Formal systems are standardly envisaged in terms of a grammar specifying well-formed formulae together with a set of axioms and rules. Derivations are ordered lists of formulae each of which is either an axiom or is generated from earlier items on the list by means of the rules of the system; the theorems of a formal system are simply those formulae for which there are derivations. Here we outline a set of alternative and explicitly visual ways of envisaging and analyzing (...) at least simple formal systems using fractal patterns of infinite depth. Progressively deeper dimensions of such a fractal can be used to map increasingly complex wffs or increasingly complex 'value spaces', with tautologies, contradictions, and various forms of contingency coded in terms of color. This and related approaches, it turns out, offer not only visually immediate and geometrically intriguing representations of formal systems as a whole but also promising formal links (1) between standard systems and classical patterns in fractal geometry, (2) between quite different kinds of value spaces in classical and infinite-valued logics, and (3) between cellular automata and logic. It is hoped that pattern analysis of this kind may open possibilities for a geometrical approach to further questions within logic and metalogic. (shrink)

It is not certain when or by whom S0009838800011642_inline1 and S0009838800011642_inline2 were first technically distinguished as genus and species. The distinction does not appear in Plato's extant writings, whereas Aristotle seems to take it for granted in the Topics, which is usually regarded as among his earliest treatises. In his dialogues Plato seems able to use S0009838800011642_inline3 interchangeably to denote any group or division in a diairesis, including the group that is to be divided.

It is not certain when or by whomandwere first technically distinguished asgenusandspecies. The distinction does not appear in Plato's extant writings, whereas Aristotle seems to take it for granted in theTopics, which is usually regarded as among his earliest treatises. In his dialogues Plato seems able to useinterchangeably to denote any group or division in a diairesis, including the group that is to be divided.

I. Logica. Philosophia naturalis.--II. Metaphysica. Ethica.

The origins of European thought about the body, the mind, the soul,the world, time, and fate: new interpretations of Greek, Roman and kindred evidence, also of some basic Jewish and Christian beliefs.

Hermann Weyls "Philosophie der Mathematik und Naturwissenschaft" erschien erstmals 1928 als Beitrag zu dem von A. Bäumler und M. Schröter herausgegebenen "Handbuch der Philosophie". Die amerikanische Ausgabe, auf der die deutsche Übersetzung von Gottlob Kirschmer beruht, erschien 1949 bei Princeton University Press. Das nunmehr bereits in der 8. Auflage vorliegende Werk ist längst auch in Deutschland zum Standardwerk geworden.

There is no general agreement among scholars that Aristotle had a unified concept of phantasia. That is evident from the most cursory glance through the literature. Freudenthal speaks of the contradictions into which Aristotle seems to fall in his remarks about phantasia, and explains the contradictions as due to the border position which phantasia occupies between Wahrnehmung and thinking. Ross, in Aristotle, p. 143, talks of passages on phantasia in De Anima 3. 3 which constitute ‘a reversal of his doctrine (...) of sensation’ and perhaps do not ‘represent his deliberate view’. This is a serious state of affairs, since De Anima 3. 3 is Aristotle’s main discussion of phantasia. Of passages on phantasia, appearances and images in De Anima 3. 3, Hamlyn says: ‘There is clearly little consistency here’. Even Schofield, who is more optimistic about saving the unity of Aristotle’s concept than the last two scholars, grants that ‘some of the inconsistencies of Aristotle’s account seem more than merely apparent’.1 He thinks of Aristotle’s phantasia as a ‘loose-knit, family concept’. My purpose here is to suggest that Aristotle is more consistent in his use of phantasia than his critics will allow him to be. The translation of the term as imagination frequently adds unnecessarily to the confusion, so I shall avoid it and use transliteration instead. (shrink)

In Infinity and the Mind, Rudy Rucker leads an excursion to that stretch of the universe he calls the "Mindscape," where he explores infinity in all its forms: potential and actual, mathematical and physical, theological and mundane. Here Rucker acquaints us with Gödel's rotating universe, in which it is theoretically possible to travel into the past, and explains an interpretation of quantum mechanics in which billions of parallel worlds are produced every microsecond. It is in the realm of infinity, he (...) maintains, that mathematics, science, and logic merge with the fantastic. By closely examining the paradoxes that arise from this merging, we can learn a great deal about the human mind, its powers, and its limitations. Using cartoons, puzzles, and quotations to enliven his text, Rucker guides us through such topics as the paradoxes of set theory, the possibilities of physical infinities, and the results of Gödel's incompleteness theorems. His personal encounters with Gödel the mathematician and philosopher provide a rare glimpse at genius and reveal what very few mathematicians have dared to admit: the transcendent implications of Platonic realism. (shrink)

