Frankel Set Theory, is still quite weak, although the gap between it and analysis is large, in the sense that many mathematical results require MORE than ordinary analysis but LESS than the whole of ZF for their proofs.

 It should be noted that Peano's Postulates (P1-P4 in Andrews, "An Introduction to Mathematical Logic and Type Theory") assert the existence of an infinitely large set. Since it is not possible to simple ‘exhibit’ an infinite model, any such assertion must simply assume that it is possible to speak about such things without the possibility of contradictions. And indeed the Axiom of Infinity is one of the axioms of ZF. Hence the assertion that PA can be “proved” within ZF can be taken as nothing more than a joke. Much of this depends on the status of induction. Normally we are careful to distinguish between common induction and mathematical induction.  The former consists of drawing general conclusions empirically from a finite set of specific examples.  The latter is understood to be an exact mathematical technique that, combined with the other axioms of arithmetic, can be used to rigorously prove things about infinite sets of integers. For example, by examining the square number 25 we might observe that it equals the sum of the first five odd integers, i.e.,  52 = 1+3+5+7+9.  We might then check a few more squares and by common induction draw the general conclusion that the square of N is always equal to the sum of the first N odd numbers.  In contrast, mathematical induction would proceed by first noting that the proposition is trivially true for the case N=1.  Moreover, if it's true for any given integer n it is also true for n+1 because (n+1)2 n2 equals 2n+1, which is the (n+1)th odd number.  Thus, by mathematical induction it follows that the proposition is true for all N. Understandably, many mathematicians take it as an insult to have mathematical induction confused with common induction.  The crucial difference is that MI requires a formal

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implicative relation connecting all possible instances of the proposition, whereas CI leaps to a general conclusion simply from the fact that the proposition is true for a finite number of specific instances.  Of course, it's easy to construct examples where CI leads to a wrong conclusion but, significantly, CI often leads to correct conclusions.  We could devote an entire discussion to "the unreasonable effectiveness of common thought processes", but suffice it to say that for a system of limited complexity the possibilities can often be "spanned" by a finite number of instances. In any case, questions about the consistency of arithmetic may cause us to view the distinction between MI and CI in a different light.  How do we know that (n+1)2 n2 always equals 2n+1?  Of course this is a trivial example; in advanced proofs the formal implicative connection can be much less self-evident.  Note that when challenged as to the absolute consistency of formal arithmetic, one response was to speak of "the supreme confidence that a century of working with ZF has given us".  This, of course, is nothing but common induction.  So too are claims that arithmetic must be absolutely consistent because otherwise bridges couldn't stand up and check books wouldn't balance.  (These last two are not only common induction, they are bad common induction.) Based on these reactions, we may wonder whether, ultimately, the two kinds of induction really are as distinct as is generally supposed.  It would seem more accurate to say that mathematical induction reduces a problem to a piece of common induction in which we have the very highest confidence, because it represents the pure abstracted essence of predictability, order, and reason that we've been able to infer from our existential experience.  Nevertheless, this inference is ultimately nothing more (or less) than common induction. It's clear that many people are highly disdainful of attempts to examine the fundamental basis of knowledge.  In particular, some mathematicians evidently take it as an affront to the dignity and value of their profession (not to mention their lives) to have such questions raised.  (One professional mathematician objected to my quoting from Morris Kline's "Mathematics and the Loss of Certainty", advising me that it is “a very very very very very very pathetic and ignorant book”.)  In general, I think people have varying thresholds of tolerance for self-doubt.  For many people the exploration of philosophical questions reaches its zenith at the point of adolescent sophistry, as in "did you ever think that maybe none of this is real, and some demon is just messing with our minds?"  Never progressing further, for the rest of their lives whenever they encounter an issue of fundamental doubt they project their own adolescent interpretation onto the question and dismiss it accordingly.  In any case, this discussion has provided some nice examples of reactions to such questions, including outrage, condescension, bafflement, fascination, and complete disinterest. The most controversial point seems to have been my contention that every formal system is inconsistent. I was therefore interested to read in Harry Kessler’s “Diaries of a Cosmopolitan” about a discussion that Kessler had had at a dinner party in Berlin in 1924. 

