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Mathematics: The Loss of Certainty
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.
384 pages, Paperback
First published January 1, 1980
37 people are currently reading
1885 people want to read
About the author
Morris Kline
76 books104 followersMorris Kline was a Professor of Mathematics, a writer on the history, philosophy, and teaching of mathematics, and also a popularizer of mathematical subjects.
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December 30, 2009
For what it's worth, the correct title is Mathematics: The Loss of Certainty, not just "Mathematics" -- and omitted line matters as it is this, truly that the book is about. Kline exposes the gradual discovery by mathematicians that their great endeavor did not lead to perfect truth, as was once supposed, but to "truth" of a very different sort, truths that in truth we are still learning to understand.
This is a truly awesome book. It well deserves its rediscovery and recent appearance back on the tables and shelves of bookstores -- it is likely that the world was literally not quite ready for it the first time it was released.
Kline takes the lay reader (that is, no particular skill or deep knowledge of mathematics or physics required) on a historical tour of the bleeding edge of axiomatic mathematics, epistemology, logic, set theory, and metaphysical philosophy -- nothing less than a whirlwind review of the human mind and how it has developed its highest and most reliable knowledge of the real world for the last 2600 years. This is a subject dear to my own heart; I'm working on a book that presents a very similar analysis in the specific context of developing an axiomatically sound worldview. Kline's work provides a compelling and convincing picture of how the grand intellectual problem of developing certain knowledge of any sort (but one) led to complete and utter failure, and indeed to the confounding of the premise that such a thing is at all possible.
To summarize the conclusion of this work in a nutshell -- there is no such thing as a priori knowledge of truth, not even in mathematics (where for most of the last 2500 years, it was believed that certain truths in mathematics were ineluctable, true without any contingency). All "truths" in mathematics turn out to be contingent truths, contingent upon axioms that are not themselves "self-evident truths" but instead are unprovable assumptions. Kline reviews the grand discoveries of the Enlightenment that inevitably led to this inescapable conclusion: the discovery the plane geometry is only one of an infinity of curved space geometries, that there are similarly many kinds of numbers, many algebras, many mathematical groups, all defined and specified by their axioms, which cannot be said to be "true" or "false" but are rather assumptions made to facilitate the development of the contingent theories.
Kline goes beyond this, however, and explores the gradual discovery of the problems with paradoxes in formal theories, paradoxes that had long been known in ordinary discourse but that turned out to be a serious problem when Cantor, Russell, and others developed formal set theory. It turned out that set theory, extended in trivial ways to reference itself, was capable of generating unprovable, paradoxical statements; one could easily define set universes that could not be partitioned into sets (e.g. the Russell paradox), one could define the largest number that can be described in 100 characters or less of English, and then note that a larger number can always be defined in 100 characters or less. This process led, a step at a time, to Poincare giving up on his grand plan of axiomatizing mathematics and to Godel's incompleteness theorems that proved that most formal mathematical systems contained precisely this kind of poison provided only that they were sufficient to encompass ordinary arithmetic. Mathematics could not be made both consistent and complete, even allowing for "self-evident" axioms that (examined carefully) are nothing of the sort.
While Kline focuses on the development of mathematics, he does not ignore the discovery of physics and development of an empirically supported description of the physical universe that led the invention of ever more complex forms of mathematics, discoveries that confounded the more or less religious beliefs that prevailed that mathematical truths were perfect truths. The development of calculus and the systematic evolution of number theory from natural numbers, through rational numbers, irrational numbers, real numbers, complex numbers, and on to the modern theory of geometric algebras (e.g. quaternions) were largely motivated by physics. Kline points out that mathematicians did not generally believe in negative numbers until roughly the latter 1800's, where of course today we cannot easily imagine a mathematics without them.
Kline's conclusions form much of the basis for my (independently developed) thesis presented in Axioms. Ultimately, if even the elegance and formal structure of pure mathematics does not lead one to non-contingent truth, seekers of truth need to get used to developing contingent truths, truths that are know certain knowledge but knowledge that can be doubted. Kline delicately avoids tackling the work of David Hume in his book (while making short work of most of Hume's contemporaries and successors) but in the end one can only be left with the feeling that Hume truly was the "seal of the philosophers" as Mohammed never was the seal of the prophets.
To conclude, this book is a tremendous work, a classic only to be compared to gems such as Russell's Problems in Philosophy. To be honest, it should be required reading for anyone who wishes to consider themselves well-read and well-informed about any sort of philosophy: students of mathematics, physics, philosophy, religion without question, but it is so eminently readable and informative that it any high school student would benefit from reading it, as would (of course) any literate adult. I strongly, strongly recommend it, especially to people who want to participate in metaphysical arguments and not make a fool out of themselves by falling into one of the many, many traps of "pure reason" that this book lays bare.
rgb
This is a truly awesome book. It well deserves its rediscovery and recent appearance back on the tables and shelves of bookstores -- it is likely that the world was literally not quite ready for it the first time it was released.
Kline takes the lay reader (that is, no particular skill or deep knowledge of mathematics or physics required) on a historical tour of the bleeding edge of axiomatic mathematics, epistemology, logic, set theory, and metaphysical philosophy -- nothing less than a whirlwind review of the human mind and how it has developed its highest and most reliable knowledge of the real world for the last 2600 years. This is a subject dear to my own heart; I'm working on a book that presents a very similar analysis in the specific context of developing an axiomatically sound worldview. Kline's work provides a compelling and convincing picture of how the grand intellectual problem of developing certain knowledge of any sort (but one) led to complete and utter failure, and indeed to the confounding of the premise that such a thing is at all possible.
To summarize the conclusion of this work in a nutshell -- there is no such thing as a priori knowledge of truth, not even in mathematics (where for most of the last 2500 years, it was believed that certain truths in mathematics were ineluctable, true without any contingency). All "truths" in mathematics turn out to be contingent truths, contingent upon axioms that are not themselves "self-evident truths" but instead are unprovable assumptions. Kline reviews the grand discoveries of the Enlightenment that inevitably led to this inescapable conclusion: the discovery the plane geometry is only one of an infinity of curved space geometries, that there are similarly many kinds of numbers, many algebras, many mathematical groups, all defined and specified by their axioms, which cannot be said to be "true" or "false" but are rather assumptions made to facilitate the development of the contingent theories.
Kline goes beyond this, however, and explores the gradual discovery of the problems with paradoxes in formal theories, paradoxes that had long been known in ordinary discourse but that turned out to be a serious problem when Cantor, Russell, and others developed formal set theory. It turned out that set theory, extended in trivial ways to reference itself, was capable of generating unprovable, paradoxical statements; one could easily define set universes that could not be partitioned into sets (e.g. the Russell paradox), one could define the largest number that can be described in 100 characters or less of English, and then note that a larger number can always be defined in 100 characters or less. This process led, a step at a time, to Poincare giving up on his grand plan of axiomatizing mathematics and to Godel's incompleteness theorems that proved that most formal mathematical systems contained precisely this kind of poison provided only that they were sufficient to encompass ordinary arithmetic. Mathematics could not be made both consistent and complete, even allowing for "self-evident" axioms that (examined carefully) are nothing of the sort.
While Kline focuses on the development of mathematics, he does not ignore the discovery of physics and development of an empirically supported description of the physical universe that led the invention of ever more complex forms of mathematics, discoveries that confounded the more or less religious beliefs that prevailed that mathematical truths were perfect truths. The development of calculus and the systematic evolution of number theory from natural numbers, through rational numbers, irrational numbers, real numbers, complex numbers, and on to the modern theory of geometric algebras (e.g. quaternions) were largely motivated by physics. Kline points out that mathematicians did not generally believe in negative numbers until roughly the latter 1800's, where of course today we cannot easily imagine a mathematics without them.
Kline's conclusions form much of the basis for my (independently developed) thesis presented in Axioms. Ultimately, if even the elegance and formal structure of pure mathematics does not lead one to non-contingent truth, seekers of truth need to get used to developing contingent truths, truths that are know certain knowledge but knowledge that can be doubted. Kline delicately avoids tackling the work of David Hume in his book (while making short work of most of Hume's contemporaries and successors) but in the end one can only be left with the feeling that Hume truly was the "seal of the philosophers" as Mohammed never was the seal of the prophets.
To conclude, this book is a tremendous work, a classic only to be compared to gems such as Russell's Problems in Philosophy. To be honest, it should be required reading for anyone who wishes to consider themselves well-read and well-informed about any sort of philosophy: students of mathematics, physics, philosophy, religion without question, but it is so eminently readable and informative that it any high school student would benefit from reading it, as would (of course) any literate adult. I strongly, strongly recommend it, especially to people who want to participate in metaphysical arguments and not make a fool out of themselves by falling into one of the many, many traps of "pure reason" that this book lays bare.
rgb
December 6, 2009
Overall, very interesting point of view, but was severely disappointed with Morris' self-serving diatribe against so-called "pure mathematics" at the end of the book. The sharp turn in the chapter "The Isolation of Mathematics" Morris analyzes modern mathematics and prescribes (rather than describes) a rift between applied and pure mathematics. Pure mathematics, he says, has proven itself useless intellectual endeavor. He quotes endlessly other mathematicians who agree with his view, and makes exactly two quotes from opposing viewpoints which he rebuts using the aforementioned quotes. He is staunchly anti-abstraction, anti-algebra, anti-topology except where it applies to a specific real-world problem. Yet he has apparently made no inquiry into the many ways these theories actually _are_ applied to fields like mathematical physics, biology, modeling, statistics, and so forth. He admits he is offended by "mathematics for mathematics sake" which is a reasonable position; but he makes the mistake of ascribing this opinion to other prominent men in his field, when in reality those men lived in a time where the distinction between mathematics and physical sciences was far blurrier than it is now.
