Matematikens filosofi

Matematiken i sina senaste utvecklingsfaser är en angelägenhet för ett fåtal specialister. Långt fler berörs av den som arbetsredskap eller undervisningsämne, men det finns också aspekter av matematiken som angår och påverkar oss alla. Matematiska modeller förutsätts i en mängd tekniska applikationer som inverkar på snart sagt varje område av det moderna livet. Ämnet hör dessutom till vårt kulturarv, och sett till sina mest elementära yttringar, i dess anslag eller uppkomst i oss själva, kan matematiken rent av förstås som en aspekt av det mänskliga. Det som är en del av vår natur tenderar att vara ogenomskinligt för sunda förnuftet och kräva en filosofisk analys för att genomlysas. När filosofin gör matematiken till sitt speciella studieobjekt uppstår "matematikens filosofi", som utmärker sig genom att ställa frågor som matematiker traditionellt varit obenägna att ställa om sitt ämne: Vad möjliggör eller förutsätts egentligen i matematikernas verksamhet? Med vilken rätt kan man tala om matematiken som oberoende gentemot oss själva? I vad består meningen hos matematiska tecken? Vad är egentligen ett tal? Till en början inriktades ansträngningarna inom matematikens filosofi på att återföra hela matematikens vitt skilda problemområden på några få grundläggande principer. Denna bok förbereder läsaren för och presenterar den matematiska grundvalsforskningens ledande filosofer, deras bärande idéer, förhoppningsfulla framstötar och oväntade svårigheter. Mot slutet redogör boken också för de senare och nutida strömningar som uppstått som en reaktion på grundvalsforskningens resultat, och berör tankar om framtidens matematik.

Ludwig Wittgenstein was one of the greatest philosophers of the modern era, while his pupil and friend Elizabeth Anscombe is increasingly recognized as being an important philosopher in her own right. Over a number of years, Anscombe kept a series of notebooks labeled "Anecdotes about Wittgenstein," documents which are uniquely valuable for their ability to shed light on the intellectual and personal character of Wittgenstein, as well as Anscombe's own views on a wide range of topics--cultural, historical and psychological. While the original manuscript was divided into sections of varying lengths and jumped around among topics, Anscombe on Wittgenstein reorders them thematically so that readers might more vividly perceive the contours of her subjects. Between 1946-1947, Anscombe, Wittgenstein, and Wasfi Hijab routinely had philosophical conversations, which Anscombe wrote down immediately afterward while they were still fresh in her mind. Transcribed here as they were originally written--similar to a play, with the voices of the participants indicated by their initials--readers can experience these philosophers grappling with topics such as rule-following, necessity, and the notion of the indefinite, all subjects that were at the heart of Wittgenstein's later philosophy. This book also offers a previously unpublished essay by Anscombe, as well as two essays by the book's editors, leading scholars on Anscombe, and biographies of the characters mentioned in the manuscripts. What emerges from these accounts is not only a picture of Wittgenstein, the man and philosopher, but also a memoir of a close friendship between two of the most significant philosophers of the twentieth century.

While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913). This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schroder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Godel. A strong American theme runs though the story, beginning with the mathematician E. H.Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GodeI. Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials. Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since.

