A suspicious mind.

Sextus Empiricus (ca. AD 160–210), exponent of scepticism and critic of the Dogmatists, was a Greek physician and philosopher, pupil and successor of the medical sceptic Herodotus (not the historian) of Tarsus. He probably lived for years in Rome and possibly also in Alexandria and Athens. His three surviving works are Outlines of Pyrrhonism (three books on the practical and ethical scepticism of Pyrrho of Elis, ca. 360–275 BC, as developed later, presenting also a case against the Dogmatists); Against the Dogmatists (five books dealing with the Logicians, the Physicists, and the Ethicists); and Against the Professors (six books: Grammarians, Rhetors, Geometers, Arithmeticians, Astrologers, and Musicians). These two latter works might be called a general criticism of professors of all arts and sciences. Sextus’ work is a valuable source for the history of thought especially because of his development and formulation of former sceptic doctrines.

The Loeb Classical Library edition of Sextus Empiricus is in four volumes.

“Inspiring and informative…deserves to be widely read.”
—Wall Street Journal

“This fun book offers a philosophical take on number systems and revels in the beauty of math.”
—Science News

Because we have ten fingers, grouping by ten seems natural, but twelve would be better for divisibility, and eight is well suited to repeated halving. Grouping by two, as in binary code, has turned out to have its own remarkable advantages.

Paul Lockhart presents arithmetic not as rote manipulation of numbers—a practical if mundane branch of knowledge best suited for filling out tax forms—but as a fascinating, sometimes surprising intellectual craft that arises from our desire to add, divide, and multiply important things. Passionate and entertaining, Arithmetic invites us to experience the beauty of mathematics through the eyes of a beguiling teacher.

“A nuanced understanding of working with numbers, gently connecting procedures that we once learned by rote with intuitions long since muddled by education…Lockhart presents arithmetic as a pleasurable pastime, and describes it as a craft like knitting.”
—Jonathon Keats, New Scientist

“What are numbers, how did they arise, why did our ancestors invent them, and how did they represent them? They are, after all, one of humankind’s most brilliant inventions, arguably having greater impact on our lives than the wheel. Lockhart recounts their fascinating story…A wonderful book.”
—Keith Devlin, author of Finding Fibonacci

The first history of postwar mathematics, offering a new interpretation of the rise of abstraction and axiomatics in the twentieth century.

Why did abstraction dominate American art, social science, and natural science in the mid-twentieth century? Why, despite opposition, did abstraction and theoretical knowledge flourish across a diverse set of intellectual pursuits during the Cold War? In recovering the centrality of abstraction across a range of modernist projects in the United States, Alma Steingart brings mathematics back into the conversation about midcentury American intellectual thought. The expansion of mathematics in the aftermath of World War II, she demonstrates, was characterized by two opposing tendencies: research in pure mathematics became increasingly abstract and rarified, while research in applied mathematics and mathematical applications grew in prominence as new fields like operations research and game theory brought mathematical knowledge to bear on more domains of knowledge. Both were predicated on the same abstractionist conception of mathematics and were rooted in the same approach: modern axiomatics. 

For American mathematicians, the humanities and the sciences did not compete with one another, but instead were two complementary sides of the same epistemological commitment. Steingart further reveals how this mathematical epistemology influenced the sciences and humanities, particularly the postwar social sciences. As mathematics changed, so did the meaning of mathematization. 

Axiomatics focuses on American mathematicians during a transformative time, following a series of controversies among mathematicians about the nature of mathematics as a field of study and as a body of knowledge. The ensuing debates offer a window onto the postwar development of mathematics band Cold War epistemology writ large. As Steingart’s history ably demonstrates, mathematics is the social activity in which styles of truth—here, abstraction—become synonymous with ways of knowing. 

Mathematics holds a central place in the traditional liberal arts. The four mathematical disciplines of the quadrivium-arithmetic, geometry, music, and astronomy-reveal their enduring significance in this work, which offers the first unified, textbook treatment of these four subjects. Drawing on fundamental sources including Euclid, Boethius, and Ptolemy, this presentation respects the proper character of each discipline while revealing the relations among these liberal arts, as well as their connections to later mathematical and scientific developments. This book makes the quadrivium newly accessible in a number of ways. First, the careful choice of material from ancient sources means that students receive a faithful, integral impression of the classical quadrivium without being burdened or confused by an unwieldy mass of scattered results. Second, the terminology and symbols that are used convey the real insights of older mathematical approaches without introducing needless archaism. Finally, and perhaps most importantly, the book is filled with hundreds of exercises. Mathematics must be learned actively, and the exercises structured to complement the text, and proportioned to the powers of a learner to offer a clear path by which students make quadrivial knowledge their own. Many readers can profit from this introduction to the quadrivium. Students in high school will acquire a sense of the nature of mathematical proof and become confident in using mathematical language. College students can discover that mathematics is more than procedure, while also gaining insight into an intellectual current that influenced authors they are already reading: authors such as Plato, Aristotle, Augustine, Thomas Aquinas, and Dante. All will find a practical way to grasp a body of knowledge that, if long neglected, is never out of date.

