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
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”.
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…
Gabrielle found her grandfather’s diaries after her mother’s death, only to discover that he had been a Nazi. Born in Berlin in 1942, she and her mother fled the city in 1945, but Api, the one surviving male member of her family, stayed behind to work as a doctor in a city 90% destroyed.
Gabrielle retraces Api’s steps in the Berlin of the 21st century, torn between her love for the man who gave her the happiest years of her childhood and trying to come to terms with his Nazi membership, German guilt, and political responsibility.
Api's Berlin Diaries: My Quest to Understand My Grandfather's Nazi Past
"This is not a book I will forget any time soon."
Story Circle Book Reviews
Moving and provocative, Api's Berlin Diaries offers a personal perspective on the fall of Berlin 1945 and the far-reaching aftershocks of the Third Reich.
After her mother's death, Robinson was thrilled to find her beloved grandfather's war diaries-only to discover that he had been a Nazi.
The award-winning memoir shows Api, a doctor in Berlin, desperately trying to help the wounded in cellars without water or light. He himself was reduced to anxiety and despair, the daily diary his main refuge. As Robinson retraces Api's…
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