*A Financial Times and Waterstones Book of the Year 2025* Many artists are unaware of the mathematics that bubble beneath their craft, while some consciously use it for inspiration. Our instincts might tell us that these two subjects are incompatible forces with nothing in common, but what if we’re wrong? Marcus du Sautoy, acclaimed mathematician and Simonyi Professor for the Public Understanding of Science at the University of Oxford, looks to art, music, design and literature to uncover the key mathematical structures that underpin both human creativity and the natural world.Blueprints takes us from the earliest stone circles to the modernist architecture of Le Corbusier, from Bach’s circular compositions to Radiohead’s disruptive soundscapes, and from Shakespeare’s hidden numerical clues to the Dada artists who embraced randomness. Instead of polar opposites we find a complementary relationship that spans a vast historical and geographic landscape.Whether we are searching for meaning in an abstract painting or deciphering poetry, there are blueprints everywhere: prime numbers, symmetry, fractals and the weirder worlds of Hamiltonian cycles and hyperbolic geometry. Nature similarly exploits these structures to achieve the wonders of our universe.In this innovative and delightfully bold exploration of creativity, Marcus explains how we make art, why a creative mindset is vital for discovering new mathematics and how a fundamental connection to the natural world intrinsically links these two subjects.‘Blueprints is an extraordinary book which shows us how mathematics and art are connected through structures. Du Sautoy shows us how to bridge the divide of science and the humanities and proves that we can only face and solve the big challenges of the twenty-first century if we go beyond the fear of pooling knowledge’ Hans Ulrich Obrist, Artistic Director, Serpentine Galleries

I Wish They'd Taught Me That: Overlooked and Omitted Topics in Mathematics concerns the topics which every undergraduate mathematics student "should" know but has probably never encountered. These topics are not the ones which dominate every syllabus, but those magnificent secrets that are beautiful, useful and accessible but which are inexplicably hidden away from the mainstream curriculum.Each chapter of this book concerns a different topic which students will almost certainly be unfamiliar with. Written in a lively, conversational style, by the end of each section the reader should feel equipped with the knowledge to explore the area more fully elsewhere.FeaturesTopics from a variety of areas of mathematics, including geometry, logic, analysis, algebra, numerical analysis, and topologyNumerous examples, diagrams, and exercisesCollections of resources where an interested reader can learn more about each topicNontechnical introductions to each chapter.For a full list of errata, maintained by the authors, please visit: https://juliangould.github.io/pages/IWTTMT_corrections.html

In this third installment of his classic 'Foundations' trilogy, Michel Serres takes on the history of geometry and mathematics. Even more broadly, Geometry is the beginnings of things and also how these beginnings have shaped how we continue to think philosophically and critically. Serres rejects a traditional history of mathematics which unfolds in a linear manner, and argues for the need to delve into the past of maths and identify a series of ruptures which can help shed light on how this discipline has developed and how, in turn, the way we think has been shaped and formed.This meticulous and lyrical translation marks the first ever English translation of this key text in the history of ideas.

This book investigates the relationships between modern mathematics and science (in particular, quantum mechanics) and the mode of theorizing that Arkady Plotnitsky defines as "nonclassical" and identifies in the work of Bohr, Heisenberg, Lacan, and Derrida. Plotinsky argues that their scientific and philosophical works radically redefined the nature and scope of our knowledge. Building upon their ideas, the book finds a new, nonclassical character in the "dream of great interconnections" Bohr described, thereby engaging with recent debates about the "two cultures" (the humanities and the sciences).Plotnitsky highlights those points at which the known gives way to the unknown (and unknowable). These points are significant, he argues, because they push the boundaries of thought and challenge the boundaries of disciplinarity. One of the book's most interesting observations is that key figures in science, in order to push toward a framing of the unknown, actually retreated into a conservative disciplinarity. Plotnitsky's informed, interdisciplinary approach is more productive than the disparaging attacks on postmodernism or scientism that have hitherto characterized this discourse.Arkady Plotnitsky is Professor of English and Director, Theory and Cultural Studies Program, Purdue University. Trained in both mathematics and literary theory, he is author of several books, including In the Shadow of Hegel: Complementarity, History and the Unconscious and Reconfigurations: Critical Theory and General Economy.

A foundational work on historical and social studies of quantificationWhat accounts for the prestige of quantitative methods? The usual answer is that quantification is desirable in social investigation as a result of its successes in science. Trust in Numbers questions whether such success in the study of stars, molecules, or cells should be an attractive model for research on human societies, and examines why the natural sciences are highly quantitative in the first place. Theodore Porter argues that a better understanding of the attractions of quantification in business, government, and social research brings a fresh perspective to its role in psychology, physics, and medicine. Quantitative rigor is not inherent in science but arises from political and social pressures, and objectivity derives its impetus from cultural contexts. In a new preface, the author sheds light on the current infatuation with quantitative methods, particularly at the intersection of science and bureaucracy.