I talked for quite awhile to Albert Einstein at a banker's jubilee banquet where we both felt rather out of place. In reply to my question what problems he was working on now, he said that he was engaged in thinking. Giving thought to almost any scientific proposition almost invariably brings progress with it. For without exception, every scientific proposition was wrong. That was due to human inadequacy of thought

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and inability to comprehend nature, so that every abstract formulation about it was always inconsistent somewhere. Therefore, every time he checked a scientific proposition, his previous acceptance of it broke down and led to a new, more precise formulation. This was again inconsistent in some respects and consequently resulted in fresh formulations, and so on indefinitely.

http://www.mathpages.com/home/kmath347/kmath347.htm

Sunday, August 28, 2005Shameless Plug Here's a brief description of my (mathematical) novel that will be out in the next 12-18 months (2 of you sent mail asking for it and I don't need much more incentive than that!):

The human heart yearns for absolute truth and certainty. But can we be truly certain about anything—or is everything we believe accidental and meaningless, shaped by the happenstance of genetic and social inheritance? Perhaps mathematics alone, with its uncompromising rigor, can lead us to certainty. In our 90,000 word novel, we examine where mathematics can and cannot take us in the quest for certainty.

Our book will show the reader the following: First, that mathematics can be deeply beautiful—in this regard it is not unlike music or painting; second, that mathematics has profound things to say about whether absolute truth is obtainable; and lastly, that a novel is the best medium through which to convey the excitement and meaning of doing mathematics

Our protagonist, Vijay Sahni, an Indian mathematician, has glimpsed the certainty that mathematics can provide and does not see why its methods cannot be extended to all branches of human knowledge, including religion. Arriving to pursue his academic career in a small New Jersey town in 1919, his outspoken views land him in jail, charged under a little-known Blasphemy law (on the state statute books to this day). His beliefs are challenged by Judge John Taylor, who does not believe that mathematical deduction can be applied to matters of faith. In their discussions the two men discover the power—and the fallibility—of Euclid's axiomatic treatment of geometry, long considered the gold standard in human certainty. In the end both Vijay and Judge Taylor come to understand that doubt must always accompany knowledge. posted by Gaurav at 2:39 PM - Gaurav Suri Location:California, United States

I'm a management consultant by vocation and a mathematician by avocation. I've authored a mathematical novel that touches on several topics discussed on these pages. It has been accepted for publication and will be out next year.

Here's a brief description of a mathematics-based novel written by Gaurav Suri and Hartosh Singh Bal. It's called A Certain Ambiguity and will be published in 2007 by Princeton. They've offered books to Mathforge for review--hopefully they'll send an advanced copy. Stay tuned...

Two Indians take a novel approach to maths Sachin Kalbag Tuesday, August 08, 2006 00:21 IST

Gaurav Suri, Hartosh Bal impress Princeton with unique tale.

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WASHINGTON: When Princeton University decides to publish your book, it usually sends you an author response form. Like any other form, it has the usual zzzz questions about your name, address, and background. But Q22 is interesting. It asks you to list the prizes and awards your book will qualify for.San Francisco-based management consultant and Stanford University alumnus Gaurav Suri answered that question in one word: Pulitzer. He then added a smiley for good measure.Did he mean that in jest? Maybe he did, maybe not. There is, like the title of his groundbreaking mathematical novel, a certain ambiguity to it. Or, as Suri puts it, “Perhaps the search for absolute certainty is destined to remain just a search.”Written by Suri and his childhood friend and New York University graduate Hartosh Singh Bal, A Certain Ambiguity, a mathematical novel, will be published by Princeton in 2007. It tells the story of an Indian mathematician, who comes to the US in 1919 and gets caught in the web of a little-known blasphemy law, and how his grandson — a Stanford student — discovers his story.The book is the result of long instant messenger chats between Suri and Bal. The latter, a former mathematics teacher, is a freelance journalist and wildlife photographer based in New Delhi. “We often talked late into the night,” says Suri. “First about the mathematics and then about the plot-lines.” They did that for over five years, and got down to sending the finished book to publishers only late last year.Their love for maths, however, is older. “We were 17 when we fell in love with mathematics. The book that did that was George Gamow’s One, Two, Three… Infinity.”Suri hopes his readers will learn to love mathematics, because “life and mathematics are closely related”. Suri told Princeton: “As the two main characters in the book struggle to determine how certain they are about what they believe, they come to find that the dilemmas of mathematics and ordinary life cannot be separated.”Therein, perhaps, lies an ambiguous certainty.

