❮Epub❯ ➝ Mathematics: The Loss of Certainty Author Morris Kline – Capitalsoftworks.co.uk 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 Cer Most intelligent Loss of MOBI ò 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.
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10 thoughts on “Mathematics: The Loss of Certainty”

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 th 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 evercomplex forms of mathematics, discoveries that confounded theor 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

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 e 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

328 Books The author reminds us that the public is the bus driver who takes the young scientist home at the end of his laboratory slog, the lady who takes care of his laboratory animals, the man who sells him his lunchtime sandwich most importantly, the public are those whose science teachers are the media Goodfield examines the roles of communicators from the printed and televised media and of researchers She takes as examples the memorable case of Summerlin s painted mouse at Memorial 328 Books The author reminds us that the public is the bus driver who takes the young scientist home at the end of his laboratory slog, the lady who takes care of his laboratory animals, the man who sells him his lunchtime sandwich most importantly, the public are those whose science teachers are the media Goodfield examines the roles of communicators from the printed and televised media and of researchers She takes as examples the memorable case of Summerlin s painted mouse at Memorial SloanKettering Institute, the 1975 Asilomar conference on the ethical inhibitions related to DNA research, the publication by Lippincott of David Rorvik s book on the cloning of an unidentified boy, and the deontological legal involvement of London s The Sunday Times in the western European thalidomide scandal during the 1960s and 1970s The author chronicles these in admirably summary form and finds the researchers involved guilty of half truths or deceit in their relations with the public Goodfield s analysis does not, however, make heroes of the media She admonishes that, to improve communication about science, at least some writers must becomelike responsible political commentators, adding analysis, judgment and criticism to their reporting To do this best, professional communicators and scientists need to elaborate together an effective modus vivendi because maintaining a level of scientific literacy in the public is as difficult a task as doing science itself, and the media cannot do this alone Currently adjunct professor at Cornell University Medical School, Goodfield has also written and directed scientific films She shows an excellent grasp of multi media impacts on the public when scientific or technical information is vernacularized and of the risks of interpretation She asks if a report on science should be treated in the traditional Anglo American fashion who what when where of presenting news How willing are editors or producers to lend continuity to coverage, from day to day or week to week, treating information as cultural process rather than ad hoc event To what extent is the public itself willing to overcome its scientific illiteracy, and what efforts must researchers and communicators make towards this goal Must they trick the public into an unwary concession to some implausible assumption, as H G Wells counseled in his preface to a volume of novels in 1895, to get on with the story while the illusion holds These are questions the popularizer must constantly review This little book is well worth reading AAAS s executive director, William D Carey, writes in the preface to Goodfield s volume, The point ofthe essay lies in what roads are open on which science and the media can journey glimpsing the changing face of science for a concerned and pre occupied society The zoologist author has ably shown the way Mathematics The Loss of Certainty Morris Kline Oxford University Press, Oxford, 1982 400 pp., illus Paper, 7.95 ISBN 0 19 503085 0 Reviewed by J Guberman Morris Kline is a well known applied mathematician who worked on electromagnetic field theory in the 1940s and 50s He has written several successful semi popular books on the interpretation of scientific doctrines in terms of our understanding of the physical and conceptual worlds Kline s encyclopedic history of mathematical thought Mathematical Thoughtfrom Ancient to Modern Times, Oxford University Press, Oxford, 1972 covered mathematical thinking from ancient to modern times His current book examines what has been a problem in the conceptual foundations of mathematics for eighty years the loss of certainty Kline poses the problem in a concise and readable form that appeals not only to those familiar with physics and mathematics, but to the intelligent layperson interested in the broad meaning of scientific discovery The loss of certainty is one of the most important unsolved issues in logic and pure mathematics Until the turn of the century, everybody believed with one or two notable exceptions that mathematics was a logically consistent formal system Around the latter part of the nineteenth century, Georg Cantor, in his endeavour to make some branches of mathematicsrigorous, found that certain formal problems called into question mathematical reasoning in even the most accepted fields of mathematics, such as arithmetic