Morris is simply railing against the academy that left him behind, that modern mathematics is not what he believes it should be and does not conform to his standards, and therefore is of no use to him, nor anyone else. As a pure mathematician, I can say with certainty it is plain ignorance to say "Algebraic topology is useless" (p.361) when it a perfectly useful in the description of, say, force fields in a curved space. Morris himself describes many ways abstraction has led to physical models; but in his mind, abstraction is only useful if it is taken to advance science. If an abstraction finds its use after it is a mathematical theory, then by his argument it is a waste of time.
To be fair, many of these "useless" branches of mathematics have been found extremely useful in mathematical physics since Morris wrote the book in 1980. It would be interesting to see what his opinion of modern applications of abstract geometry, algebraic topology, and other "useless" subjects in theories such as quantum field theory, theory of computing, dynamical systems, models for speciation and mutation, DNA and protein geometry, modeling the nervous system, and so forth.
I suppose it is common for scientists to publish books authenticated by their their credentials in order to get their points of view out to the public. And to an extent, I do agree; modern mathematics is often too self-serving and disconnected from other sciences. I won't get into that here; but I definitely disagree with his thesis for the last three chapters that "Pure mathematicians have ruined mathematics". I would contend that "Pure mathematicians have found that mathematics can be beautiful and interesting, and hs developed itself into its own identity". It is a shame that Morris cannot see mathematics the same way I or my colleagues can; on the other hand, I know many Morris' too, and we can all agree to disagree because no matter what we all love the subject, and disagree on what's worthy of our study.
Morris is simply railing against the academy that left him behind, that modern mathematics is not what he believes it should be and does not conform to his standards, and therefore is of no use to him, nor anyone else. As a pure mathematician, I can say with certainty it is plain ignorance to say "Algebraic topology is useless" (p.361) when it a perfectly useful in the description of, say, force fields in a curved space. Morris himself describes many ways abstraction has led to physical models; but in his mind, abstraction is only useful if it is taken to advance science. If an abstraction finds its use after it is a mathematical theory, then by his argument it is a waste of time.
To be fair, many of these "useless" branches of mathematics have been found extremely useful in mathematical physics since Morris wrote the book in 1980. It would be interesting to see what his opinion of modern applications of abstract geometry, algebraic topology, and other "useless" subjects in theories such as quantum field theory, theory of computing, dynamical systems, models for speciation and mutation, DNA and protein geometry, modeling the nervous system, and so forth.
I suppose it is common for scientists to publish books authenticated by their their credentials in order to get their points of view out to the public. And to an extent, I do agree; modern mathematics is often too self-serving and disconnected from other sciences. I won't get into that here; but I definitely disagree with his thesis for the last three chapters that "Pure mathematicians have ruined mathematics". I would contend that "Pure mathematicians have found that mathematics can be beautiful and interesting, and hs developed itself into its own identity". It is a shame that Morris cannot see mathematics the same way I or my colleagues can; on the other hand, I know many Morris' too, and we can all agree to disagree because no matter what we all love the subject, and disagree on what's worthy of our study.
July 4, 2009
This book was about the history of mathematics, its foundations and the fact that those foundations are shakier than we have all been lead to believe. Unfortunately, the author is incredibly long winded. This book is much longer than it should have been. While I enjoyed the historical sections and the premise is interesting, it takes him hundreds of pages to get to the point. It's almost as if he had no editor. I would have liked a Cliff's Notes version of this book. The full enchilada?? Not so much.
March 18, 2014
Picked this up on a whim in a Seattle used book store, because I'm addicted to buying books and because I thoroughly enjoyed Kline's Mathematics for the Nonmathematician. I thought I knew this story well: mathematicians believed Euclid's 5 axioms were the law and couldn't conceive of changing them. Then, one day, people realized one of those axioms was malleable!
I thought wrong; the story is much more interesting.
This book is a detailed guide to the history of the philosophy of mathematics. It covers 4 schools of thought:
* logicism - mathematics can be reduced to logic, and therefore is a part of logic.
* intuitionism - mathematics is a mental human activity irregardless of what happens in the physical world.
* formalism - mathematics is the manipulation of formal systems of axioms.
* set-theorists - mathematics starts with Zermelo-Fraenkel set theory with the axiom of choice.
My disrespectful definitions would enrage adult mathematicians but, luckily, I don't work w/ any. Lucky because this boils down to whose kung fu is the best. In a martial arts flick, the opponents will taunt each other, dance around, and trade blows, but there can only be one winner. In math, there's lots of taunting and dancing, but no winners. Not joking about the taunts, there are plenty. These schools are not compatible, and it is a wonder that these great men got any work done when they devoted so much time to insulting each other.
Math started out as a quest to uncover God's design, the pursuit of science as worship. Soon math's successes are are so grand, its practitioners wonder if they can do away w/ God entirely. As time-honored truths crumble under scrutiny, fear grows that math is entirely faith-based, a religion whose adherents' best path to success is the extermination of the unclean.
The book ends w/ a diatribe against number theory and abstract topology, and a plea to return mathematics to its roots of addressing human concerns. In 1980, when this book was written, number theory had been for thousands of years little more than mental masturbation. Today, number theory underpins literally hundreds of billions of dollars of online commerce in the US alone. Buying and selling is indeed a human concern! Despite Kline's extensive knowledge of both math and history, even he missed seeing that it is so very, very hard to predict what will and won't be useful.
This is not a light, quick read. I only recommend it to anyone already committed to studying history and mathematics. I would definitely not recommend it to high schoolers, because it might turn them away from math altogether.
I thought wrong; the story is much more interesting.
This book is a detailed guide to the history of the philosophy of mathematics. It covers 4 schools of thought:
* logicism - mathematics can be reduced to logic, and therefore is a part of logic.
* intuitionism - mathematics is a mental human activity irregardless of what happens in the physical world.
* formalism - mathematics is the manipulation of formal systems of axioms.
* set-theorists - mathematics starts with Zermelo-Fraenkel set theory with the axiom of choice.
My disrespectful definitions would enrage adult mathematicians but, luckily, I don't work w/ any. Lucky because this boils down to whose kung fu is the best. In a martial arts flick, the opponents will taunt each other, dance around, and trade blows, but there can only be one winner. In math, there's lots of taunting and dancing, but no winners. Not joking about the taunts, there are plenty. These schools are not compatible, and it is a wonder that these great men got any work done when they devoted so much time to insulting each other.
Math started out as a quest to uncover God's design, the pursuit of science as worship. Soon math's successes are are so grand, its practitioners wonder if they can do away w/ God entirely. As time-honored truths crumble under scrutiny, fear grows that math is entirely faith-based, a religion whose adherents' best path to success is the extermination of the unclean.
The book ends w/ a diatribe against number theory and abstract topology, and a plea to return mathematics to its roots of addressing human concerns. In 1980, when this book was written, number theory had been for thousands of years little more than mental masturbation. Today, number theory underpins literally hundreds of billions of dollars of online commerce in the US alone. Buying and selling is indeed a human concern! Despite Kline's extensive knowledge of both math and history, even he missed seeing that it is so very, very hard to predict what will and won't be useful.
This is not a light, quick read. I only recommend it to anyone already committed to studying history and mathematics. I would definitely not recommend it to high schoolers, because it might turn them away from math altogether.
June 12, 2021
pessimistic in tone but fascinating in content looks at mathematical history as a series of crises. Where the field isolates and becomes fractured as it progresses. It covers not only the foundational crisis I go on about all the time but the euclidean controversy, the controversy of the infinitesimal, and the of course the infinite and the continuum. This shows that math often is crisis-prone at the frontier. The way of the world. Pessimistic in attitude which appeals to a certain caste of mind but a straightforward history of math.
February 10, 2020
Excellent review of the history of mathematics, mostly Western mathematics from the Greeks to the 20th century. Kline reviews the many historical milestones in how mathematics developed with the major cultural changes focusing on the modern era. I was lucky to read this book together with my math, philosophy, and engineering faculty colleagues at Saint Martin's University. Thanks so much to Joe M. whose math expertise and leading helped us understand much more.
Among the many things Kline discusses, how controversial many mathematical concepts were such as irrationals, negatives, and complex numbers is fascinating to learn about. He outlines "the first debacle" of "the withering of truth" and follows with four chapters of "illogical developments." This is where things get really interesting: in the late 1700s/early 1800s with the rise of non-Euclidean geometries and new arithmetics. I learned from this part of the book how deep the crisis in math was in the 19th century, how disorganized, ungrounded, and in disarray it was, and how all this led to a huge effort to bring order and solid grounding to math in the late 1800s/early 1900s - a project that ultimately failed with Godel's Incompleteness Theorem. This and other development thus relate to the book's subtitle, "the loss of certainty."
