Mathematics

• Number, The Language of Science by Tobias Dantzig.

  "This is beyond doubt the most interesting book on the evolution of mathematics which has ever fallen into my hands. If people know how to treasure the truly good, this book will attain a lasting place in the literature of the world. The evolution of mathematical thought from the earliest times to the latest constructions is presented here with admirable consistency and originality and in a wonderfully lively style."

  – Albert Einstein

• Achilles, the Tortoise, and the Objectivity of Mathematics by Pat Corvini.

  Philosophers and mathematicians alike have long misunderstood the relation of mathematics to the real world and to the rest of human knowledge. A proper theory of concepts is essential to an understanding of the nature of math. Thanks to Ayn Rand, we now have the basis for such an understanding.

  In this course, Dr. Corvini draws on Objectivist epistemology to offer a new identification of how mathematical concepts are related to physical concretes, including a new formulation of the concept of infinity. She uses the easy-to-visualize example of Achilles and the tortoise to make the ideas accessible to a general audience. In the process, she also identifies the fundamental error that underlies Zeno's famous paradox and that has long obstructed men's understanding of mathematical abstractions. Her analysis underscores the importance and power of Ayn Rand's theory of concepts.

  (Of interest to anyone interested in epistemology; no prior mathematics background assumed.)

• The Foundations of Mathematics by Ronald Pisaturo and Glenn Marcus. Essays on the philosophic basis of mathematical ideas such as number, sequence and quantity. Published in The Intellectual Activist volume 8 numbers 4 and 5. See also Mathematics in One Lesson in volume 12 numbers 9 and 10, and Undermining Reason: The 20^th Century Assault on the Philosophy of Mathematics in volume 14 number 10.
• Mathematics for the Million, How to Master the Magic of Numbers by Lancelot Hogben. Taking the most elementary knowledge for granted, Lancelot Hogben leads readers of this famous book through the whole course from simple arithmetic to calculus. His illuminating explanation is addressed to the person who wants to understand the place of mathematics in modern civilization but who has been intimidated by its supposed difficulty. Mathematics is the language of size, shape, and order – a language Hogben shows one can both master and enjoy.
• Calculus Made Easy by Silvanus P. Thompson. Preferably get the third edition. Originally published in 1910, it is not full of the hierarchical-inversions brought on by the "new math", nor is it anywhere near as rationalistic as standard math textbooks. Instead, it is a progression of essentials, beginning with identifying the essence of calculus, then working simple examples, then gradually increasing the complexity. Anyone wanting to understand calculus, and use it in real, applied situations, will find this book to be a vast improvement over standard textbooks.
• Against the Gods, The Remarkable Story of Risk by Peter L. Bernstein. Against the Gods, a narrative that reads like a novel, chronicles the remarkable intellectual adventure that liberated humanity from the oracles and soothsayers by means of the powerful tools of risk management that are available to us today. This is a richly-woven tale of Greek philosophers and Arab mathematicians, of merchants and scientists, gamblers and philosophers, world-renowned intellects and obscure but inspired amateurs who helped discover the modern methods of putting the future at the service of the present, replacing helplessness before the fates with choice and decision.
• A History of Pi by Petr Beckman. An excellent history of mathematics from ancient Greece to the present, from the point of view of the discovery and use of pi, showing that advancements in knowledge occurred in cultures which prized reason and languished in irrational ones.
• Div, Grad, Curl, and All That, An Informal Text on Vector Calculus by Harry M. Schey. Even though the book treats a relatively advanced topic, this book may be the best math text I've ever used. I am recommending the book for its clear treatment of a difficult mathematical subject. I'm not making any claim about the text's underlying philosophy. -- Glenn Marcus

Links

• Archimedes's Palimpsest – The oldest surviving manuscript containing the work of Archimedes.
• Eric Weisstein's World of Mathematics