In Contradiction advocates and defends the view that there are true contradictions, a view that flies in the face of orthodoxy in Western philosophy since Aristotle. The book has been at the center of the controversies surrounding dialetheism ever since its first publication in 1987. This second edition of the book substantially expands upon the original in various ways, and also contains the author’s reflections on developments over the last two decades. Further aspects of dialetheism are discussed in the companion (...) volume, Doubt Truth to be a Liar, also published by Oxford University Press in 2006. (shrink)

As Aristotle himself says, A.Po. 2.13 is an attempt to provide some rules to hunt out the items predicated in what something is, namely to discover definitions. Since most of this chapter is devoted to the discussion of some rules of division , it may be inferred that somehow division plays a central role in the discovery of definitions. However, in the following pages I shall not discuss what this role is. Nor shall I discuss what place division has in (...) the wider discussion of definition and explanation as it emerges from A.Po. 2. 1 shall rather focus on the argument that Aristotle reports and discusses in A.Po. 2.13.97a6–22, and which our extant sources ascribe to Speusippus. As will become clear later on, this argument undermines the possibility of giving any definition, and Aristotle deals with it here because he can block it by exploiting some properties of the method of division. (shrink)

As Aristotle himself says, A.Po. 2.13 is an attempt to provide some rules to hunt out the items predicated in what something is, namely to discover definitions. Since most of this chapter is devoted to the discussion of some rules of division, it may be inferred that somehow division plays a central role in the discovery of definitions. However, in the following pages I shall not discuss what this role is. Nor shall I discuss what place division has in the (...) wider discussion of definition and explanation as it emerges from A.Po. 2. 1 shall rather focus on the argument that Aristotle reports and discusses in A.Po. 2.13.97a6–22, and which our extant sources ascribe to Speusippus. As will become clear later on, this argument undermines the possibility of giving any definition, and Aristotle deals with it here because he can block it by exploiting some properties of the method of division. (shrink)

Considered one of the most important works of one of France's foremost philosophers, and long-awaited in English, "The Logic Of Sense" is an essay in literary and psychoanalytic theory, and philosophy, and helps to illuminate such works as "Anti-Oedipus".

Cet ouvrage examine tous les textes du philosophe en rapport avec la dialectique, qui ne peut se réduire chez Platon à un simple art de converser ni à une procédure logique. Elle est la forme du savoir, dont la puissance est de se métamorphoser en fonction du problème posé.

A new edition of a classic work compares Plato's dialogues to Aristotle's depiction of them. Reprint.

In this paper the claim that Zeno's paradoxes have been solved is contested. Although "no one has ever touched Zeno without refuting him" (Whitehead), it will be our aim to show that, whatever it was that was refuted, it was certainly not Zeno. The paper is organised in two parts. In the first part we will demonstrate that upon direct analysis of the Greek sources, an underlying structure common to both the Paradoxes of Plurality and the Paradoxes of Motion can (...) be exposed. This structure bears on a correct - Zenonian - interpretation of the concept of “division through and through”. The key feature, generally overlooked but essential to a correct understanding of all his arguments, is that they do not presuppose time. Division takes place simultaneously. This holds true for both PP and PM. In the second part a mathematical representation will be set up that catches this common structure, hence the essence of all Zeno's arguments, however without refuting them. Its central tenet is an aequivalence proof for Zeno's procedure and Cantor's Continuum Hypothesis. Some number theoretic and geometric implications will be shortly discussed. Furthermore, it will be shown how the “Received View” on the motion-arguments can easely be derived by the introduction of time as a (non-Zenonian) premiss, thus causing their collapse into arguments which can be approached and refuted by Aristotle's limit-like concept of the “potentially infinite”, which remained — though in different disguises - at the core of the refutational strategies that have been in use up to the present. Finally, an interesting link to Newtonian mechanics via Cremona geometry can be established. (shrink)