Here are 100 books that What Is Mathematics, Really? fans have personally recommended if you like
What Is Mathematics, Really?.
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I'm a mathematician but an unusual one because I am interested in how mathematics is created and how it is learned. From an early age, I loved mathematics because of the beauty of its concepts and the precision of its organization and reasoning. When I started to do research I realized that things were not so simple. To create something new you had to suspend or go beyond your rational mind for a while. I realized that the learning and creating of math have non-logical features. This was my eureka moment. It turned the conventional wisdom (about what math is and how it is done) on its head.
I’m interested in how mathematicians create mathematics but this book made me realize that learning mathematics is also a form of creativity. Each of us has created our understanding of mathematics as we were growing up. We are all creative!
What is amazing about this book is that even children as young as six months possess rudimentary mathematical concepts, in particular, the concept of number. (Actually, Carey shows children have two distinct ways of thinking about numbers). The concept of number is built-in. That’s amazing to me! The mastery of counting numbers, 1,2,3,… is a great creative leap in the development of the child. This leap is followed by a series of further amazing accomplishments, for example, the insight that a fraction like 2/3, is a completely new kind of number (and not just a problem in division). How do kids manage to accomplish such radical changes in their concept…
Only human beings have a rich conceptual repertoire with concepts like tort, entropy, Abelian group, mannerism, icon and deconstruction. How have humans constructed these concepts? And once they have been constructed by adults, how do children acquire them? While primarily focusing on the second question, in The Origin of Concepts , Susan Carey shows that the answers to both overlap substantially.
Carey begins by characterizing the innate starting point for conceptual development, namely systems of core cognition. Representations of core cognition are the output of dedicated input analyzers, as with perceptual representations, but these core representations differ from perceptual representations…
I'm a mathematician but an unusual one because I am interested in how mathematics is created and how it is learned. From an early age, I loved mathematics because of the beauty of its concepts and the precision of its organization and reasoning. When I started to do research I realized that things were not so simple. To create something new you had to suspend or go beyond your rational mind for a while. I realized that the learning and creating of math have non-logical features. This was my eureka moment. It turned the conventional wisdom (about what math is and how it is done) on its head.
Lots of people have a priori ideas about what mathematics is all about but Lakatos had the brilliant idea of looking at what actually happened. His book is all about one famous theorem: “for all regular polyhedra, V – E + F =2, where V is the number of vertices, E is the number of edges, and F is the number of faces. Think of a cube where V=8, E = 12, F = 6.
We tend to think that mathematics proceeds from a well-defined hypothesis to conclusion. But that is only the finishing step. Along the way the definitions keep changing as do the hypotheses and even the conclusion. Everything is moving! This is what makes doing mathematics so exciting!
Imre Lakatos's Proofs and Refutations is an enduring classic, which has never lost its relevance. Taking the form of a dialogue between a teacher and some students, the book considers various solutions to mathematical problems and, in the process, raises important questions about the nature of mathematical discovery and methodology. Lakatos shows that mathematics grows through a process of improvement by attempts at proofs and critiques of these attempts, and his work continues to inspire mathematicians and philosophers aspiring to develop a philosophy of mathematics that accounts for both the static and the dynamic complexity of mathematical practice. With a…
I experienced being a parent as a return to my own childhood. As much as I enjoyed teaching my children, I loved learning from them as well. That got me thinking about how one might recapture the joys and insights of childhood. As a philosopher interested in education, I have long wondered whether we leave childhood behind or somehow carry it with us into old age. I discovered that several important philosophers, such as Aristotle, Augustine, and Rousseau have keen insights about the relation of childhood to adulthood. And the biblical Jesus seems to have been the first person to suggest that adults can learn from children.
What if children are not little adults but a different species? Perhaps children are butterflies who develop into caterpillars? Child psychologist Allision Gopnik asks wonderful questions about human development. She notes that most of us produce our best art and ask our deepest questions (“Why is the sky blue?”) as small children.
Childhood, she says, is our time of basic research; adulthood is the time for practical applications. Like Jean-Jacques Rousseau, she celebrates the unique gifts of childhood, but she does not offer suggestions about how we might recapture those gifts.
In the last decade there has been a revolution in our understanding of the minds of infants and young children. We used to believe that babies were irrational, and that their thinking and experience were limited. Now Alison Gopnik ― a leading psychologist and philosopher, as well as a mother ― explains the cutting-edge scientific and psychological research that has revealed that babies learn more, create more, care more, and experience more than we could ever have imagined. And there is good reason to believe that babies are actually smarter, more thoughtful, and more conscious than adults. In a lively…
I'm a mathematician but an unusual one because I am interested in how mathematics is created and how it is learned. From an early age, I loved mathematics because of the beauty of its concepts and the precision of its organization and reasoning. When I started to do research I realized that things were not so simple. To create something new you had to suspend or go beyond your rational mind for a while. I realized that the learning and creating of math have non-logical features. This was my eureka moment. It turned the conventional wisdom (about what math is and how it is done) on its head.
