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Mathematics and the Physical World
"Kline is a first-class teacher and an able writer. . . . This is an enlarging and a brilliant book." ― Scientific American
"Dr. Morris Kline has succeeded brilliantly in explaining the nature of much that is basic in math, and how it is used in science." ― San Francisco Chronicle Since the major branches of mathematics grew and expanded in conjunction with science, the most effective way to appreciate and understand mathematics is in terms of the study of nature. Unfortunately, the relationship of mathematics to the study of nature is neglected in dry, technique-oriented textbooks, and it has remained for Professor Morris Kline to describe the simultaneous growth of mathematics and the physical sciences in this remarkable book. In a manner that reflects both erudition and enthusiasm, the author provides a stimulating account of the development of basic mathematics from arithmetic, algebra, geometry, and trigonometry, to calculus, differential equations, and the non-Euclidean geometries. At the same time, Dr. Kline shows how mathematics is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, and other phenomena. Historical and biographical materials are also included, while mathematical notation has been kept to a minimum.
This is an excellent presentation of mathematical ideas from the time of the Greeks to the modern era. It will be of great interest to the mathematically inclined high school and college student, as well as to any reader who wants to understand ― perhaps for the first time ― the true greatness of mathematical achievements.
"Dr. Morris Kline has succeeded brilliantly in explaining the nature of much that is basic in math, and how it is used in science." ― San Francisco Chronicle Since the major branches of mathematics grew and expanded in conjunction with science, the most effective way to appreciate and understand mathematics is in terms of the study of nature. Unfortunately, the relationship of mathematics to the study of nature is neglected in dry, technique-oriented textbooks, and it has remained for Professor Morris Kline to describe the simultaneous growth of mathematics and the physical sciences in this remarkable book. In a manner that reflects both erudition and enthusiasm, the author provides a stimulating account of the development of basic mathematics from arithmetic, algebra, geometry, and trigonometry, to calculus, differential equations, and the non-Euclidean geometries. At the same time, Dr. Kline shows how mathematics is used in optics, astronomy, motion under the law of gravitation, acoustics, electromagnetism, and other phenomena. Historical and biographical materials are also included, while mathematical notation has been kept to a minimum.
This is an excellent presentation of mathematical ideas from the time of the Greeks to the modern era. It will be of great interest to the mathematically inclined high school and college student, as well as to any reader who wants to understand ― perhaps for the first time ― the true greatness of mathematical achievements.
482 pages, Paperback
First published January 1, 1959
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About the author
Morris Kline
78 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 - 4 of 4 reviews
April 10, 2019
Read this as a kid.
I remember being mesmerized by his diagrams describing physics problems.
This was before you could look up everything on Wikipedia and watch YouTube videos on every conceivable subject.
I remember being mesmerized by his diagrams describing physics problems.
This was before you could look up everything on Wikipedia and watch YouTube videos on every conceivable subject.
October 14, 2025
An excellent exploration of mathematics and how it relates to science (physics in particular). The book explored the history of math and science, covering the lives and discoveries of many figures in history from the Renaissance to the Victorian era who shaped how we now understand the world. As is usual with these types of books, the author claims in the introduction that minimal mathematics is needed and it was a book for the general reader. Although he did go over some fundamentals, this was certainly not for the general public, but someone with at least a solid grade 12 math base and likely some university math to understand everything. For me it was a perfect level, but it wouldn't have been for everyone. The philosophical aspects, however, are accessible to the general public, and it is amazing how the author delves into even simple math and science topics to pull out their real meaning.
There were definitely some dated parts to the book, not much in terms of the science, since it covers primarily long established math and science rather than new discoveries, but the language and, dare I say, some misogynistic examples and phrasing. But given that it was written in the 1950s, it's really not bad in that respect.
Overall, this was a great read and a real eye-opener about how math is entwined in our physical world and how scientists have strove to discover its secrets through experimentation and logic.
There were definitely some dated parts to the book, not much in terms of the science, since it covers primarily long established math and science rather than new discoveries, but the language and, dare I say, some misogynistic examples and phrasing. But given that it was written in the 1950s, it's really not bad in that respect.
Overall, this was a great read and a real eye-opener about how math is entwined in our physical world and how scientists have strove to discover its secrets through experimentation and logic.
April 11, 2025
I loved this book.
I've rarely heard anyone talk about why mathematics is important, about why people care to use it. In primary and secondary schools, teachers (at least the ones I and most other people seem to have had) emphasize techniques and only occasionally do they give explanations about the technique's source. Techniques are taught like they're eternal truths and that if you don't 'get' it there must be something wrong with your brain.
I'd argue that this book is about more than just mathematics, it's about the scientific enterprise in general.
