Mathematics

The history of mathematics is centered on the struggle to come to terms with the relationship between discreteness and continuity, i.e. to understand the uniqueness and mutual coherence of number and space. Unfortunately throughout its history mathematics opted for one-sided extremes. It started out in Greek culture with the Pythagorean claim that “everything is number.” The discovery of incommensurability by Hippasus of Metapont (450 B.C.) caused a crisis because within the assumed form-giving function of number the formless (infinite) was revealed. This resulted in a switch to space – the geometrization of mathematics. Moreover, it inspired a space metaphysics lasting until Descartes and Kant. During the nineteenth century, however, Cauchy, Weierstrass, Dedekind and Cantor once again pursued the path of an arithmeticistic approach. Of particular significance in this regard is set theory as it was developed by Cantor (including his theory of transfinite arithmetic. When Russell and Zermelo independently in 1900 and 1901 discovered the fundamental inconsistency of Cantor's set theory, mathematics gave birth to three schools of thought, the logicist school (Russell, Gödel), the intuitionist school (Poincaré, Brouwer, Heyting, Weyl and Dummett), and the axiomatic formalist school (guided by the foremost mathematician of the twentieth century, David Hilbert and still largely dominating the scene of contemporary mathematics). By exploring questions, such as whether 2+2=4? and whether a line can be defined as the shortest distance between two points, systematic distinctions are introduction on the basis of a third option never explored in the history of mathematics: accept the uniqueness and irreducibility of number and space, and on that basis, analyze the mutual coherence between these two aspects of reality.

Physics

The history of the concept of matter is employed in order to show how different modes of explanation dominated the scene, starting with the Pythagorean thesis that everything is number and the subsequent switch to space as an orientation point. The long legacy of a space metaphysics lasted well into the modern era, for both Descartes and Kant still characterized material things in term of space. It was Galileo who introduced motion as a new mode of explanation in the formulation of the law of inertia. This opened the way for the world view of classical physics – the world as an inter-play of particles in motion. The law of inertia implies that motion is something given, and that therefore instead of trying to deduce or explain it, it should be accepted as a mode of explanation in its own right. Motion is original and unique and indeed embodies a distinct mode of explanation different from those used by the Pythagoreans (number) and the Eleatic school of Parmenides (space). If motion does not need a causing force, then at most it is possible to speak of a change of motion (acceleration or deceleration) – and this does need a physical force. The dominance of the mechanistic main tendency of classical physics lasted up to Heinrich Hertz (end of the 19th century). The 20th century explored the meaning of (physical) energy-operation and thus paved the way for a different understanding of matter. Problems regarding natural laws, individualization, the structure of physical entities and their interlacement in molecular structures are dealt with in terms of the systematic distinctions entailed in a non-reductionist ontology.