All throughout the book he uses language such as "debacle," "disaster," "morass," "withering," "loss," "shock," and so on to describe the slow realization among mathematicians and philosophers that certain assumptions about truth and reality inherited from the Greeks onward were falling apart and dissolving before their eyes. On p95 Kline writes:
Thus the sad conclusion which mathematicians were obliged to draw is that there is no truth in mathematics, that is, truth in the sense of laws about the real world. The axioms of the basic structures of arithmetic and geometry are suggested by experience, and the structures as a consequence have a limited applicability. Just where they are applicable can be determined only by experience. The Greeks' attempt to guarantee the truth of mathematics by starting with self-evident truths and by using only deductive proof proved futile. To many thoughtful mathematicians the fact that mathematics is not a body of truth was too repugnant to swallow. It seems as though God had sought to confound them with several geometries and several algebras just as He had confounded the people of Babel with different languages. Hence they refused to accept the new creations.
Somehow in the book, he fails to discuss much how important and controversial the introduction of zero was to mathematics - there's very little in the book of how central zero was to the development of math. A good book on this is Zero, The Biography of a Dangerous Idea by Charles Seife. Also, Kline's treatment of non-Western math is pretty spare and seems ignorant of the advancements and contributions of the Hindus, the Chinese, and other non-Western cultures. Because these cultures approaches to math were so different, he fails to see how significant their contributions were. I'm thinking of the system of the Hindu-Arabic numerals along with the concept of zero as a cardinal entity, not just as a place value holder. The Hindus were the first to operate arithmetically on zero as an entity. A good book on non-Western contributions to math history is History of Mathematics, vols 1 and 2 by David Smith.
Another thing Kline leaves out of the book is the enormous importance of the mathematics of probability that grew up in the 19th century and, with statistics, has arguably become as important to mathematics as anything coming out of pure math such as set theory. A good book on the singular importance of the mathematics of probability is The Empire of Chance: How Probability Changed Science and Everyday Life by Gigerenzer, Swijtink, et al (1989). [Goodread's "insert book/author" function is currently not working].
Also the fact that this book was written in the late 70s and published in 1980 means that it couldn't have factored into its presentation the many developments in math and science that came soon thereafter such as the rise of computationalism, nonlinear analysis, discreet math, fractal geometry and the discoveries of dynamical chaos, parallel processing, complexity theory plus new understandings from cognitive science and consciousness studies of how the brain works and how all this has folded back on our understanding of the role of mathematics. These developments serve to ground our classical/modernist understanding of mathematical realities in embodied minds, living beings, and their environments/communities. Dozens of books explore how these developments reorient our view of mathematics. Here are a couple: The Dreams of Reason by Heinz Pagels; Where Mathematics Comes From by Lakoff and Nunez.
Despite these shortcomings, Kline's book gives one an excellent understanding of the most important developments in the history of Western mathematics from a classical mathematics standpoint.
Among the many things Kline discusses, how controversial many mathematical concepts were such as irrationals, negatives, and complex numbers is fascinating to learn about. He outlines "the first debacle" of "the withering of truth" and follows with four chapters of "illogical developments." This is where things get really interesting: in the late 1700s/early 1800s with the rise of non-Euclidean geometries and new arithmetics. I learned from this part of the book how deep the crisis in math was in the 19th century, how disorganized, ungrounded, and in disarray it was, and how all this led to a huge effort to bring order and solid grounding to math in the late 1800s/early 1900s - a project that ultimately failed with Godel's Incompleteness Theorem. This and other development thus relate to the book's subtitle, "the loss of certainty."
All throughout the book he uses language such as "debacle," "disaster," "morass," "withering," "loss," "shock," and so on to describe the slow realization among mathematicians and philosophers that certain assumptions about truth and reality inherited from the Greeks onward were falling apart and dissolving before their eyes. On p95 Kline writes:
Thus the sad conclusion which mathematicians were obliged to draw is that there is no truth in mathematics, that is, truth in the sense of laws about the real world. The axioms of the basic structures of arithmetic and geometry are suggested by experience, and the structures as a consequence have a limited applicability. Just where they are applicable can be determined only by experience. The Greeks' attempt to guarantee the truth of mathematics by starting with self-evident truths and by using only deductive proof proved futile. To many thoughtful mathematicians the fact that mathematics is not a body of truth was too repugnant to swallow. It seems as though God had sought to confound them with several geometries and several algebras just as He had confounded the people of Babel with different languages. Hence they refused to accept the new creations.
Somehow in the book, he fails to discuss much how important and controversial the introduction of zero was to mathematics - there's very little in the book of how central zero was to the development of math. A good book on this is Zero, The Biography of a Dangerous Idea by Charles Seife. Also, Kline's treatment of non-Western math is pretty spare and seems ignorant of the advancements and contributions of the Hindus, the Chinese, and other non-Western cultures. Because these cultures approaches to math were so different, he fails to see how significant their contributions were. I'm thinking of the system of the Hindu-Arabic numerals along with the concept of zero as a cardinal entity, not just as a place value holder. The Hindus were the first to operate arithmetically on zero as an entity. A good book on non-Western contributions to math history is History of Mathematics, vols 1 and 2 by David Smith.
Another thing Kline leaves out of the book is the enormous importance of the mathematics of probability that grew up in the 19th century and, with statistics, has arguably become as important to mathematics as anything coming out of pure math such as set theory. A good book on the singular importance of the mathematics of probability is The Empire of Chance: How Probability Changed Science and Everyday Life by Gigerenzer, Swijtink, et al (1989). [Goodread's "insert book/author" function is currently not working].
Also the fact that this book was written in the late 70s and published in 1980 means that it couldn't have factored into its presentation the many developments in math and science that came soon thereafter such as the rise of computationalism, nonlinear analysis, discreet math, fractal geometry and the discoveries of dynamical chaos, parallel processing, complexity theory plus new understandings from cognitive science and consciousness studies of how the brain works and how all this has folded back on our understanding of the role of mathematics. These developments serve to ground our classical/modernist understanding of mathematical realities in embodied minds, living beings, and their environments/communities. Dozens of books explore how these developments reorient our view of mathematics. Here are a couple: The Dreams of Reason by Heinz Pagels; Where Mathematics Comes From by Lakoff and Nunez.
Despite these shortcomings, Kline's book gives one an excellent understanding of the most important developments in the history of Western mathematics from a classical mathematics standpoint.
November 15, 2010
La matematica è la quintessenza del ragionamento deduttivo, ed è l'unica scienza che ci dia delle certezze, no? Falso. In questo saggio di filosofia della matematica Morris Kline mostra non solo che la matematica non ha certezze, come forse molti hanno intuito cercando di capire il significato del teorema di incompletezza di Gödel, ma che il suo sviluppo è stato molto meno "matematico" di quello che ci vogliono far credere, che Euclide si è dimenticato una mezza dozzina di postulati, che le stagioni del rigore sono state brevi e che il rigore di ieri non è quello di oggi e quello di oggi non sarà quello di domani, che non è vero che la matematica modelli il mondo reale e che curiosamente la perdita della fede in Dio si sia riflessa sulla perdita della fede nella matematica stessa.
Nella prima parte del libro Kline mostra come la matematica da un lato aveva un'aria di sicurezza di sé ma in realtà era basata su fondamenta fragilissime - si pensi a come nacque l'analisi matematica, con tutti che assumevano tacitamente che una funzione continua fosse anche differenziabile, o il trattamento delle serie infinite, con risultati formali assolutamente senza senso reale. Parallelamente c'era il mito della rigorosità della geometria greca, che rappresentava proprietà intrinsecamente vere e reali pur se nel mondo c'erano solo loro approssimazioni. Man mano però che la matematica imparò ad essere più precisa nel XIX e XX secolo questa rigorosità e rappresentazione del mondo reale si sfaldò, proprio mentre si sfaldava il pensiero religioso che vedeva tutto ciò come segno dell'operato divino.
L'ultima parte del libro è una tirata contro i matematici iperformalistici che a partire dal 1900 o giù di lì hanno fatto matematica basata su sé stessa e non a partire dalla fisica, cioè da problemi del mondo reale; secondo lui la fecondità della matematica sta nel modellare, ancorché imperfettamente, il mondo. In definitiva, un libro piuttosto pessimista, anche se c'è una nota positiva sul fatto che si può fare matematica (e la si fa) anche in maniera più intuitiva.
Nella prima parte del libro Kline mostra come la matematica da un lato aveva un'aria di sicurezza di sé ma in realtà era basata su fondamenta fragilissime - si pensi a come nacque l'analisi matematica, con tutti che assumevano tacitamente che una funzione continua fosse anche differenziabile, o il trattamento delle serie infinite, con risultati formali assolutamente senza senso reale. Parallelamente c'era il mito della rigorosità della geometria greca, che rappresentava proprietà intrinsecamente vere e reali pur se nel mondo c'erano solo loro approssimazioni. Man mano però che la matematica imparò ad essere più precisa nel XIX e XX secolo questa rigorosità e rappresentazione del mondo reale si sfaldò, proprio mentre si sfaldava il pensiero religioso che vedeva tutto ciò come segno dell'operato divino.