By explaining the sources and the thought processes of some of the greatest scientists (Newton, Galileo, Decartes, Maxwell, and Gauss among others), this books shows and tells at the same time.
What's the deal with all the math symbols?
- "The use of special symbols enables one to write mathematical statements concisely and precisely. The symbolism also contributes to clarity. The same statements in ordinary words would often be long, complicated sentences, too difficult for the mind to grasp..." and later, "There is no doubt that language of algebra is enormously effective in recording and in operating with mathematical concepts. Nevertheless, many people complain about the need to learn this special language. These people would be more justified, however, in complaining that the French people insist on using their own language and the Germans, theirs... Certainly insofar as the language of algebra is concerned, no subsititute serves the purposes so well... On the other hand there are some legitimate objections to the language of algebra and of mathematics generally... There is no doubt, too, that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgment has caused mathematicians to equate elegance and conciseness at the cost of intelligibility."
Why do people use math to study the world?
- "To confine our activity entirely to the physical investigations or observations may lead to getting lost in a jungle of physical facts; it may require traversing impassable territory, as in measuring the size of the earth; it may involve climbing mountains so high that the atmosphere becomse too rare to support physical activity; or it may mean the at present [1959] impossible task of getting to the moon...
"The mathematician... with concepts and axioms in mind, whether solely mathematical or a combination of physical and mathematical ones, the mathematician retires to a corner and deduces new conclusions about the physical world... [The conclusions] yield such knowledge as the distance to the sun and sometimes such totaly unexpected phenomea as the existence of radio waves."
Why is the world explained by math?
- "Mathematics is commonly regarded as highly abstract and remote from the real world... but every abstraction that even the greatest mathematician has introduced is ultimately derived from and is therefore understandable in terms of intuitively meaningful objects or phenomena...
"[Man] has one more means at his disposial to make his mathematics fit the physical world. If his theorems do not fit, he is free to change his axioms. This recources seems farfetched, but scientists have adopted this procedure in our time."
_____
Once again, this is a fantasic book. Now I just need some friends or strangers (I'm not picky!) to recommend this to.
I've rarely heard anyone talk about why mathematics is important, about why people care to use it. In primary and secondary schools, teachers (at least the ones I and most other people seem to have had) emphasize techniques and only occasionally do they give explanations about the technique's source. Techniques are taught like they're eternal truths and that if you don't 'get' it there must be something wrong with your brain.
I'd argue that this book is about more than just mathematics, it's about the scientific enterprise in general.
By explaining the sources and the thought processes of some of the greatest scientists (Newton, Galileo, Decartes, Maxwell, and Gauss among others), this books shows and tells at the same time.
What's the deal with all the math symbols?
- "The use of special symbols enables one to write mathematical statements concisely and precisely. The symbolism also contributes to clarity. The same statements in ordinary words would often be long, complicated sentences, too difficult for the mind to grasp..." and later, "There is no doubt that language of algebra is enormously effective in recording and in operating with mathematical concepts. Nevertheless, many people complain about the need to learn this special language. These people would be more justified, however, in complaining that the French people insist on using their own language and the Germans, theirs... Certainly insofar as the language of algebra is concerned, no subsititute serves the purposes so well... On the other hand there are some legitimate objections to the language of algebra and of mathematics generally... There is no doubt, too, that mathematicians are generally overzealous about conciseness, and in their passion for brevity indulge in symbols even where these seem no better than a familiar English word or phrase. A faulty judgment has caused mathematicians to equate elegance and conciseness at the cost of intelligibility."
Why do people use math to study the world?
- "To confine our activity entirely to the physical investigations or observations may lead to getting lost in a jungle of physical facts; it may require traversing impassable territory, as in measuring the size of the earth; it may involve climbing mountains so high that the atmosphere becomse too rare to support physical activity; or it may mean the at present [1959] impossible task of getting to the moon...
"The mathematician... with concepts and axioms in mind, whether solely mathematical or a combination of physical and mathematical ones, the mathematician retires to a corner and deduces new conclusions about the physical world... [The conclusions] yield such knowledge as the distance to the sun and sometimes such totaly unexpected phenomea as the existence of radio waves."
Why is the world explained by math?
- "Mathematics is commonly regarded as highly abstract and remote from the real world... but every abstraction that even the greatest mathematician has introduced is ultimately derived from and is therefore understandable in terms of intuitively meaningful objects or phenomena...
"[Man] has one more means at his disposial to make his mathematics fit the physical world. If his theorems do not fit, he is free to change his axioms. This recources seems farfetched, but scientists have adopted this procedure in our time."
_____
Once again, this is a fantasic book. Now I just need some friends or strangers (I'm not picky!) to recommend this to.
July 24, 2020
It's good to get an idea of the amount of work and non obviousness of what you are taught in school
Displaying 1 - 4 of 4 reviews