A history of mathematics with a focus on the crisis of foundations that developed in the 19th century and culminated in Godel s Incompleteness theorem in 1930 It is a story about the rock solid certainty of mathematics being shaken by uncertainties in its foundations This is an undertold story about how the queen of the sciences and the epitome of the dream of reason has been called into question in the times it was supposed to triumph In a time where so much of our civilization depends on m A history of mathematics with a focus on the crisis of foundations that developed in the 19th century and culminated in Godel s Incompleteness theorem in 1930 It is a story about the rock solid certainty of mathematics being shaken by uncertainties in its foundations This is an undertold story about how the queen of the sciences and the epitome of the dream of reason has been called into question in the times it was supposed to triumph In a time where so much of our civilization depends on mathematics, it is a bit worrisome that it isn t as solid as it was believed to be in the past I think the author overplays the drama a bit in the narrative still it is a big development

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 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

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 muchinteresting.This book is a detailed guide to the history of the philosophy of mathematics It c 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 muchinteresting.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 littlethan 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

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 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

Rereading this book I see it is a tragedy The foundations of mathematics can never be complete It is always open and some truths in it we will never reach This is microcosm of human limitations written in the queen of the sciences The surest thing in the world ,mathematics, can never rest on ultimate certainty This is just oneadditional limitation that comes with a birth certificate The real tragedy of the book is that Mathematics has sealed itself off from the general public with o Rereading this book I see it is a tragedy The foundations of mathematics can never be complete It is always open and some truths in it we will never reach This is microcosm of human limitations written in the queen of the sciences The surest thing in the world ,mathematics, can never rest on ultimate certainty This is just oneadditional limitation that comes with a birth certificate The real tragedy of the book is that Mathematics has sealed itself off from the general public with only a few initiates entering into its mysteries I love math it is a wonderful discipline filled with beauty that unlike the beauty of the world has a taste of the infinite and eternal no where else except religion or maybe art do you get to taste such fruits

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.

