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The classification of abstract and imaginary objects is a major source of contradictions among philosophers, who mistakenly believe that these objects are non-physical. Descartes thought that there were fundamental differences between physical and mental objects, and many of his contemporary philosophers also believed in supernatural spiritual objects. In contrast, the outstanding mathematician Gottlob Frege postulated that numbers belong to an abstract realm that has only negative properties. He acerbically criticized John Stuart Mill for confusing pure arithmetical propositions with the practical use of arithmetic. However, the written and archeological evidence from early arithmetic and geometry, and the developmental studies of Piaget, contradict Frege's ideas. In addition, neuroscience has demonstrated that abstract, imaginary, and metaphysical objects are the product of abstraction, which is physically realized in the frontal lobes of the brain.

DOI: 10.1007/978-3-642-27198-4_8

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