L'ultima parte del libro è una tirata contro i matematici iperformalistici che a partire dal 1900 o giù di lì hanno fatto matematica basata su sé stessa e non a partire dalla fisica, cioè da problemi del mondo reale; secondo lui la fecondità della matematica sta nel modellare, ancorché imperfettamente, il mondo. In definitiva, un libro piuttosto pessimista, anche se c'è una nota positiva sul fatto che si può fare matematica (e la si fa) anche in maniera più intuitiva.
February 17, 2018
A history of mathematics for the educated public centered on mathematicians' evolving views of the foundations of certain and consistency of their discipline. Beginning with its identification with truth among some Greeks, it moves to its identification with the workings of the divine mind in the 1400-1600s and then with the truths of nature in the Enlightenment and early 1800s. In the 1800s mathematics becameseen as a human invention, a reflection of human psychology, and an effort was made to base it on axioms in the way Euclid did with his geometry. This effort failed in the 20th century, and math broke into several incompatible schools which tended to turn away from nature so that most mathematicians deal with abstract problems arising in math itself.
The book goes through 1980s, when it was published. It is clearly written and understandable without a math background, being a history of ideas rather than a discussion of the content of mathematics. I didn't care for his treatment of ancient mathematics, but his concern with the justification of mathematics, its claim to truth or some source of inherent certainty as opposed to utility, explains his approach to this period, and indeed throughout. I could only skim the last three chapters as it was an interlibrary loan that couldn't be renewed -- these were on math's turning in on itself and where the subject might be headed after the 1980s.
An interesting and accessible read, even for a person like me who had no math after high school.
The book goes through 1980s, when it was published. It is clearly written and understandable without a math background, being a history of ideas rather than a discussion of the content of mathematics. I didn't care for his treatment of ancient mathematics, but his concern with the justification of mathematics, its claim to truth or some source of inherent certainty as opposed to utility, explains his approach to this period, and indeed throughout. I could only skim the last three chapters as it was an interlibrary loan that couldn't be renewed -- these were on math's turning in on itself and where the subject might be headed after the 1980s.
An interesting and accessible read, even for a person like me who had no math after high school.
April 11, 2012
First part was whiz-bang. Last four chapters went on-and-on-and-on about math not having a sound basis and abstract math is only for math sake. Blech. I slogged through hoping for redemption. Should have skipped the last four.
December 22, 2014
The best popular book on the truly tragic achievement of Godel's incompleteness theorem, and the path to it.
January 20, 2016
People, this book really helped me trying to convince my students that Mathematics is so much more than just numbers, it is pure art.
June 1, 2012
Before I read “Mathematics: Loss of Certainty”, I assumed mathematics represented an a priori set of ideas, a rigorous set of universal truths that existed outside of human experience. Synonyms for the word mathematical include “exact, precise, meticulous, and rigorous.” When we refer to something as precise, exact, logical, we call it “mathematical.” What a shock to discover that at its core mathematics is fragmented into many different camps with many different standards for rigor, that there is no understanding of why mathematics applies to the real world, that for large swaths of history intuition has held sway over logic, proof and reason in mathematics. Many of the algebraic concepts taken for granted in my high school class were not really supported for hundreds of years. These were not esoteric ideas- unless you consider things like negative or irrational numbers (such as pi) to be esoteric. "Mathematics: Loss of Certainty” describes nothing less than a history of these once presumed a priori ideas being uncovered as little more than ad hoc intuition.
This volume is surprisingly graspable to a lay-person. As a history of mathematics, there must necessarily be some equations and diagrams, but happily there is nothing too lengthy or arduous. One does not need to fully grasp every nuance of math to understand Klein. I also think given those facts, the success this book has in describing the field of mathematics is remarkable. This is a history of many technical ideas- as the eras shift mathematics tries to find root in such diverse fields as geometry, algebra, and logic. Klein narrates the astonishing failure of mathematics to find shelter in any of them. To be able to make the history of such a diverse, abstruse and abstracted field accessible requires both great knowledge about the field and a talent for teaching. This book is also an invaluable teaching aid for what it does teach. Klein enlarged my understanding of mathematics, and the fundamental problems and changes that lead it to its present state.
That is not to say there weren’t times I felt that I was over my head in terms of technical knowledge, wondering if this particular digression would be on the test. Kline has to bear some responsibility, but I think my impatience near the end and the difficulty of the ideas is mostly to blame. Perhaps there could’ve been less block quotes from mathematicians opining on things as well. I do understand that Klein needs to support his opinions, but it got pretty tedious reading the quotes. One slightly sour note for me was that there seemed to be preferential treatment given to Greek and European mathematicians over Babylonian, Egyptian, Arab and Hindu mathematics. These civilizations made inroads into geography, astronomy and optics and seemed like the shining examples of intuitive insight Klein most admired. Also, I wish he could’ve wrapped things up a little quicker at the end of the book instead of delivering a sermon about how mathematics needs to become practical again. I sort of doubt that anyone who needs this history is a mathematician anyway.
Overall, Klein wrote a good paean to scientific culture and the triumphs of mathematics. If you want to know what mathematicians have been doing all these thousands of years but not so badly you'll pick up a textbook, I recommend it. The book ends with a simple question: If mathematics is nothing more than an ad hoc set of contingent truths, then how come it applies so well to the physical world? It turns out no one really knows. To quote Einstein “The most incomprehensible thing about the universe is that it is at all comprehensible.” For me, the same holds true for mathematics.
This volume is surprisingly graspable to a lay-person. As a history of mathematics, there must necessarily be some equations and diagrams, but happily there is nothing too lengthy or arduous. One does not need to fully grasp every nuance of math to understand Klein. I also think given those facts, the success this book has in describing the field of mathematics is remarkable. This is a history of many technical ideas- as the eras shift mathematics tries to find root in such diverse fields as geometry, algebra, and logic. Klein narrates the astonishing failure of mathematics to find shelter in any of them. To be able to make the history of such a diverse, abstruse and abstracted field accessible requires both great knowledge about the field and a talent for teaching. This book is also an invaluable teaching aid for what it does teach. Klein enlarged my understanding of mathematics, and the fundamental problems and changes that lead it to its present state.
That is not to say there weren’t times I felt that I was over my head in terms of technical knowledge, wondering if this particular digression would be on the test. Kline has to bear some responsibility, but I think my impatience near the end and the difficulty of the ideas is mostly to blame. Perhaps there could’ve been less block quotes from mathematicians opining on things as well. I do understand that Klein needs to support his opinions, but it got pretty tedious reading the quotes. One slightly sour note for me was that there seemed to be preferential treatment given to Greek and European mathematicians over Babylonian, Egyptian, Arab and Hindu mathematics. These civilizations made inroads into geography, astronomy and optics and seemed like the shining examples of intuitive insight Klein most admired. Also, I wish he could’ve wrapped things up a little quicker at the end of the book instead of delivering a sermon about how mathematics needs to become practical again. I sort of doubt that anyone who needs this history is a mathematician anyway.
Overall, Klein wrote a good paean to scientific culture and the triumphs of mathematics. If you want to know what mathematicians have been doing all these thousands of years but not so badly you'll pick up a textbook, I recommend it. The book ends with a simple question: If mathematics is nothing more than an ad hoc set of contingent truths, then how come it applies so well to the physical world? It turns out no one really knows. To quote Einstein “The most incomprehensible thing about the universe is that it is at all comprehensible.” For me, the same holds true for mathematics.
July 5, 2015
This book did more than live up to my expectations. It is really a collection of discussions around the theme of the search for certain knowledge. Each chapter through to the very last opens up quite new and interesting threads. As a result, it is a demanding read but never dry.
It surveys a remarkable variety of candidates for achieving secure knowledge: Induction, Deduction, Analogy, Intuition, Revelation, Innate wisdom (placed in us by God), recovered memory from a past life (Plato), the heart (Pascal: The heart has its reasons, which reason does not know).... Very soon the book concentrates on the way these different conceptions have been deployed in mathematics. Kline achieves this without demanding of the reader very much mathematical experience at all, beyond a willingness to follow the clear explanations, which are supplied only where really necessary. There is nothing to frighten the non mathematician. (My maths education stopped 45 years ago at the age of 16.)
Mathematics generally, and Euclid's axiomatic system of geometry in particular, have long been represented as evidence of our capacity to make true and certain statements about nature. Kline surveys the history of this claim, and notes how it rises to its peak in Kant's philosophy at the very time when mathematicians were disclosing evidence of the opposite. Kline investigates theories about how Mathematics has been thought to achieve its undoubted success, with a number of very different approaches to explaining the foundations of mathematics amounting to quite different accounts of what mathematics actually is. He describes the very convincing process by which mathematicians have been obliged to concede that it is as fallible and uncertain a path to knowledge as any of the sciences.
Finally, Kline offers an intriguing discussion about the nature of reality in the light of all this. If we are to avoid descending into a destructive scepticism about the possibility of knowledge, then we need constructive ways to function in the face of uncertainty and doubt.
This is an attractive book for many reasons. Kline patiently - and in very clear, non technical language - dissects and shatters many illusions without becoming cynical or seeming arrogant. He salvages from every failed attempt an assurance that there remains much to value and appreciate. He proposes in the end an attitude that I felt was optimistic and uplifting. Mathematics and science have achieved remarkable things together and have every prospect of continuing in that way. The lack of certain knowledge does not make science futile; instead it really ensures that it need never be closed to new discoveries.