The fascinating story of an important lottery that flourished in France from 1757 to 1836 and its role in transforming our understanding of the nature of risk.

In the 1750s, at the urging of famed adventurer Giacomo Casanova, the French state began to embrace risk in adopting a new Loterie. The prize amounts paid varied, depending on the number of tickets bought and the amount of the bet, as determined by each individual bettor. The state could lose money on any individual Loterie drawing while being statistically guaranteed to come out on top in the long run. In adopting this framework, the French state took on risk in a way no other has, before or after. At each drawing the state was at risk of losing a large amount; what is more, that risk was precisely calculable, generally well understood, and yet taken on by the state with little more than a mathematical theory to protect it.

Stephen M. Stigler follows the Loterie from its curious inception through its hiatus during the French Revolution, its renewal and expansion in 1797, and finally to its suppression in 1836, examining throughout the wider question of how members of the public came to trust in new financial technologies and believe in their value. Drawing from an extensive collection of rare ephemera, Stigler pieces together the Loterie’s remarkable inner workings, as well as its implications for the nature of risk and the role of lotteries in social life over the period 1700–1950. 

Both a fun read and fodder for many fields, Casanova's Lottery shines new light on the conscious introduction of risk into the management of a nation-state and the rationality of playing unfair games.

“A must-read.” —Joel Mokyr, Nobel Prize–winning author of A Culture of Growth: The Origins of the Modern Economy

An illuminating, fine-grained account of how Indo-Arabic numerals facilitated the spread of practical knowledge in Western Europe—reshaping both commerce and mathematics in the process.

In the thirteenth-century Mediterranean, commerce transformed as merchants shifted from Roman to Indo-Arabic numerals—an alternative that better facilitated complex calculations. It has long been known that this transition stemmed from Europe’s increasing exchanges with India, Persia, and the Arabic world. Yet much remains to be understood about how Indo-Arabic numerals—and the practical arithmetic they enabled—actually spread across Europe. As Raffaele Danna shows, it was hundreds of ordinary merchants, schoolmasters, and artisans who nurtured these changes, thereby driving key advances in both commerce and mathematics.

Drawing on an original catalog of more than 1,200 practical arithmetic manuals, Danna charts the incremental spread of the new figures with unprecedented precision. While Italian merchants were the early adopters, it took nearly three centuries for Indo-Arabic numerals to become established in northern Europe. As Danna shows, adoption did not follow the routes of maritime trade. Rather, Indo-Arabic numerals moved gradually across the continent through inland networks of practitioners. Everywhere they went, the ten figures enhanced commercial practices and facilitated the emergence of a coherent language of mathematical craft. The growing social circulation of this knowledge, in turn, had a lasting impact on the economic trajectory of Western Europe. By the late sixteenth century, even academics were absorbing lessons from the vernacular tradition—a development that led to the first major breakthroughs in European mathematical theory since antiquity.

Combining economic history with the social history of mathematics, The Craft of Indo-Arabic Numerals illuminates the integral role of practical arithmetic in both intellectual and commercial transformations across Western Europe.

“A brilliant travel guide to the coming world of AI.”
—Jeanette Winterson

What does it mean to be creative? Can creativity be trained? Is it uniquely human, or could AI be considered creative?

Mathematical genius and exuberant polymath Marcus du Sautoy plunges us into the world of artificial intelligence and algorithmic learning in this essential guide to the future of creativity. He considers the role of pattern and imitation in the creative process and sets out to investigate the programs and programmers—from Deep Mind and the Flow Machine to Botnik and WHIM—who are seeking to rival or surpass human innovation in gaming, music, art, and language. A thrilling tour of the landscape of invention, The Creativity Code explores the new face of creativity and the mysteries of the human code.

