Extension is probably the most general natural property. Is it a fundamental property? Leibniz claimed the answer was no, and that the structureless intuition of extension concealed more fundamental properties and relations. This paper follows Leibniz's program through Herbart and Riemann to Grassmann and uses Grassmann's algebra of points to build up levels of extensions algebraically. Finally, the connection between extension and measurement is considered.

This paper deals with an attempt of the mathematician Riemann to develop an outstandingly broad view of the philosophy of nature encompassing basic phenomena of both the material and the mental world. Riemann's draft is traced in its main aspects, and is accompanied by a comparison with certain chapters in the philosophical writings of Herbart that were particularly relevant to Riemann's conception of mathematics and science on the whole. This applies, in particluar, to the epistemological background and to Herbart's theory (...) of the Self. (shrink)

Neutral monism is a position in metaphysics defended by Mach, James, and Russell in the early twentieth century. It holds that minds and physical objects are essentially two different orderings of the same underlying neutral elements of nature. This paper sets out some of the central concepts, theses and the historical background of ideas that inform this doctrine of elements. The discussion begins with the classic neutral monism of Mach, James, and Russell in the first part of the paper, then (...) considers recent neo-Russellian versions in the second half. The chances for a revival of neutral monism are probably slight; its key ideas and starting points lie far from those in contemporary philosophy of mind. A better route might be through the philosophy of science and a deeper understanding of causation. (shrink)

Both biosemiotics and evolutionary epistemology are concerned with how knowledge evolves. (Applied) Evolutionary Epistemology thereby focuses on identifying the units, levels, and mechanisms or processes that underlie the evolutionary development of knowing and knowledge, while biosemiotics places emphasis on the study of how signs underlie the development of meaning. We compare the two schools of thought and analyze how in delineating their research program, biosemiotics runs into several problems that are overcome by evolutionary epistemologists. For one, by emphasizing signs, biosemiotics (...) needs to delineate a semiotic threshold, which is a problem not encountered by evolutionary epistemologists. Instead, the latter recognizes that all organisms are knowers that evolve knowledge, which they recognize to extend toward phenomena produced by organisms such as behavior, cognition, language, culture, science, and technology. Secondly, biosemiotics attempts at continuing adaptationist notions on how organisms relate to their environment, while especially Applied Evolutionary Epistemology comes to redefine the nature of the organism–environment relationship in such a way that it recognizes the spatiotemporal boundedness of existence, which in turn makes adaptationist accounts obsolete. (shrink)

Albert Lewis's article analysing the influence of Friedrich Schleiermacher on Hermann Grassmann, stimulated many different studies on the founder of n-dimensional outer algebra.Following a brief outline of the various, sometimes diverging, analyses of Grassmann's creative thinking, new research is presented which confirms Lewis's original contribution and widens it considerably. It will be shown that:i. Grassmann, although a self-taught mathematician, was at the centre of a hitherto understated intellectual trend, which was defining for Germany. Initiated by Pestalozzi's concept of elementary mathematical (...) education and culminating in the modern mathematics of the late 19th Century, it was reflected in the contributions of Grassmann, Riemann, Jacobi and Eisenstein.ii. Hermann Grassmann, his father Justus, and his brother Robert were all demonstrably influenced by Schleiermacher's dialectic; however the two brothers responded to it in very different ways.iii. Whilst the more philosophical parts of Hermann's 1844 Extension Theory are characterised by the influence of Schleiermacher and also by the mathematical knowledge of his father, the entire development of this work is the unfolding of a single idea based on the father's interpretation of combinatorial multiplication as a ‘chemical conjunction‘, which was developed largely dialectically by Hermann. (shrink)

According to Kant’s philosophy of geometry, Euclidean geometry is synthetic _a priori_. The advent of non-Euclidean geometries proved this position at least problematic, if not obsolete. However, based on Nash’s embedding theorems we show that a weaker notion of _preeminence_ supports the view that Euclidean geometry, even though not strictly _a priori_, enjoys a more fundamental status than non-Euclidean geometries.

In this essay, I term the dominant educational paradigm of our time as scientific education and subject it to historical analysis in order to bring its tacit racial, colonial and Eurocentric biases into view. I subsume this cluster of problems under the general heading of “the problem of scientific education”, a problem simultaneously submerged deeply in the invisible background of current education and across its foreground inasmuch as it conditions daily educational practices beyond educators’ awareness. The delicate question to be (...) answered is: enclosed as we are within a scientific framing of our educational system, how can we find an alternative way of looking at this educational system that will help us resolve the problem of colonial, racial and cultural subordinations inherent in its scientific framing? (shrink)