A "fascinating romp" (Johnjoe McFadden) and stirring account of the mathematicians who went looking for the bedrock philosophical foundations of their field and witnessed a house of cards collapse instead As the nineteenth century ended, mathematicians were celebrating a century of triumphs that-surprisingly-made clear how little they knew: What is the nature of infinity? Is math free from self-contradiction? And what does math have to do with reality? This was the Foundational Crisis in mathematics. In The Great Math War, Jason Socrates Bardi tells the story of three competing efforts by mathematicians to resolve it-and the firefight that ensued. Bertrand Russell thought we could achieve certainty if we treated math as an extension of logic. David Hilbert believed redemption lay in accepting mathematics as a formal game of arbitrary rules, no different from the moves and pieces in chess. And L. E. J. Brouwer argued math is entirely rooted in human intuition-and that math is not based on logic but rather logic is based on math. It was a bitter struggle, intellectually and personally, as the three vied to set the course for mathematics in the twentieth century. Set against the backdrop of international warfare unfolding alongside it, The Great Math War brings the Foundational Crisis to radiant life-and shows how it indelibly shaped twentieth-century intellectual life.

A fascinating look into how the transformative joys of mathematical experience are available to everyone, not just specialists  Math has a reputation for being inaccessible. People think that it requires a special gift or that comprehension is a matter of genes. Yet the greatest mathematicians throughout history, from René Descartes to Alexander Grothendieck, have insisted that this is not the case. Like Albert Einstein, who famously claimed to have “no special talent,” they said that they had accomplished what they did using ordinary human doubts, weaknesses, curiosity, and imagination.  David Bessis guides us on an illuminating path toward deeper mathematical comprehension, reconnecting us with the mental plasticity we experienced as children. With simple, concrete examples, Bessis shows how mathematical comprehension is integral to the great learning milestones of life, such as learning to see, to speak, to walk, and to eat with a spoon.  Focusing on the deeply human roots of mathematics, Bessis dispels the myths of mathematical genius. He offers an engaging initiation into the experience of math not as a series of discouragingly incomprehensible logic problems but as a physical activity akin to yoga, meditation, or a martial art. This perspective will change the way you think not only about math but also about intelligence, intuition, and everything that goes on inside your head.

_______________'Beguiling ... I was riveted. Olsson is evocative on curiosity as an appetite of the mind, on the pleasure of glutting oneself on knowledge' - New York Times'A wonderful book ... Reading it is akin to kicking a can along the road of higher learning' - Patti Smith'A remarkable tour de force' - Standpoint_______________Simone Weil: philosopher, political activist, mystic – and sister to André, one of the most influential mathematicians of the twentieth century. These two extraordinary siblings formed an obsession for Karen Olsson, who studied mathematics at Harvard, only to turn to writing as a vocation.When Olsson got hold of the 1940 letters between the siblings, she found they shared a curiosity about the inception of creative thought – that flash of insight – that Olsson experienced as both a mathematics student, and later, a novelist. Following this thread of connections, The Weil Conjectures explores the lives of Simone and André, the lore and allure of mathematics, and its significance in Olsson’s own life.

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.

The Quadrivium consists of the four Liberal Arts of Number, Geometry, Music, and Cosmology, studied from antiquity to the Renaissance as a way of glimpsing the nature of reality. They synthesize number, space, and time. Geometry is number in space, music is number in time, and the cosmos expresses number in space and time. Number, music, and geometry are metaphysical truths, good and beautiful everywhere at all times. Life across the universe investigates them. They foreshadow the physical sciences. This is the first volume to bring together the Quadrivium for many hundreds of years.