http://www.dnaindia.com/report.asp?NewsID=1046029

Numbers and Experience It is often argued that while geometry is unable to adequately describe the world around us, numbers are more reliable, more certain. 2 cows plus 2 cows, always equal 4 cows. Unlike non euclidean geometries, there are no non standard arithmetics. Gauss, for a time at least, believed that ‘truth resides in number.’ In a similar vein Jacobi said “God ever arithmetizes” (as opposed to eternally geometrizing).However, as Kline observes in Mathematics, The Loss of Certainty, the sharpest attack on the truth of arithmetic came from Hermann von Helmoholtz, a superb physicist and mathematician. In his Counting and Measuring he observed that the problem in arithmetic lay in the automatic application of arithmetic to physical phenomena. Some kinds of experiences suggest whole numbers and fractions, while others don’t: one raindrop added to another does not make two raindrops. Two pools of water, one at 40◦ another pool of water at 50◦ when mixed together do not make a pool of water at 90◦. Lebesgue facetiously pointed out that if one puts a lion and a rabbit in a cage, one will not find two animals an hour later! Helmoholtz gives many (more serious) examples but his overarching point is that only experience can tell us where to apply, and not apply, standard arithmetic.

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Like Euclidean Geometry, arithmetic is not absolutely applicable to the physical world. posted by Gaurav at 11:27 PM

Monday, June 13, 2005

Evolutionary Mathematics Chaitin, as he often does, has got me thinking. He writes:

Von Neumann also said that we ought to have a general mathematical theory of the evolution of life... But we want it to be a very general theory, we don't want to get involved in low-level questions like biochemistry or geology... He insisted that we should do things in a more general way, because von Neumann believed, and I guess I do too, that if Darwin is right, then it's probably a very general thing.For example, there is the idea of genetic programming, that's a computer version of this. Instead of writing a program to do something, you sort of evolve it by trial and error. And it seems to work remarkably well, but can you prove that this has got to be the case? Or take a look at Tom Ray's Tierra... Some of these computer models of biology almost seem to work too well---the problem is that there's no theoretical understanding why they work so well. If you run Ray's model on the computer you get these parasites and hyperparasites, you get a whole ecology. That's just terrific, but as a pure mathematician I'm looking for theoretical understanding, I'm looking for a general theory that starts by defining what an organism is and how you measure its complexity, and that proves that organisms have to evolve and increase in complexity. That's what I want, wouldn't that be nice?

And if you could do that, it might shed some light on how general the phenomenon of evolution is, and whether there's likely to be life elsewhere in the universe. Of course, even if mathematicians never come up with such a theory, we'll probably find out by visiting other places and seeing if there's life there... But anyway, von Neumann had proposed this as an interesting question, and at one point in my deluded youth I thought that maybe program-size complexity had something to do with evolution... But I don't think so anymore, because I was never able to get anywhere with this idea...

Tons of interesting stuff to chew on, but I'll limit myself to this: Imagine a simulation where you have two entities: organisms and resources. The organisms are just data structures which reproduce when they have been getting enough resources. The resources are re-generable and are of various types.

Now let's add on a few complexities: Assume that an organism 'eats' only certain types of resources. So Organism 42 can only live on Resource 118 for example. Further assume that the quantity of Resources stays relatively stable...with exceptions of rare time units of plenty and others (also rare) of drought. Also assume that there can be more than one type of Organism that consumes a certain type of Resource, and also that there are Resources that are not consumed by any organism when the simulation starts.

