Brendan Larvor

The metaphor of scaffolding has become current in discussions of the cognitive help we get from artefacts, environmental affordances and each other. Consideration of mathematical tools and representations indicates that in these cases at least, scaffolding is the wrong picture, because scaffolding in good order is immobile, temporary and crude. Mathematical representations can be manipulated, are not temporary structures to aid development, and are refined. Reflection on examples from elementary algebra indicates that Menary is on the right track with his ‘enculturation’ view of mathematical cognition. Moreover, these examples allow us to elaborate his remarks on the uniqueness of mathematical representations and their role in the emergence of new thoughts.

Hyde claims that the trickster spirit is necessary for the renewal of culture, and that he lives only in the ‘complex terrain of polytheism’. Fortunately for those of us in monotheistic cultures, Weber gives reasons for thinking that polytheism is making a return, albeit in a new, disenchanted form. The plan of this paper is to elaborate some basic notions from Weber, to explore Hyde’s thesis in more detail and then to take up the question of the plurality of spirits both around and within us and whether the trickster is one of them. Weber has three roles in this argument. First, he theorises rationalisation, disenchantment and bureaucracy; second, he offers an argument that in a certain sense polytheism is returning ; and third, he presents a way to translate the mytho-poetic register in which Hyde works into terms acceptable to social science of a more materialist bent. The claim of the paper is that polytheism as a practical attitude means recognising that there are diverse and contradictory ethical orders built into the world around us and active with our psyches. Weber explains why this is especially difficult for us, and Hyde offers us the hope that we may be tricky enough to cope.

This paper assumes the success of arguments against the view that informal mathematical proofs secure rational conviction in virtue of their relations with corresponding formal derivations. This assumption entails a need for an alternative account of the logic of informal mathematical proofs. Following examination of case studies by Manders, De Toffoli and Giardino, Leitgeb, Feferman and others, this paper proposes a framework for analysing those informal proofs that appeal to the perception or modification of diagrams or to the inspection or imaginative manipulation of mental models of mathematical phenomena. Proofs relying on diagrams can be rigorous if it is easy to draw a diagram that shares or otherwise indicates the structure of the mathematical object, the information thus displayed is not metrical and it is possible to put the inferences into systematic mathematical relation with other mathematical inferential practices. Proofs that appeal to mental models can be rigorous if the mental models can be externalised as diagrammatic practice that satisfies these three conditions.

After the publication of The structure of scientific revolutions, Kuhn attempted to fend off accusations of extremism by explaining that his allegedly “relativist” theory is little more than the mundane analytical apparatus common to most historians. The appearance of radicalism is due to the novelty of applying this machinery to the history of science. This defence fails, but it provides an important clue. The claim of this paper is that Kuhn inadvertently allowed features of his procedure and experience as an historian to pass over into his general account of science. Kuhn’s familiar claims, that science is directed in part by extra-scientific influences; that the history of science is divided by revolutionary breaks into periods that cannot be easily compared; that there is no ahistorical standard of rationality by which past episodes may be judged; and that science cannot be shown to be heading towards the Truth—these now appear as methodological commitments rather than historico–philosophical theses.Author Keywords: Kuhn; Koyré; Butterfield; Historicism; Revolution.

It is difficult to imagine mathematics without its symbolic language. It is especially difficult to imagine doing mathematics without using mathematical notation. Nevertheless, that is how mathematics was done for most of human history. It was only at the end of the sixteenth century that mathematicians began to develop systems of mathematical symbols. It is startling to consider how rapidly mathematical notation evolved. Viète is usually taken to have initiated this development with his Isagoge of 1591, and a recognisably modern symbol system was available by the 1630s. In little more than a generation, mathematicians went from writing mathematics in natural language to manipulating symbols much as we do today. The manipulation of symbols became both a source of new mathematics and a mode of mathematical argument. This required profound conceptual changes in both mathematics and philosophy; antique conceptions of number, proof, language, and mathematics had to be replaced or at least suspended. Thus, the development of mathematical symbolism was sufficiently rapid and profound to justify the word ‘revolution’.Moreover, what began as a technical innovation in mathematics soon took on a wider philosophical significance. The last line of Viète's introduction reads ‘To solve every problem’. Viète presumably meant that his algebra would solve every mathematical problem. However, two generations after Descartes, the teenage Leibniz articulated the characteristically modern dream of a general algebra of thought. In this ‘universal characteristic’, conceptual errors would …

