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Mathematical Thought from Ancient to Modern Times #1
Mathematical Thought from Ancient to Modern Times, Vol. 2
This comprehensive history traces the development of mathematical ideas and the careers of the men responsible for them. Volume 1 looks at the discipline's origins in Babylon and Egypt, the creation of geometry and trigonometry by the Greeks, and the role of mathematics in the medieval and early modern periods. Volume 2 focuses on calculus, the rise of analysis in the 19th century, and the number theories of Dedekind and Dirichlet. The concluding volume covers the revival of projective geometry, the emergence of abstract algebra, the beginnings of topology, and the influence of Godel on recent mathematical study.
480 pages, Paperback
First published September 29, 1972
92 people are currently reading
1278 people want to read
About the author
Morris Kline
79 books103 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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Displaying 1 - 23 of 23 reviews
December 9, 2015
Perhaps the best and most exaustive ,médium to high level,history of mathematics and mathematical though ever written
August 6, 2009
This is a great, smooth read, if you're comfortable with general math. It is more difficult (though arguably more rewarding) read if you are not. Working through all of the problems/equations is well worth it. The inclusion of non-modern notations is fascinating.
In short, I wish that I had been taught math through this lens rather than the rather dry "learn these things by rote and if you're lucky, we'll mention Newton and Leibniz in passing." Not that the individuals are the important factor in this history, but rather that the history and progression of math lends mental stickiness to the ideas.
Favorite Quote:
"...the Romans [...:] entire role in the history of mathematics was that of an agent of destruction."
In short, I wish that I had been taught math through this lens rather than the rather dry "learn these things by rote and if you're lucky, we'll mention Newton and Leibniz in passing." Not that the individuals are the important factor in this history, but rather that the history and progression of math lends mental stickiness to the ideas.
Favorite Quote:
"...the Romans [...:] entire role in the history of mathematics was that of an agent of destruction."
December 27, 2013
Kline is a very biased author. He gives no credit to the non-European civilizations. His language about the ancient Egyptians and Babylonians is, for example, very demeaning. The book involves some pretty complicated math, but it was the degrading language that made it more annoying to read. Our professor wanted us to see that some people still hold on to a very Eurocentric view of the way mathematics developed.
June 29, 2017
After Descartes comes math. I'm reading this from the library without really paying too much attention, and vaguely pretending it will solve my coding problems in xna.
A helper text for Manuel DeLanda.
A helper text for Manuel DeLanda.
December 17, 2015
A few more, closer, readings of this book are in order. Thus far I have only really read closely the parts that mattered to me (foundations, function concept, complex function theory, infinite series, algebra leading up to Galois), and perhaps one day I'll be able to get something more valuable out of the calculus and ODE/PDE parts.
Gian Carlo-Rota says that the three volumes (this is just volume 2) are probably the best account of the history of modern mathematics up till the early 20th century in the English language (I have heard that there are even superior accounts published in Russian and German). I am inclined to agree.
The accounts are mathematicobiographical -- there is certainly much said (and much can be inferred) about the mathematicians themselves, but of course there is still a much heavier emphasis on their mathematical legacy. While I have only skimmed through vols 1 and 3, I can say that vol 2 and vol 3 are not very accessible to a general audience (if one wishes to get a lot out of it). Most of volume 1 is easily accessible by a relatively attentive mathematics undergraduate.
On the other hand, vols 2 and 3 probably aren't even accessible to some graduate students, though this is not really due to the difficulty to the material, but more of the scope. Much is said about the calculus of variations and ODE/PDEs in vol 2, as well as Galois theory, elimination theory, and so on. The topics concerned in vol 3 are even more advanced; if "birational geometry" or "ideal theory" sound like gibberish to you, it is likely that little else in vol 3 will be coherent at all.
Of course, the sophistication (compared to its popular counterparts) of this historical account is its primary merit. The bibliography alone would be useful to certain mathematical researchers. The exposition found here still manages to capture the whole essence of the mathematics done by our ancestors, without deforming it all such that it loses all precision and meaning.
Of particular interest to virtually all readers would be a discussion of the metaphysical arguments used by many prominent mathematicians in their work, and a brief (two centuries max?) loss of primacy of rigorous proof in mathematics. Euler himself apparently agreed with the assessment that mathematics would have been exhausted and that in the future, the important problems will only come from the natural sciences.
