André du Toit

Abstract

In this commentary, I suggest that Sam Vice’s exploratory and communicative goals in ‘How Do I Live in This Strange Place?’ are not appropriately advanced by the form of her writing. In particular, her analysis and recommendations—which I regard as perceptive and profound—would have been better presented in a non-argumentative format, and by employing a less direct mode of communication.     

Poincaré against the logicians

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicist's conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type.The author would like to thank the Alexander von Humboldt Foundation for its generous financial support. Thanks also to Dick Foley for useful comments on an earlier draft.     

Experimenter naiveté and imaginal judgments

Finke and Kurtzman report that fields of resolution increase with increases in the diameter of both perceived and imagined circular patterns. Their study apparently was conducted by an experimenter who was aware of the hypotheses under test. In contrast to these results, we find no such increase for imagined circular patterns when the experimenter is not aware of the experimental predictions, even though our subjects received imagery training as extensive as that used by Finke and Kurtzman. Our results cannot readily be attributed to differences between the experiments, other than those associated with experimenter naiveté, for we replicated Finke and Kurtzman's observations that (a) the fields of resolution in perception increase with both the diameter and the relative brightness of the patterns and (b) the fields of resolution in imagery do not increase with the relative brightness of the patterns. Our results also challenge Finke and Kurtzman's contentions that imagery and perception are treated as functionally equivalent at area-processing levels of the visual system but not at contrast-processing levels.     

Harré on Piaget's Sociological Studies

Piaget's Sociological Studies is largely taken up with questions of sociology, epistemology, moral, political and legal theory. Roughly, these are by-passed in Harré's (2000, this journal) interpretation of Piaget's social account. Our critique is in two parts. In the first part, we explain the basis of our disagreement with Harré's interpretation of six specific issues. These are Piaget's model of social exchange, sociology, concept of egocentrism, response to Wallon, psycho-social parallelism, distinction between the concrete and the abstract. In the second part, we challenge two central theses invoked in Harré's review, namely (A) all psychological activity is a joint activity, and (B) any society is based on irreducible differences in psychological activity. We have tried to set the record straight as far as Piaget's social account is concerned. Harré's review shows a general lack of acquaintance with Piaget's account and our aim has been to compensate for this.     

Laïcité et diversité religieuse: l’approche québécoise

The Conseil des relations interculturelles (CRI) is a research and advisory council whose mission is to advise the Minister of Citizens relations and Immigration on primary matters concerning integration of immigrants and intercultural relations. In the summer of 2002, the members of the Conseil produced an advisory report on religious diversity. The purpose of this work was to propose a framework to reflect on the subject and suggest avenues which could facilitate the recognition of religious diversity in citizen public civic spaces. This document is a revision of the original abridged version, published in March 2004.     

Responsabilité Sociale, Gestion de la Diversité et Lutte Contre les Discriminations: Des Perspectives Convergentes

Signatory States of international agreements have to report regularly to international bodies on the measures and monitoring they apply to comply with and enforce their commitments on human rights and the fight against racism and discrimination. All the same, national governments do not always adequately assess the prevailing measures and practices in organisations in all sectors of society within their jurisdiction. This article discusses the accountability and social responsibility of organisations in terms of non-discrimination, which are rarely questioned despite legal obligations to comply with charters and legislation on employment equality and equity, as in Canada. The issue of the current effectiveness of rights, which underlines the legitimacy and “ethics” of institutions, companies and governments, requires that obligation in the strict legal sense fits in with a broader and normative perspective, approach and discussion. Today, this issue is prominent at both national and international levels in the social and scientific debate on “citizen companies”, indicators of respect for human rights, social and environmental reports, “ethical” investments, diversity management, equity and access to equality, and the fight against systemic discrimination.     

Duhem and Koyré on Domingo de Soto

Galileo's view of science is indebted to the teaching of the Jesuit professors at the Collegio Romano, but Galileo's concept of mathematical physics also corresponds to that of Giovan Battista Benedetti. Lacking documentary evidence that would connect Benedetti directly with the Jesuits, or the Jesuits with Benedetti, I infer a common source: the Spanish connection, that is, Domingo de Soto. I then give indications that the fourteenth-century work at Oxford and Paris on calculationes was transmitted via Spain and Portugal to Rome and other centers where Jesuits had colleges, and figured in the rise of mathematical physics at the beginning of the seventeenth century. A result of these researches is their vindication of Duhem, as contrasted with Koyré, on the origins of modern mechanics.     

Rescuing Poincaré from Richard’s Paradox

Poincaré in a 1909 lecture in Göttingen proposed a solution to the apparent incompatibility of two results as viewed from a definitionist perspective: on the one hand, Richard’s proof that the definitions of real numbers form a countable set and, on the other, Cantor’s proof that the real numbers make up an uncountable class. Poincaré argues that, Richard’s result notwithstanding, there is no enumeration of all definable real numbers. We apply previous research by Luna and Taylor on Richard’s paradox, indefinite extensibility and unrestricted quantification to evaluate Poincaré’s proposal. We emphasize that Poincaré’s solution involves an early recourse to indefinite extensibility and argue that his proposal, if it is to completely avoid Richard’s paradox, requires rejecting absolutely unrestricted quantification: Richard’s paradox provides a context in which paradox seems inescapable if unrestricted quantification is possible. In proposing his solution to the apparent conflict between Richard’s and Cantor’s results, Poincaré employs temporal expressions whose exact meaning he does not clarify. We suggest an interpretation of these expressions in terms of order of availability and briefly discuss its explanatory power in topics like paradoxes, limitation theorems and indefinite extensibility.