The ancients and the Scholastics fought about abstractions. After the fifteenth century, though, metaphysics was compelled to respond to physics. Curiously, the first intimation of the actual infinite appeared in music. During the s, musicians began tempering musical intervals, using string lengths that corresponded to irrational numbers. This provoked a crisis in mathematics as well as metaphysics.

Aristotle knew that an irrational number could be represented as an infinite series of rational numbers, and that the irrationals therefore implied the existence of an actual infinite, which of course could not be grasped by the senses. He rejected the concept, and under his influence, fifteenth- and sixteenth-century mathematicians and music theorists agonized over whether to admit the irrationals into musical tuning.

The calculus gives us the exact sum of an infinite number of infinitesimal quantities, which by definition are imperceptible.

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We cannot perceive vanishingly small quantities, yet in the calculus their sum is a definite number. The physics that issued from the work of Newton and Leibniz transformed the world. That made it more difficult if not quite impossible to dismiss infinitesimals as the mere imaginings of mathematicians. At issue was not a mythical bird, but rather the precise calculation of ballistic trajectories and planetary orbits. If God is nature, there can be nothing in nature except God, and individual objects cannot exist.

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Leibniz removed God from nature and re-situated Him outside it, where He creates an endless multiplicity of infinitesimal monads that comprise a coherent world through a pre-established harmony. As Leach points out, there still are dissenters among mathematicians. But the revolution in mathematical physics and physics made for a different sort of debate than had occurred among the ancients or the Schoolmen. The infinitesimals, by contrast, were not simply a new sort of Platonic number mysticism, but rather a working principle that transformed the world.

The new mathematics of the sixteenth century roused the philosophers to explain the existence of objects in the mind that were not in the senses.

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There is nothing entirely new under the sun, to be sure: Leach gets this backwards. This vision says that our physical senses can apprehend all there exists. The philosophy that attended the first stirrings of modern science came from Descartes, Leibniz, and Kant, all of whom shared the premise that some faculty of the mind must transcend the senses. That is why Kant triumphed over the Scholastics and empiricists: His followers quickly learned how to use his theory to explain Newtonian natural science. The neo-Kantian school that dominated Continental academic philosophy from the last quarter of the nineteenth century through the first quarter of the twentieth hung its hat on the problem of infinitesimals.

And it got stranger still. Different orders, or densities, of infinity do in fact exist. The latter seems rather more interesting.

Painters, poets, musicians, and architects brought about a scientific revolution that eluded the philosopher-scientists of the day. A History in Sum. In the twentieth century, American mathematicians began to make critical advances in a field previously dominated by Europeans. Harvard's mathematics department was at the center of these developments.

A History in Sum is an inviting account of the pioneers who trailblazed a distinctly American tradition of mathematics--in algebraic geometry, complex analysis, and other esoteric subdisciplines that are rarely written about outside of journal articles or advanced textbooks. The heady mathematical concepts that emerged, and the men and women who shaped them, are described here in lively, accessible prose.

The story begins in , when a precocious sixteen-year-old freshman, Benjamin Peirce, arrived at the College. He would become the first American to produce original mathematics--an ambition frowned upon in an era when professors largely limited themselves to teaching. Peirce's successors transformed the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff.

Influential figures soon flocked to Harvard, some overcoming great challenges to pursue their elected calling. A History in Sum elucidates the contributions of these extraordinary minds and makes clear why the history of the Harvard mathematics department is an essential part of the history of mathematics in America and beyond. Philosophy of Mathematics in the Twentieth Century. In these selected essays, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the past century: From the ancients' first readings of the innards of birds to your neighbor's last bout with the state lottery, humankind has put itself into the hands of chance.

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## Mathematics and Religion: Our Languages of Sign and Symbol - Javier Leach - Google Книги

Today life itself may be at stake when probability comes into play--in the chance of a false negative in a medical test, in the reliability of DNA findings as legal evidence, or in the likelihood of passing on a deadly congenital disease--yet as few people as ever understand the odds.

This book is aimed at the trouble with trying to learn about probability. A story of the misconceptions and difficulties civilization overcame in progressing toward probabilistic thinking, Randomness is also a skillful account of what makes the science of probability so daunting in our own day. The great strength of this book is the way it uses history and even prehistory of probability to chart its present territory and cast light on its core point of contention: Chances are high that reading this book will clear up your misconceptions about randomness and probabilities.