(The following concerns less about vol 2 but more about the scope of the whole work, especially in regards to certain accusations of the works being too Euro-centric)
Another important note is that the 3 volumes are not meant to be exhaustive at all. The objective of the work is to provide an account on what Kline calls the "main line" of mathematical thought. Other readers have noted that this is too "Euro-centric". Kline perhaps does not help his case by writing:
"To keep the material within bounds I have ignored several civilisations such as the Chinese, Japanese, and Mayan because their work had no material impact on the main line of mathematical thought." [emphasis mine].
Yet any serious reading of the volumes will show that any Euro-centric bias, other than the desire to not create an exhaustive account, is not present. The main line of this mathematical thought is not an imaginary Euro-centric fiction, but an abbreviation of a historical account of interplaying ideas. For example, the mathematics of the Hindus and Arabs, which (deservedly) take up several chapters in volume 1, for their specific contributions directly affected the major mathematicians that worked in the ensuing centuries. On the other hand, results of the Chinese, for example, though some of which were certainly ahead of their time, were virtually, if not completely, unknown to the developers of the main-line of mathematics.
The significance of this main-line of thought is evidenced in volume 3. Two thousand years and 800 pages later, Euclidean geometry continues to haunt the late 19th-early 20th century mathematicians. The main-line is not an arbitrary thread of history, but a coherent, omni-present braid that underlies much, if not all of mathematical thought, up till the present day.
Otherwise, readers still interested in an account of the non-European involvement in the main line of thought need not look towards the past. For if a volume 4 ever to be written (unfortunately, Kline is dead), many contributions from non-European mathematicians will surely be elaborated on. For example, one should expect that any account written in the next 50 years concerning the development of modern algebra and algebraic geometry would expect many chapters dedicated to the works of Japanese mathematicians such as Hironaka, Nagata, Taniyama and Shimura (of Taniyama-Shimura conjecture fame), Nakayama, Mochizuki, etc.
Gian Carlo-Rota says that the three volumes (this is just volume 2) are probably the best account of the history of modern mathematics up till the early 20th century in the English language (I have heard that there are even superior accounts published in Russian and German). I am inclined to agree.
The accounts are mathematicobiographical -- there is certainly much said (and much can be inferred) about the mathematicians themselves, but of course there is still a much heavier emphasis on their mathematical legacy. While I have only skimmed through vols 1 and 3, I can say that vol 2 and vol 3 are not very accessible to a general audience (if one wishes to get a lot out of it). Most of volume 1 is easily accessible by a relatively attentive mathematics undergraduate.
On the other hand, vols 2 and 3 probably aren't even accessible to some graduate students, though this is not really due to the difficulty to the material, but more of the scope. Much is said about the calculus of variations and ODE/PDEs in vol 2, as well as Galois theory, elimination theory, and so on. The topics concerned in vol 3 are even more advanced; if "birational geometry" or "ideal theory" sound like gibberish to you, it is likely that little else in vol 3 will be coherent at all.
Of course, the sophistication (compared to its popular counterparts) of this historical account is its primary merit. The bibliography alone would be useful to certain mathematical researchers. The exposition found here still manages to capture the whole essence of the mathematics done by our ancestors, without deforming it all such that it loses all precision and meaning.
Of particular interest to virtually all readers would be a discussion of the metaphysical arguments used by many prominent mathematicians in their work, and a brief (two centuries max?) loss of primacy of rigorous proof in mathematics. Euler himself apparently agreed with the assessment that mathematics would have been exhausted and that in the future, the important problems will only come from the natural sciences.
(The following concerns less about vol 2 but more about the scope of the whole work, especially in regards to certain accusations of the works being too Euro-centric)
Another important note is that the 3 volumes are not meant to be exhaustive at all. The objective of the work is to provide an account on what Kline calls the "main line" of mathematical thought. Other readers have noted that this is too "Euro-centric". Kline perhaps does not help his case by writing:
"To keep the material within bounds I have ignored several civilisations such as the Chinese, Japanese, and Mayan because their work had no material impact on the main line of mathematical thought." [emphasis mine].
