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Where Mathematics Come From: How The Embodied Mind Brings Mathematics Into Being Paperback – 16 Aug 2001
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About the Author
George Lakoff is Professor of Linguistics at the University of California, Berkeley. He was a founder of the generative semantics movements in linguistics in the 1960s and of the field of cognitive linguistics in the 1970s, and one of the developers of the neural theory of language in the 1980s and'90s. He is the co-author, with Mark Johnson, of Metaphors We Live By and Philosophy in the Flesh.Rafael Nuñez is currently at the Department of Psychology of the University of Freiburg, and is a research associate of the University of California, Berkeley. He is the co-editor of Reclaiming Cognition: The Primacy of Action, Intention and Emotion.
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Some elements of the authors' approach were new to me (a science student in the 1960s), and they offer a more practical and far less abstract method of considering the higher forms of maths. If you have no mathematical background at all, then it is definitely not for you; but if you spent several years enjoying the struggle with maths at school and mostly succeeding with a brilliant teacher, but then getting lost at Uni because of deadly dull soporific lecturers, then it could well be of benefit.
It is still hard work, and I was tempted to give up several times over a couple of months. But something kept drawing me back to it, and I managed to find the time to work on it sufficiently to appreciate why I previously had problems, and why with a slightly different approach it could all have been so much easier. I'm still no expert, but some of the mysteries have been revealed. Now, if only they could apply the same lucid reasoning to Quantum Mechanics...
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I first read this book hoping it might give me some insight into how to discuss a variety of things with my brother who's study of math and physics has prevented him from imagining much value in poetry. The book marks a trail beginning with an inventory of cognitive science experiments that set the ground for basic math as cognitive metaphor, and leads to the heart warming jewel of mathematical beauty in Euler's equation. Quite the ride! When I shared it with my brother he was hooked.
Obviously, we don't all think alike, but cognitive science has begun to shed light on the basic patterns of our variety and a shared mindfulness at the heart of being human. This book is an important indication of the power of that revelation which should be shared.
In the Platonic view, mathematics is essentially completely disconnected from the physical world, but can be perceived by the mind. In the "math in the universe" view, it's "out there" outside the individual mind. The embodied mind viewpoint may be thought of as locating mathematics in the interface between individual minds and the universe around them. (Of course, there is no sharp boundary between individuals and their environments.)
The essence of this book may be summarised by a single paragraph on page 9.
"Mathematics as we know it is human mathematics, a product of the human mind. Where does mathematics come from? It comes from us! We create it, but it is not arbitrary---not a mere historically contingent social construction. What makes mathematics nonarbitrary is that it uses the basic conceptual mechanisms of the embodied human mind as it has evolved in the real world. Mathematics is a product of the neural capacities of our brains, the nature of our bodies, our evolution, our environment, and our long social and cultural history."
The rest of the book gives evidence for this claim and develops the consequences for some areas of mathematics. My favourite part of the book is the evidence for arithmetic in six-month-old human infants, using puppets, on pages 16-19. This shows clearly that some basic mathematics is not purely a matter of culture. It is innate.
Chapters 2 to 11 present a series of "metaphors" and "schemas" for various mathematical concepts --- arithmetic, algebra, logic, sets, real numbers, infinity, transfinite numbers, infinitesimals. Initially it all seems quite plausible. Integers come from adding and removing objects from collections of objects. Real numbers come from measuring sticks. But the "metaphors" get less and less credible. My marginal notes pencilled in the margins changed from very positive to slightly positive, to lukewarm, to skeptical, to strong disagreement, to utter derision. My negative marginal notes started at page 91, concerning complex numbers interpreted as rotations. At page 108, my comments are "Huh?" and "infantile" and "Why not?". On pages 109 to 111, I commented "more nonsense", "unjustified conclusion", "nonsense", "total twaddle" and "nonsense".
The metaphors are mostly like what are used to teach mathematics. In fact, a useful application of this book would be to mathematics teaching. That's what it seems like really.
I do think that this is a very good book, but the authors could have said everything much better in 150 pages instead of 493 pages. After the first couple of chapters, the rest may be considered "metaphors for teaching mathematics". The first 4 chapters are well worth reading as an antidote to the Platonic "universe of forms" philosophy of mathematics. Modern mathematics is taught as if it were "analytic" truth, i.e. as absolute, universal truth which even life-forms in other galaxies would agree with.
The fact that the basic concepts of mathematics are wired into the brain is very important. However, that is only the launch point for mathematics. The detailed content of modern mathematics cannot be mapped one-to-one with the "embodied mind".
Maybe each individual component of mathematical thinking may be identified with innate capabilities of the human mind. But all cooking uses pretty much the same set of ingredients and the same set of techniques, and yet the art of great cooks cannot be reduced to mere ingredients and techniques. Painters all use the same colours, tools, concepts of perspective and composition etc., but that does not explain all of the art in the art galleries.
The basic components of mathematical activity are indubitably in-born in the sensory/motor system of the human mind, but it is very doubtful that "cognitive science" (if it is indeed a science) can identify which mental process is being applied in each mathematical thinking-process. Probably each individual uses different innate brain functions to other individuals for the same mathematical concepts. For example, I use diagrams for almost all mathematics, whereas some mathematicians I have met say that they do all mathematics symbolically and algebraically without any visualisation.
The "embodied mind" theory also does not explain how mathematicians at some points in history totally rejected concepts which at other times were accepted as self-evident. Examples are zero, negative numbers, irrational numbers, transcendental numbers, complex numbers, infinite sets, transfinite ordinal numbers and non-Euclidean geometry. The vicious debates between intuitionists and formalists in the late 19th and early 20th centuries show that even at one point in history, there can be strong disagreement on the most fundamental ideas of what constitutes mathematics. So it seems unlikely that the "embodied mind" theory can do explain anything more than the ingredients and utensils of mathematics. I cannot explain the recipes in the cookbooks.