“As machines outsmart us in ever more domains, we can at least comfort ourselves that one area will remain sacrosanct and uncomputable: human creativity. Or can we?…In his fascinating exploration of the nature of creativity, Marcus du Sautoy questions many of those assumptions.”
—Financial Times

“Fascinating…If all the experiences, hopes, dreams, visions, lusts, loves, and hatreds that shape the human imagination amount to nothing more than a ‘code,’ then sooner or later a machine will crack it. Indeed, du Sautoy assembles an eclectic array of evidence to show how that’s happening even now.”
—The Times

In the fog of a Paris dawn in 1832, Évariste Galois, the 20-year-old founder of modern algebra, was shot and killed in a duel. That gunshot, suggests Amir Alexander, marked the end of one era in mathematics and the beginning of another.Arguing that not even the purest mathematics can be separated from its cultural background, Alexander shows how popular stories about mathematicians are really morality tales about their craft as it relates to the world. In the eighteenth century, Alexander says, mathematicians were idealized as child-like, eternally curious, and uniquely suited to reveal the hidden harmonies of the world. But in the nineteenth century, brilliant mathematicians like Galois became Romantic heroes like poets, artists, and musicians. The ideal mathematician was now an alienated loner, driven to despondency by an uncomprehending world. A field that had been focused on the natural world now sought to create its own reality. Higher mathematics became a world unto itself—pure and governed solely by the laws of reason.In this strikingly original book that takes us from Paris to St. Petersburg, Norway to Transylvania, Alexander introduces us to national heroes and outcasts, innocents, swindlers, and martyrs–all uncommonly gifted creators of modern mathematics.

The mysteries of the physical world speak to us through equations--compact statements about the way nature works, expressed in nature's language, mathematics. In this book by the renowned Dutch physicist Sander Bais, the equations that govern our world unfold in all their formal grace--and their deeper meaning as core symbols of our civilization.

Trying to explain science without equations is like trying to explain art without illustrations. Consequently Bais has produced a book that, unlike any other aimed at nonscientists, delves into the details--historical, biographical, practical, philosophical, and mathematical--of seventeen equations that form the very basis of what we know of the universe today. A mathematical objet d'art in its own right, the book conveys the transcendent excitement and beauty of these icons of knowledge as they reveal and embody the fundamental truths of physical reality.

These are the seventeen equations that represent radical turning points in our understanding--from mechanics to electrodynamics, hydrodynamics to relativity, quantum mechanics to string theory--their meanings revealed through the careful and critical observation of patterns and motions in nature. Mercifully short on dry theoretical elaborations, the book presents these equations as they are--with the information about their variables, history, and applications that allows us to chart their critical function, and their crucial place, in the complex web of modern science.

Reading The Equations, we can hear nature speaking to us in its native language.

The use of perspective in Renaissance painting caused a revolution in the history of seeing, allowing artists to depict the world from a spectator’s point of view. But the theory of perspective that changed the course of Western art originated elsewhere—it was formulated in Baghdad by the eleventh-century mathematician Ibn al Haithan, known in the West as Alhazen. Using the metaphor of the mutual gaze, or exchanged glances, Hans Belting—preeminent historian and theorist of medieval, Renaissance, and contemporary art—narrates the historical encounter between science and art, between Arab Baghdad and Renaissance Florence, that has had a lasting effect on the culture of the West.

In this lavishly illustrated study, Belting deals with the double history of perspective, as a visual theory based on geometrical abstraction (in the Middle East) and as pictorial theory (in Europe). How could geometrical abstraction be reconceived as a theory for making pictures? During the Middle Ages, Arab mathematics, free from religious discourse, gave rise to a theory of perspective that, later in the West, was transformed into art when European painters adopted the human gaze as their focal point. In the Islamic world, where theology and the visual arts remained closely intertwined, the science of perspective did not become the cornerstone of Islamic art. Florence and Baghdad addresses a provocative question that reaches beyond the realm of aesthetics and mathematics: What happens when Muslims and Christians look upon each other and find their way of viewing the world transformed as a result?

Mark Peterson makes an extraordinary claim in this fascinating book focused around the life and thought of Galileo: it was the mathematics of Renaissance arts, not Renaissance sciences, that became modern science. Galileo's Muse argues that painters, poets, musicians, and architects brought about a scientific revolution that eluded the philosopher-scientists of the day, steeped as they were in a medieval cosmos and its underlying philosophy.

According to Peterson, the recovery of classical science owes much to the Renaissance artists who first turned to Greek sources for inspiration and instruction. Chapters devoted to their insights into mathematics, ranging from perspective in painting to tuning in music, are interspersed with chapters about Galileo's own life and work. Himself an artist turned scientist and an avid student of Hellenistic culture, Galileo pulled together the many threads of his artistic and classical education in designing unprecedented experiments to unlock the secrets of nature.

In the last chapter, Peterson draws our attention to the Oratio de Mathematicae laudibus of 1627, delivered by one of Galileo's students. This document, Peterson argues, was penned in part by Galileo himself, as an expression of his understanding of the universality of mathematics in art and nature. It is "entirely Galilean in so many details that even if it is derivative, it must represent his thought," Peterson writes. An intellectual adventure, Galileo’s Muse offers surprising ideas that will capture the imagination of anyone—scientist, mathematician, history buff, lover of literature, or artist—who cares about the humanistic roots of modern science.