Yet any serious reading of the volumes will show that any Euro-centric bias, other than the desire to not create an exhaustive account, is not present. The main line of this mathematical thought is not an imaginary Euro-centric fiction, but an abbreviation of a historical account of interplaying ideas. For example, the mathematics of the Hindus and Arabs, which (deservedly) take up several chapters in volume 1, for their specific contributions directly affected the major mathematicians that worked in the ensuing centuries. On the other hand, results of the Chinese, for example, though some of which were certainly ahead of their time, were virtually, if not completely, unknown to the developers of the main-line of mathematics.
The significance of this main-line of thought is evidenced in volume 3. Two thousand years and 800 pages later, Euclidean geometry continues to haunt the late 19th-early 20th century mathematicians. The main-line is not an arbitrary thread of history, but a coherent, omni-present braid that underlies much, if not all of mathematical thought, up till the present day.
Otherwise, readers still interested in an account of the non-European involvement in the main line of thought need not look towards the past. For if a volume 4 ever to be written (unfortunately, Kline is dead), many contributions from non-European mathematicians will surely be elaborated on. For example, one should expect that any account written in the next 50 years concerning the development of modern algebra and algebraic geometry would expect many chapters dedicated to the works of Japanese mathematicians such as Hironaka, Nagata, Taniyama and Shimura (of Taniyama-Shimura conjecture fame), Nakayama, Mochizuki, etc.
October 1, 2015
Definitely reading part 2. A very riveting account of exactly the title. All the anecdotes, the myths, the stories, the hard facts, personalities, rivalries, everything. Oh, and math! It's refreshing to read elementary math woven seamlessly into the text. He doesn't shy away from it like most. Wonderful for the math literate, charming and coherent even for the rest.
December 5, 2024
This was a suggested reference for my class on the History and Philosophy of Mathematics. It was a fine read. I thought this was better than "History of Mathematics: An Introduction" (3rd ed. 2008) by Victor Katz, even though this book was originally published in 1972 and did not cover anything after 1950, as the author Morris Kline stated in the introduction.
June 8, 2019
Some results were very much abbreviated, as in : "next, newton proved that...". Much of the book therefore, amounted to a big collection of formulas without real insight into how they were proved or derived. Other than that it was quite enjoyable, if at times hard to understand.
August 8, 2020
[Reseña de la edición unificada de la obra]
El objetivo de este libro es presentar el desarrollo del pensamiento matemático así como el desarrollo de esta disciplina y su aplicación a lo largo de la historia de la humanidad. El libro abarca desde las civilizaciones de Mesopotamia y Babilonia hasta el primer tercio del siglo pasado, pues fue escrito en 1972 y en palabras del autor, era entonces pronto para analizar qué lugar en la historia tendrían los desarrollos producidos más allá de 1930. Así, a lo largo de sus 51 capítulos y 1600 páginas, el autor presenta un estudio pormenorizado, exhaustivo y (esencialmente) cronológico sobre los desarrollos y las corrientes matemáticas que considera que tuvieron una mayor influencia a lo largo de la historia y en el estado actual de esta ciencia. Los primeros capítulos están dedicados al estudio de las matemáticas en la Antigüedad con numerosos planteamientos filosóficos y nada complicados desde el punto de vista técnico. En contraste, hacia el capítulo 15 empezará a abordar los avances matemáticos en la Europa posmedieval, con un texto mucho más técnico y algo más independiente del contexto histórico. Las matemáticas de los siglos XVII, XVIII y XIX constituyen el núcleo del libro, incluyendo áreas como álgebra, cálculo, geometría diferencial, y a medida que avanzamos en el tiempo, también geometría algebraica, teoría de números, topología o lógica, y por último se introducen las prespectivas de avance de principios del siglo XX.