It’s a little weird that this book should find a place on my list. It’s a book about how society has become resistant to anything that is difficult and painful and the kinds of people that we have become as a result. But mathematics is difficult! To understand mathematics you have to think hard, sometimes for a long time. Moreover understanding something hard is discontinuous, it requires a leap to a new way of thinking. You have to start with a problem and this problem might be an ambiguity or a contradiction. A is true and Bis true but A and B seem to contradict one another. When you sort out this problem you will have learned something.
The moral here is to embrace things that are difficult if you want to learn significant new things. “No pain, no gain.” You don’t have to worry about some super AI…
Our societies today are characterized by a universal algophobia: a generalized fear of pain. We strive to avoid all painful conditions - even the pain of love is treated as suspect. This algophobia extends into society: less and less space is given to conflicts and controversies that might prompt painful discussions. It takes hold of politics too: politics becomes a palliative politics that is incapable of implementing radical reforms that might be painful, so all we get is more of the same.
Faced with the coronavirus pandemic, the palliative society is transformed into a society of survival. The virus enters…
As a professor, I see students fascinated by science, but petrified to take a science class. This is in part because we have dehumanized science, removed the story, edited out the human, deleted the parts that allow people to connect with it. Science does not get delivered by gods, but is created by people: smart, quirky, sometimes immoral people. As a writer, my hope is to be able to reinsert life into readers’ understanding of our greatest advances. As a reader myself, I am deeply appreciative when other authors do it too.
David Hilbert was the most important mathematician at the dawn of the 20th century. In 1900, he gave the mathematical community its homework for the next 100 years setting out the list of open problems that had to be solved by 2000. While to the rest of the mathematicians, he may have appeared as their professor, he was also the class clown. As colorful and funny as he was brilliant, you cannot but come away loving this great mathematical genius.
"It presents a sensitive portrait of a great human being. It describes accurately and intelligibly on a nontechnical level the world of mathematical ideas in which Hilbert created his masterpieces. And it illuminates the background of German social history against which the drama of Hilberts life was played. Beyond this, it is a poem in praise of mathematics." -SCIENCE
I studied Greek philosophy in college and graduate school and wrote my Ph.D. dissertation on Plato. In response to the environmental crisis, first widely recognized in the 1960s, I turned my philosophical attention to that contemporary challenge, which, with the advent of climate change, has by now proved to be humanity’s greatest. I taught the world’s first course in environmental ethics at the University of Wisconsin-Stevens Point in 1971 and, with a handful of other philosophers, helped build a literature in this new field over the course of the next decade—a literature that has subsequently grown exponentially. With Greek Natural Philosophy, I rekindled the romance with my first love.
A renowned historian of science, Clagett carries the story of Greek science forward all the way to the sixth century CE—a span of 1200 years. From that point in time, Greek science passed into the hands of Islamic scholars who advanced it further, especially the mathematical sciences.
This book is not, like ours, organized chronologically and developmentally but according to modern scientific domains—biology and medicine, mathematics, physics, and astronomy. And it focuses on specific scientific inquiries, while we focus on more general and fundamental things like ontology (what exists), cosmology (the overall structure of the universe), the laws of nature, and the drivers of change and motion.
This book is thus a complement to ours in its wide historical sweep and in what it highlights.
Greek Science in Antiquity is a comprehensive book written by Marshall Clagett that explores the scientific advancements made by ancient Greeks. The book covers a wide range of topics, including mathematics, astronomy, mechanics, and medicine, and provides a detailed account of the theories and discoveries made by Greek scientists from the 6th century BCE to the 5th century CE. The book begins by examining the early Greek philosophers and their contributions to the development of science, including Thales, Pythagoras, and Aristotle. It then delves into the mathematical achievements of the Greeks, such as the invention of geometry and the discovery…
Having majored in both philosophy and physics as an undergraduate, I specialized in the philosophy of science in graduate school–with a focus on the possibility of a “logic of scientific discovery.” Most philosophers of science have been skeptical about such a sub-discipline, restricting their theories of scientific method to the justification of already-formulated hypotheses. Others (including myself) have held that there is also a logic to the generation of hypotheses.
This is a fascinating analysis of the works of Copernicus, Kepler, Galileo, Descartes, Hobbes, Gilbert, Boyle, and Newton. It not only establishes the reasons for the triumph of the modern perspective but also accounts for certain limitations in this view that continue to characterize contemporary scientific thought.