Elemental learning.

The splendid achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I (LCL 335) contains the divisions of mathematics; mathematics in Greek education; calculation; arithmetical notation and operations, including square root and cube root; Pythagorean arithmetic, including properties of numbers; the square root of 2; proportion and means; algebraic equations; Proclus; Thales; Pythagorean geometry; Democritus; Hippocrates of Chios; duplicating the cube and squaring the circle; trisecting angles; Theaetetus; Plato; Eudoxus of Cnidus (pyramid, cone); Aristotle (the infinite, the lever); Euclid.

Volume II (LCL 362) contains Aristarchus (distances of sun and moon); Archimedes (cylinder, sphere, cubic equations; conoids; spheroids; spiral; expression of large numbers; mechanics; hydrostatics); Eratosthenes (measurement of the earth); Apollonius (conic sections and other works); later development of geometry; trigonometry (including Ptolemy’s table of sines); mensuration: Heron of Alexandria (mensuration); Diophantus (algebra, determinate and indeterminate equations); Pappus (the revival of geometry).

Elemental learning.

The splendid achievement of Greek mathematics is here illustrated in two volumes of selected mathematical works. Volume I (LCL 335) contains the divisions of mathematics; mathematics in Greek education; calculation; arithmetical notation and operations, including square root and cube root; Pythagorean arithmetic, including properties of numbers; the square root of 2; proportion and means; algebraic equations; Proclus; Thales; Pythagorean geometry; Democritus; Hippocrates of Chios; duplicating the cube and squaring the circle; trisecting angles; Theaetetus; Plato; Eudoxus of Cnidus (pyramid, cone); Aristotle (the infinite, the lever); Euclid.

Volume II (LCL 362) contains Aristarchus (distances of sun and moon); Archimedes (cylinder, sphere, cubic equations; conoids; spheroids; spiral; expression of large numbers; mechanics; hydrostatics); Eratosthenes (measurement of the earth); Apollonius (conic sections and other works); later development of geometry; trigonometry (including Ptolemy’s table of sines); mensuration: Heron of Alexandria (mensuration); Diophantus (algebra, determinate and indeterminate equations); Pappus (the revival of geometry).

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 and topology, complex analysis, number theory, and a host of esoteric subdisciplines that have rarely been 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 1825, 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—William Fogg Osgood and Maxime Bôcher—undertook the task of transforming the math department into a world-class research center, attracting to the faculty such luminaries as George David Birkhoff. Birkhoff produced a dazzling body of work, while training a generation of innovators—students like Marston Morse and Hassler Whitney, who forged novel pathways in topology and other areas. Influential figures from around the world 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.

How Cold War urgency, academic ambition, and federal power converged to make math the cornerstone of American education policy.

The Math Mafia by Emily T. Hamilton is a historical account of mathematics education reform in twentieth-century America and how that movement played a role in shaping the broader contours of national education policy. Hamilton contends that a cohesive and well-organized community of advocates was instrumental in the rise of standardized testing and curriculum reforms, as well as federal involvement in classrooms, especially during the Cold War. By positioning mathematics as central to national strength and security, these reformers established math education as a key area of federal focus.

The “mafia” at the center of Hamilton’s story includes mathematicians, educators, and researchers from universities, as well as members of government-funded organizations such as the National Science Foundation and various educational psychology circles. Prominent figures such as Edward Begle, who led the School Mathematics Study Group during the “new math” movement, played key roles. Other influential participants include federal education officials, advocates from professional organizations such as the National Council of Teachers of Mathematics, and cognitive scientists, who contributed both research and theoretical frameworks that informed curriculum design and assessment methods.

The Math Mafia offers historians, educators, policymakers, and other interested readers a compelling perspective on the legacy of the math education reforms on federal education policy in the United States and is a welcome contribution to ongoing discourse around “teaching toward the test,” educational equity, school performance, and other hot-button issues in US education today.

To borrow a phrase from Galileo: What does it mean that the story of the creation is “written in the language of mathematics?” This book is an attempt to understand the natural world, its consistency, and the ontology of what we call laws of nature, with a special focus on their mathematical expression. It does this by arguing in favor of the Essentialist interpretation over that of the Humean and Anti-Humean accounts. It re-examines and critiques Descartes’ notion of laws of nature following from God’s activity in the world as mover of extended bodies, as well as Hume’s arguments against causality and induction. It then presents an Aristotelian-Thomistic account of laws of nature based on mathematical abstraction, necessity, and teleology, finally offering a definition for laws of nature within this framework.