It surveys a remarkable variety of candidates for achieving secure knowledge: Induction, Deduction, Analogy, Intuition, Revelation, Innate wisdom (placed in us by God), recovered memory from a past life (Plato), the heart (Pascal: The heart has its reasons, which reason does not know).... Very soon the book concentrates on the way these different conceptions have been deployed in mathematics. Kline achieves this without demanding of the reader very much mathematical experience at all, beyond a willingness to follow the clear explanations, which are supplied only where really necessary. There is nothing to frighten the non mathematician. (My maths education stopped 45 years ago at the age of 16.)
Mathematics generally, and Euclid's axiomatic system of geometry in particular, have long been represented as evidence of our capacity to make true and certain statements about nature. Kline surveys the history of this claim, and notes how it rises to its peak in Kant's philosophy at the very time when mathematicians were disclosing evidence of the opposite. Kline investigates theories about how Mathematics has been thought to achieve its undoubted success, with a number of very different approaches to explaining the foundations of mathematics amounting to quite different accounts of what mathematics actually is. He describes the very convincing process by which mathematicians have been obliged to concede that it is as fallible and uncertain a path to knowledge as any of the sciences.
Finally, Kline offers an intriguing discussion about the nature of reality in the light of all this. If we are to avoid descending into a destructive scepticism about the possibility of knowledge, then we need constructive ways to function in the face of uncertainty and doubt.
This is an attractive book for many reasons. Kline patiently - and in very clear, non technical language - dissects and shatters many illusions without becoming cynical or seeming arrogant. He salvages from every failed attempt an assurance that there remains much to value and appreciate. He proposes in the end an attitude that I felt was optimistic and uplifting. Mathematics and science have achieved remarkable things together and have every prospect of continuing in that way. The lack of certain knowledge does not make science futile; instead it really ensures that it need never be closed to new discoveries.
November 1, 2019
The Twilight of Applied Mathematics and the End of Science
Renowned physicist John L. Synge wrote in 1944 that, "Our science started with mathematics and will surely end not long after mathematics is withdrawn from it ..." (p. 291) Hermann Weyl, one of the greatest mathematicians of the 20th century, predicted in 1949 that "science would perish without a supporting transcendental faith in truth and reality..." (p. 348) Author Morris Kline establishes in this book that mathematicians have in fact lost their faith in transcendental truth and reality, and are now withdrawing from science in droves.
Kline begins with classical Greece and writes, "Of all the triumphs of the speculative thought of the Greeks, the most truly novel was their conception of the cosmos operating in accordance with mathematical laws discoverable by human thought." (18) Anaxagoras proclaimed that "Reason rules the world" and Plutarch reported Plato's famous "God eternally geometrizes." (11, 16) The great Pythagoras proceeded upon "the all-important doctrines that nature was built according to mathematical principles and that number relationships underlie, unify, and reveal the order in nature." (15) To the Pythagoreans, "There was no question this world was mathematically structured." (16)
Following the destruction of the Greek civilization, scientific investigation of the natural world lay largely inactive until awakened from its Scholastic and Aristotelian slumber. Galileo declared, "When we have the decree of nature, authority goes for nothing." (48) Added to the Greek concept of mathematical design in nature was identification of the designer, the Christian God. "The search for the mathematical laws of nature was an act of devotion which would reveal the glory and grandeur of His handiwork." (34-35) Kepler's book on planetary motion proclaimed: "Sun, moon, and all the planets glorify Him in your ineffable language! Celestial harmonies, all ye who comprehend His marvelous works, praise Him. ... That which we know best is comprised in Him, as well as in our vain science." (38-39) Sir Isaac Newton wrote, "This most beautiful system of sun, planets, and comets could only proceed from the counsel and dominion of an intelligent and powerful Being ... This Being governs all things, not as the Soul of the world, but as Lord over all..." (59)
The a priori conviction that God incorporated mathematical laws into the construction of the universe produced the courage and tenacity to discover and declare those truths in the face of substantial opposition. "The strength of Copernicus's and Kepler's conviction that God must have designed the world harmoniously and simply can be judged by the objections with which they had to contend." (39) Their opponents included not only religious authorities but also the predominant scientific and intellectual establishment. "Only a mathematician convinced that the universe was mathematically and simply designed would have had the mental fortitude to disregard the prevailing philosophical, religious, and scientific counter-arguments and to appreciate the mathematics of such a revolutionary astronomy. Only one possessed of unshakable convictions about the importance of mathematics in the design of the universe would have dared to affirm the new theory against the mass of powerful opposition it met." (40-41)
Today's mathematicians lack the convictions of Kepler, Copernicus and Newton, and even those of Pythagoras and Anaxagoras. While mathematicians "prefer to believe that they create the nourishment on which philosophers feed," it was the philosophers "in the van in denying truths about the physical world." (74) Who guarantees that there is a permanently existing world of solid objects? Hume's skepticism gained ground among the mathematicians and stripped away the inevitability of the laws of nature, their eternality and their inviolability. We know neither mind nor matter; both are fictions. (74)
"Science is rationalized fiction, rationalized by mathematics," writes Kline. Albert Einstein emphasized the fictional character of modern science in 1931: "According to Newton's system, physical reality is characterized by the concepts of space, time, material point, and force... After Maxwell they conceived physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations. ... The view I have just outlined of the purely fictitious character of the fundamentals of scientific theory was by no means the prevailing one in the eighteenth and nineteenth centuries. But it is steadily gaining ground from the fact that the distance in thought between the fundamental concepts and laws on one side and, on the other, the conclusions which have to be brought into relation with our experience grows larger and larger..." (337)
Mathematics is "a human construction and any attempt to find an absolute basis for it is probably doomed to failure." (312) Nobel physicist Percy Bridgman flatly rejected any objective world of mathematics writing in 1946, "It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention." (325)
"Today," Kline writes, "the belief in the mathematical design of nature seems far-fetched." (349) There is no longer correspondence between the mathematics within man's mind and the observable universe outside it, only an inexplicable correlation. Erwin Schrödinger said that the miracle of man's discovering laws of nature may well be beyond human understanding, and Einstein remarked, "The most incomprehensible thing about the world is that it is comprehensible." (349)
The upshot of this post-modern skepticism is that mathematicians are abandoning applied mathematics for "pure" mathematics. Few, it seems, are willing to invest their lives and stake their reputations on the search for truths about the real world when they have no confidence that there are any truths out there to be found, or that the universe won't arbitrarily produce some new fact that contradicts and destroys their life's work. Safer to escape from what Henri Poincare described as the "tyranny of the external world" and retreat to the idealized realm of pure mathematics. After all, "human effort and ingenuity are limited and should therefore be devoted to good risks." (285) Some rejoice at this development. One of Britain's leading mathematicians, Geofrey Hardy, is reported to have made the toast: "Here's to pure mathematics! May it never have any use"; and Leonard Dickson, an authority at the University of Chicago, used to say: "Thank God that number theory is unsullied by any applications." (295)
The thesis of Kline's book is that the "loss of truth, the constantly increasing complexity of mathematics and science, and the uncertainty about which approach is secure have caused most mathematicians to abandon science." (7) He states that "about ninety percent of the mathematicians active today are ignorant of science and are quite content to remain in that blissful state." (303) Others put the percentage even higher. Distinguished professor John Slater of MIT said that for every one mathematician interested in areas of physics "there are twenty who have no interest." (302) Leading French mathematician Laurent Schwartz said that the most active fields of mathematics have no application. (299)
Returning then to John Synge's observation that science "will surely end not long after mathematics is withdrawn from it" and Hermann Weyl's prediction that "science would perish without a supporting transcendental faith in truth and reality," are we to conclude that the end is near?
Renowned physicist John L. Synge wrote in 1944 that, "Our science started with mathematics and will surely end not long after mathematics is withdrawn from it ..." (p. 291) Hermann Weyl, one of the greatest mathematicians of the 20th century, predicted in 1949 that "science would perish without a supporting transcendental faith in truth and reality..." (p. 348) Author Morris Kline establishes in this book that mathematicians have in fact lost their faith in transcendental truth and reality, and are now withdrawing from science in droves.
