This book presents a detailed analysis of three ancient models of spatial magnitude, time, and local motion. The Aristotelian model is presented as an application of the ancient, geometrically orthodox conception of extension to the physical world. The other two models, which represent departures from mathematical orthodoxy, are a "quantum" model of spatial magnitude, and a Stoic model, according to which limit entities such as points, edges, and surfaces do not exist in (physical) reality. The book is unique in its (...) discussion of these ancient models within the context of later philosophical, scientific, and mathematical developments. (shrink)

It is not very surprising that it was no less true in antiquity than it is today that adult human beings are held to be responsible for most of their actions. Indeed, virtually all cultures in all historical periods seem to have had some conception of human agency which, in the absence of certain responsibility-defeating conditions, entails such responsibility. Few philosophers have had the temerity to maintain that this entailment is trivial because such responsibility-defeating conditions are always present. Another not (...) very surprising fact is that ancient thinkers tended to ascribe integrality to "what is" (to on). That is, they typically regarded "what is" as a cosmos or whole with distinguishable parts that fit together in some coherent or cohesive manner, rather than either as a "unity" with no parts or as a collection containing members (ta onta or "things that are") standing in no "natural" relations to one another. 1 The philoso phical problem of determinism and responsibility may, I think, best be characterized as follows: it is the problem of preserving the phenomenon of human agency (which would seem to require a certain separateness of individual human beings from the rest of the cosmos) when one sets about the philosophical or scientific task of explaining the integrality of "what is" by means of the development of a theory of causation or explanation ( concepts that came to be lumped together by the Greeks under the term "aitia"). (shrink)

This paper begins by pointing out that the Aristotelian conception of continuity (synecheia) and the contemporary topological account share the same intuitive, proto-topological basis: the conception of a ?natural whole? or unity without joints or seams. An argument of Aristotle to the effect that what is continuous cannot be constituted of ?indivisibles? (e.g., points) is examined from a topological perspective. From that perspective, the argument fails because Aristotle does not recognize a collective as well as a distributive concept of a (...) multiplicity of points. It is the former concept that allows contemporary topology to identify some point sets with spatial regions (in the proto-topological sense of this term). This identification, in turn, allows contemporary topology to do what Aristotle was unwilling to do: to conceive the property of continuity, as well as the properties of having measure greater than zero and having n- dimension, as emergent properties. Thus, a point set can be continuous (connected) although none of its subsets of sufficiently smaller cardinality can be. Finally, the paper discusses the manner in which a topological principle, viz., the principle that none of the singletons of points of a continuum can be open sets of that continuum, captures certain aspects of the Aristotelian proto-topological conception of the relation between points and continua. E.g., for both Aristotle and contemporary topology, points in a continuum exist simple as limits of the remainder of the continuum: their singletons have empty ?interiors? and, hence, they are not ?chunks? (topologically, regular closed set) of the continuum. (shrink)

This chapter contains sections titled: Aristotle on the Infinite (to apeiron): From Cosmological Principle to Mathematical Operation Aristotle on Space: Magnitude (megethos) and Place (topos) Aristotle on Time: The “Number of Motion” and “Ever‐rolling Stream” Bibliography.

This paper develops an interpretation of the fourth account of conditionals in Sextus Empiricus's Outlines of Pyrrhonism that conceptually links it with contemporary ?relevance? interpretations of entailment. It is argued that the third account of conditionals, which analyzes the truth of a conditional in terms of the joint impossibility of antecedent and denial of consequent, should not be interpreted in terms of a relative incompatibility of antecedent and denial of consequent because of Stoic acceptance of the truth of some conditionals (...) of the form p ? ?p and its converse. Rather, it is suggested, ancient attempts to avoid the so-called paradoxes of implication involve the fourth account of conditionals. I hypothesize that this account is related to Stoic attempts to define truth conditions for conditionals in terms of a theory of the concludency (validity) of arguments in opposition to the more common procedure (represented by the first three accounts of conditionals) of specifying truth conditions for conditionals ?semantically? and using those truth conditions in the development of a theory of argument validity. (shrink)

Partisan or Neutral? critically examines the Rawlsian ideal of a public, supposedly neutral, political theory meant to justify contemporary constitutional democracies. Placing this ideal-appealed to by neo-natural law theorists and advocates of "public theology" as well as by political theorists-against the background of the history of political liberalism, White shows its contradictory nature. He argues that any such legitimating theory will be 'partisan,' in the sense of appealing to convictions concerning the human good that will not be universally accepted. He (...) concludes that all politics must be imperfect-a matter of pragmatism and prudence in forming the most workable compromises possible and in acquiescing, where our principles allow us to do so, in situations that are often far from optimal. (shrink)

