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What is mathematics about? Is there a mathematical universe glimpsed by a mathematical intuition? Or is mathematics an arbitrary game of symbols, with no inherent meaning, that somehow finds application to life on earth? Robert Knapp holds, on the contrary, that mathematics is about the world. His book develops and applies its alternative viewpoint, first, to elementary geometry and the number system and, then, to more advanced topics, such as topology and group representations. Its theme is that mathematics, however abstract, arises from and is shaped by requirements of indirect measurement. Eratosthenes, in 200 BC, demonstrated the power of indirect measurement when he estimated the circumference of the earth by measuring a shadow at noon, in Alexandria, on the day of the summer solstice. Establishing geometric relationships, solving equations, finding approximations, and, generally, discovering quantitative relationships are tools of indirect measurement: They are the core of mathematics, the drivers of its development, and the heart of its power to enhance our lives.
- Sales Rank: #947627 in Books
- Published on: 2014-08-14
- Original language: English
- Dimensions: 9.00" h x 1.22" w x 6.00" l, 1.86 pounds
- Binding: Paperback
- 538 pages
About the Author
Robert Knapp earned his Ph.D. in mathematics from Princeton University in 1972. He has published work on differential geometry and partial differential equations, and, after a year at the Institute for Advanced Studies in Princeton, taught graduate and undergraduate mathematics at Purdue University. His study and appreciation of abstract mathematics began in high school and his conviction that mathematics, including abstract mathematics, is about the world began then, as well. Although he retired from the profession in the late 1970s, his study of the content, history and application of mathematics continues to this day. In recent years he has presented his unique perspectives on geometry and the number system in a series of lectures at Objectivist Summer Conferences organized by the Ayn Rand Institute. He has lived in the Philadelphia area for almost 30 years.
Most helpful customer reviews
66 of 72 people found the following review helpful.
Brilliant Breakthrough in Mathematical Theory
By Ray Shelton
Just as Ayn Rand corrected, completed, and extended the Aristotelian project into a new, brilliant, organic - and true - philosophic whole; so too retired professor of mathematics, Robert Knapp - applying Ayn Rand's theory of knowledge - has corrected, completed, and extended the Euclidian postulates into a new, brilliant, organic - and true - mathematic whole.
Dr. Knapp has synthesized his decades of research in modern academic mathematical theory together with Ayn Rand's theory of knowledge into a stunning work in which he has created the framework for a new philosophy of mathematics.
Chapter I, "Euclid's Method" lays the groundwork for a reinterpretation of Euclid's method that is solidly grounded in reality. After more than 2,000 years, Euclid's mathematical treatise has finally received the proper objective foundation necessary for modern mathematical theory to be set aright. This single chapter deepened my understanding of both the foundations of geometry and AR's theory of concepts.
While I do teach math to children, I found that having a mathematical background was not necessary to follow Dr. Knapp's arguments. Moreover, no prior knowledge of Ayn Rand's theory of knowledge was required either. Dr. Knapp's writing is crystal clear as he shuttles back and forth between abstract points and concrete examples. I found the book's illustrations extremely helpful in grasping the material.
My favorite chapter was "Geometry and Human Cognition" for its insightful application of AR's idea of "measurement omission" to mathematical diagrams and the resulting cognitive rewards. Surprisingly, I found my mind wandering to AR's theory of stylization and art while considering Dr. Knapp's views. There is much here for the serious thinker.
If you are interested in the philosophy of science in general, or the philosophy of mathematics specifically - this is the book that all future mathematical theorists will have to take into account. As for students of AR's philosophy, here we have an application of her view of knowledge to heretofore unexplored territory - a territory rich with untapped wealth.
Dr. Knapp's book can but only be the necessary beginning of a philosophy of mathematics for the 21st century - and a brilliant beginning it is.
23 of 24 people found the following review helpful.
a very readable, philosophical, objective defense of Euclid
By Molly Johnson
As a homeschooling mom I was delighted to see this defense of Euclid's geometry. The author explains what angles, line segments, and circles are actually measuring, how Euclid relates one to another to bootstrap his way into measuring one thing by means of another, and how these abstractions symbolize real objects. It is both a philosophical defense of the useful, reality-based, practical worth of mathematics and an explanation of the math itself. I'm neither a math geek nor a philosopher, so a lot of it is over my head, but the author's explanations are very clear and thorough and non-technical (as far as is possible). The subject is becoming slowly more clear to me. So, I'm enjoying the book and highly recommend it even to other laymen.
27 of 30 people found the following review helpful.
The value of being worldly
By Amazon Customer
I'm excited by this book. While I'm not through it entirely, it warrants a review already.
Prof. Knapp's integration, in the early sections of the book, of Euclid's great "Elements" and Ayn Rand's "Introduction to Objectivist Epistemology" is a pleasure to see and read. His discussion of the concepts of perfection and imperfection in shapes, and in relation to perception is eye-opening (in an intellection sense). And explaining the concept "precision", in the service of one's purpose for using geometry and measurement, adds great clarity to how one should think about mathematics and its relation to the world.
His whole project here, to extend Ayn Rand's epistemological approach to measurement and mathematics (as opposed to Aristotle's metaphysical approach, or Plato's approach, or the modern symbolist approach) into a new philosophy of mathematics is done with great clarity (I have to use that word again because clarity stands out compared to many other books on math theory that I've seen.)
Overall this book is a great tonic and antidote to the absurd Hilbert/Russellian flights of fancy in mathematics or the deeply embedded Platonic mathematical "universe" that so many mathematicians have mistakenly embedded in their way of thinking.
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