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An Organism will then have the following data elements: Its type [corresponds to the species it belongs to]; its number [i.e. its name]; the Resource number(s) it consumes; its wellness number - a measure of how well fed the organism is - if the wellness number goes over a limit the organism will reproduce; an organism competitive index which will measure how well the individual competes within his species; and a species competitive number that measures how well the species competes with other species vying for the same resource. Reproduction passes on the competitive indices to the progeny. When the wellness index falls below a certain level, the organism dies.

Now also imagine that you have random mutations. A random mutation could change the type of resources an individual consumes and/or its competitive indices (either up or down).

These are only the barest details...but I hope you believe that it is possible to capture the main points of Darwin theory in a reasonable simulation.

Hit start and run the simulation: You will probably see organisms dying and being born; species will be created by the right mutations - they will also thrive or struggle - but eventually all will die out. The world itself may reach some kind of stable equilibrium, but more likely than not...at some point we'd hit zero organisms or zero resources.

All this is worth doing in its own right (in fact I'd be shocked if someone hasn't already done it), but now, just for fun, imagine one last externality: Say that organisms of a certain complexity level can perceive a proportional complexity of mathematical truths. So for example an organism of complexity index 1088 could really 'get' that there can be no largest prime (but other, more difficult theorems are beyond it), and an organism of complexity index 4063 could 'get' the prime number theorem ('get' = a deep understandig that does not allow for the result not be true. Similar, but not equal to proof).

It seems to me then that there will always be mathematical statements that we humans couldn't get, no matter what.

This is far from air tight, but there may be something to chew on here.

--Gaurav Suri posted by Gaurav at 5:21 PM

http://meaningofmath.blogspot.com/

Tuesday, June 07, 2005

The Voynich Manuscript The Voynich manuscript is a very old 230+ page manuscript written in a code that no one has been able to crack. Here's the Wikipedia entry:

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The Voynich manuscript is a mysterious illustrated book of unknown contents, written some 600 years ago by an anonymous author in an unidentified alphabet and unintelligible language.Over its recorded existence, the Voynich manuscript has been the object of intense study by many professional and amateur cryptographers — including some top American and British codebreakers of World War II fame — who all failed to decipher a single word. This string of egregious failures has turned the Voynich manuscript into the Holy Grail of historical cryptology; but it has also given weight to the theory that the book is nothing but an elaborate hoax — a meaningless sequence of random symbols.The book is named after the Russian-American book dealer Wilfrid M. Voynich, who acquired it in 1912. It is presently item MS 408 in the Beinecke Rare Book Library of Yale University.

The book has strange drawings of flowers, alien looking plants and naked women. Its history is utterly fascinating. Find out more here. Almost as fascinating as the book itself is the history of the men who have attempted to decipher the symbols. In many cases they have bolted on semi-plausible theories even though they were few supporting facts to be had.

All this forces us me to ask - where does the meaning of anything lie? Surely it is not in the symbols we use to communicate the ideas. No, meaning must lie in the mind of the humans deciphering the language/symbol. The human mind has the power to give anything meaning; it also has the power to force meaning where there is none to be had. Be it the notion of the color red, the undecidability of the Continuum Hypothesis or the rhythms of the Voynich manuscript.

--Gaurav posted by Gaurav at 11:03 PM

Friday, May 20, 2005

First thoughts on Rebecca Goldstein’s, Incompleteness: The proof and paradox of Kurt Gödel I bought this book despite myself. I’ve carefully studied Gödel’s Incompleteness Theorems and expected Goldstein to give a soft, non rigorous, largely biographical treatment which wouldn’t teach me anything new. I bought the book almost out of duty – it is after all in the subject I care most deeply about – I should read it just in case. I am glad I got took the chance. This is a great book, and I am not one to use the term loosely.

