Since the discovery of incommensurability in ancient Greece, arithmeticism and geometricism constantly switched roles. After ninetieth century arithmeticism Frege eventually returned to the view that mathematics is really entirely geometry. Yet Poincaré, Brouwer, Weyl and Bernays are mathematicians opposed to the explication of the continuum purely in terms of the discrete. At the beginning of the twenty-first century ‘continuum theorists’ in France (Longo, Thom and others) believe that the continuum precedes the discrete. In addition the last 50 years witnessed the (...) revival of infinitesimals (Laugwitz and Robinson—non-standard analysis) and—based upon category theory—the rise of smooth infinitesimal analysis and differential geometry. The spatial whole-parts relation is irreducible (Russell) and correlated with the spatial order of simultaneity. The human imaginative capacities are connected to the characterization of points and lines (Euclid) and to the views of Aristotle (the irreducibility of the continuity of a line to its points), which remained in force until the ninetieth century. Although Bolzano once more launched an attempt to arithmetize continuity, it appears as if Weierstrass, Cantor and Dedekind finally succeeded in bringing this ideal to its completion. Their views are assessed by analyzing the contradiction present in Grünbaum’s attempt to explain the continuum as an aggregate of unextended elements (degenerate intervals). Alternatively a line-stretch is characterized as a one-dimensional spatial subject, given at once in its totality (as a whole) and delimited by two points—but it is neither a breadthless length nor the (shortest) distance between two points. The overall aim of this analysis is to account for the uniqueness of discreteness and continuity by highlighting their mutual interconnections exemplified in the nature of a line as a one-dimensional spatial subject, while acknowledging that points are merely spatial objects which are always dependent upon an extended spatial subject. Instead of attempting to reduce continuity to discreteness or discreteness to continuity, a third alternative is explored: accept the irreducibility of number and space and then proceed by analyzing their unbreakable coherence. The argument may be seen as exploring some implications of the view of John Bell, namely that the “continuous is an autonomous notion, not explicable in terms of the discrete.” Bell points out that initially Brouwer, in his dissertation of 1907, “regards continuity and discreteness as complementary notions, neither of which is reducible to each other.”. (shrink)

‘Facts have no independent existence in science, or in any human endeavor; theories grant differing weights, values, and descriptions, even to the most empirical and undeniable of observations’ . All academic disciplines have access to undeniable states of affairs that require meaningful and constructive accounts of them. Oftentimes such an account reflect diverging theoretical views of reality. Wittgenstein’s view ‘that only connexions that are subject to law are thinkable’ paves the way for a discussion of the state of affairs that (...) theoretical thinking is characterised by a ‘special abstraction’, namely ‘aspect-abstraction’ resulting in ‘aspect-disciplines’. As an example of diverging ontological commitments Cantor’s famous diagonal proof of the non-denumerability of the real numbers is briefly discussed. Another example is given by means of a brief analysis of the wave-particle duality, focused on distinguishing different modes of explanation, designated as representations. Occam’s razor provides a further example of a state of affairs shared by different philosophical orientations, albeit that some ascribe a purely logical nature to it while someone else may discern in it an analogical link between the logical-analytical aspect and the economic aspects of ontic reality. The problem of persistence and change and the state of affairs that change can only be detected on the basis of constancy permeated the history of philosophy and the various academic disciplines, including the discipline of paleontology. The dominant pattern of the paleontological record, namely stasis, poses empirical and theoretical limits to the definition of evolution as ‘continuous change’. Apart from showing that special science always operate on the basis of philosophical assumptions, the underlying aim is to advance a non-reductionist approach to undeniable states of affairs. (shrink)

This paper deals with the following questions: What features of modern natural science are responsible for the fact that, of all forms of science, this form is technologically exploitable? The three notions: concept of nature, epistemic ideal, and experiment, suggest the most important components of my answer. I will argue, first, that only the peculiar interplay of the modern concept of nature with an epistemic ideal attuned to it can cast experiment in the specific, highly central role it plays in (...) the pursuit of knowledge about nature. It will then become clear that the form of science in which experiment plays such a role will, necessarily, prove technologically exploitable. (shrink)