Kline begins with classical Greece and writes, "Of all the triumphs of the speculative thought of the Greeks, the most truly novel was their conception of the cosmos operating in accordance with mathematical laws discoverable by human thought." (18) Anaxagoras proclaimed that "Reason rules the world" and Plutarch reported Plato's famous "God eternally geometrizes." (11, 16) The great Pythagoras proceeded upon "the all-important doctrines that nature was built according to mathematical principles and that number relationships underlie, unify, and reveal the order in nature." (15) To the Pythagoreans, "There was no question this world was mathematically structured." (16)
Following the destruction of the Greek civilization, scientific investigation of the natural world lay largely inactive until awakened from its Scholastic and Aristotelian slumber. Galileo declared, "When we have the decree of nature, authority goes for nothing." (48) Added to the Greek concept of mathematical design in nature was identification of the designer, the Christian God. "The search for the mathematical laws of nature was an act of devotion which would reveal the glory and grandeur of His handiwork." (34-35) Kepler's book on planetary motion proclaimed: "Sun, moon, and all the planets glorify Him in your ineffable language! Celestial harmonies, all ye who comprehend His marvelous works, praise Him. ... That which we know best is comprised in Him, as well as in our vain science." (38-39) Sir Isaac Newton wrote, "This most beautiful system of sun, planets, and comets could only proceed from the counsel and dominion of an intelligent and powerful Being ... This Being governs all things, not as the Soul of the world, but as Lord over all..." (59)
The a priori conviction that God incorporated mathematical laws into the construction of the universe produced the courage and tenacity to discover and declare those truths in the face of substantial opposition. "The strength of Copernicus's and Kepler's conviction that God must have designed the world harmoniously and simply can be judged by the objections with which they had to contend." (39) Their opponents included not only religious authorities but also the predominant scientific and intellectual establishment. "Only a mathematician convinced that the universe was mathematically and simply designed would have had the mental fortitude to disregard the prevailing philosophical, religious, and scientific counter-arguments and to appreciate the mathematics of such a revolutionary astronomy. Only one possessed of unshakable convictions about the importance of mathematics in the design of the universe would have dared to affirm the new theory against the mass of powerful opposition it met." (40-41)
Today's mathematicians lack the convictions of Kepler, Copernicus and Newton, and even those of Pythagoras and Anaxagoras. While mathematicians "prefer to believe that they create the nourishment on which philosophers feed," it was the philosophers "in the van in denying truths about the physical world." (74) Who guarantees that there is a permanently existing world of solid objects? Hume's skepticism gained ground among the mathematicians and stripped away the inevitability of the laws of nature, their eternality and their inviolability. We know neither mind nor matter; both are fictions. (74)
"Science is rationalized fiction, rationalized by mathematics," writes Kline. Albert Einstein emphasized the fictional character of modern science in 1931: "According to Newton's system, physical reality is characterized by the concepts of space, time, material point, and force... After Maxwell they conceived physical reality as represented by continuous fields, not mechanically explicable, which are subject to partial differential equations. ... The view I have just outlined of the purely fictitious character of the fundamentals of scientific theory was by no means the prevailing one in the eighteenth and nineteenth centuries. But it is steadily gaining ground from the fact that the distance in thought between the fundamental concepts and laws on one side and, on the other, the conclusions which have to be brought into relation with our experience grows larger and larger..." (337)
Mathematics is "a human construction and any attempt to find an absolute basis for it is probably doomed to failure." (312) Nobel physicist Percy Bridgman flatly rejected any objective world of mathematics writing in 1946, "It is the merest truism, evident at once to unsophisticated observation, that mathematics is a human invention." (325)
"Today," Kline writes, "the belief in the mathematical design of nature seems far-fetched." (349) There is no longer correspondence between the mathematics within man's mind and the observable universe outside it, only an inexplicable correlation. Erwin Schrödinger said that the miracle of man's discovering laws of nature may well be beyond human understanding, and Einstein remarked, "The most incomprehensible thing about the world is that it is comprehensible." (349)
The upshot of this post-modern skepticism is that mathematicians are abandoning applied mathematics for "pure" mathematics. Few, it seems, are willing to invest their lives and stake their reputations on the search for truths about the real world when they have no confidence that there are any truths out there to be found, or that the universe won't arbitrarily produce some new fact that contradicts and destroys their life's work. Safer to escape from what Henri Poincare described as the "tyranny of the external world" and retreat to the idealized realm of pure mathematics. After all, "human effort and ingenuity are limited and should therefore be devoted to good risks." (285) Some rejoice at this development. One of Britain's leading mathematicians, Geofrey Hardy, is reported to have made the toast: "Here's to pure mathematics! May it never have any use"; and Leonard Dickson, an authority at the University of Chicago, used to say: "Thank God that number theory is unsullied by any applications." (295)
The thesis of Kline's book is that the "loss of truth, the constantly increasing complexity of mathematics and science, and the uncertainty about which approach is secure have caused most mathematicians to abandon science." (7) He states that "about ninety percent of the mathematicians active today are ignorant of science and are quite content to remain in that blissful state." (303) Others put the percentage even higher. Distinguished professor John Slater of MIT said that for every one mathematician interested in areas of physics "there are twenty who have no interest." (302) Leading French mathematician Laurent Schwartz said that the most active fields of mathematics have no application. (299)
Returning then to John Synge's observation that science "will surely end not long after mathematics is withdrawn from it" and Hermann Weyl's prediction that "science would perish without a supporting transcendental faith in truth and reality," are we to conclude that the end is near?
January 28, 2012
This was a very different look at the history of mathematics development. I have taken history of mathematics classes but the textbooks did not treat the history in a manner that Kline does, by exposing the revelations in their order.I especially enjoyed the discussion of math concepts that were thought to have come out of pure interest in math but later proved to be important for their application, such the ellipse example. Kline writes that the pure mathematicians write that the Greeks investigated parabolas, hyperbolas, and ellipses out of pure curiosity, but in fact Kepler used the ellipse to describe the motions of planets around the sun.
September 30, 2018
"When a mathematician asks himself why some result should hold, the answer he seeks is some intuitive understanding. In fact, a rigorous proof means nothing to him if the results do not make sense intuitively." (313)
A wonderful text in the history of mathematics. Learning the of many errors "the greats" have made (in our privileged retrospect) humanizes them all. Learning of the controversies surrounding new definitions / conceptual approaches puts the air of authority that mathematical instruction often has in perspective.
A wonderful text in the history of mathematics. Learning the of many errors "the greats" have made (in our privileged retrospect) humanizes them all. Learning of the controversies surrounding new definitions / conceptual approaches puts the air of authority that mathematical instruction often has in perspective.
January 20, 2022
Mathematics must be one of the most difficult subjects the human mind has ever had to grapple with. I, having used mathematics quite a bit in my life, still find the more advanced topics way too abstract and convoluted to understand. Apparently, it's not my fault (at least not entirely). It seems mathematics has suffered a disastrous revelation in the last century that put it in a state of crisis - a crisis that haunts mathematicians to this day. The purpose of this book is to outline the historical development of mathematical ideas and the understanding of their inherent nature so that by having this broad view of the evolution of thought we can explain how the current state of mathematics along with its crisis came to be.
Kline wrote several books on the topic of the history of mathematics (which I've read) and his analysis on that subject is as great here as it is in all of his other works. But in contrast to his previous writings, here he focuses more on how people perceived mathematical ideas and how that perception has changed throughout the centuries. I commend Kline's storytelling here - I think he does a great job at highlighting the key features of mathematical thinking in each period of human history. More than that, the sheer number of citations gathered here is huge and is more than enough to account for the accuracy of Kline's arguments.
Mathematics as a field in itself was first developed (or is it discovered?) by the ancient Greeks. They were so mesmerized by its beauty and simplicity that they assigned it a divine origin. This belief has held for up to the late Renascence when the greatest minds of that period would echo the same sentiment in their writings. Then something interesting happened - the mathematical objects that seemed contradictory to the mind and thus were always kept isolated from the pure body of mathematics (mainly geometry), suddenly became so useful that they were no longer possible to ignore. Not having fully realized what to do with these findings, the mathematicians marched on and found new theorems, discovered new laws, explained new phenomena - all while using the paradoxical objects of imaginary numbers and infinitesimal lengths. They had hoped that geometry was the true foundation of all mathematics and arithmetic while providing correct results, had become too detached from reality. By the end of the 19th century, with the discovery of non-Euclidean geometry, mathematicians' outlook had completely reversed - now number theory was taken as the base of all things mathematical. There was one problem, however - arithmetic was never axiomised. And so started the race to find the foundations of all mathematics.
Three schools of thought had naturally arisen to take up the challenge: logicians, formalists and intuitionists. Logicians tried to root mathematics in an extended form of Aristotelian logic. Mathematical equations have the same kind of certainty as logical statements so the one must arise from the other, they postured. Formalists partially agreed with logicians view, but they weren't so sure which arose from which - logic from mathematics, or mathematics from logic. Thusly, a blend of both domains was used to form the formalist's foundations. Intuitionists' view was the most superfluous. Its members rejected different aspects of the other two schools: some disliked the law of excluded middle, some - the axioms of choice and continuity. The deciding factor for an intuitionist is the innate feeling of what's reasonable to assume when solving a mathematical problem.
Turn the time up to the present day, no school have managed to construct a satisfactory theory. Logicians admitted to having used some hard-to-intuit axioms (the axiom of reducibility), formalists' endeavour was utterly destroyed by Gödel's work, and intuitionists… well, they didn't even work on the problem as mathematics to them was good as it was.
And so we find ourselves now in this weird position, where this grand question that was tackled by the smartest minds of the last century has never been properly solved. Set theory has taken the place as the ultimate explanation of the nature of mathematics, but it's based on a mixture of different ideas from the three aforementioned schools with no attempt to address the paradoxes that they pose. Changing the configuration of the axioms, mathematicians created new kind of systems, some of which doesn't necessarily have any practical use. Mathematicians have begun to play with symbols in hope of stumbling on something useful, while their field has grown wide beyond any one person's understanding. This is what Kline finds disturbing - the turn away from nature as a scientist's source of inspiration.