The best popular book on the truly tragic achievement of Godel s incompleteness theorem, and the path to it.
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 th 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 evercomplex forms of mathematics, discoveries that confounded theor 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
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 e 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
328 Books The author reminds us that the public is the bus driver who takes the young scientist home at the end of his laboratory slog, the lady who takes care of his laboratory animals, the man who sells him his lunchtime sandwich most importantly, the public are those whose science teachers are the media Goodfield examines the roles of communicators from the printed and televised media and of researchers She takes as examples the memorable case of Summerlin s painted mouse at Memorial 328 Books The author reminds us that the public is the bus driver who takes the young scientist home at the end of his laboratory slog, the lady who takes care of his laboratory animals, the man who sells him his lunchtime sandwich most importantly, the public are those whose science teachers are the media Goodfield examines the roles of communicators from the printed and televised media and of researchers She takes as examples the memorable case of Summerlin s painted mouse at Memorial SloanKettering Institute, the 1975 Asilomar conference on the ethical inhibitions related to DNA research, the publication by Lippincott of David Rorvik s book on the cloning of an unidentified boy, and the deontological legal involvement of London s The Sunday Times in the western European thalidomide scandal during the 1960s and 1970s The author chronicles these in admirably summary form and finds the researchers involved guilty of half truths or deceit in their relations with the public Goodfield s analysis does not, however, make heroes of the media She admonishes that, to improve communication about science, at least some writers must becomelike responsible political commentators, adding analysis, judgment and criticism to their reporting To do this best, professional communicators and scientists need to elaborate together an effective modus vivendi because maintaining a level of scientific literacy in the public is as difficult a task as doing science itself, and the media cannot do this alone Currently adjunct professor at Cornell University Medical School, Goodfield has also written and directed scientific films She shows an excellent grasp of multi media impacts on the public when scientific or technical information is vernacularized and of the risks of interpretation She asks if a report on science should be treated in the traditional Anglo American fashion who what when where of presenting news How willing are editors or producers to lend continuity to coverage, from day to day or week to week, treating information as cultural process rather than ad hoc event To what extent is the public itself willing to overcome its scientific illiteracy, and what efforts must researchers and communicators make towards this goal Must they trick the public into an unwary concession to some implausible assumption, as H G Wells counseled in his preface to a volume of novels in 1895, to get on with the story while the illusion holds These are questions the popularizer must constantly review This little book is well worth reading AAAS s executive director, William D Carey, writes in the preface to Goodfield s volume, The point ofthe essay lies in what roads are open on which science and the media can journey glimpsing the changing face of science for a concerned and pre occupied society The zoologist author has ably shown the way Mathematics The Loss of Certainty Morris Kline Oxford University Press, Oxford, 1982 400 pp., illus Paper, 7.95 ISBN 0 19 503085 0 Reviewed by J Guberman Morris Kline is a well known applied mathematician who worked on electromagnetic field theory in the 1940s and 50s He has written several successful semi popular books on the interpretation of scientific doctrines in terms of our understanding of the physical and conceptual worlds Kline s encyclopedic history of mathematical thought Mathematical Thoughtfrom Ancient to Modern Times, Oxford University Press, Oxford, 1972 covered mathematical thinking from ancient to modern times His current book examines what has been a problem in the conceptual foundations of mathematics for eighty years the loss of certainty Kline poses the problem in a concise and readable form that appeals not only to those familiar with physics and mathematics, but to the intelligent layperson interested in the broad meaning of scientific discovery The loss of certainty is one of the most important unsolved issues in logic and pure mathematics Until the turn of the century, everybody believed with one or two notable exceptions that mathematics was a logically consistent formal system Around the latter part of the nineteenth century, Georg Cantor, in his endeavour to make some branches of mathematicsrigorous, found that certain formal problems called into question mathematical reasoning in even the most accepted fields of mathematics, such as arithmetic
A history of mathematics with a focus on the crisis of foundations that developed in the 19th century and culminated in Godel s Incompleteness theorem in 1930 It is a story about the rock solid certainty of mathematics being shaken by uncertainties in its foundations This is an undertold story about how the queen of the sciences and the epitome of the dream of reason has been called into question in the times it was supposed to triumph In a time where so much of our civilization depends on m A history of mathematics with a focus on the crisis of foundations that developed in the 19th century and culminated in Godel s Incompleteness theorem in 1930 It is a story about the rock solid certainty of mathematics being shaken by uncertainties in its foundations This is an undertold story about how the queen of the sciences and the epitome of the dream of reason has been called into question in the times it was supposed to triumph In a time where so much of our civilization depends on mathematics, it is a bit worrisome that it isn t as solid as it was believed to be in the past I think the author overplays the drama a bit in the narrative still it is a big development
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 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
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 muchinteresting.This book is a detailed guide to the history of the philosophy of mathematics It c 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 muchinteresting.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 littlethan 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
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 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
Rereading this book I see it is a tragedy The foundations of mathematics can never be complete It is always open and some truths in it we will never reach This is microcosm of human limitations written in the queen of the sciences The surest thing in the world ,mathematics, can never rest on ultimate certainty This is just oneadditional limitation that comes with a birth certificate The real tragedy of the book is that Mathematics has sealed itself off from the general public with o Rereading this book I see it is a tragedy The foundations of mathematics can never be complete It is always open and some truths in it we will never reach This is microcosm of human limitations written in the queen of the sciences The surest thing in the world ,mathematics, can never rest on ultimate certainty This is just oneadditional limitation that comes with a birth certificate The real tragedy of the book is that Mathematics has sealed itself off from the general public with only a few initiates entering into its mysteries I love math it is a wonderful discipline filled with beauty that unlike the beauty of the world has a taste of the infinite and eternal no where else except religion or maybe art do you get to taste such fruits
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.
The best popular book on the truly tragic achievement of Godel s incompleteness theorem, and the path to it.