Mathematical Discourses: The Heart of Mathematical Science
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MATHEMATICAL DISCORSES:
THE HEART OF MATHEMATICAL SCIENCE
© 1964 by Carroll V. Newsom
PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N. J.
Boek met harde bruine linnen kaft met bruine letteropdruk in perfecte (onbelezen) conditie;
126 pagina's met tientallen schema's, illustraties enz.;
THE MATHEMATICIZING OF SCIENCE has led to revolutionary changes in our civilization, especially in recent years.
In addition, new approaches to the study of many aspects of human knowledge have been made possible by what scholars have learned through their struggle to understand the nature of mathematics. Thus a person living in our time can hardly be characterized as educated unless he has ome understanding of the fundamental concepts of mathematical science, their meaning, and their proper utilization.
Yet, hardly any expositions treating these subjects are available to the inquisitive reader who possesses limited mathematical background. Some excellent books and articles upon the foundations of mathematics have been written for the professional mathematician or logician, but the sophistication of their approach is such that only a few members of the reading public can understand them. The concern of most popular books on mathematics, on the other hand, is merely with interesting problems and anecdotes that appear upon the fringe of mathematical knowledge; little attention is given to mathematics as a systematic discipline.
This little treatise, therefore, written for the nonmathematician, is conerned with mathematical discourses, which are central to all mathematical study. Unless a person has some familiarity with the concept of mathematical discourse, he is not prepared to understand even the rudiments of modern mathematics and its significance. An understanding of the subject of this book, in other words, is essential to mathematical literacy.
It is the hope of the author that the development of his thesis has been accomplished without too much compromise with the tenets of mathematical rigor. He is quite aware that a popular presentation of a very difficult idea will rarely satisfy the experts. Perhaps, however, even with its faults, the present work will help some students, young and old, to a somewhat better comprehension of mathematics, what it is, and how it is being employed to assist man in his continuing effort to discover his place in the universe.
This book had its inception in an after-dinner conversation during which Mrs. Alvin C. Eurich urged me to undertake its writing; it is a pleasure now to be able to present her the completed product. The manuscript was read critically in its initial state by Professor Howard Eves, my friend and collaborator on many projects; I am deeply grateful to him for his constructive suggestions. And, again, I acknowledge with appreciation the assistance of my wife, Frances Higley Newsom, who may feel that she has spent a substantial part of her married life in typing and editing my writings.
I desire to express my special gratitude to the University of Toronto Press for permission to quote the axiomatic foundation for Euclidean geometry developed by Professor Gilbert de B. Robinson in his book. The Foundations of Geometry. It is a pleasure also to thank the St. John's College Press, Dover Publications, Inc., The University of Chicago Press, the Oxford University Press, John Wiley & Sons, Inc., Science, Daedalus, and Professor Anthony F. C. Wallace for graciously granting me permission to quote from certain works.
C. V. Newsom
CONTENTS
ONE
The Earliest Mathematical Ideas
TWO
Introduction of the Concept of Mathematical Discourse:
THE CONTRIBUTION OF THE ANCIENT GREEKS
THREE
The Concept of Mathematical Discourse is Perfected:
CONTRIBUTIONS
SINCE THE TIME OF THE RENAISSANCE
FOUR
Typical Mathematical Discourses
FIVE
The Utilization of
Mathematical Discourses in Man's Intellectual Endeavors
INDEX
THE HEART OF MATHEMATICAL SCIENCE
© 1964 by Carroll V. Newsom
PRENTICE-HALL, INC., ENGLEWOOD CLIFFS, N. J.
Boek met harde bruine linnen kaft met bruine letteropdruk in perfecte (onbelezen) conditie;
126 pagina's met tientallen schema's, illustraties enz.;
THE MATHEMATICIZING OF SCIENCE has led to revolutionary changes in our civilization, especially in recent years.
In addition, new approaches to the study of many aspects of human knowledge have been made possible by what scholars have learned through their struggle to understand the nature of mathematics. Thus a person living in our time can hardly be characterized as educated unless he has ome understanding of the fundamental concepts of mathematical science, their meaning, and their proper utilization.
Yet, hardly any expositions treating these subjects are available to the inquisitive reader who possesses limited mathematical background. Some excellent books and articles upon the foundations of mathematics have been written for the professional mathematician or logician, but the sophistication of their approach is such that only a few members of the reading public can understand them. The concern of most popular books on mathematics, on the other hand, is merely with interesting problems and anecdotes that appear upon the fringe of mathematical knowledge; little attention is given to mathematics as a systematic discipline.
This little treatise, therefore, written for the nonmathematician, is conerned with mathematical discourses, which are central to all mathematical study. Unless a person has some familiarity with the concept of mathematical discourse, he is not prepared to understand even the rudiments of modern mathematics and its significance. An understanding of the subject of this book, in other words, is essential to mathematical literacy.
It is the hope of the author that the development of his thesis has been accomplished without too much compromise with the tenets of mathematical rigor. He is quite aware that a popular presentation of a very difficult idea will rarely satisfy the experts. Perhaps, however, even with its faults, the present work will help some students, young and old, to a somewhat better comprehension of mathematics, what it is, and how it is being employed to assist man in his continuing effort to discover his place in the universe.
This book had its inception in an after-dinner conversation during which Mrs. Alvin C. Eurich urged me to undertake its writing; it is a pleasure now to be able to present her the completed product. The manuscript was read critically in its initial state by Professor Howard Eves, my friend and collaborator on many projects; I am deeply grateful to him for his constructive suggestions. And, again, I acknowledge with appreciation the assistance of my wife, Frances Higley Newsom, who may feel that she has spent a substantial part of her married life in typing and editing my writings.
I desire to express my special gratitude to the University of Toronto Press for permission to quote the axiomatic foundation for Euclidean geometry developed by Professor Gilbert de B. Robinson in his book. The Foundations of Geometry. It is a pleasure also to thank the St. John's College Press, Dover Publications, Inc., The University of Chicago Press, the Oxford University Press, John Wiley & Sons, Inc., Science, Daedalus, and Professor Anthony F. C. Wallace for graciously granting me permission to quote from certain works.
C. V. Newsom
CONTENTS
ONE
The Earliest Mathematical Ideas
TWO
Introduction of the Concept of Mathematical Discourse:
THE CONTRIBUTION OF THE ANCIENT GREEKS
THREE
The Concept of Mathematical Discourse is Perfected:
CONTRIBUTIONS
SINCE THE TIME OF THE RENAISSANCE
FOUR
Typical Mathematical Discourses
FIVE
The Utilization of
Mathematical Discourses in Man's Intellectual Endeavors
INDEX
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