Descartes held that practicing mathematics was important for developing the mental faculties necessary for science and a virtuous life. Otherwise, he maintained that the proper uses of mathematics were extremely limited. This article discusses his reasons which include a theory of education, the metaphysics of matter, and a psychologistic theory of deductive reasoning. It is argued that these reasons cohere with his system of philosophy.

Professor Grosholz packs a lot into her interesting and suggestive paper “Formal Unities and Real Individuals” (Grosholz 1990b). In the limited space available I can comment briefly on its several parts, or direct more substantive comments at a single issue. I will opt for the latter; specifically, I want to address her critique of mathematical structuralism, as found especially in the writings of Michael Resnik.I begin with a brief, hence necessarily caricatured, summary of Resnik’s influential view. According to structuralism, the (...) subject matter of a mathematical theory is a given pattern, or structure, and the objects of the theory are intrinsically unstructured points, or positions, within that pattern. Mathematical objects thus have no identity, and no intrinsic features, outside of the patterns in which they occur. Hence, they cannot be given in isolation but only in their role within an antecedently given pattern, and are distinguishable from one another only in virtue of the relations they bear to one another in the pattern (see, e.g., Resnik (1981)). (shrink)

This article examines Hobbes’s conception of mathematical method, situating his methodological writings in the context of disputed mathematical issues of the seventeenth century. After a brief exposition of the Hobbesian philosophy of mathematics, it investigates Hobbes’s attempts to resolve three important mathematical controversies of the seventeenth century: the debates over the status of analytic geometry, disputes over the nature of ratios, and the problem of the “angle of contact” between a curve and tangent. In the course of these investigations, Hobbes’s (...) account of mathematics and its method is contrasted with the those of Descartes, Isaac Barrow, John Wallis, and Christopher Clavius. (shrink)

The relationship between the objects of mathematics and physics has been a recurrent source of philosophical debate. Rationalist philosophers can minimize the distance between mathematical and physical domains by appealing to transcendental categories, but then are left with the problem of where to locate those categories ontologically. Empiricists can locate their objects in the material realm, but then have difficulty explaining certain peculiar “transcendental” features of mathematics like the timelessness of its objects and the unfalsifiability of (at least some of) (...) its truths. During the past twenty years, the relationship between mathematics and physics has come to seem particularly problematic, in part because of a strong interest in “naturalized epistemology” among American philosophers. The tendency to construe epistemological relations in causal and materialist terms seems to enforce a sharp distinction between mathematical and physical entities, and makes the former seem at best uncomfortably inaccessible and at worst irrelevant. (shrink)

Philosophy and Memory Traces defends two theories of autobiographical memory. One is a bewildering historical view of memories as dynamic patterns in fleeting animal spirits, nervous fluids which rummaged through the pores of brain and body. The other is new connectionism, in which memories are 'stored' only superpositionally, and reconstructed rather than reproduced. Both models, argues John Sutton, depart from static archival metaphors by employing distributed representation, which brings interference and confusion between memory traces. Both raise urgent issues about control (...) of the personal past, and about relations between self and body. Sutton demonstrates the role of bizarre body fluids in moral physiology, as philosophers from Descartes and Locke to Coleridge struggled to control their own innards and impose cognitive discipline on 'the phantasmal chaos of association'. Going on to defend connectionism against Fodor and critics of passive mental representations, he shows how problems of the self are implicated in cognitive science. (shrink)

Reduction is a concept first introduced by Descartes in explaining his view of the rationalization of philosophy through mathematics. He seeks to consider length, breadth, and depth for phenomena so that reducing the phenomenon to his own analytical geometric apparatus; thus shrinking the whole world into a small machine. In the present study, the authors took into account the deficiency in defining the reduction of phenomena to a mathematically sound system as the reason for a large group of problems and (...) therefore they came to redefine the Cartesian reductionism of phenomena by removing the search space through a learning system. In due definition, it is possible to reduce the NP problems to P space without using a quantum algorithm that requires a quantum computer to exist. The present study points out that the problems arising from the mathematical modeling of the Covid-19 pandemic are due to a deficiency in the definition of Cartesian reduction, which leads to an increase in the computational complexity of its diagnosis and treatment using computational tools. (shrink)

Leibniz frequently argued that reasons are to be weighed against each other as in a pair of scales, as Professor Marcelo Dascal has shown in his article "The Balance of Reason." In this kind of weighing it is not necessary to reach demonstrative certainty – one need only judge whether the reasons weigh more on behalf of one or the other option However, a different kind of account about rational decision-making can be found in some of Leibniz's writings. In his (...) article "Was Leibniz's Deity an Akrates?" Professor Jaakko Hintikka has argued that Leibniz developed a new vectorial model for rational decisions which is better suited to complicated decisions, where values are complementary to each other. This model, related closely to his work in metaphysics and the philosophy of mind, is a heuristic device which helps in finding rational combinations - and in an ideal case an optimum - between plural inclinations to the good. I shall argue that Leibniz applies more or less implicitly both of these models in his practical rationality. In simple situations he applied the pair of scales model and in more complicated situations he applied the vectorial model. (shrink)

Descartes is usually taken to be a strict reductionist, and he frequently describes his work in reductionist terms. This dissertation, however, makes the case that he is a nonreductionist in certain areas of his philosophy and natural philosophy. This might seem like simple inconsistency, or a mismatch between Descartes's ambitions and his achievements. I argue that here it is more than that: nonreductionism is compatible with his wider commitments, and allowing for irreducibles increases the explanatory power of his system. Moreover, (...) Descartes depends on nonreductionist knowledge and on irreducibles in order to maintain his epistemology and metaphysics. That is, there is more in the Cartesian world than mind, matter, and God. (shrink)