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Visual Thinking in Mathematics, by Marcus Giaquinto
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Visual thinking - visual imagination or perception of diagrams and symbol arrays, and mental operations on them - is omnipresent in mathematics. Is this visual thinking merely a psychological aid, facilitating grasp of what is gathered by other means? Or does it also have epistemological functions, as a means of discovery, understanding, and even proof? By examining the many kinds of visual representation in mathematics and the diverse ways in which they are used, Marcus Giaquinto argues that visual thinking in mathematics is rarely just a superfluous aid; it usually has epistemological value, often as a means of discovery. Drawing from philosophical work on the nature of concepts and from empirical studies of visual perception, mental imagery, and numerical cognition, Giaquinto explores a major source of our grasp of mathematics, using examples from basic geometry, arithmetic, algebra, and real analysis. He shows how we can discern abstract general truths by means of specific images, how synthetic a priori knowledge is possible, and how visual means can help us grasp abstract structures.
Visual Thinking in Mathematics reopens the investigation of earlier thinkers from Plato to Kant into the nature and epistemology of an individual's basic mathematical beliefs and abilities, in the new light shed by the maturing cognitive sciences. Clear and concise throughout, it will appeal to scholars and students of philosophy, mathematics, and psychology, as well as anyone with an interest in mathematical thinking.
- Sales Rank: #1793736 in Books
- Published on: 2011-11-14
- Released on: 2011-11-14
- Original language: English
- Number of items: 1
- Dimensions: 6.20" h x .70" w x 9.10" l, .95 pounds
- Binding: Paperback
- 304 pages
Review
`Review from previous edition The author shows great ambition in tackling such an interesting topic with an interdisciplinary spirit.'
Sun-Joo Shin, Notre Dame Philosophical Reviews
About the Author
Marcus Giaquinto is a Reader in Philosophy at University College London.
Most helpful customer reviews
6 of 6 people found the following review helpful.
Extraordinary Text, Far More than Philosophy
By Let's Compare Options Preptorial
The subtitle of this fine volume mentions Epistemology, and the author is promoted as a philosopher by the publisher. This is an extreme disservice to mathematicians, educators, engineers, psychologists and indeed ANY author of works with mathematical pedagogy. Epistemology is generally the study of sources and extent of knowledge, so by description alone, one would assume this book would cover the ways we learn to visualize math, or the sources of that visualization "philosophically" -- as other have, with insider semantics beginning and ending with words that represent a significant contribution to solving insomnia. I was all ready to read for the umpteenth time how counting on our fingers is the etiology of partial differential equations in a certain holistically connected neural region.
Not at all! This book is lively, intuitive, and filled with both detailed math and excellent illustrations. Sure, the author does cover little "blurbs" of philosophy, but in an unexpected way. First, he shows with detailed formulas and illustrations how we visualize a topic in shapes, geometry, number lines, generalization of analysis, calculation, pattern manipulation, with a wonderful wrap up of algebra vs. geometry. As each subject is painstakingly argued and explored, the author then gives a related cognition and etiology based snapshot (not heavy, but extremely insightful, and devoid of philosophy buzzwords), not covering general issues of value or philosophical controversy as much as practical questions about generality, induction, instantiation and other very useful discussions for students, educators, and practicing mathematicians, physicists, engineers and interested math autodidacts.
I'm a postdoc in applied math and work animating math concepts for educators. What are the benefits and caveats of using visuals in math, not "philosophically" but realistically-- IOW, where do visuals fall short if overextended, as in the area of proofs? The author deftly uses the counterpoint between the abstractness of analysis and the visual wealth of it's child (or in some senses parent)-- calculus. He carefully parses where visuals are of great value in understanding, defining, and sometimes discovery (he's less firm on that!), but not proof, validation or even generalization and induction.
The bottom line of this book is balance. There is a LOT of good math, and over 200 of the 280 or so pages are dense formula explorations with the aforementioned etiology discussions for a page or two afterward. Ironically, the "visuals" are mostly little line charts, without the firewords of MatLab, MathCad or GNU Octave to blow us away with color 3D plates. The author's "conclusion," if there is one, (author is very balanced and a clear pro/con thinker) is to be careful with visuals, so this in some ways follows his pov. The analysis (as in logical, not Hilbert spaces) is wonderful, with many examples from research including far more neuro and psych than philosophy, although there are sprinklings of it to please the promoted audience.
One of the most extraordinary conclusions is about the lack of a good, structured visualization taxonomy in math. The value of visualization is widely accepted, yet rarely studied-- the "math thinking" literature is far more heavily dominated by questions of what can and can't be solved, with new taxonomies about disciplines springing up daily (the 2010 MSC lists 6,500 math study areas). And although the author in earlier chapters warns about limits in visuals in several areas, especially discovery, he reveals an enthusiasm about discovery, and the beauty of mathematic visualization, when he visualizes a future with such a taxonomy taken seriously.
Highly recommended, but for far more than philosophy-- any math thinker or educator must read this, and any math entusiast will thoroughly enjoy it, as well as psychologists and digital artists with a math bent like myself. The author is a good writer, and the logical flow reads a little like a detective story, so this text is NOT a good insomnia cure, despite the promotion as a philosophy text! Psychologists would likely read this and say... philosophy? Since when did they move into details of cognition? To be fair, Oxford is organized a little differently than MIT in this regard. Don't you UK readers get in a huff... yeah, I'm an American, but my British friends still think of us as one of your more successful colonies.
A final but key point: at $20 to $30 US, this is an outstanding buy. This level and quantity of unique and original content, as well as relevant and up to date material/ bib/ research citations would be over $100 from Springer!
Library Picks reviews only for the benefit of Amazon shoppers and has nothing to do with Amazon, the authors, manufacturers or publishers of the items we review. We always buy the items we review for the sake of objectivity, and although we search for gems, are not shy about trashing an item if it's a waste of time or money for Amazon shoppers. If the reviewer identifies herself, her job or her field, it is only as a point of reference to help you gauge the background and any biases.
2 of 2 people found the following review helpful.
This is a seminal work by a member of the ...
By Arthur Conroy
This is a seminal work by a member of the species who is at least two decades ahead of his peers. Combine this research and narrative with Visual Language Theory (Marriott & Meyer), Conceptual Spaces and Geometry of Meaning (Gardenfors), and top it off with From Mathematics to Generic Programming (Stepanov and Rose), and you have the basis for navigating everyday change with ease. My research at Virginia Tech is in diagramming prior knowledge and Giaquinto was the remaining puzzle piece in understanding Kant's assertion that we bring with us synthetic a priori knowledge to the lived human experience. Read it at least a dozen times and everything he has written becomes accessible.
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