In this very entertaining little book, simply written but intended for careful readers, some of the most common mistakes people make about chance are carefully analyzed. While describing interesting aspects of the mathematics of probability, the author takes frequent detours into the history of humanity's understanding and misunderstanding of the laws of chance, touching on subjects as diverse as chance in decision-making and the fairness of those decisions, gambling and our intuitive understanding of chance, the likelihood of the extremely rare, the existence of true randomness and how computers have helped shape modern thinking about probabilities An insightful chapter is "Chance or Necessity?

The author describes the problem beautifully: Whether well-educated in mathematics or not, people have always been fascinated by randomness and intrigued by the fundamental question of the real nature of randomness, of how you can tell randomness from something that is not. Rial, American Scientist Reviews of this book: I can cheerfully recommend it to anyone who is a total beginner when it comes to probability, what it means, why it is desperately puzzling, and what it can do for us despite that It is fascinating to read about the pioneers of probability, such as Pierre Simon de Laplace with his "normal distribution"--now more familiar as the notorious bell curve--and Adolphe Quetelet, perhaps the first to realise that there are statistical patterns in human behaviour.

And I applaud the blunt reminder that when it comes to the real world the 'normal' distribution is actually highly abnormal Randomness , by mathematician Deborah J. Bennett, was obviously a labor of love. The result is an interesting book that combines a well-researched, anecdotally presented survey of the history of chance, probability and randomness along with some elementary instruction in probability It includes a wide-ranging and rich bibliography that reflects the passion of the author for the subject.

## Can We Know What We Know?

Anybody interested in gaming, random numbers, the Monte Carlo method and so on will find nice anecdotal descriptions of these topics, together with detailed notes and references to the bibliography for more detailed study. It is a good book to have. The fact that randomness, agency, and holiness can readily displace each other in phenomenological explanations of human action is the central concern that might draw students of consciousness to Bennett's book.

Bennett does an excellent job, explaining and drawing out the major questions that swirl around the randomness-agency-holiness issue. Draper, Journal of Consciousness Studies Reviews of this book: In this book, Bennett seeks to account for the centuries-long lapse between early uses of chance in decision making and the more technical studies of probability first undertaken in the seventeenth century.

At the same time, she explores the confusions and misunderstandings about probability that persist today. She argues that the notion of randomness played a crucial role in inhibiting conceptual progress in probability and that it also accounts for present-day struggles to come to terms with the subject Bennett's book is written in a lucid, engaging style and provides an entertaining introduction to some questions in probability. This volume is exceptionally readable. It takes away much of the mystery of probability while adding to our sense of wonder.

In Charles Hailey and David Helfand reported their calculations of the odds of a commercial airliner being struck by a meteor, in response to speculation about TWA flight They conclude that, in over 30 years of air travel, the probability that a commercial flight would have been hit by a meteor big enough to crash it is 1 in This bit of probability trivia is an indication of human beings continuous struggle to understand probability and chance through the ages, and Deborah Bennett captures the fascination with numbers in this pocket-sized volume.

The book is filled with Clearly, the computation of probabilities is not just an arid game As Deborah Bennett shows in her excellent little book on the mathematics of chance, the concept has been controversial for thousands of years.. Mathematics is its own language, and sometimes it doesn't translate readily into other human tongues. But Bennett is brilliantly bilingual, well able to put mathematical concepts into clear, expressive English. Her topic is intrinsically fascinating, for who has not felt buffeted by random events, and who has not sought to see when the wheel of fortune may turn up good luck?

More than an intriguing exploration of a peculiarly fascinating part of mathematics, its coverage, ranging from ancient games of chance to modern probability mind-games, makes it comprehensive as well as compulsively readable. A clear and detailed examination of the role of pure chance, with fascinating historical asides. This book puts risk and chance into perspective for the airline passenger and the lottery player alike.

How Engineers Get from Thought to Thing A careful and well-written treatment of an intriguing subject. Beginners will find themselves welcomed and well led. The Seven Pillars of Statistical Wisdom.

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