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 wrote
How Mathematicians Think: Using Ambiguity, Contradiction, and Paradox to Create Mathematics
Reuben Hersh is responsible for a revolution in the way we look at mathematics. His main idea is very simple: mathematics is something that is created by human beings. Isn’t that obvious, you say? Not if you believe that mathematics is there even before life itself, that it is built into the nature of reality in some way. In philosophy, this view is called Platonism. Hersh had the radical but obvious idea that if we want to understand what mathematics is we should look at what mathematicians actually do when they create mathematics. Like all great ideas it can be stated very simply but the implications are enormous. His ideas are what got me started writing my own books about math and science.
This book tackles the important questions which have engaged mathematicians, scientists, and philosophers for thousands of years and which are still being asked today. It does so with clarity and with scholarship born of first-hand experience; a knowledge both of the ideas and of the people who have pronounced on them. The main purpose of the book is to confront philosophical problems: In what sense do mathematical objects exist? How can we have knowledge of them? Why do mathematicians think mathematical entities exist for ever, independent of human action and knowledge? The book proposes an unconventional answer: mathematics has existence…
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’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 of number? If we could answer this we might be able to say what creativity is.
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…
This is another book about the new research into how babies think. I am excited about this research because of its implications for how people learn mathematics and how researchers create math. This book taught me something important about how we all think. Gopnik distinguishes between what she calls flashlight consciousness and lantern consciousness. Flashlight is the way adults think. You focus on one thing at a time and give it your full attention. But babies, she claims, use their minds differently. Their lantern consciousness is unfocused and is aware of the big picture all at once.
So what happens to lantern consciousness when you grow up? The answer is that creative individuals use it and alternate between lantern and flashlight consciousness. When we are creating or learning something new, we have to drop back to lantern consciousness. Logic comes from flashlight consciousness and, by itself, will never produce anything original. So creativity in math and elsewhere comes from using your mind in a way that seems new but is actually the way we all thought when we were babies.
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…
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 program taking over the world and making you redundant. It takes human beings to work with situations that don’t seem to make sense, find a new way to think about the situation and, as a result, expand mathematical knowledge or personal understanding.
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…
If you love mathematics then this book will show you where the beauty and profundity that you love comes from. Most people mistakenly think that mathematics is nothing but logic, something like an AI program. This book demonstrates that something very different is going on. Mathematics makes use of non-logical features like ambiguity, contradiction, and paradox. It is precisely these non-logical features that make math profound. The book demonstrates this with fascinating examples from all levels of math.
Profundity comes from being able to look at an idea from more than one point of view. Profound ideas often come from resolving situations of conflict, for example, zero resolves the conflict of having something that stands for nothing. Maybe I should have called the book, “mathematics beyond logic”.
I’ve become passionate about telling parents how to raise happy, resilient, creative, confident, entrepreneurial children who are doing something that gives them joy. So many young people are unhappy; parents don’t understand how to help. They think their children should follow their path, but that no longer works for many. For the last 10 years, I’ve been speaking to parent groups; I was an Advisor to EQ Generation, an after-school program that gives children the skills to succeed; on the Advisory Board of MUSE School, preparing young people with passion-based learning; and on the Board of Spark the Journey, mentoring low-income high school students to achieve college and career success.
This book shakes longstanding assumptions of parenting.
Through 99 stories of people who are now changing the world, it shows how to raise creative, confident, resilient children who are filled with joy and purpose. Based on interviews with top entrepreneurs and their parents, it guides you to help your children identify their passion and figure out how they can spend their professional lives doing something they love.
Parents' well-intentioned efforts often boomerang. By ignoring their children’s skills and interests, parents can inadvertently create pressure and anxiety, thwarting their children's ability to excel and find happiness. Too often, following your heart…
Raising an Entrepreneur: How to Help Your Children Achieve Their Dreams - 99 Stories from Families Who Did
Learn how successful entrepreneurs were raised! Could your children start a company that disrupts existing industries? Or a non-profit that helps people around the world? Or follow their passion as an artist or activist? And most important, lead a life of joy and purpose, to be happy and fulfilled? Margot Machol Bisnow, mother of two thriving entrepreneurs, reveals how to raise creative, confident, resilient, fearless kids who achieve their dreams, through 99 stories of families who did it.
Read stories from 70 families who raised true game changers. See family photos of these thriving entrepreneurs, both when they were young…
