Where Mathematics Come From How The Embodied Mind Brings Mathematics Into Being
When you think about it, it seems obvious: The only mathematical ideas that human beings can have are ideas that the human brain allows. We know a lot about what human ideas are like from research in Cognitive Science. Most ideas are unconscious, and that is no less true of the mathematical ones. Abstract ideas, for the most part, arise via conceptual metaphor-a mechanism for projecting embodied (that is, sensory-motor) reasoning to abstract reasoning. This book argues that conceptual metaphor plays a central, defining role in mathematical ideas within the cognitive unconscious-from arithmetic and algebra to sets and logic to infinity in all of its forms: transfinite numbers, points at infinity, infinitesimals, and so on. Even the real numbers, the imaginary numbers, trigonometry, and calculus are based on metaphorical ideas coming out of the way we function in the everyday physical world.This book is about mathematical ideas, about what mathematics means-and why. The authors believe that understanding the metaphors implicit in mathematics will make mathematics make more sense. Moreover, understanding mathematical ideas and how they arise from our bodies and brains will make it clear that the brain's mathematics is mathematics, the only mathematics we know or can know.
Why Cognitive Science Matters to Mathematics
The Brains Innate Arithmetic
A Brief Introduction to the Cognitive Science of the Embodied Mind
18 other sections not shown
abstract actual infinity addition algebra Basic Metaphor bers brain branches of mathematics calculus Cartesian plane Chapter characterize closure cognitive mechanisms cognitive science Commutative law complex numbers complex plane conceptual blend conceptual metaphors Container schemas corresponding curve Dedekind defined disc discrete elements embodied Entailment entities equation essence everyday example exponential function finite geometry granular numbers grounding metaphors human hyperreals image schemas infinite decimal infinite sequence infinite set infinitesimals innate arithmetic integers inversive geometry least upper bound length limit logic mathe mathematical idea analysis mathematicians matics means Metaphor of Infinity motion multiplication natural numbers negative numbers neural number line Number-Line blend object collection one-to-one operations ordered pairs pairs of numbers physical segments point-locations properties rational numbers real numbers rotation set of ordered set theory Source Domain space-filling curves structure subitizing subtraction symbols Target Domain tion transcendent mathematics transfinite understanding unique unit circle Weierstrass zero