Paul Lockhart’s Mathematician’s Lament outlined how we introduce math to students in the wrong way. Measurement explains how math should be done. With plain English and pictures, Lockhart makes complex ideas about shape and motion intuitive and graspable, and offers a solution to math phobia by introducing us to math as an artful way of thinking and living.

In conversational prose that conveys his passion for the subject, Lockhart makes mathematics accessible without oversimplifying. He makes no more attempt to hide the challenge of mathematics than he does to shield us from its beautiful intensity. Favoring plain English and pictures over jargon and formulas, he succeeds in making complex ideas about the mathematics of shape and motion intuitive and graspable. His elegant discussion of mathematical reasoning and themes in classical geometry offers proof of his conviction that mathematics illuminates art as much as science.

Lockhart leads us into a universe where beautiful designs and patterns float through our minds and do surprising, miraculous things. As we turn our thoughts to symmetry, circles, cylinders, and cones, we begin to see that almost anyone can “do the math” in a way that brings emotional and aesthetic rewards. Measurement is an invitation to summon curiosity, courage, and creativity in order to experience firsthand the playful excitement of mathematical work.

A joyful and intimate celebration of the beauty and creativity of algebra from one of the foremost math educators of our time.

For many of us, algebra conjures up memories of dull classes spent wondering when we’d ever have to solve a system of equations or factor a polynomial. Indeed, most of the time, if we need to plan a budget or figure out how early to leave the house for work, common sense or a quick, seat-of-the-pants approximation is good enough.

But as mathematician Paul Lockhart argues in The Mending of Broken Bones, once we stop thinking of algebra in terms of its practical applications, we can appreciate it for the beautiful and gratifying subject that it is. In his hands, algebra is the delicate craft of untangling numerical puzzles to reveal the hidden patterns and often surprising behaviors of the numbers themselves. As Lockhart traverses numerical systems, slips into and out of higher-dimensional space, and delights in the intimate connections between algebra and geometry, we come to see the discipline from his viewpoint: not as the mundane exercises of our school years, but instead as an art form whose beauty lies in its elegant simplicity.

Written in Lockhart’s charming and conversational prose, The Mending of Broken Bones is an impassioned and deeply personal celebration of algebra that helps us experience the profound joys of mathematical discovery.

There is no question that native cultures in the New World exhibit many forms of mathematical development. This Native American mathematics can best be described by considering the nature of the concepts found in a variety of individual New World cultures. Unlike modern mathematics in which numbers and concepts are expressed in a universal mathematical notation, the numbers and concepts found in native cultures occur and are expressed in many distinctive ways. Native American Mathematics, edited by Michael P. Closs, is the first book to focus on mathematical development indigenous to the New World.

Spanning time from the prehistoric to the present, the thirteen essays in this volume attest to the variety of mathematical development present in the Americas. The data are drawn from cultures as diverse as the Ojibway, the Inuit (Eskimo), and the Nootka in the north; the Chumash of Southern California; the Aztec and the Maya in Mesoamerica; and the Inca and Jibaro of South America. Among the strengths of this collection are this diversity and the multidisciplinary approaches employed to extract different kinds of information. The distinguished contributors include mathematicians, linguists, psychologists, anthropologists, and archaeologists.

A classic on mind, matter, and the unified world model

C. G. Jung's work in his later years suggested that the seemingly divergent sciences of psychology and modern physics might, in fact, be approaching a unified world model in which the dualism of matter and psyche would be resolved into “one world” or Unus Mundus. Jung believed that the natural integers are the archetypal patterns that regulate the unitary realm of psyche and matter, and that number serves as a special instrument for man's becoming conscious of this unity.

Written in a clear style and replete with illustrations which help make the mathematical ideas visible, Number and Time is a piece of original scholarship which introduces a view of how "mind" connects with "matter" at the most fundamental level.
 

Books four and five of a landmark seven-volume work of medieval scientific study of optics.

Ibn al-Haytham was perhaps the greatest mathematician and physicist of the medieval Arabic/Islamic world. The most famous book in which he applied his scientific method is his Optics, through which he dealt with both the mathematics of rays of light and the physical aspects of the eye in seven comprehensive books. His rethinking of the entire science of optics set the scene for the whole of the subsequent development of the subject, influencing figures such as William of Ockham, Kepler, Descartes, and Christaan Huygens. The immense work of editing, translating into English, and commenting on this work was undertaken by Abdelhamid I. Sabra. This English translation of Books IV–V was completed by Sabra just before his death in 2013 with an introduction and critical analysis. It has been extensively revised by Jan Hogendijk.

