Brendan Larvor

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 paper discusses the connection between the actual history of mathematics and Lakatos's philosophy of mathematics, in three parts. The first points to studies by Lakatos and others which support his conception of mathematics and its history. In the second I suggest that the apparent poverty of Lakatosian examples may be due to the way in which the history of mathematics is usually written. The third part argues that Lakatos is right to hold philosophy accountable to history, even if Lakatos's own view of mathematics fails that test.

The limits of ‘criterial rationality’ (that is, rationality as rule‐following) have been extensively explored in the philosophy of science by Kuhn and others. In this paper I attempt to extend this line of enquiry into mathematics by means of a pair of case studies in early algebra. The first case is the Ars Magna (Nuremburg 1545) by Jerome Cardan (1501–1576), in which a then recently‐discovered formula for finding the roots of some cubic equations is extended to cover all cubics and proved. The second is the formulation by Albert Girard (1595–1632) of an early version of the fundamental theorem of algebra in his L'invention nouvelle en l'algèbre (Amsterdam, 1629). I conclude that in these cases at least, the questions raised in the philosophy of science debate can also be asked of the history of mathematics, and that a modest methodological anarchism is the appropriate stance.

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

Peter Williams complains that Richard Dawkins wraps his naturalism in ‘a fake finery of counterfeit meaning and purpose’. For his part, Williams has wrapped his complaint in an unoriginal and inapt analogy. The weavers in Hans Christian Andersen's fable announce that the Emperor's clothes are invisible to stupid people; almost the whole population pretends to see them for fear of being thought stupid . Fear of being thought stupid does not seem to trouble Richard Dawkins. Moreover, Williams offers no reason to think that such fear motivates any of Dawkins' readers. Perhaps all we are supposed to take from the fable is that Dawkins' naturalism is obviously lacking in meaning and purpose. If that is the intended reading, then by using this analogy, Williams has given himself an unnecessarily difficult task. Surely, it would be achievement enough for him to show that Dawkins' naturalism lacks meaning and purpose. There is no reason for Williams to make the extra claim that it obviously lacks meaning and purpose. After all, there is an obvious difficulty with arguing over several pages that something is obviously the case.

Logical theory – and philosophical theory generally – is just that, theory. Generations of logic students felt a sort of unease about it without knowing what to do about it. Nowadays, students of mathematical logic feel a similar unease when faced with the fact that in standard predicate calculus, “All unicorns are sneaky” is true precisely because there are no unicorns. Blanché’s analysis reminds us that such feelings of unease may indicate a shortcoming in the theory rather than in the student’s understanding.

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

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.

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.