Pasando a valorar los contenidos del libro, es de justicia reconocer el inmenso mérito del autor por la elaboración de esta obra a todas luces colosal. En mi opinión consigue su objetivo de proporcionar una visión global de la evolución del pensamiento matemático, consiguiendo conectar las primeras ideas documentadas en las civilizaciones de Mesopotamia, Babilonia y Egipto con las matemáticas que he aprendido. Además, pese a la complejidad de muchas de los planteamientos presentados, están escritos con mucha claridad y exhaustividad, lo cual hace de esta obra un libro especializado y no de divulgación. Por otra parte, pienso que esta circunstancia hace extraordinariamente difícil la tarea de hacer una lectura comprensiva de muchos de los capítulos. Así, ha habido muchos momentos en los que la experiencia ha sido bastante agotadora. La estructura de los capítulos, sin dejar de ser flexible, estaba bien compartimentada por siglos, algo que tiene especial mérito teniendo en cuenta el volumen descomunal de los contenidos a presentar.
Por enumerar algunas de las carencias de este libro, tiene una marcada visión eurocentrista, restringiendo a Occidente su análisis en la mayor parte de la obra (algo que por otra parte el autor justifica en el prólogo aduciendo que las matemáticas que ha dejado de lado no han sido tan relevantes). Digna de mención me parece también la infrarrepresentación de las mujeres en la obra, siendo las únicas mencionadas Hipatia de Alejandría, Sophie Germain, Sofia Kovalevskaya y Emmy Noether, si no me falla la memoria. Es tremendamente sintomático de la falta de perspectiva de género en este libro que se presente a Hipatia como "la hija de Teon de Alejandría" sin ni siquiera presentar sus aportaciones, o que no se mencionen los primos de Germain. Esto puede ser debido en buena parte a la escasa participación de las mujeres en matemáticas y en ciencias en general a lo largo de la historia, y también a que la conciencia de género en ciencias era bastante inferior a la actual en la segunda mitad del siglo pasado. Una tercera carencia no es culpa del autor, y es la falta de las matemáticas de los últimos casi 100 años. Ya no solo por los grandes hitos que se han conseguido, como la demostración del último teorema de Fermat; sobre todo por el desarrollo de las corrientes de fundamentación de las matemáticas y la perspectiva actual del papel de las matemáticas en el mundo. Creo que la falta de relevancia en el libro de la teoría de probabilidades y las matemáticas aplicadas tal y como las conocemos hoy en día pueden tener su origen en esta carencia.
De este libro he aprendido muchas cosas. En un plano más anecdótico, he aprendido numerosas curiosidades como que el signo = tiene su origen en que quien lo introdujo por primera vez consideraba que no hay nada más igual que dos rectas paralelas. Pero más importante, he aprendido sobre el origen de las ideas que hoy nos enseñan en el grado en matemáticas y el encomiable esfuerzo que se ha hecho para llegar a las mismas. Es esencial para cualquiera que se quiera dedicar a la investigación en matemáticas tener presente que grandes matemáticos no estuvieron exentos de cometer graves errores y que detrás del paso de un teorema al siguiente hay multitud de intentos infructuosos. En esto, pienso que el autor también consigue su objetivo.
Para terminar, pienso que para abordar la lectura de este libro hacen falta estudios superiores en matemáticas o ingeniería. Dicho lo cual, me parece un libro muy recomendable para aprender historia de las matemáticas, aunque quizás para leerlo en un intervalo de tiempo muy prolongado o para consultar determinados capítulos.
El objetivo de este libro es presentar el desarrollo del pensamiento matemático así como el desarrollo de esta disciplina y su aplicación a lo largo de la historia de la humanidad. El libro abarca desde las civilizaciones de Mesopotamia y Babilonia hasta el primer tercio del siglo pasado, pues fue escrito en 1972 y en palabras del autor, era entonces pronto para analizar qué lugar en la historia tendrían los desarrollos producidos más allá de 1930. Así, a lo largo de sus 51 capítulos y 1600 páginas, el autor presenta un estudio pormenorizado, exhaustivo y (esencialmente) cronológico sobre los desarrollos y las corrientes matemáticas que considera que tuvieron una mayor influencia a lo largo de la historia y en el estado actual de esta ciencia. Los primeros capítulos están dedicados al estudio de las matemáticas en la Antigüedad con numerosos planteamientos filosóficos y nada complicados desde el punto de vista técnico. En contraste, hacia el capítulo 15 empezará a abordar los avances matemáticos en la Europa posmedieval, con un texto mucho más técnico y algo más independiente del contexto histórico. Las matemáticas de los siglos XVII, XVIII y XIX constituyen el núcleo del libro, incluyendo áreas como álgebra, cálculo, geometría diferencial, y a medida que avanzamos en el tiempo, también geometría algebraica, teoría de números, topología o lógica, y por último se introducen las prespectivas de avance de principios del siglo XX.
