Where Mathematics Comes From: How the Embodied Mind Brings Mathematics into Being

Rating:
Summary: Inspiring and Flawed
Review: Where Mathematics Comes From is an inspiring book on one hand and flawed on the other. The book brings the cognitive theory of mathematics to the masses on one hand, and on the other risks boring the reader to death. The text of the book features sections too dry and boring for the average person to handle; scientists and the more advanced science reader will find the more scientific sections too limited to use as the basis for their own research. To give the book credit though, the each of each section provides a number of references.

Where mathematics comes from seeks to explain how mathematics can be explained as an extended metaphor of the world. It seeks to explain mathematical abstractions as metaphors for the events that we witness in our ordinary lives. The development of these metaphors historically is also chronicled.

A large part of the book is devoted to explaining the consequences of the authors' "Basic Metaphor of Infinity". Here the book touches on many topics that will most likely have not been seen before by readers of popular science books such as Robinson's non-standard analysis and the formation of the irrationals from Dedekind Cuts. For this I give the book an extra star.

However, as pointed out in previous reviews, the book suffers from a few noticible mathematics errors that readers who closely follow the details of the text will quickly find. In addition, many parts of the book contain lists connecting various metaphors to develop their theory at a more scientific level. Unfortunately, these metaphorical lists are presented in a manner that is likely to bore the average reader. Once again, the book most likely does not contain information that is not known to people who have already been performing research in the field.

I give this book four stars because of its try-the-middle approach that does not succeed in forming a book suitable for the casual or the serious scientific readers. However, most readers should find the ideas in this book sufficiently enthralling for them to read to the end.

Rating:
Summary: Inspiring and Flawed
Review: Where Mathematics Comes From is an inspiring book on one hand and flawed on the other. The book brings the cognitive theory of mathematics to the masses on one hand, and on the other risks boring the reader to death. The text of the book features sections too dry and boring for the average person to handle; scientists and the more advanced science reader will find the more scientific sections too limited to use as the basis for their own research. To give the book credit though, the each of each section provides a number of references.

Where mathematics comes from seeks to explain how mathematics can be explained as an extended metaphor of the world. It seeks to explain mathematical abstractions as metaphors for the events that we witness in our ordinary lives. The development of these metaphors historically is also chronicled.

A large part of the book is devoted to explaining the consequences of the authors' "Basic Metaphor of Infinity". Here the book touches on many topics that will most likely have not been seen before by readers of popular science books such as Robinson's non-standard analysis and the formation of the irrationals from Dedekind Cuts. For this I give the book an extra star.

However, as pointed out in previous reviews, the book suffers from a few noticible mathematics errors that readers who closely follow the details of the text will quickly find. In addition, many parts of the book contain lists connecting various metaphors to develop their theory at a more scientific level. Unfortunately, these metaphorical lists are presented in a manner that is likely to bore the average reader. Once again, the book most likely does not contain information that is not known to people who have already been performing research in the field.

I give this book four stars because of its try-the-middle approach that does not succeed in forming a book suitable for the casual or the serious scientific readers. However, most readers should find the ideas in this book sufficiently enthralling for them to read to the end.

Rating:
Summary: Systems of math and how they arise from the brain
Review: Where Mathematics Comes From represents a unique collaboration between linguist Lakoff and psychologist Nunez as they study ideas and how mathematics has evolved. Links between consciousness, math, metaphor and daily life provide new and important details on the systems of math and how they arise from the brain. An excellent, detailed survey.

Rating:
Summary: Mathematics as a product of the human mind
Review: While I agree with the previous reviewer that the authors may at times suggest a little too much credit for a work that does have its predecessors, I still consider this a great book.

By attacking the transcendental nature of mathematics, and elaborating the grounding of mathematical thought in the metaphorical mapping of the mind, many important implications arise ranging from the meaning of mathematics, the way mathematics is practiced and proofs are formulated, to the way mathematics should be taught. The authors formulate their intention to link the fields of mathematical thought and cognitive sciences to generate the field of mathematical idea analysis. They stress the point that their work should be considered as an initial step and in no way as the final word. In the analysis of the thought process a number important aspects of mathematical thought get visited. Having recently read Aczel's book about Cantor and Infinity- I now feel I over-rated it at 2 stars- Lakoff and Nunez give a treatment of the concept of infinity based on the basic metaphor of infinity (BMI) that simply ridicules Aczel's. Masterful.

Is this book perfect? It's excellent, but could (and will) be improved.

Little attention is paid to the idea of linearization that is such a central concept in much of mathematics. In attempt to save the best for last, the authors conclude with a detailed analysis of the ideas behind Euler's famous formula: e^ip = -1. They claim that such a treatment would be very helpful to develop a better understanding of the formula, than a more standard approach. It may be that my former Dutch high school education, blessed with a great math teacher, deviates from the current US standard. Yet, I must say that the analysis of Lakoff and Nunez is simply not as clear and thorough as the one I received in my teens. Not only did my high school analysis include all the metaphors but a much clearer link between the e^ip and the sin(t) + i sin(t) functions based on the Taylor expansions. It is especially in this last section that the authors undermine their cause, by making statements that an expression e^p would be devoid of implicit meaning.

While I agree with the author's central dogma of mathematics as one of the human mind's most beautiful and enduring products they sometimes take their argument just a little too far. By a careful analysis and conceptualization of simple ideas mathematics has generated formalized concepts that allowed extrapolation into conclusions that initially appeared non- or even counter-intuitive. I think, that this process has been so crucial in establishing the magic or romance of mathematics.

No matter what the authors may say, wherever in the Universe any group of beings draw the line connecting the series of points that share the same distance, r, to this center, the resulting circle will always have a 2pr circumference. They may conceptualize it completely differently, but will come to the same conclusion.