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Arithmetic
Paul Lockhart
Harvard University Press, 2017

“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

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Axiomatics
Mathematical Thought and High Modernism
Alma Steingart
University of Chicago Press, 2023
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. 
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Before Voltaire
The French Origins of “Newtonian” Mechanics, 1680-1715
J.B. Shank
University of Chicago Press, 2017
We have grown accustomed to the idea that scientific theories are embedded in their place and time. But in the case of the development of mathematical physics in eighteenth-century France, the relationship was extremely close. In Before Voltaire, J.B. Shank shows that although the publication of Isaac Newton’s Principia in 1687 exerted strong influence, the development of calculus-based physics is better understood as an outcome that grew from French culture in general.
 
Before Voltaire explores how Newton’s ideas made their way not just through the realm of French science, but into the larger world of society and culture of which Principia was an intertwined part. Shank also details a history of the beginnings of calculus-based mathematical physics that integrates it into the larger intellectual currents in France at the time, including the Battle of the Ancients and the Moderns, the emergence of wider audiences for science, and the role of the newly reorganized Royal Academy of Sciences. The resulting book offers an unprecedented cultural history of one the most important and influential elements of Enlightenment science.
 
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Between Raphael and Galileo
Mutio Oddi and the Mathematical Culture of Late Renaissance Italy
Alexander Marr
University of Chicago Press, 2011

Although largely unknown today, during his lifetime Mutio Oddi of Urbino (1569–1639) was a highly esteemed scholar, teacher, and practitioner of a wide range of disciplines related to mathematics. A prime example of the artisan-scholar so prevalent in the late Renaissance, Oddi was also accomplished in the fields of civil and military architecture and the design and retail of mathematical instruments, as well as writing and publishing. 

In Between Raphael and Galileo, Alexander Marr resurrects the career and achievements of Oddi in order to examine the ways in which mathematics, material culture, and the book shaped knowledge, society, and the visual arts in late Renaissance Italy. Marr scrutinizes the extensive archive of Oddi papers, documenting Oddi’s collaboration with prominent intellectuals and officials and shedding new light on the practice of science and art during his day. What becomes clear is that Oddi, precisely because he was not spectacularly innovative and did not attain the status of a hero in modern science, is characteristic of the majority of scientific practitioners and educators active in this formative age, particularly those whose energetic popularization of mathematics laid the foundations for the Scientific Revolution. Marr also demonstrates that scientific change in this era was multivalent and contested, governed as much by friendship as by principle and determined as much by places as by purpose.

Plunging the reader into Oddi’s world, Between Raphael and Galileo is a finely wrought and meticulously researched tale of science, art, commerce, and society in the late sixteenth and early seventeenth century. It will become required reading for any scholar interested in the history of science, visual art, and print culture of the Early Modern period.

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Blaise Pascal
Miracles and Reason
Mary Ann Caws
Reaktion Books, 2017
Few people have had as many influences on as many different fields as true Renaissance man Blaise Pascal. At once a mathematician, philosopher, theologian, physicist, and engineer, Pascal’s discoveries, experiments, and theories helped usher in a modern world of scientific thought and methodology. In this singular book on this singular genius, distinguished scholar Mary Ann Caws explores the rich contributions of this extraordinary thinker, interweaving his writings and discoveries with an account of his life and career and the wider intellectual world of his time.
            Caws takes us back to Pascal’s youth, when he was a child prodigy first engaging mathematics through the works of mathematicians such as Father Mersenne. She describes his early scientific experiments and his construction of mechanical calculating machines; she looks at his correspondence with important thinkers such as René Descartes and Pierre de Fermat; she surveys his many inventions, such as the first means of public transportation in Paris; and she considers his later religious exaltations in works such as the “Memorial.” Along the way, Caws examines Pascal’s various modes of writing—whether he is arguing with the strict puritanical modes of church politics, assuming the personality of a naïve provincial trying to understand the Jesuitical approach, offering pithy aphorisms in the Pensées, or meditating on thinking about thinking itself.
            Altogether, this book lays side by side many aspects of Pascal’s life and work that are seldom found in a single volume: his religious motivations and faith, his scientific passions, and his practical savvy. The result is a comprehensive but easily approachable account of a fascinating and influential figure.
 