This Element aims to present an outline of mathematics and its history, with particular emphasis on events that shook up its philosophy. It ranges from the discovery of irrational numbers in ancient Greece to the nineteenth- and twentieth-century discoveries on the nature of infinity and proof. Recurring themes are intuition and logic, meaning and existence, and the discrete and the continuous. These themes have evolved under the influence of new mathematical discoveries and the story of their evolution is, to a large extent, the story of philosophy of mathematics.

A NEW SCIENTIST BOOK OF THE YEAR'Clear, clever and friendly...even at her most whimsical, she is rigorous and insightful' ALEX BELLOS'Eugenia Cheng has humour, grace and a natural gift' DANIEL LEVITIN'Playful and deeply serious ... shows how exploring equality in maths may help the real world' NEW SCIENTISTWhen we see an equals sign, we usually see something clear cut: problem on one side, solution on the other. We might need to shuffle some things around - balance the equation - to get there, but it's all pretty clear cut...isn't it?As it turns out, between those parallel lines lies something far more exciting than questions and answers. An equals sign is an invitation to a mathematical playground of choice and abstraction. By opening your mind to different mathematical choices and perspectives, you can unlock far greater insight than you could have dreamed.Eugenia Cheng explores the rich and rewarding interplay between sameness and difference, and offers us a new way to see the world based on the choices and interpretations we make. A glorious celebration of mathematics, Unequal will change the way you think - in maths, and in life.

Think of a number between one and ten.No, hang on, let's make this interesting. Between zero and infinity. Even if you stick to the whole numbers, there are a lot to choose from - an infinite number in fact. Throw in decimal fractions and infinity suddenly gets an awful lot bigger (is that even possible?) And then there are the negative numbers, the imaginary numbers, the irrational numbers like pi which never end. It literally never ends.The world of numbers is indeed strange and beautiful. Among its inhabitants are some really notable characters - pi, e, the "imaginary" number i and the famous golden ratio to name just a few. Prime numbers occupy a special status. Zero is very odd indeed: is it a number, or isn't it?How Numbers Work takes a tour of this mind-blowing but beautiful realm of numbers and the mathematical rules that connect them. Not only that, but take a crash course on the biggest unsolved problems that keep mathematicians up at night, find out about the strange and unexpected ways mathematics influences our everyday lives, and discover the incredible connection between numbers and reality itself.ABOUT THE SERIESNew Scientist Instant Expert books are definitive and accessible entry points to the most important subjects in science; subjects that challenge, attract debate, invite controversy and engage the most enquiring minds. Designed for curious readers who want to know how things work and why, the Instant Expert series explores the topics that really matter and their impact on individuals, society, and the planet, translating the scientific complexities around us into language that's open to everyone, and putting new ideas and discoveries into perspective and context.

A WATERSTONES BEST BOOK OF 2023A NEW SCIENTIST BEST BOOK OF 2023WINNER OF THE LA TIMES SCIENCE & TECHNOLOGY 2023 BOOK PRIZE'A generous tour of mathematics for anyone whose instincts tend less towards "Just tell me the answer" and more towards "Wait, but why?"' JORDAN ELLENBERGTo many, maths feels like an unmapped wilderness. Between abstract concepts like imaginary numbers and infinity, it can sometimes feel like a lot of maths is just made up. Why, for example, is 1 not a prime? Why do two negatives cancel each other out? Where does trigonometry come from? Is maths even real? Abstract mathematician Eugenia Cheng shows that curiosity is the best teacher. Is Maths Real? takes us on a scintillating tour of the simple questions that provoke mathematics' deepest insights. 'Intriguing...celebrates the dizziness and disorientation engendered by childlike questions that hint at the deep mysteries beneath'NEW SCIENTIST'Masterfully uncovers what's simply profound in the profoundly simple'FRANCIS SU'Discover what it feels like to be a real mathematician'DAILY TELEGRAPH