This brief overview of the main ideas mentioned in the book doesn't do justice to all the material that Kline has covered. The underlying mathematics - the axioms, the paradoxes, and their outcomes - all the most important milestones of the history of mathematics are covered and explained simply and beautifully. This point is what I appreciate the most in this book. In all my years of formal study, never had I had such a clear understanding of what mathematics is, where its roots lie, and how its different branches interlink with each other. Although I had spent my life studying the sciences, it feels like only now, after reading this book, I've started to get a faint sense of its methods. And it’s a joyous feeling to have.
Wonderful book. Highly recommended.
Kline wrote several books on the topic of the history of mathematics (which I've read) and his analysis on that subject is as great here as it is in all of his other works. But in contrast to his previous writings, here he focuses more on how people perceived mathematical ideas and how that perception has changed throughout the centuries. I commend Kline's storytelling here - I think he does a great job at highlighting the key features of mathematical thinking in each period of human history. More than that, the sheer number of citations gathered here is huge and is more than enough to account for the accuracy of Kline's arguments.
Mathematics as a field in itself was first developed (or is it discovered?) by the ancient Greeks. They were so mesmerized by its beauty and simplicity that they assigned it a divine origin. This belief has held for up to the late Renascence when the greatest minds of that period would echo the same sentiment in their writings. Then something interesting happened - the mathematical objects that seemed contradictory to the mind and thus were always kept isolated from the pure body of mathematics (mainly geometry), suddenly became so useful that they were no longer possible to ignore. Not having fully realized what to do with these findings, the mathematicians marched on and found new theorems, discovered new laws, explained new phenomena - all while using the paradoxical objects of imaginary numbers and infinitesimal lengths. They had hoped that geometry was the true foundation of all mathematics and arithmetic while providing correct results, had become too detached from reality. By the end of the 19th century, with the discovery of non-Euclidean geometry, mathematicians' outlook had completely reversed - now number theory was taken as the base of all things mathematical. There was one problem, however - arithmetic was never axiomised. And so started the race to find the foundations of all mathematics.
Three schools of thought had naturally arisen to take up the challenge: logicians, formalists and intuitionists. Logicians tried to root mathematics in an extended form of Aristotelian logic. Mathematical equations have the same kind of certainty as logical statements so the one must arise from the other, they postured. Formalists partially agreed with logicians view, but they weren't so sure which arose from which - logic from mathematics, or mathematics from logic. Thusly, a blend of both domains was used to form the formalist's foundations. Intuitionists' view was the most superfluous. Its members rejected different aspects of the other two schools: some disliked the law of excluded middle, some - the axioms of choice and continuity. The deciding factor for an intuitionist is the innate feeling of what's reasonable to assume when solving a mathematical problem.
Turn the time up to the present day, no school have managed to construct a satisfactory theory. Logicians admitted to having used some hard-to-intuit axioms (the axiom of reducibility), formalists' endeavour was utterly destroyed by Gödel's work, and intuitionists… well, they didn't even work on the problem as mathematics to them was good as it was.
And so we find ourselves now in this weird position, where this grand question that was tackled by the smartest minds of the last century has never been properly solved. Set theory has taken the place as the ultimate explanation of the nature of mathematics, but it's based on a mixture of different ideas from the three aforementioned schools with no attempt to address the paradoxes that they pose. Changing the configuration of the axioms, mathematicians created new kind of systems, some of which doesn't necessarily have any practical use. Mathematicians have begun to play with symbols in hope of stumbling on something useful, while their field has grown wide beyond any one person's understanding. This is what Kline finds disturbing - the turn away from nature as a scientist's source of inspiration.
This brief overview of the main ideas mentioned in the book doesn't do justice to all the material that Kline has covered. The underlying mathematics - the axioms, the paradoxes, and their outcomes - all the most important milestones of the history of mathematics are covered and explained simply and beautifully. This point is what I appreciate the most in this book. In all my years of formal study, never had I had such a clear understanding of what mathematics is, where its roots lie, and how its different branches interlink with each other. Although I had spent my life studying the sciences, it feels like only now, after reading this book, I've started to get a faint sense of its methods. And it’s a joyous feeling to have.
Wonderful book. Highly recommended.
November 17, 2024
Personally, I was suspicious of this notion of a priori "knowledge" independent from any experience: mathematics, tautologies and deduction from "pure reason". (A posteriori knowledge depends on empirical evidence as in most fields of science and aspects of personal knowledge.) This convincing and well-researched book drawing on many cited sources has me now solid in camp of questioning mathematics a truth discovering technique as much as a concept generating process.
Of course, history is full of examples where mathematics has proved and been disproved.
....
....
However, many more mathematicians are pessimistic. Hermann Weyl, one of the greatest mathematicians of this century, said in 1944:
We find really it is an act of faith, a will to believe....
Many have instead taken an empirical approach.
I am fine with this "loss" of "certainty" and find comfort in placing mathematics along the greatest IMO human achievements like art.
Some summary:
...and finally from the great Karl Popper with a cheeky attitude:
Of course, history is full of examples where mathematics has proved and been disproved.
From the standpoint of the search for truths, it is noteworthy that Ptolemy, like Eudoxus, fully realized that his theory was just a convenient mathematical description which fit the observations and was not necessarily the true design of nature. For some planets he had a choice of alternative schemes and he chose the mathematically simpler one. Ptolemy says in Book XIII of his Almagest that in astronomy one ought to seek as simple a mathematical model as possible. But Ptolemy's mathematical model was received as the truth by the Christian world.
....
There are mathematicians who believe that the differing views on what can be accepted as sound mathematics will some day be reconciled. Prominent among these is a group of leading French mathematicians who write under the pseudonym of Nicholas Bourbaki...
....
Since the earliest times, all critical revisions of the principles of mathematics as a whole, or of any branch of it, have almost invariably followed periods of uncertainty, where contradictions did appear and had to be resolved.. There are now twenty-five centuries during which the mathematicians have had the practice of correcting their errors and thereby seeing their science enriched, not impoverished; this gives them the right to view the future with serenity.
However, many more mathematicians are pessimistic. Hermann Weyl, one of the greatest mathematicians of this century, said in 1944:
The question of the foundations and the ultimate meaning of mathematics remains open; we do not know in what direction it will find its final solution or even whether a final objective answer can be expected at all. "Mathematizing" may well be a creative activity of man, like language or music, of primary originality, whose historical decisions defy complete objective rationalization.
We find really it is an act of faith, a will to believe....
...in fact that these theorems use the axioms. However, they must be used to derive a large part of classical mathematics. In the second edition of his Principles (1937), Russell backtracked still more. He said that "The whole question of what are logical principles becomes to a very considerable extent arbitrary." The axioms of infinity and choice "can only be proved or disproved by empirical evidence." Nevertheless, he insisted that logic and mathematics are a unity.
However, the critics could not be stilled. In his Philosophy of Mathematics and Natural Science (1949), Hermann Weyl said the Principia based mathematics not on logic alone, but on a sort of logician's paradise, a universe endowed with an "ultimate furniture" of rather complex structure. Would any realistically-minded man dare say he believes in this transcendental world?... This complex structure taxes the strength of our faith hardly less than the doctrines of the early Fathers of the Church or of the Scholastic philosophers of the Middle Ages.
Many have instead taken an empirical approach.
The upshot of these views is that sound mathematics must be deter- mined not by any one foundation which may some day prove to be right. The "correctness" of mathematics must be judged by its applicability to the physical world. Mathematics is an empirical science much as Newtonian mechanics. It is correct only to the extent that it works and when it does not, it must be modified. It is not a priori knowledge even though it was so regarded for two thousand years. It is not absolute or unchangeable.
I am fine with this "loss" of "certainty" and find comfort in placing mathematics along the greatest IMO human achievements like art.
Weierstrass endorsed this thought with the words, "The true mathematician is a poet." And Ludwig Wittgenstein (1889-1951), a student of Russell and an authority in his own right, believed that the mathematician is an inventor not a discoverer...
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As Weyl stated, mathematics is an activity of thought, not a body of exact knowledge. It is best viewed historically. The rational constructions and reconstructions of the foundations appear now only as a trav esty of the history.
The most extreme view was expressed by Karl Popper, a notable philosopher of science, in The Logic of Scientific Discovery. Mathematical reasoning is never verifiable but only falsifiable. Mathematical theorems are not guaranteed in any way. One may continue to use the existent theory in the absence of a better one, just as Newton's theory of mechanics was used for two hundred years before relativity, or as Euclidean geometry was before Riemannian geometry. But assurance of correctness is not attainable...
Some summary:
Arthur Stanley Eddington once said, "Proof is an idol before whom the mathematician tortures himself." Why should they continue to do so? We might well ask what mathematicians accomplish with their stress on reasoning if they no longer know that their subject is consistent and if, especially, they no longer agree on what correct proof is. Should they rather become indifferent to rigor, throw up their hands and say that mathematics as a soundly established body of knowledge is an illusion? Should they abandon deductive proof and resort merely to convincing, intuitively sound arguments? After all, the physical sciences use such arguments, and even where they use mathematics they are not too concerned with the mathematician's passion for rigor. Abandonment is not the advisable path. Anyone who has looked into the contributions of mathematics to human thought would not sacrifice the concept of proof.
...and finally from the great Karl Popper with a cheeky attitude:
To these gibes at proof we may add the words of a leading student of the logic of mathematics, Karl Popper: "There are three levels of understanding of a proof. The lowest is the pleasant feeling of having grasped the argument; the second is the ability to repeat it; and the third or top level is that of being able to refute it."