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A Brief Quadrivium
Peter Ulrickson
Catholic University of America Press, 2023
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.
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Casanova's Lottery
The History of a Revolutionary Game of Chance
Stephen M. Stigler
University of Chicago Press, 2022
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.
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The Creativity Code
Art and Innovation in the Age of AI
Marcus du Sautoy
Harvard University Press, 2020

“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

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The Cult of Pythagoras
Math and Myths
Alberto A. Martinez
University of Pittsburgh Press, 2012
In this follow-up to his popular Science Secrets, Alberto A. Martínez discusses various popular myths from the history of mathematics: that Pythagoras proved the hypotenuse theorem, that Archimedes figured out how to test the purity of a gold crown while he was in a bathtub, that the Golden Ratio is in nature and ancient architecture, that the young Galois created group theory the night before the pistol duel that killed him, and more. Some stories are partly true, others are entirely false, but all show the power of invention in history. Pythagoras emerges as a symbol of the urge to conjecture and “fill in the gaps” of history. He has been credited with fundamental discoveries in mathematics and the sciences, yet there is nearly no evidence that he really contributed anything to such fields at all. This book asks: how does history change when we subtract the many small exaggerations and interpolations that writers have added for over two thousand years?

The Cult of Pythagoras is also about invention in a positive sense. Most people view mathematical breakthroughs as “discoveries” rather than invention or creativity, believing that mathematics describes a realm of eternal ideas. But mathematicians have disagreed about what is possible and impossible, about what counts as a proof, and even about the results of certain operations. Was there ever invention in the history of concepts such as zero, negative numbers, imaginary numbers, quaternions, infinity, and infinitesimals?

Martínez inspects a wealth of primary sources, in several languages, over a span of many centuries. By exploring disagreements and ambiguities in the history of the elements of mathematics, The Cult of Pythagoras dispels myths that obscure the actual origins of mathematical concepts. Martínez argues that an accurate history that analyzes myths reveals neglected aspects of mathematics that can encourage creativity in students and mathematicians.

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Duel at Dawn
Heroes, Martyrs, and the Rise of Modern Mathematics
Amir Alexander
Harvard University Press, 2011
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.
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Florence and Baghdad
Renaissance Art and Arab Science
Hans Belting
Harvard University Press, 2011

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?

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Galileo's Muse
Renaissance Mathematics and the Arts
Mark A. Peterson
Harvard University Press, 2011

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.

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Greek Mathematical Works, Volume I
Thales to Euclid
Ivor Thomas
Harvard University Press

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).

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Greek Mathematical Works, Volume II
Aristarchus to Pappus
Ivor Thomas
Harvard University Press

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).

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A History in Sum
150 Years of Mathematics at Harvard (1825–1975)
Steve Nadis and Shing-Tung Yau
Harvard University Press, 2013

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.

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The Invention of Imagination
Aristotle, Geometry and the Theory of the Psyche
Justin Humphreys
University of Pittsburgh Press, 2023

A Provocative Examination of the Origin of Imagination

Aristotle was the first philosopher to divide the imagination—what he called phantasia—from other parts of the psyche, placing it between perception and intellect. A mathematician and philosopher of mathematical sciences, Aristotle was puzzled by the problem of geometrical cognition—which depends on the ability to “produce” and “see” a multitude of immaterial objects—and so he introduced the category of internal appearances produced by a new part of the psyche, the imagination. As Justin Humphreys argues, Aristotle developed his theory of imagination in part to explain certain functions of reason with a psychological rather than metaphysical framework. Investigating the background of this conceptual development, The Invention of Imagination reveals how imagery was introduced into systematic psychology in fifth-century Athens and ultimately made mathematical science possible. It offers new insights about major philosophers in the Greek tradition and significant events in the emergence of ancient mathematics while offering space for a critical reflection on how we understand ourselves as thinking beings. 
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Mathematics and Religion
Our Languages of Sign and Symbol
Javier Leach
Templeton Press, 2010

Mathematics and Religion: Our Languages of Sign and Symbol is the sixth title published in the Templeton Science and Religion Series, in which scientists from a wide range of fields distill their experience and knowledge into brief tours of their respective specialties. In this volume, Javier Leach, a mathematician and Jesuit priest, leads a fascinating study of the historical development of mathematical language and its influence on the evolution of metaphysical and theological languages.

