The Neurological Foundation of Geometry

Geometry is based on vision. Hence, the visual information processing of the nervous system regulates the structure of geometry. In this paper, we shall construct geometry following the process of visual information processing in the nervous system. Firstly, photons are captured by photoreceptor cells in the retina. At this stage, the retinal bitmap image is constructed using photo- receptor cells as pixels. The retinal bitmap is the foundation of quantitative properties of images throughout the visual processing. Secondly, the edge of the object is extracted in the primary visual cortex. When a three-dimensional object is projected in two dimensions, the edge of the object is ideally a line without width. While Euclid defined the line as the length without width. Surprisingly, a type of cells in the primary visual cortex react the Euclidean line. At this stage, Euclidean geometry without curves is constructed. Thirdly, curves are recognized in the visual area V4. At this stage, Euclidean geometry with curves is constructed. The next problem is the compatibility of these stages. The problem of the compatibility between the first and second stages is that there are irrational lengths in Euclidean geometry. Ancient Greeks used the double contradiction to solve the compatibility problem. An irra- tional number is defined as a number that divides rational numbers into larger and smaller rational numbers. The double contradiction is a method of defining non-rational numbers using rational numbers. Also, double contra- diction is used to solve the compatibility problem between the second and third stages. Even though the length of the curve is not defined in Elements, the length of a curve can be defined by the length of straight lines. Similarly, properties of curves are defined by straight lines. In differentiation, the slope of the curve is defined by the slope of the line. In integration, the area under the curve is defined by the total area of thin rectangles. Finally, as a logical basis for calculus, the double contradiction should be rethought.