The Purpose in Life Test is a measure of purpose in life widely used in many cultures and countries; however, cross-cultural assessments are scarce. The present study aimed to evaluate the cross-cultural measurement invariance of the PIL in the general population of seven Latin American countries. A total of 4306 people participated, selected by non-probabilistic convenience sampling, where Uruguay has the highest mean age ; while Ecuador has the lowest mean age. Furthermore, in each country, there is a higher proportion (...) of women than men. Using Multi-Group Confirmatory Factor Analysis, the factorial structure does not show evidence of invariance among the included countries. However, based on the Multi-Group Factor Analysis Alignment, there is evidence that a three-dimensional structure of the PIL is the same in the participating countries. Results based on item response theory indicate that most PIL items can significantly differentiate responses according to the level of life purpose. In addition, people with low life purpose will tend to choose the lower response alternatives on the PIL; while people with higher life purpose will choose higher response alternatives. The findings indicate that the PIL has the potential to increase knowledge about how people conceive and experience their purpose in life in different countries. (shrink)

There is what might be called a ‘majority position’ in the history of Western philosophy according to which causes are sufficient for or ‘necessitate’ their effects. However, there is also a singificant ‘minority position’ according to which causes are necessary relative to their effects. The second/third century A.D. Peripatetic Alexander of Aphrodisias is an ancient representative of the minority position. He attributes his own view — with some justification, I shall suggest – to Aristotle. This paper has two, somewhat loosely (...) connected purposes. The first is to explore the origin of the conception of ‘causes’ as necessary conditions in Aristotle, particularly in On Generation and Corruption 2.11 and Posterior Analytics 2.12, and the development and use of the conception in Alexander's De fato. The second purpose of the paper is to explore and criticize a sophisticated contemporary version of the conception of causes as ‘necessary in the circumstances,’ that of J.L. Mackie. (shrink)

In early 1864, disappointed by the response to his previous work, the young Manchester academic W. Stanley Jevons announced that he was undertaking a study of the so-called coal question: ‘A good publication on the subject would draw a good deal of attention … it is necessary for the present at any rate to write on popular subjects’. When Jevons's The Coal Question was published in April 1865, however, it received comparatively little attention and sales were slow. Jevons and his (...) publisher, Alexander Macmillan, then began sending complimentary copies to luminaries such as Sir John Herschel and Alfred Tennyson. In February 1866 the marketing campaign produced its first substantial return. Macmillan had sent CQ to William Gladstone who responded with letters to both Macmillan and Jevons, noting that the book had strengthened his ‘conviction’ on the necessity for reducing the National Debt. In April, John Stuart Mill praised CQ in the House of Commons, calling for action on the Debt and, three weeks later, as Chancellor of the Exchequer, Gladstone introduced the budget using half his speech to examine the Debt situation and referring to CQ in support for a proposed measure of Debt reduction. With the extensive publicity given to CQ following Mill's speech and the budget, Jevons had achieved his objective in writing the text which went into a second edition in 1866. On the face of it, CQ 's success was due to its effect of introducing a change in budget policy and this is the impression given by some accounts of the episode. (shrink)

DESPITE THE WELL-KNOWN historical significance of Aristotle’s doctrine of the productive or active intellect it is not unusual to find contemporary discussions treating the doctrine as an excrescence on the text of the De anima, a work, it is frequently nowadays supposed, in which an otherwise securely naturalistic epistemology and rational psychology are developed. Although the doctrine of the intellectus agens is found only in one place in Aristotle’s extant texts, the third book of the De anima, I shall nonetheless (...) maintain that an argument can be ferreted out of Aristotle’s discussion that establishes that the doctrine can be seen to follow from principles that are fundamental to Aristotle’s thought. I shall call this an “Aristotelian” argument for nous poiêtikos because of the fact that the argument obviously is not found, as I state it, on the surface of Aristotle’s text. I claim that there is a significant sense in which it is there, but that one must dig for it or, as I just put it, ferret it out. In a schematic form, the Aristotelian argument goes as follows. (shrink)

A well-known epigram by Callimachus on the philosopher Diodorus Cronus reads as follows:The question of the third line, while perhaps recondite from a contemporary perspective, was clear in antiquity. The crows are asking ‘What follows ?’, in allusion to the Hellenistic disputes concerning the truth conditions of conditional propositions , disputes in which the views of Diodorus figured prominently.I agree with Sedley that the question of the last line is ‘much more problematic’. The common interpretation has been to read the (...) αθι as a form of αθις and to interpret it temporally. The result, in Pfeiffer's estimation, is ‘quomodo posthac erimus?’.This interpretation derives from Sextus Empiricus' discussion at M. 1.309–12 of the last two lines of the epigram. After crediting the grammarian with the ability to understand the allusion in the crows' first question , he proceeds to argue that the philosopher has a better chance than the grammarian of understanding the second question. But, to quote Sedley, Sextus ‘makes a ghastly mess of it’ when he attempts his own elucidation. According to an argument of Diodorus, a living thing does not die in the time in which it lives nor in a time in which it does not live. Hence, Sextus concludes, it must be the case that it never dies and, ‘if this is the case, we are always living and, according to him, we shall come to be hereafter ’. (shrink)

This chapter contains sections titled: Introduction: Mathematics and Philosophers – Plato, in Particular Mathematics and the Training of the Soul To Pythagoreanize or Not To Pythagoreanize Pythagoreanized Meta‐Mathematics and Ancient Mathematical Practice Mathematical Ontology Conclusion.