The power of the book doesn’t come from its treatment of the theorem itself (she does an adequate job, but others have done better. See for example Nagel and Newman’s classic, Gödel’s Proof for a fine non-technical treatment); rather the books achievement is that it puts Gödel’s work in context. Goldstein successfully (and finally) gives Gödel’s theorems the philosophical interpretation that he himself would have intended.

Before reading Incompleteness I often wondered why Gödel, an avowed Platonist, did most of his work in Mathematical Logic, the most formalist of all mathematical fields. Also, why did he join the Logical Positivists of Vienna who in their way were the most extreme kind of Formalists; and lastly why did Gödel associate himself with a group who revered the teachings

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of Wittgenstein – the very same Wittgenstein who essentially claimed that all of a mathematics was a mere tautology (a claim that was almost surely quite repulsive to Gödel, and to almost every other mathematician).

Goldstein answered all of this (and more). She gets her answers not from the mathematics, but from the story of Gödel’s life and the philosophical battles that drove him.

In brief, the story is that the Logical Positivists essentially believed that truth lived in the precise, meaning-aware use of language. According tho them, it is only possible to identify a statement as being true or false by proving or disproving it by experience. Logic and mathematics was excluded from this rule; they claimed that mathematics was a branch of logic and was for all intents and and purposes a mere tautology.

Gödel on the other hand was a Platonist; he believed that mathematicians uncovered truths about the universe, and mathematical concepts were merely communicated by—but not contained within—its equations and symbols. Yet, confusingly, Gödel belonged to a Positivist group. He largely stayed silent through their meetings, neither objecting nor agreeing, for that was not his way.

But the internal storm of disagreement that welled within him did lead him to prove that Positivists were wrong. He proved that the structural manipulation of mathematical symbols could not yield all statements that we know to be true. He demonstrated a ‘true’ statement that was not provable—which should have banished Logical Positivism for ever.

Yet it didn’t; for Godel, before Goldstein’s book, was never well understood.

I’ll have a lot more to say about all this in the coming weeks. posted by Gaurav at 11:33 AM

http://meaningofmath.blogspot.com/

BBC-Dangerous KnowledgeJuly 31, 2008 In this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

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The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity.

http://www.youtube.com/watch?v=Cw-zNRNcF90&feature=related

also available here:http://bestdocumentaries.blogspot.com/2007/09/dangerous-knowledge-full-documentary.html

excellent math visuals and links:http://faculty.etsu.edu/gardnerr/Math-Videos/Math-Videos.htm

DANGEROUS KNOWLEDGEhttp://www.bbc.co.uk/bbcfour/documentaries/features/dangerous-knowledge.shtml

BBC Two: Wednesday 11 June 2008 11.30pmIn this one-off documentary, David Malone looks at four brilliant mathematicians - Georg Cantor, Ludwig Boltzmann, Kurt Gödel and Alan Turing - whose genius has profoundly affected us, but which tragically drove them insane and eventually led to them all committing suicide.

The film begins with Georg Cantor, the great mathematician whose work proved to be the foundation for much of the 20th-century mathematics. He believed he was God's messenger and was eventually driven insane trying to prove his theories of infinity.

"MY BEAUTIFUL PROOF LIES ALL IN RUINS"Presenter David Malone reads letters which demonstrate Cantor's crumbling self-belief.

Ludwig Boltzmann's struggle to prove the existence of atoms and probability eventually drove him to suicide. Kurt Gödel, the introverted confidant of Einstein, proved that there would always be problems which were outside human logic. His life ended in a sanatorium where he starved himself to death.

Finally, Alan Turing, the great Bletchley Park code breaker, father of computer science and homosexual, died trying to prove that some things are fundamentally approvable.

The film also talks to the latest in the line of thinkers who have continued to pursue the question of whether there are things that mathematics and the human mind cannot know. They include Greg Chaitin, mathematician at the IBM TJ Watson Research Center, New York, and Roger Penrose.

Dangerous Knowledge tackles some of the profound questions about the true nature of reality that mathematical thinkers are still trying to answer today.

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