The Axiomatic-Deductive Ideal in Early Modern Thinking: A Cognitive History of Human Rationality

Abstract

Euclid completed his geometrical proofs in the Elements (c. 300BC) with the abbreviation “QED” (quod erat demonstrandum). This phrase was not only used in mathematics in the early modern period, but also in deductive proofs in physics, astronomy, philosophy, and ethics. This chapter is a cognitive- historical interpretation of the axiomatic-deductive ideal that consists of some widely accepted propositions or statements about geometry and mathematics: i) Mathematics is objective, its truths are universal, absolutely certain; it is abstract and independent of the human body, and transcends the reality of human beings; ii) Mathematics’ efficiency as a scientific tool leads to the assumption that it exists in the physical structure of the universe. To learn mathematics was simultaneously to learn the language of nature. It is the scientific ideal; and iii) Mathematics characterizes logic and rational thinking. This axiomatic-deductive ideal could, however, be studied from a cognitive perspective stating that mathematics is created, obviously, by humans, and can be seen as a product of the bio-cultural coevolution of human cognitive abilities. These views of thinking and human rationality expressed, in particular, during the scientific revolution are some of the most beautiful dreams of humanity; the dreams of objectivity, certainty, a transcendental reality independent of human beings, an ideal, reliable method of thinking. The conclusion is that these “inventions” of the human mind, that thinking could be objective, certain, universal, transcendent, abstract, and should form the ideal for science and human thinking, are an important and crucial stepping stone in the history of human rationality which occurred during the scientific revolution from the sixteenth to the eighteenth century. In this period of transformation of human rationality, new tools for thinking were invented that have since had a significant impact on human life and cognition. (Less)