Darwinian Evolution and Classical Liberalism brings together a collection of new essays that examine the multifaceted ferment between Darwinian biology and classical liberalism.

This is a collection of articles on two central topics in epistemology: certainty and surfaces. Of the ten articles, four discuss Avrum Stroll's Surfaces. The topics explored have been the subject of intense philosophical debate since ancient times.

After briefly considering the ancient Greek and nineteenth-century history of incommensurables (magnitudes that do not have a common aliquot part) and incomparables (magnitudes such that the larger can never be surpassed by any finite number of additions of the smaller to itself), this paper undertakes two tasks. The first task is to consider whether the numerical accommodation of incommensurables by means of the extension of the ordered field of rational numbers to the field of reals is `similar' or analogous to (...) the numerical accommodation of incomparables by means of the extension of the ordered field of reals to the field of hyperreals. The second task is to evaluate several contemporary attempts to use concepts and techniques of the nonstandard mathematics of hyperreals to address classical, Zenonian puzzles concerning continuous magnitudes. The result of both these undertakings is, in a certain sense, `deflationary'. (shrink)

SummaryThis paper develops a semantical account of sentences containing performative principal verbs in which these verbs are analyzed as indexical expressions: the proposition picked out by a sentence containing a performative verb depends on aspects of the context of use of the sentence; and these same aspects of context of use also determine the truth value of the proposition picked out. A two‐dimensional modal operator is utilized in analyzing non‐ performative sentences that contain principal verb which, in other contexts, have (...) performative uses. Thus, the analysis developed accounts for the performative — non‐performative distinction within a formal semantic‐pragmatic framework; the more common approach of relegating the distinction to the somewhat vague realm of ‘language use’ is avoided.RésuméCet article développe un traitement sémantique des phrases contenant un verbe principal performatif. Ces verbes sont analysés comme des expressions indicées: la proposition sélectionnée par une phrase contenant un verbe performatif dépend du contexte d'usage de cette phrase; et ces mêmes aspects du contexte d'usage déterminent également la valeur de vérité de la proposition sélectionnée. Un opérateur modal à deux dimensions est utilisé pour l'analyse de phrases non‐performatives contenant un verbe principal performatif dans d'autres contextes. L'analyse développée rend ainsi compte de la distinction performatif — non‐performatif dans un cadre sémantico‐pragmatique; on évite ainsi la solution habituelle consistant à reléguer cette distinction dans le domaine plutôt vague de l'usage linguistique.ZusammenfassungEs wird eine semantische Darstellung von Sätzen entwickelt, die performative Hauptverben enthalten, indem diese Verben als indexikale Ausdrücke analysiert werden: die Proposition, die durch einen Satz mit einem performativen Verb bezeichnet wird, hängt von Aspekten des Gebrauchskontextes des Satzes ab; und gerade diese Aspekte bestimmen auch den Wahrheitswert der Proposition. Ein zweidimensionaler Modaloperator wird verwendet, um nicht‐performative Sätze zu analysieren, die Hauptverben enthalten, welche — in andern Kontexten — einen performativen Gebrauch haben. Auf diese Weise gibt die entwickelte Analyse innerhalb eines formalen semantisch‐syntaktischen Rahmenwerkes Rechenschaft über die Unterscheidung performativ — nicht‐performativ; man vermeidet dadurch das übliche Vorgehen, die Unterscheidung in den etwas vagen Bereich des sog. 〈Sprachgebrauchs〉 abzuschieben. (shrink)

There is what might be called a ‘majority position’ in the history of Western philosophy according to which causes are sufficient for or ‘necessitate’ their effects. However, there is also a singificant ‘minority position’ according to which causes are necessary relative to their effects. The second/third century A.D. Peripatetic Alexander of Aphrodisias is an ancient representative of the minority position. He attributes his own view — with some justification, I shall suggest – to Aristotle. This paper has two, somewhat loosely (...) connected purposes. The first is to explore the origin of the conception of ‘causes’ as necessary conditions in Aristotle, particularly in On Generation and Corruption 2.11 and Posterior Analytics 2.12, and the development and use of the conception in Alexander's De fato. The second purpose of the paper is to explore and criticize a sophisticated contemporary version of the conception of causes as ‘necessary in the circumstances,’ that of J.L. Mackie. (shrink)

This paper discusses the 'master argument' of diodorus cronos from a semantic perspective. An argument is developed which suggests that proposition (1), 'every proposition true about the past is necessary', May have provided the principal motivation for diodorus denial of proposition (3), I.E., His equation of possibility with present-Or-Future truth. It is noted that (1) and (3) are jointly inconsistent only given the assumption of a linear ordering of time. It is further noted that diodorus' fatalism "could" be employed to (...) justify this additional assumption. However, To then use the conclusion of the 'master' to argue for fatalism would obviously be circular. I suspect, Rather, That diodorus' assumption of temporal linearity was implicit and uncritical. (shrink)