Leach traces three historical moments of change in this evolution: the introduction of the deductive method in Greece, the use of mathematics as a language of science in modern times, and the formalization of mathematical languages in the nineteenth and twentieth centuries. As he unfolds this fascinating history, Leach notes the striking differences and interrelations between the two languages of science and religion. Until now there has been little reflection on these similarities and differences, or about how both languages can complement and enrich each other.

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Matter and Mathematics
An Essential Account of Laws of Nature
Andrew Younan
Catholic University of America Press, 2022
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.
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Measurement
Paul Lockhart
Harvard University Press, 2012

For seven years, Paul Lockhart’s A Mathematician’s Lament enjoyed a samizdat-style popularity in the mathematics underground, before demand prompted its 2009 publication to even wider applause and debate. An impassioned critique of K–12 mathematics education, it outlined how we shortchange students by introducing them to math the wrong way. Here Lockhart offers the positive side of the math education story by showing us how math should be done. Measurement offers a permanent solution to math phobia by introducing us to mathematics 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.

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Native American Mathematics
By Michael P. Closs
University of Texas Press, 1996

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.

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The Optics of Ibn al-Haytham Books IV–V
On Reflection and Images Seen by Reflection
Abdelhamid I. Sabra
University of London Press, 2023
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.
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Performing Math
A History of Communication and Anxiety in the American Mathematics Classroom
Andrew Fiss
Rutgers University Press, 2021
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.
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Philosophy of Mathematics in the Twentieth Century
Selected Essays
Charles Parsons
Harvard University Press, 2014

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.

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Randomness
Deborah J. Bennett
Harvard University Press, 1998

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.

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Reactionary Mathematics
A Genealogy of Purity
Massimo Mazzotti
University of Chicago Press, 2023
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.
 
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The Seven Pillars of Statistical Wisdom
Stephen M. Stigler
Harvard University Press, 2016

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.

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A Source Book in Mathematics, 1200-1800
D. J. Struik
Harvard University Press

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.

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Squaring the Circle
The War between Hobbes and Wallis
Douglas M. Jesseph
University of Chicago Press, 1999
In 1655, the philosopher Thomas Hobbes claimed he had solved the centuries-old problem of "squaring of the circle" (constructing a square equal in area to a given circle). With a scathing rebuttal to Hobbes's claims, the mathematician John Wallis began one of the longest and most intense intellectual disputes of all time. Squaring the Circle is a detailed account of this controversy, from the core mathematics to the broader philosophical, political, and religious issues at stake.

Hobbes believed that by recasting geometry in a materialist mold, he could solve any geometric problem and thereby demonstrate the power of his materialist metaphysics. Wallis, a prominent Presbyterian divine as well as an eminent mathematician, refuted Hobbes's geometry as a means of discrediting his philosophy, which Wallis saw as a dangerous mix of atheism and pernicious political theory.

Hobbes and Wallis's "battle of the books" illuminates the intimate relationship between science and crucial seventeenth-century debates over the limits of sovereign power and the existence of God.
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Teaching the Quadrivium
A Guide for Instructors
Peter Ulrickson
Catholic University of America Press, 2023
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.
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Timaeus. Critias. Cleitophon. Menexenus. Epistles
Plato
Harvard University Press

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.

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Uncountable
A Philosophical History of Number and Humanity from Antiquity to the Present
David Nirenberg and Ricardo L. Nirenberg
University of Chicago Press, 2021
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.
 
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Vector
A Surprising Story of Space, Time, and Mathematical Transformation
Robyn Arianrhod
University of Chicago Press, 2024
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.
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