Pasando a valorar los contenidos del libro, es de justicia reconocer el inmenso mérito del autor por la elaboración de esta obra a todas luces colosal. En mi opinión consigue su objetivo de proporcionar una visión global de la evolución del pensamiento matemático, consiguiendo conectar las primeras ideas documentadas en las civilizaciones de Mesopotamia, Babilonia y Egipto con las matemáticas que he aprendido. Además, pese a la complejidad de muchas de los planteamientos presentados, están escritos con mucha claridad y exhaustividad, lo cual hace de esta obra un libro especializado y no de divulgación. Por otra parte, pienso que esta circunstancia hace extraordinariamente difícil la tarea de hacer una lectura comprensiva de muchos de los capítulos. Así, ha habido muchos momentos en los que la experiencia ha sido bastante agotadora. La estructura de los capítulos, sin dejar de ser flexible, estaba bien compartimentada por siglos, algo que tiene especial mérito teniendo en cuenta el volumen descomunal de los contenidos a presentar.
Por enumerar algunas de las carencias de este libro, tiene una marcada visión eurocentrista, restringiendo a Occidente su análisis en la mayor parte de la obra (algo que por otra parte el autor justifica en el prólogo aduciendo que las matemáticas que ha dejado de lado no han sido tan relevantes). Digna de mención me parece también la infrarrepresentación de las mujeres en la obra, siendo las únicas mencionadas Hipatia de Alejandría, Sophie Germain, Sofia Kovalevskaya y Emmy Noether, si no me falla la memoria. Es tremendamente sintomático de la falta de perspectiva de género en este libro que se presente a Hipatia como "la hija de Teon de Alejandría" sin ni siquiera presentar sus aportaciones, o que no se mencionen los primos de Germain. Esto puede ser debido en buena parte a la escasa participación de las mujeres en matemáticas y en ciencias en general a lo largo de la historia, y también a que la conciencia de género en ciencias era bastante inferior a la actual en la segunda mitad del siglo pasado. Una tercera carencia no es culpa del autor, y es la falta de las matemáticas de los últimos casi 100 años. Ya no solo por los grandes hitos que se han conseguido, como la demostración del último teorema de Fermat; sobre todo por el desarrollo de las corrientes de fundamentación de las matemáticas y la perspectiva actual del papel de las matemáticas en el mundo. Creo que la falta de relevancia en el libro de la teoría de probabilidades y las matemáticas aplicadas tal y como las conocemos hoy en día pueden tener su origen en esta carencia.
De este libro he aprendido muchas cosas. En un plano más anecdótico, he aprendido numerosas curiosidades como que el signo = tiene su origen en que quien lo introdujo por primera vez consideraba que no hay nada más igual que dos rectas paralelas. Pero más importante, he aprendido sobre el origen de las ideas que hoy nos enseñan en el grado en matemáticas y el encomiable esfuerzo que se ha hecho para llegar a las mismas. Es esencial para cualquiera que se quiera dedicar a la investigación en matemáticas tener presente que grandes matemáticos no estuvieron exentos de cometer graves errores y que detrás del paso de un teorema al siguiente hay multitud de intentos infructuosos. En esto, pienso que el autor también consigue su objetivo.
Para terminar, pienso que para abordar la lectura de este libro hacen falta estudios superiores en matemáticas o ingeniería. Dicho lo cual, me parece un libro muy recomendable para aprender historia de las matemáticas, aunque quizás para leerlo en un intervalo de tiempo muy prolongado o para consultar determinados capítulos.
June 11, 2018
What appealed to me in this series of books is that it seems to give the big picture of mathematics, introducing all sub-disciplines and how they are related. To be honest I did perhaps not realize that it is still a very detailed history book. While unfamiliar and a bit heavy at the beginning (I don't usually read history) I eventually got used to it and I must say that after I read this book I am truly amazed by the old greeks in particular (volume one covers from antiquity up to the invention of calculus).