June 1, 2020
Personas sumamente inteligentes siguen creyendo, hoy en día, que las matemáticas son un conjunto de verdades inquebrantables sobre el mundo físico y que el razonamiento matemático es exacto e infalible. Este libro "Matemáticas: la pérdida de la certidumbre" refuta este mito. Morris Kline pone de manifiesto que, hoy en día, no hay un concepto de las matemáticas universalmente aceptado, que de hecho hay muchos conceptos enfrentados unos a otros. Sin embargo, la capacidad de las matemáticas para describir y explorar los fenómenos físicos y sociales continúa aumentando.
Este libro, lejos de lo que podría suponerse, no es un libro que contenga fórmulas y teoremas aunque sí se requiere tener una cultura matemática somera. Puedo decir, desde mi humilde postura, que este libro es de historia y de sociología pues describe, en el recuento de los siglos, cómo la matemática es un constructo social con todas sus contradicciones, con negociación de conceptos y con cambios de paradigmas en la forma de razonar y de entender los fundamentos de esta magnífica obra humana. También debe ser leído si te consideras un filósofo pues, al final del mismo, se aborda un problema aún no resuelto: ¿por qué las matemáticas describen con tanta exactitud a la naturaleza? ¿Es acaso porque, en un acto voluntario, así fue diseñada? o ¿Acaso la matemática describe los estados naturales con más probabilidad de ocurrencia? o ¿Acaso este es un problema irresoluble para la eternidad? ¿Acaso es la feliz coincidencia de dos hechos: la armonía natural y la capacidad que posee nuestro cerebro de organizar bajo patrones? Tal vez será que «la dedicación a las matemáticas es una divina locura del espíritu humano... locura, quizá, pero con seguridad divina»
Este libro, lejos de lo que podría suponerse, no es un libro que contenga fórmulas y teoremas aunque sí se requiere tener una cultura matemática somera. Puedo decir, desde mi humilde postura, que este libro es de historia y de sociología pues describe, en el recuento de los siglos, cómo la matemática es un constructo social con todas sus contradicciones, con negociación de conceptos y con cambios de paradigmas en la forma de razonar y de entender los fundamentos de esta magnífica obra humana. También debe ser leído si te consideras un filósofo pues, al final del mismo, se aborda un problema aún no resuelto: ¿por qué las matemáticas describen con tanta exactitud a la naturaleza? ¿Es acaso porque, en un acto voluntario, así fue diseñada? o ¿Acaso la matemática describe los estados naturales con más probabilidad de ocurrencia? o ¿Acaso este es un problema irresoluble para la eternidad? ¿Acaso es la feliz coincidencia de dos hechos: la armonía natural y la capacidad que posee nuestro cerebro de organizar bajo patrones? Tal vez será que «la dedicación a las matemáticas es una divina locura del espíritu humano... locura, quizá, pero con seguridad divina»
February 13, 2019
А Ложки то совсем нет
По-моему, Карл Маркс писал, что любое учение можно считать научным, если оно стоит на математической базе. Профессор истории математики/есть, оказывается, и такая наука/ Морис Клайн разбивает в пух и прах всё основание математики: систему аксиом и весь логический аппарат, на которых жиждется математика, и которая оказалась огромным колоссом науки, стоящим на глинянных ногах.
Но поскольку практически все научные дисциплины основываются на математическом аппарате - то подвергается сомнению истинность и остальных научных дисциплин. Так и хочется сказать об этом мире словами мальчика-буддиста из фильма "Матрица".
- А Ложки то совсем нет.
Мир, который мы видим и в котором живём является лишь одной из версий описаний мира. А наука, эта квинэссенция описания мира, его высшее проявление - всего лишь система, состоящая из набора слов, сомнительных определений и абстракций, которые рисуют нам не реальный мир, а лишь его иллюзию.
По-моему, Карл Маркс писал, что любое учение можно считать научным, если оно стоит на математической базе. Профессор истории математики/есть, оказывается, и такая наука/ Морис Клайн разбивает в пух и прах всё основание математики: систему аксиом и весь логический аппарат, на которых жиждется математика, и которая оказалась огромным колоссом науки, стоящим на глинянных ногах.
Но поскольку практически все научные дисциплины основываются на математическом аппарате - то подвергается сомнению истинность и остальных научных дисциплин. Так и хочется сказать об этом мире словами мальчика-буддиста из фильма "Матрица".
- А Ложки то совсем нет.
Мир, который мы видим и в котором живём является лишь одной из версий описаний мира. А наука, эта квинэссенция описания мира, его высшее проявление - всего лишь система, состоящая из набора слов, сомнительных определений и абстракций, которые рисуют нам не реальный мир, а лишь его иллюзию.
June 14, 2023
When you hear the phrase "Philosophy of Mathematics," you probably don't think of drama, intrigue, and the tragic struggle of humanity to overcome our limitations. But if you read this book, that's exactly what you'll get. "The Loss of Certainty" is an overview of the history of the philosophy of mathematics, from the ancient Greeks and Egyptians through to modern academia, and in it Kline goes far beyond the raw facts to tell a compelling story - the story of thousands of mathematicians over thousands of years struggling to make sense of a strange and magical universe. And while the most promising attempts to achieve certainty have ended in defeat and confusion, the story of our mathematics is not yet over, and Kline leaves the reader with a sense of quiet wonder and excitement for where human thought might go next. If you read one book about mathematics, make it this one.
January 25, 2025
One of key takeaways from Mathematics and the Loss of Certainty is that there is a significant cultural difference between the West and the East. In the West, the dominant focus has been on the pursuit of science, while in the East, there has historically been a stronger inclination toward religious and philosophical beliefs. For centuries, intellectual debates in the East often revolved around the different schools of thought, such as Confucianism, Taoism, and Buddhism.
In contrast, since the medieval period, Western thinkers have relentlessly sought a fundamental understanding of the universe. Figures like Isaac Newton, who demonstrated Kepler's laws of planetary motion, and Leibniz, who further developed calculus, continuing the quest for to unravel the mysteries of the world. Their work shapes Western intellectual culture today.
In contrast, since the medieval period, Western thinkers have relentlessly sought a fundamental understanding of the universe. Figures like Isaac Newton, who demonstrated Kepler's laws of planetary motion, and Leibniz, who further developed calculus, continuing the quest for to unravel the mysteries of the world. Their work shapes Western intellectual culture today.
December 24, 2019
A tour of force in its field... Morris Kline did a great framework of his area... You will enjoy the first chapters of the book, and then... too long for his point of view! As you can guess, science is made by humans...
October 30, 2020
Най-добрата книга за история на математиката с елемент на философки разсъждения.
ПП. Руското издание е допълнено с много уместни бележки.
ПП. Руското издание е допълнено с много уместни бележки.
December 21, 2024
Would recommend to anyone with an interest in the history of mathematics
February 25, 2012
I remember vividly this discussion during an early academic phase of my life. Probably brought on by the death of the author at the time in 1980.
Much of this work I had read in journals or transcripts of presentations that then was collated in a book form.
Recently running across a new printing and looking at the bibliography with nostalgia (both fond and horror) I thought I'd get a new copy and review it again.
I'll settle down with this over the next week or two and see if I can do it justice. Probably have to search out some of the original references as well if they are not handy at the bottom of a shelf of books. Two or three of the reference books I know right where they are and the dust is a bit thick.
Three stars until I have time to see if I still feel it is worth more!
Much of this work I had read in journals or transcripts of presentations that then was collated in a book form.
Recently running across a new printing and looking at the bibliography with nostalgia (both fond and horror) I thought I'd get a new copy and review it again.
I'll settle down with this over the next week or two and see if I can do it justice. Probably have to search out some of the original references as well if they are not handy at the bottom of a shelf of books. Two or three of the reference books I know right where they are and the dust is a bit thick.
Three stars until I have time to see if I still feel it is worth more!
May 29, 2012
I'm not a mathematician, but found this book incredibly interesting. The way it chronicles the history of math and frames progress as not necessarily being our finding of absolute truth, but the discovery of ways we can approximate the reality we exist in is interesting. It was a relatively difficult read as I got deeper into the book and was faced with more kinds of math that I hadn't studied or had studied years ago. I would recommend it to anyone interested in theories that undermine a necessarily progressive narrative of history, but warn non-mathematicians that it may be frustrating at times.
September 27, 2012
Kline is a hysterically militant atheist. Despite a lack of citations, the history is interesting enough - Math, the hallmark of apriori science, is shown to at times (as long as two centuries at a time) be in a state of complete anxiety. Mathematicians could not justify the existence of negative numbers, calculus or imaginary numbers but these tools demonstrated themselves as valuable to science. Kline ends with the conclusion that Math is really just justified with respect to other sciences, a lame ending. Why not as an end in itself?
April 10, 2016
Always having struggled with mathematics and trying to tie together the foundations and theorems of the last 25,000 years into one cohesive thing, this book doesn't promise to do so although there is much discussion about it. There is no certain answers in this book, it's more about the history of the development of mathematics, logic, and science. I enjoyed it a lot and learned a lot about math w/o actually having to do math.. Win! Quaternions.
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