Mathematics and the Real World Read Online Free PDF

babies a few months old, at an age when it is reasonable to assume they could not have absorbed these characteristics from the environment. This too is not surprising. The preference for a just distribution of resources is an attribute that helps a society survive the evolutionary struggle, and it is reasonable, therefore, that it is inherent at the genetic level.
    4. MATHEMATICS THAT YIELDS AN EVOLUTIONARY ADVANTAGE
    Mathematics has many aspects. The previous section showed that the ability to perform arithmetic calculations is the result of evolution. In this section we will indicate other branches of mathematical operations that, it may reasonably be assumed, provided an advantage in the evolutionary struggle. We will present evidence that those parts of mathematics werealso incorporated in the genetic heritage. We may refer to this aspect of mathematics as natural mathematics . In the next section we will describe mathematical operations that are not natural, as they did not afford any evolutionary advantage in the hundreds of thousands of years during which the human genome was formed.
    It is reasonable to assume that the ability to recognize geometrical elements gave an evolutionary advantage. As sources of food and water have typical geometric shapes, being able to recognize those shapes correctly constituted an advantage in the competition for sources of sustenance. But is there any evidence that, as a result of evolution, the recognition of geometric shapes is carried by the genes? We will soon turn our attention to such evidence but will first introduce what is known as the golden cut, or the golden-ratio rectangle.

    The golden-ratio rectangle is one in which the ratio between the longer side and the shorter side is such that if a square with sides the length of the shorter side of the rectangle is removed, the sides of the rectangle that remains will have the same ratio as the original one. We note, although this is not relevant to our tale, that it is not difficult to calculate the numerical value of the golden ratio (and skipping the calculation below will not impair the reader's understanding of what follows).
    Denote the length of the rectangle by a , and the width by b . The relation required between the ratios is expressed by . If we denote by x the desired ratio , the unknown x satisfies the quadratic equation x 2 – x = 1, the solution of which is (recalling secondary-school mathematics) . This is the golden ratio, approximately 1.6180 in decimal numbers.

    The golden ratio appears in many instances and processes in nature, and several of its attributes were known in ancient times. It has been identified in ancient architecture. For example, the dimensions of the Parthenon in Athens are amazingly close to those defined by the golden ratio. The ratio can also be discerned in Leonardo da Vinci's paintings; he also referred to it in his mathematical writings, although he did not state that he used it in his art.
    The discovery of the golden ratio in various and sometimes-unexpected forms in nature resulted in the ancients attributing mystical properties to it, and they even referred to it as the divine proportion . For many years, a lively debate continued and is still continuing today among historians and artists on the question of whether builders and artists in ancient times made conscious use of the golden ratio in their architecture and art, or whether its frequent appearance is due to the fact that it is aesthetically pleasing. We will not join this open debate at this point but will just note that the ratio is indeed very pleasing to the eye. This has been proved in dozens of empirical studies, including studies showing that infants react with greater pleasure and calm to golden-ratio rectangles than they do to rectangles with other proportions, including those with greatly different proportions.
    This needs to be explained. We are used to the fact that the pleasantness of a drawing or a painting or a