Performing Math tells the history of expectations for math communication—and the conversations about math hatred and math anxiety that occurred in response. Focusing on nineteenth-century American colleges, this book analyzes foundational tools and techniques of math communication: the textbooks that supported reading aloud, the burnings that mimicked pedagogical speech, the blackboards that accompanied oral presentations, the plays that proclaimed performers’ identities as math students, and the written tests that redefined “student performance.” Math communication and math anxiety went hand in hand as new rules for oral communication at the blackboard inspired student revolt and as frameworks for testing student performance inspired performance anxiety. With unusual primary sources from over a dozen educational archives, Performing Math argues for a new, performance-oriented history of American math education, one that can explain contemporary math attitudes and provide a way forward to reframing the problem of math anxiety.

In this illuminating collection, Charles Parsons surveys the contributions of philosophers and mathematicians who shaped the philosophy of mathematics over the course of the past century.

Parsons begins with a discussion of the Kantian legacy in the work of L. E. J. Brouwer, David Hilbert, and Paul Bernays, shedding light on how Bernays revised his philosophy after his collaboration with Hilbert. He considers Hermann Weyl’s idea of a “vicious circle” in the foundations of mathematics, a radical claim that elicited many challenges. Turning to Kurt Gödel, whose incompleteness theorem transformed debate on the foundations of mathematics and brought mathematical logic to maturity, Parsons discusses his essay on Bertrand Russell’s mathematical logic—Gödel’s first mature philosophical statement and an avowal of his Platonistic view.

Philosophy of Mathematics in the Twentieth Century insightfully treats the contributions of figures the author knew personally: W. V. Quine, Hilary Putnam, Hao Wang, and William Tait. Quine’s early work on ontology is explored, as is his nominalistic view of predication and his use of the genetic method of explanation in the late work The Roots of Reference. Parsons attempts to tease out Putnam’s views on existence and ontology, especially in relation to logic and mathematics. Wang’s contributions to subjects ranging from the concept of set, minds, and machines to the interpretation of Gödel are examined, as are Tait’s axiomatic conception of mathematics, his minimalist realism, and his thoughts on historical figures.

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. 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.

To acquire a (correct) intuition of chance is not easy to begin with, and moving from an intuitive sense to a formal notion of probability presents further problems. Author Deborah Bennett traces the path this process takes in an individual trying to come to grips with concepts of uncertainty and fairness, and also charts the parallel path by which societies have developed ideas about chance. Why, from ancient to modern times, have people resorted to chance in making decisions? Is a decision made by random choice “fair”? What role has gambling played in our understanding of chance? Why do some individuals and societies refuse to accept randomness at all? If understanding randomness is so important to probabilistic thinking, why do the experts disagree about what it really is? And why are our intuitions about chance almost always dead wrong?

Anyone who has puzzled over a probability conundrum is struck by the paradoxes and counterintuitive results that occur at a relatively simple level. Why this should be, and how it has been the case through the ages, for bumblers and brilliant mathematicians alike, is the entertaining and enlightening lesson of Randomness.

A forgotten episode of mathematical resistance reveals the rise of modern mathematics and its cornerstone, mathematical purity, as political phenomena.
 
The nineteenth century opened with a major shift in European mathematics, and in the Kingdom of Naples, this occurred earlier than elsewhere. Between 1790 and 1830 its leading scientific institutions rejected as untrustworthy the “very modern mathematics” of French analysis and in its place consolidated, legitimated, and put to work a different mathematical culture. The Neapolitan mathematical resistance was a complete reorientation of mathematical practice. Over the unrestricted manipulation and application of algebraic algorithms, Neapolitan mathematicians called for a return to Greek-style geometry and the preeminence of pure mathematics.
 
For all their apparent backwardness, Massimo Mazzotti explains, they were arguing for what would become crucial features of modern mathematics: its voluntary restriction through a new kind of rigor and discipline, and the complete disconnection of mathematical truth from the empirical world—in other words, its purity. The Neapolitans, Mazzotti argues, were reacting to the widespread use of mathematical analysis in social and political arguments: theirs was a reactionary mathematics that aimed to technically refute the revolutionary mathematics of the Jacobins. During the Restoration, the expert groups in the service of the modern administrative state reaffirmed the role of pure mathematics as the foundation of a newly rigorous mathematics, which was now conceived as a neutral tool for modernization. What Mazzotti’s penetrating history shows us in vivid detail is that producing mathematical knowledge was equally about producing certain forms of social, political, and economic order.
 

The first of five planned volumes, this book collects Wittgenstein’s unabridged writings on mathematics and logic composed between 1941 and 1943.

In 1956, a significant but abridged portion of Ludwig Wittgenstein’s 1937–1944 writings for a continuation of what became Philosophical Investigations was published as Remarks on the Foundations of Mathematics. Now, in the five German-English facing pages volumes of Remarks on Mathematics and Logic, these writings are presented in full as Wittgenstein wrote them: in chronological order, without editorial omissions, selections, or rearrangements.

