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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 prior 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.
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