Books

GEOMETRY LOGIC CLASSICAL(dominant 2,000 years) REVOLUTIONARY(challenging the classical) CLASSICAL(dominant 2,000 years) REVOLUTIONARY(challenging the classical) SYSTEM Euclidean Non-Euclidian Aristotolean Non-Aristotolean ELEMENTS lines and points classes and individuals CRUCIAL ELEMENTS parallels opposites GENERAL DESCRIPTION OF CRUCIAL ELEMENTS Eeuclidean parallels are special kinds of lines involving at least two lines and an intervening boundary-space Aristotolian opposites are special… read more »

So far, we have only-considered cases where the modulus is finite. In these systems, infinity appears as implied (therefore potential only) by the indefinite repetition of the finite modular system. That is it is implied by the self-representation or self-reproduction of the system. The finite – modular system itself consists of the finite set of… read more »

On considering the examples just given, we are forcibly reminded of a phenomenon which occurs, just occasionally, in thousands of homes. It occurs whenever we see on our television screens a picture of the commentator sitting beside a television set which shows a picture of the commentator sitting beside a television set which shows a… read more »

Soon after the discovery of the calculus by Newton and Leibniz, problems of consistency in mathematics arose which centred around the concept of “infinity,” i.e. “infinitesimals” or “infinitely small quantities.” The inconsistencies, together with ensuing disputes among mathematicians and philosophers, were not allayed until the middle of the last century. Whether or not Weierstrass, Dedekind… read more »

Aristotle’s Principle, as we have just seen, implies the existence of some contradictions and denies the existence of others. This would account for the otherwise inexplicable fact that it is sometimes called “the Principle of Contradiction” and sometimes “the Principle of Non-contradiction,” titles which are in flat contradiction of one another! Strictly, of course, since… read more »

The limits one may set to the term “contradiction” are to some extent arbitrary, since the word is normally used in different senses. I shall use the term in the widest sense compatible with my immediate object. This follows excellent precedent. For example, Aristotle’s Principle is generally applied to “contraries” (such as “black” and “white”)… read more »