Euclid’s Elements of Geometry was a book that changed the world. In a sweeping history, Benjamin Wardhaugh traces how an ancient Greek text on mathematics – often hailed as the world’s first textbook – shaped two thousand years of art, philosophy and literature, as well as science and maths. Thirteen volumes of mathematical definitions, propositions and proofs. Writing in 300 BC, Euclid could not have known his logic would go unsurpassed until the nineteenth century, or that his writings were laying down the very foundations of human knowledge.Wardhaugh blasts the dust from Euclid’s legacy to offer not only a vibrant history of mathematics, told through people and invention, but also a broader story of culture. Telling stories from every continent, ranging between Ptolemy and Isaac Newton, Hobbes and Lewis Carrol, this is a history that dives from Ancient Greece to medieval Byzantium, early modern China, Renaissance Italy, the age of European empires, and our world today.How has geometry sat at the beating heart of sculpture, literature, music and thought? How can one unknowable figure of antiquity live through two millennia?

Euclid’s Elements of Geometry was a book that changed the world. In a sweeping history, Benjamin Wardhaugh traces how an ancient Greek text on mathematics – often hailed as the world’s first textbook – shaped two thousand years of art, philosophy and literature, as well as science and maths. Thirteen volumes of mathematical definitions, propositions and proofs. Writing in 300 BC, Euclid could not have known his logic would go unsurpassed until the nineteenth century, or that his writings were laying down the very foundations of human knowledge.Wardhaugh blasts the dust from Euclid’s legacy to offer not only a vibrant history of mathematics, told through people and invention, but also a broader story of culture. Telling stories from every continent, ranging between Ptolemy and Isaac Newton, Hobbes and Lewis Carrol, this is a history that dives from Ancient Greece to medieval Byzantium, early modern China, Renaissance Italy, the age of European empires, and our world today.How has geometry sat at the beating heart of sculpture, literature, music and thought? How can one unknowable figure of antiquity live through two millennia?

‘An astonishingly readable and informative history of the greatest mathematical bestseller of all time … The writing is vivid and the stories are gripping. Highly recommended ’ IAN STEWART, AUTHOR OF SIGNIFICANT FIGURES Euclid’s Elements of Geometry was a book that changed the world. In this sweeping history, Benjamin Wardhaugh traces how the ancient Greek text on mathematics – often hailed as the world’s first textbook – shaped two thousand years of art, philosophy and literature, as well as science and maths.With stories of influence on every continent, and encounters with the likes of Ptolemy and Isaac Newton, Hobbes and Lewis Carroll, Wardhaugh gives dramatic life to the evolution of mathematics.Previously published as The Book of Wonders

SHORTLISTED FOR THE BRITISH SOCIETY FOR THE HISTORY OF SCIENCE PRIZE 2025 SHORTLISTED FOR THE NEUMANN PRIZE 2025 WHAT DOES IT MEAN TO COUNT? WHY ARE HUMANS THE ONLY SPECIES ON EARTH THAT CAN DO IT? WHERE DID COUNTING COME FROM? HOW HAS IT SHAPED SOCIETIES ALONG THE WAY? AND WHY DOES IT MATTER? Counting is an innovative, erudite, world-wrapping journey through humanity’s marvellous ability to impose numbers on things. Acclaimed historian and mathematician Benjamin Wardhaugh draws on stories from the Stone Age to cyberspace in pursuit of the elusive, fascinating, endlessly diverse history of human counting.Starting with the roots of counting in human brains, bodies and environments, Wardhaugh tours us around the world and through time while exploring the different flavours of counting that have developed over millennia. We meet the makers of bead necklaces in ancient South Africa, the inventors of writing in the world ’ s first metropolis, and the ‘counter culture’ of classical Athens. We see counting used – and changed – by Indian scholars, Chinese peasants and Papuan shopkeepers; we meet the distinctive numerate agendas of Mayan kings, US governments and Korean vloggers.Weaving these stories together, Wardhaugh shows how cultures have shaped counting, and how counting has shaped culture, in a rich tapestry spanning thousands of years. This is the vast story of human attempts to find some order in an unruly world; or, perhaps, to impose on a reluctant world the order that humans find within themselves. It is a history as wide, deep and tangled as that of humanity itself