While the book is generally in a chronological order, it jumps a bit back and forth because it treats topic by topic (e.g. first algebra in the 1600s, then geometry in the 1600s and so forth). While this makes it better organized and probably more interesting, it is sometimes a bit hard to connect which historical event happend before one another.
The book is well written but detailed enough that you should be rather passionate about the subject. If you've never solved differential equations it's probably not for you. But if you're up for this sort of literature, I would definitely recommend it.
While the book is generally in a chronological order, it jumps a bit back and forth because it treats topic by topic (e.g. first algebra in the 1600s, then geometry in the 1600s and so forth). While this makes it better organized and probably more interesting, it is sometimes a bit hard to connect which historical event happend before one another.
The book is well written but detailed enough that you should be rather passionate about the subject. If you've never solved differential equations it's probably not for you. But if you're up for this sort of literature, I would definitely recommend it.
August 29, 2018
È un interessante e dettagliato excursus sulla storia della Matematica e della evoluzione del pensiero matematico. Personalmente ho trovato affascinante il primo volume, molto più del secondo, in quanto ha descritto le radici dei numeri per come lo conosciamo e usiamo oggigiorno nella cultura occidentale, a partire dagli albori e dal dilemma dello zero e della notazione posizionale.
In ogni caso un lavoro davvero entusiasmante, ricco di spunti di approfondimento e riflessione e di tanta sana Matematica!
In ogni caso un lavoro davvero entusiasmante, ricco di spunti di approfondimento e riflessione e di tanta sana Matematica!
July 24, 2020
If you want to follow step by step some of the reasoning behind important mathematical subjects, this is good... But it seems like an enormous effort to reproduce something already done. If you only want to grasp some intuitions and general ideas about mathematics and scientific thinking, which was my purpose, you'll have to read between lines
March 18, 2024
Astounding account of mathematical history!
September 22, 2025
Skimmed through chapter 4 on Euclid. Was fascinating to see how Greek math (e.g. Euclidean geometry) is so similar to how we do modern math in terms of axioms and proofs and method.
May 14, 2020
A great introduction to the history of mathematics! To me it seems to go quite in depth into the details, maybe a bit too much at times. Some of the Classical Greek proofs are tricky to follow due to the completely different way they had to doing mathematics, compared to our modern version.
The book covers the history of mathematics from its beginnings in Mesopotamia and Egypt, to the discovery of Calculus by Newton and Leibniz in the 17th century.
I do have to say that the book seems slightly biased towards Europeans, as while going over the mathematics of Babylonians, Egyptians, Arabs, and Hindus, the author just makes it seem like their advancements were not huge, due to the fact that they mostly worked on applied mathematics. The book also completely ignores East Asian and American mathematics, as they did not influence the main branch of mathematics which is the European one (I do have to say this is a fair point).
Overall a great read that I would recommend to all lovers of mathematics.
The book covers the history of mathematics from its beginnings in Mesopotamia and Egypt, to the discovery of Calculus by Newton and Leibniz in the 17th century.
I do have to say that the book seems slightly biased towards Europeans, as while going over the mathematics of Babylonians, Egyptians, Arabs, and Hindus, the author just makes it seem like their advancements were not huge, due to the fact that they mostly worked on applied mathematics. The book also completely ignores East Asian and American mathematics, as they did not influence the main branch of mathematics which is the European one (I do have to say this is a fair point).
Overall a great read that I would recommend to all lovers of mathematics.
May 27, 2020
Great book for a lockdown.
May 15, 2015
very useful
May 27, 2020
Well worth the effort.
September 12, 2020
I only read about one third of this.
A bit too mathematical for casual reading after a long day spent doing... math.
This does seem like a fine book and I might very well finish it later in life.
A bit too mathematical for casual reading after a long day spent doing... math.
This does seem like a fine book and I might very well finish it later in life.
dont-want-to-read
August 23, 2020Jeg har også bind 3 som ikke er på Goodreads. Jeg kommer ikke til at læse dem.
March 18, 2024
Astounding account of mathematical history!
Read
October 29, 2012putos
Read
September 13, 2018I bailed out half way through volume 2, as the math got too technical for me. Thoroughly enjoyed volume 1.
Displaying 1 - 23 of 23 reviews