This article canvasses five senses in which one might introduce an historical element into the philosophy of mathematics: 1. The temporal dimension of logic; 2. Explanatory Appeal to Context rather than to General Principles; 3. Heraclitean Flux; 4. All history is the History of Thought; and 5. History is Non-Judgmental. It concludes by adapting Bernard Williams’ distinction between ‘history of philosophy’ and ‘history of ideas’ to argue that the philosophy of mathematics is unavoidably historical, but need not and must not merge with historiography.

It is argued in this study that (i) progress in the philosophy of mathematical practice requires a general positive account of informal proof; (ii) the best candidate is to think of informal proofs as arguments that depend on their matter as well as their logical form; (iii) articulating the dependency of informal inferences on their content requires a redefinition of logic as the general study of inferential actions; (iv) it is a decisive advantage of this conception of logic that it accommodates the many mathematical proofs that include actions on objects other than propositions; (v) this conception of logic permits the articulation of project-sized tasks for the philosophy of mathematical practice, thereby supplying a partial characterisation of normal research in the field

The Chances of Explanation: Causal Explanation in the Social, Medical, and Physical Sciences Paul Humphreys, 1989 Princeton University Press x+170 pp., £12.95 (paperback) ISBN 0 691 020286 8; £25.00 (hardback) ISBN 0 69107353 8In Search of a Better World: Lectures and Essays from Thirty Years Karl Popper London, Routledge £25.00 (hardback)Artificial Morality: Virtuous Robots for Virtual Games Peter Danielson, 1992 London, Routledge £35.00 (hardback) ISBN 0 415 034841; £10.99 (paperback) ISBN 0 415 07691 9A System of Pragmatic Idealism Nicholas Rescher, 1992 Princeton, Princeton University Press 335pp., $37.50 (hardback)

Dans cet article, j’explore dans un premier temps la conception que se fait Lautman de la dialectique en examinant ses références à Platon et Heidegger. Je compare ensuite les structures dialectiques identifiées par Lautman dans les mathématiques contemporaines avec celles qui émergent de ses sources philosophiques. Enfin, je soutiens que les structures qu’il a découvertes dans les mathématiques sont plus riches que le suggère son modèle platonicien, et que la distinction « ontologique » de Heidegger est moins utile que semblait le penser Lautman. In this paper, I first explore Lautman’s conception of dialectics by a consideration of his references to Plato and Heidegger. I then compare the dialectical structures that he found in contemporary mathematics with the model that emerges from his philosophical sources. Finally, I argue that the structures that he discovered in mathematics are richer than his Platonist model suggests, and that Heidegger’s “ontological” distinction is less useful than Lautman seemed to believe

The schism between analytic and continental philosophy resists repair because it is not confined to philosophers. It is a local manifestation of a far more profound and pervasive division. In 1959 C.P. Snow lamented the partition of intellectual life in to `two cultures': that of the scientist and that of the literary intellectual. If we follow the practice of most universities and bundle historical and literary studies together in the faculty of humanities on the one hand, and count pure mathematics among the sciences on the other, then it is fair to say that the mutual ignorance and occasional hostility between scientists and humanists decried by Snow is still with us. And it runs through the middle of philosophy. Philosophy aspires to say something about everything, so it is unsurprising that philosophers have reproduced in miniature the division between the arts and the sciences. What is worrying is that we have failed to overcome it.

Abstract: Particularism is usually understood as a position in moral philosophy. In fact, it is a view about all reasons, not only moral reasons. Here, I show that particularism is a familiar and controversial position in the philosophy of science and mathematics. I then argue for particularism with respect to scientific and mathematical reasoning. This has a bearing on moral particularism, because if particularism about moral reasons is true, then particularism must be true with respect to reasons of any sort, including mathematical and scientific reasons.