Brendan Larvor

This paper is an exercise in the phenomenology of science. It examines the tendency to prefer formal accounts in a familiar body of experimental psychology. It will argue that, because of this tendency, psychologists of this school neglect those forms of human cognition typical of the humanities disciplines. This is not a criticism of psychology, however. Such neglect is compatible with scientific rigour, provided it does not go unnoticed. Indeed, reflection on the case in hand allows us to refine the characterisation of the formalising tendency.

It would be a mistake to imagine that the problem of the Cartesian circle lies in Descartes’ suggestion that we cannot know anything unless we know God. It is true that this thought seems fatal to his enterprise; for if we cannot know anything prior to knowing that God exists, then it follows that we cannot know the arguments that prove God’s existence. However the problem of the Cartesian circle does not consist in this logical error. It consists, rather, in the fact that Descartes’ attempts to deal with the charge of circular reasoning seem so inadequate. It is implausible that Descartes simply failed to appreciate the point, for the objection is a very simple one, requiring no special vocabulary nor any advanced logical apparatus. He addressed the problem twice in the replies to his critics and went over it again in the Principles of Philosophy. Despite these qualifications, Descartes was not able to lay out his position with sufficient clarity to satisfy his reviewers or to prevent the publication of a bewildering variety of interpretations. The question then becomes, if Descartes understood the objection, why did he not deal with it more effectively.

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.