A criticism as well as a history of the change that made possible the rise of modern science, this volume is also a guide to understanding the methods and accomplishments of the great philosopher-scientists of the sixteenth and seventeenth centuries.
s/t: A Historical & Critical Essay Many books well received when originally published ultimately fail the test of time & seem outdated to future generations. Occasionally, a book seen as a solid effort when written is found later to be the definitive work on the subject. The Metaphysical Foundations of Modern Science by Edwin Arthur Burtt is such. Burtt investigates the origins of the modern scientific worldview, a view that's only a few centuries old. Concepts used to describe the world--mass, velocity, energy, time etc--form the substratum of so many modern ideas that their very ubiquity has made it hard…
I am a Reader in the Mathematics Education Centre at Loughborough University in the UK. I have always loved mathematics and, when I became a PhD student and started teaching, I realized that how people think about mathematics is fascinating too. I am particularly interested in demystifying the transition to proof-based undergraduate mathematics. I believe that much of effective learning is not about inherent genius but about understanding how theoretical mathematics works and what research tells us about good study strategies. That is what these books, collectively, are about.
Mathematics requires accurate calculation, and students sometimes think that getting the right answer is enough. But mathematics is also about valid logical arguments, and the demand for clear communication increases through an undergraduate degree. Students, therefore, need to learn to write professionally, with attention to general issues like good grammar, and mathematics-specific issues like accuracy in notation, precision in logical language, and structure in extended arguments. Vivaldi’s book has a great many examples and exercises, and students could benefit from studying it systematically or from dipping into it occasionally and reflecting on small ways to improve.
This book teaches the art of writing mathematics, an essential -and difficult- skill for any mathematics student.
The book begins with an informal introduction on basic writing principles and a review of the essential dictionary for mathematics. Writing techniques are developed gradually, from the small to the large: words, phrases, sentences, paragraphs, to end with short compositions. These may represent the introduction of a concept, the abstract of a presentation or the proof of a theorem. Along the way the student will learn how to establish a coherent notation, mix words and symbols effectively, write neat formulae, and structure a…
I am a Reader in the Mathematics Education Centre at Loughborough University in the UK. I have always loved mathematics and, when I became a PhD student and started teaching, I realized that how people think about mathematics is fascinating too. I am particularly interested in demystifying the transition to proof-based undergraduate mathematics. I believe that much of effective learning is not about inherent genius but about understanding how theoretical mathematics works and what research tells us about good study strategies. That is what these books, collectively, are about.
Many undergraduate mathematics books – even those aimed at new students – are dense, technical, and difficult to read at any sort of speed. This is a natural feature of books in a deductive science, but it can be very discouraging, even for dedicated students. Houston’s book covers many ideas useful at the transition to proof-based mathematics, and he has worked extensively and attentively with students at that stage. Consequently, his book maintains high mathematical integrity and has lots of useful exercises while also being an unusually friendly read.
Looking for a head start in your undergraduate degree in mathematics? Maybe you've already started your degree and feel bewildered by the subject you previously loved? Don't panic! This friendly companion will ease your transition to real mathematical thinking. Working through the book you will develop an arsenal of techniques to help you unlock the meaning of definitions, theorems and proofs, solve problems, and write mathematics effectively. All the major methods of proof - direct method, cases, induction, contradiction and contrapositive - are featured. Concrete examples are used throughout, and you'll get plenty of practice on topics common to many…
My name is Susan Marie Chapman and I am an award-winning Children’s Book Author. I have written over fourteen children’s books. I grew up on a farm surrounded by animals and nature and my seven sisters and brothers. Wow!! My goal is to get as many books into the hands of children that I possibly can. You see, reading books, especially picture books, is a way for a child to see the world through the pictures and words of a book. It creates imagination and excitement and fun and questions which lead to answers which makes you smarter. So read, read, read, until you run out of books, which will never happen.
What child isn’t curious about the night sky and all the stars that live up there? Did you know that the Sun is a giant star? This book is full of fun facts, not just about stars but about our planet. It helps to put things into perspective, so to speak. It talks about gravity and how many miles away the moon is from the earth. I think kids will learn a lot from reading this book and will even be able to impress their friends with all of their newly acquired knowledge. Did you know the earth looks green because it’s covered in 3,000,000,000,000 trees?? I love this book because learning new things is fun and this book is all about fun. I felt very smart after reading this book.
A Boston Globe–Horn Book Honor Book * Winner of the Mathical Book Prize
Perfect for curious children, classrooms eager for STEM content, and readers who have devoured Ada Twist, Scientist and How Much Is a Million?
Did you know that the earth is covered in three trillion trees? And that seven billion people weigh about the same as ten quadrillion ants? Our world is full of constantly changing numbers, from a hundred billion trillion stars in space to thirty-seven billion rabbits on Earth. Can you imagine that many of anything?
The playful illustrations from New York Times–bestselling artist Isabel Greenberg…