Volume IV contains materials from 1941 to 1943, including one previously unpublished manuscript and three manuscripts from which selections were made for the previous abridgment. The topics covered in this volume include the philosophy of mathematics, mathematical foundationalism, surveyability in mathematics, language-games and proof-networks, Gödelian undecidability, pictures in mathematics, Cantorian diagonal proofs, applied mathematics without pure mathematics, and mathematics and chess, among others.

This German-English facing pages edition presents Wittgenstein’s writings in their original context—as Wittgenstein wrote them, in chronological order, and without editorial omissions, selections, or rearrangements.

What gives statistics its unity as a science? Stephen Stigler sets forth the seven foundational ideas of statistics—a scientific discipline related to but distinct from mathematics and computer science.

Even the most basic idea—aggregation, exemplified by averaging—is counterintuitive. It allows one to gain information by discarding information, namely, the individuality of the observations. Stigler’s second pillar, information measurement, challenges the importance of “big data” by noting that observations are not all equally important: the amount of information in a data set is often proportional to only the square root of the number of observations, not the absolute number. The third idea is likelihood, the calibration of inferences with the use of probability. Intercomparison is the principle that statistical comparisons do not need to be made with respect to an external standard. The fifth pillar is regression, both a paradox (tall parents on average produce shorter children; tall children on average have shorter parents) and the basis of inference, including Bayesian inference and causal reasoning. The sixth concept captures the importance of experimental design—for example, by recognizing the gains to be had from a combinatorial approach with rigorous randomization. The seventh idea is the residual: the notion that a complicated phenomenon can be simplified by subtracting the effect of known causes, leaving a residual phenomenon that can be explained more easily.

The Seven Pillars of Statistical Wisdom presents an original, unified account of statistical science that will fascinate the interested layperson and engage the professional statistician.

The Source Book contains 75 excerpts from the writings of Western mathematics from the thirteenth to the end of the eighteenth century. The selection has been confined to pure mathematics or to those fields of applied mathematics that had a direct bearing on the development of pure mathematics.

The authors range from Al-Khwarizmi (a Latin translation of whose work was much used in Europe), Viète, and Oresme, to Newton, Euler, and Lagrange. The selections are grouped in chapters on arithmetic, algebra, geometry, and analysis. All the excerpts are translated into English. Some of the translations have been newly made by Mr. and Mrs. Struik; if a translation was already available it has been used, but in every such case it has been checked against the original and amended or corrected where it seemed necessary. The editor has taken considerable pains to put each selection in context by means of introductory comments and has explained obscure or doubtful points in footnote wherever necessary.

The Source Book should be particularly valuable to historians of science, but all who are concerned with the origins and growth of mathematics will find it interesting and useful.

Reviving an educational tradition involves a double task. A new generation of students must be taught, and at the same time the teachers themselves must learn. This book addresses the teachers who seek to hand on the quadrivium-the four mathematical liberal arts of arithmetic, geometry, music, and astronomy-at the same time as they acquire it. Two components run in parallel throughout the book. The first component is practical. Weekly overviews and daily lesson plans explain how to complete the study of A Brief Quadrivium in the course of a single school year, and suggestions for weekly assessments make it easy to plan tests and monitor student progress. The second component is directed to the continuing education of the teacher. Short essays explore the history, philosophy, and practice of mathematics. The themes of these essays are coordinated with the simultaneous mathematical work being done by students, allowing the teacher to instruct more reflectively. Some users of this book are confident in their grasp of mathematics and natural science. For them, the essays will clarify the unity of mathematical activity over time and reveal the old roots of new developments. Other users of this book, including some parents who school their children at home, find mathematics intimidating. The clear structure of the lesson plans, and the support of the companion essays, give them the confidence to lead students through a demanding but doable course of study. The British mathematician John Edensor Littlewood remarked that one finds in the ancient mathematicians not “clever schoolboys” but rather “Fellows of another College.” This guide invites all teachers of the quadrivium to join the enduring mathematical culture of Littlewood and his predecessors, and to witness for themselves the significance and vitality of a tradition as old as Pythagoras.

On the creation of the world, and the destruction of Atlantis.

Plato, the great philosopher of Athens, was born in 427 BC. In early manhood an admirer of Socrates, he later founded the famous school of philosophy in the grove Academus. Much else recorded of his life is uncertain; that he left Athens for a time after Socrates’ execution is probable; that later he went to Cyrene, Egypt, and Sicily is possible; that he was wealthy is likely; that he was critical of “advanced” democracy is obvious. He lived to be 80 years old. Linguistic tests including those of computer science still try to establish the order of his extant philosophical dialogues, written in splendid prose and revealing Socrates’ mind fused with Plato’s thought.

