Much of the mathematics with which Felix Klein and Sophus Lie are now associated (Klein’s Erlangen Program and Lie’s theory of transformation groups) is rooted in ideas they developed in their early work: the consideration of geometric objects or properties preserved by systems of transformations. As early as 1870, Lie studied particular examples of what he later called contact transformations, which preserve tangency and which came to play a crucial role in his systematic study of transformation groups and differential equations. (...) This note examines Klein’s efforts in the 1870s to interpret contact transformations in terms of connexes and traces that interpretation (which included a false assumption) over the decades that follow. The analysis passes from Klein’s letters to Lie through Lindemann’s edition of Clebsch’s lectures on geometry in 1876, Lie’s criticism of it in his treatise on transformation groups in 1893, and the careful development of that interpretation by Dohmen, a student of Engel, in his 1905 dissertation. The now-obscure notion of connexes and its relation to Lie’s line elements and surface elements are discussed here in some detail. (shrink)

L. D. Harris; IV*—Solving the “Naval Battle”, Proceedings of the Aristotelian Society, Volume 78, Issue 1, 1 June 1978, Pages 45–62, https://doi.org/10.1093/ari.

Provability logics with many modal operators for progressions of theories obtained by iterating their consistency statements are introduced. The corresponding arithmetical completeness theorem is proved.

The liar paradox can be shown semantically defective if we distinguish the /sentence/ ''snow is white' is true' from the /string/ that constitutes it. This paper develops the String-to-Sentence Theory of Truth---for short, String Theory---according to which, while the /string/ contains the string 'true', the /sentence/ is merely 'snow is white', which contains no such occurrence: more generally, a string like 'S is true' constitutes, relative to an assessor, the sentence which, to the assessor, means the same as S. So (...) suppose we attempt to define a singular term 'L' referring to the sentence 'not: L is true'. Relative to an assessor, 'L' refers to the sentence negating the assessor's sentence meaning the same as the referent of 'L'. So the referent of 'L' means the same as its negation. But no sentence means the same as its negation, so 'L' does not refer. The act of naming with which the liar paradox commences is semantically defective; so there can be no liar paradox. (shrink)

A general equation is derived describing the concentration of all possible complexes of a central molecule with a set of ligands bound to the central molecule. This deduction allows the reaction rate constants for the binding of a given molecule to the central molecule to depend on the species of molecules already bound and the location of the molecules already bound. The model thus allows for structural alteration of the central molecule by binding. Functions describing the concentration dependence of any (...) effect whatever depending on the distribution of complexes are deduced. Possible applications and methods of application are indicated.II est dérivé une équation qui décrit la concentration des toutes les complexes d'une molécule centrale avec une collection de ligands connectés à la molécule centrale. La déduction montre comme les coefficients de la rapidité de reaction entre la molécule centrale et les molécules de la collection dépend de l'espèce et de la location des molécules déjà connectées.La modèle dérivée de l'équation montre les changements structurelles par la connection. Les functions qui décrivent la dependence de chaque effet à la concentration, sont dérivés. Ces effects dépendent aussi à la distribution des complexes. Des applications et des méthodes sont indiqués.Eine allgemeine Vergleichung hat man bekommen welche die Konzentration aller möglichen Komplexen eines zentralen Moleküles mit einer Reihe von „Ligands” verbunden mit dem zentralen Molekül beschreibt. Diese Folgerung gestattet dass die Reaktiongeschwindigkeit Konstanten für die Bindung eines gegebenen Moleküles mit dem zentralen Molekül abhÄngt von den Arten schon gegebenen Molekülen und die Ortsbestimmung der schon gebundenen Molekülen. Das Model zeigt die strukturelle Änderungen des zentralen Moleküls durch Bindung. Funktionen, die die Konzentrationband jedes Effektes beschreiben dass abhÄngt von der Distribution der Komplexen, sind abgeleitet.Mögliche Anwendungen und Methoden von Anwendung sind dargelegt. (shrink)

Scepticism about ought simpliciter is the view that there is no such thing as what one ought simpliciter to do. Instead, practical deliberation is governed by a plurality of normative standpoints, each authoritative from their own perspective but none authoritative simpliciter. This paper aims to resist such scepticism. After setting out the challenge in general terms, I argue that scepticism can be resisted by rejecting a key assumption in the sceptic’s argument. This is the assumption that standpoint-relative ought judgments bring (...) with them a commitment to act in accordance with those judgments. Instead, I propose an alternative account of our normative concepts according to which only ought simpliciter judgments commit one to acting in accordance with those judgments. In addition to answering the sceptical challenge, the proposal offers an independently motivated account of what makes a concept normatively authoritative. (shrink)

Carole Pearce's argument against African philosophy is founded on a set of factual flaws and the fallacious assumption that African philosophy is equivalent to ethnophilosophy, which she defines as a form of intellectual apartheid founded on irrational belief systems. I argue that African philosophy is in no way qualitatively different from, say, French or Chinese philosophy, and that ethnophilosophy is merely one aspect of it But ethnophilosophy could play the important role of critically evaluating African ethnic belief systems and the (...) interpretation of such by Western anthropologists. In general, African philoso phy is nothing more than the philosophical research (of whatever order) carried out by African philosophers in the context of African research paradigms. (shrink)

This work examines the claim to scienific status made by supporters and practitioners of neoclassical economics. The approach taken is that of the history and philosophy of science. Analysis points to the conclusion that theories of economic choice are necessarily normative, essentially because of the nature of human behavior.

Clocks and Rulers Play Tricks. 6. Work Changes Mass. 7. Summing Up. Index. Unabridged republication of the edition published by Basic Books, Inc.

Neoclassical economic theory in its pretensions to scientific status is founded on one of the variants of a now discredited positivism. Neoclassical economic theory claims that there are two distinct areas of economic research: positive economics and normative economics. The former is assumed to deal with the cognitive as scientific content of economics while the later focuses on welfare or equity issues. I argue that the reliance of the whole theoretical structure of economics on the normative postulate of rationality renders (...) neoclassical economics a normative discipline. I also argue that neoclassical economics should thus be viewed as an instance of applied ethics rather than as applied mathematics, say. Finally, it is suggested that neoclassical economics, if its scientific pretensions are to be taken seriously, should be absorbed theoretically into general anthropology. (shrink)