The uses and abuses of mathematics in early modern philosophy: introduction

Diagrammatic carriers and the acceptance of Newton’s optical theory Abstract A permissivist framework is developed to include images in the reconstruction of the evidential base and of the theoretical content. The paper uses Newton’s optical theory as a case study to discuss mathematical idealizations and depictions of experiments (‘equidiametric’ and ‘harmonic’ spectral images, ray-paths in drawings of prism-experiments), together with textual correlates of diagrams. Instead of assuming some specific type of theoretical content, focus is on novel traits that are delineable when studying the carriers of a theory. The framework is developed to trace elliptic and ambiguous message design (polysemy), and utilizes variegated acceptance (heterogeneous uptake and polyphony) as an asset. Newton’s resources (novel diagrammatic carriers, mathematical idealizations, and neologisms) allowed for various framing modes and reconstructions, entailing various judgements concerning the theoretical content, the evidence base, and Newton’s use (and/or abuse) of mathematics. Elliptic presentation of the theory’s proof-structure and ambiguities (also pertaining to the function and material setup of the crucial experiment) influenced uptake, contributed to the process of opinion-polarization, and the acceptance/rejection of the theory. The study suggests that the analysed carriers of theoretical content (including visuals) have an argumentative function, and one of their uses is to adjust the burden of proof.

Basing for the Bayesian Abstract There is a distinction between merely having the right belief, and further basing that belief on the right reasons. Any adequate epistemology needs to be able to accommodate the basing relation that marks this distinction. However, trouble arises for Bayesianism. I argue that when we combine Bayesianism with the standard approaches to the basing relation, we get the result that no agent forms their credences in the right way; indeed, no agent even gets close. This is a serious problem, for it prevents us from making epistemic distinctions between agents that are doing a reasonably good job at forming their credences and those that are forming them in clearly bad ways. I argue that if this result holds, then we have a problem for Bayesianism. However, I show how the Bayesian can avoid this problem by rejecting the standard approaches to the basing relation. By drawing on recent work on the basing relation, we can develop an account of the relation that allows us to avoid the result that no agent comes close to forming their credences in the right way. The Bayesian can successfully accommodate the basing relation.

Existentialism, aliens and referentially unrestricted worlds Abstract Existentialism claims that propositions that directly refer to individuals depend on those individuals for their existence. I argue for two points regarding Existentialism. First, I argue that recent accounts of Existentialism run into difficulties accommodating the possibility of there being a lonely alien electron. This problem is distinct from one of the better-known alien problems—concerning iterated modal properties of aliens—and can’t be solved using a standard response to the iterated case. Second, though the lonely alien electron problem might seem to be reason to reject the sort of Existentialist view at hand, there’s a plausible way to preserve the view: accept the existence of possible worlds that directly refer to individuals that don’t exist in those worlds. Such a solution might seem incompatible with Existentialism, but I show that Existentialists can avoid the incompatibility and should find the resulting view plausible.

Vital anti-mathematicism and the ontology of the emerging life sciences: from Mandeville to Diderot Abstract Intellectual history still quite commonly distinguishes between the episode we know as the Scientific Revolution, and its successor era, the Enlightenment, in terms of the calculatory and quantifying zeal of the former—the age of mechanics—and the rather scientifically lackadaisical mood of the latter, more concerned with freedom, public space and aesthetics. It is possible to challenge this distinction in a variety of ways, but the approach I examine here, in which the focus on an emerging scientific field or cluster of disciplines—the ‘life sciences’, particularly natural history, medicine, and physiology (for ‘biology’ does not make an appearance at least under this name or definition until the late 1790s)—is, not Romantically anti-scientific, but resolutely anti-mathematical. Diderot bluntly states, in his Thoughts on the interpretation of nature (1753), that “We are on the verge of a great revolution in the sciences. Given the taste people seem to have for morals, belles-lettres, the history of nature and experimental physics, I dare say that before a hundred years, there will not be more than three great geometricians remaining in Europe. The science will stop short where the Bernoullis, the Eulers, the Maupertuis, the Clairauts, the Fontaines and the D’Alemberts will have left it.... We will not go beyond.” Similarly, Buffon in the first discourse of his Histoire naturelle (1749) speaks of the “over-reliance on mathematical sciences,” given that mathematical truths are merely “definitional” and “demonstrative,” and thereby “abstract, intellectual and arbitrary.” Earlier in the Thoughts, Diderot judges “the thing of the mathematician” to have “as little existence in nature as that of the gambler.” Significantly, this attitude—taken by great scientists who also translated Newton (Buffon) or wrote careful papers on probability theory (Diderot), as well as by others such as Mandeville—participates in the effort to conceptualize what we might call a new ontology for the emerging life sciences, very different from both the ‘iatromechanism’ and the ‘animism’ of earlier generations, which either failed to account for specifically living, goal-directed features of organisms, or accounted for them in supernaturalistic terms by appealing to an ‘anima’ as explanatory principle. Anti-mathematicism here is then a key component of a naturalistic, open-ended project to give a successful reductionist model of explanation in ‘natural history’ (one is tempted to say ‘biology’), a model which is no more vitalist than it is materialist—but which is fairly far removed from early modern mechanism.