In Laches, Charmides, and Lysis, Socrates and others discuss separate ethical conceptions. Protagoras, Ion, and Meno discuss whether righteousness can be taught. In Gorgias, Socrates is estranged from his city’s thought, and his fate is impending. The Apology (not a dialogue), Crito, Euthyphro, and the unforgettable Phaedo relate the trial and death of Socrates and propound the immortality of the soul. In the famous Symposium and Phaedrus, written when Socrates was still alive, we find the origin and meaning of love. Cratylus discusses the nature of language. The great masterpiece in ten books, the Republic, concerns righteousness (and involves education, equality of the sexes, the structure of society, and abolition of slavery). Of the six so-called dialectical dialogues Euthydemus deals with philosophy; metaphysical Parmenides is about general concepts and absolute being; Theaetetus reasons about the theory of knowledge. Of its sequels, Sophist deals with not-being; Politicus with good and bad statesmanship and governments; Philebus with what is good. The Timaeus seeks the origin of the visible universe out of abstract geometrical elements. The unfinished Critias treats of lost Atlantis. Unfinished also is Plato’s last work, Laws, a critical discussion of principles of law which Plato thought the Greeks might accept.

The Loeb Classical Library edition of Plato is in twelve volumes.

The volume presents the material of the first Oxford-Budapest Conference on Truth, Reference and Realism held at CEU in 2005. The problem addressed by the conference, famously formulated by Paul Benacerraf in a paper on Mathematical Truth, was how to understand truth in the semantics of discourses about abstract domains whose objects and properties cannot be observed by sense perception. The papers of the volume focus on this semantic issue in four major fields: logic, mathematics, ethics and the metaphysics of properties in general. Beyond marking an important event, the collected papers are also substantial contributions to the above topic, from the most distinguished authors in these areas.

Ranging from math to literature to philosophy, Uncountable explains how numbers triumphed as the basis of knowledge—and compromise our sense of humanity.

Our knowledge of mathematics has structured much of what we think we know about ourselves as individuals and communities, shaping our psychologies, sociologies, and economies. In pursuit of a more predictable and more controllable cosmos, we have extended mathematical insights and methods to more and more aspects of the world. Today those powers are greater than ever, as computation is applied to virtually every aspect of human activity. Yet, in the process, are we losing sight of the human? When we apply mathematics so broadly, what do we gain and what do we lose, and at what risk to humanity?

These are the questions that David and Ricardo L. Nirenberg ask in Uncountable, a provocative account of how numerical relations became the cornerstone of human claims to knowledge, truth, and certainty. There is a limit to these number-based claims, they argue, which they set out to explore. The Nirenbergs, father and son, bring together their backgrounds in math, history, literature, religion, and philosophy, interweaving scientific experiments with readings of poems, setting crises in mathematics alongside world wars, and putting medieval Muslim and Buddhist philosophers in conversation with Einstein, Schrödinger, and other giants of modern physics. The result is a powerful lesson in what counts as knowledge and its deepest implications for how we live our lives.
 

A celebration of the seemingly simple idea that allowed us to imagine the world in new dimensions—sparking both controversy and discovery. 
 
The stars of this book, vectors and tensors, are unlikely celebrities. If you ever took a physics course, the word “vector” might remind you of the mathematics needed to determine forces on an amusement park ride, a turbine, or a projectile. You might also remember that a vector is a quantity that has magnitude and (this is the key) direction. In fact, vectors are examples of tensors, which can represent even more data. It sounds simple enough—and yet, as award-winning science writer Robyn Arianrhod shows in this riveting story, the idea of a single symbol expressing more than one thing at once was millennia in the making. And without that idea, we wouldn’t have such a deep understanding of our world.

Vector and tensor calculus offers an elegant language for expressing the way things behave in space and time, and Arianrhod shows how this enabled physicists and mathematicians to think in a brand-new way. These include James Clerk Maxwell when he ushered in the wireless electromagnetic age; Einstein when he predicted the curving of space-time and the existence of gravitational waves; Paul Dirac, when he created quantum field theory; and Emmy Noether, when she connected mathematical symmetry and the conservation of energy. For it turned out that it’s not just physical quantities and dimensions that vectors and tensors can represent, but other dimensions and other kinds of information, too. This is why physicists and mathematicians can speak of four-dimensional space-time and other higher-dimensional “spaces,” and why you’re likely relying on vectors or tensors whenever you use digital applications such as search engines, GPS, or your mobile phone.

In exploring the evolution of vectors and tensors—and introducing the fascinating people who gave them to us—Arianrhod takes readers on an extraordinary, five-thousand-year journey through the human imagination. She shows the genius required to reimagine the world—and how a clever mathematical construct can dramatically change discovery’s direction.