A puzzle about desire Abstract This paper develops a novel puzzle about desire consisting of three independently plausible but jointly inconsistent propositions: (1) all desires are dispositional states, (2) we have privileged access to some of our desires, and (3) we do not have privileged access to any dispositional state. Proponents of the view that all desires are dispositional states might think the most promising way out of this puzzle is to deny (3). I argue, however, that such attempts fail because the most plausible accounts of self-knowledge of desires do not explain how we possess privileged access to dispositional desires. I conclude by offering what I take to be a more promising solution to the puzzle, one that involves the rejection of (1) on the grounds that some desires possess phenomenology.

Descartes on the limited usefulness of mathematics Abstract 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.

Imagination, metaphysics, mathematics: Descartes’s arguments for the Vortex Hypothesis Abstract In this paper, I examine the manner in which Descartes defends his Vortex Hypothesis in Part III of the Principles of Philosophy (1644), and expand on Ernan McMullin’s characterization of the methodology that Descartes uses to support his planetary system. McMullin illuminates the connection between the deductive method of Part III and the method Descartes uses in earlier portions of the Principles, and he brings needed light to the role that imaginative constructions play in Descartes’s explanations of the phenomena. I develop McMullin’s reading by bringing further attention to the constraints that Descartes places on the imagination in Part III. I focus in particular on the way in which Descartes uses metaphysical truths concerning God’s nature to support his general description of the planetary system, and on the way he relies on a mathematical standard of intelligibility to defend his proposals about the configuration of matter. Attending to the role of metaphysics and mathematics in Part III shows that Descartes’s arguments for the explanatory power of the Vortex Hypothesis are more effective than McMullin suggests. The reading I forward also offers important perspective on how Descartes’s hypotheses in Part III can be seen as both metaphysically and mathematically well-grounded.

A metarepresentational theory of intentional identity Abstract Geach points out that some pairs of beliefs have a common focus despite there being, apparently, no object at that focus. For example, two or more beliefs can be directed at Vulcan even though there is no such planet. Geach introduced the label ‘intentional identity’ to pick out the relation that holds between attitudes in these cases; Geach says that ’[w]e have intentional identity when a number of people, or one person on different occasions, have attitudes with a common focus, whether or not there actually is something at that focus’ (Geach in J Philos 64(20):627–632, 1967). In this paper, I propose a novel theory of intentional identity, the triangulation theory, and argue that it has considerable advantages over its principal rivals. My approach centers on agents’ metarepresentational beliefs about what it takes for intentional attitudes to be about particular objects.

Constraining (mathematical) imagination by experience: Nieuwentijt and van Musschenbroek on the abuses of mathematics Abstract Like many of their contemporaries Bernard Nieuwentijt (1654–1718) and Pieter van Musschenbroek (1692–1761) were baffled by the heterodox conclusions which Baruch Spinoza (1632–1677) drew in the Ethics. As the full title of the Ethics—Ethica ordine geometrico demonstrata—indicates, these conclusions were purportedly demonstrated in a geometrical order, i.e. by means of pure mathematics. First, I highlight how Nieuwentijt tried to immunize Spinoza’s worrisome conclusions by insisting on the distinction between pure and mixed mathematics. Next, I argue that the anti-Spinozist underpinnings of Nieuwentijt’s distinction between pure and mixed mathematics resurfaced in the work of van Musschenbroek. By insisting on the distinction between pure and mixed mathematics, Nieuwentijt and van Musschenbroek argued that Spinoza abused mathematics by making claims about things that exist in rerum natura by relying on a pure mathematical approach (type 1 abuse). In addition, by insisting that mixed mathematics should be painstakingly based on mathematical ideas that correspond to nature, van Musschenbroek argued that René Descartes’ (1596–1650) natural-philosophical project (and that of others who followed his approach) abused mathematics by introducing hypotheses, i.e. (mathematical) ideas, that do not correspond to nature (type 2 abuse).