Aristotle’s philosophy of geometry is widely interpreted as a reaction against a Platonic realist conception of mathematics. Here I argue to the contrary that Aristotle is concerned primarily with the methodological question of how universal inferences are warranted by particular geometrical constructions. His answer hinges on the concept of abstraction, an operation of “taking away” certain features of material particulars that makes perspicuous universal relations among magnitudes. On my reading, abstraction is a diagrammatic procedure for Aristotle, and it is through (...) imagined variation of diagrams that one can universalize spatial inferences. In order to shift the discussion of Aristotle’s work on geometry from ontological to methodological questions, I argue in the first section that geometrical properties for Aristotle are necessary accidents of the particulars, canonically diagrams, in which they inhere. I then consider a line of research that shows how diagrams in ancient geometry are not illustrations of the textual proof but are in part constitutive of geometrical inference, an insight I claim can shed light on Aristotle’s philosophical project. Next I substantiate this assertion by giving an alternative interpretation of abstraction according to which it is a diagrammatic procedure that allows a part of a particular diagram to stand in for a universal. In the last section, I consider Aristotle’s account of mathematical cognition, arguing that there is evidence for the view that imagination is necessary both for the construction of figures, and for their presentation as abstractions subject to universal inference. (shrink)

Intellectualists hold that knowledge-how is a species of knowledge-that, and consequently that the knowledge involved in skill is propositional. In support of this view, the intentional action argument holds that since skills manifest in intentional action and since intentional action necessarily depends on propositional knowledge, skills necessarily depend on propositional knowledge. We challenge this argument, and suggest that instructive representations, as opposed to propositional attitudes, can better account for an agent’s reasons for action. While a propositional-causal theory of action, according (...) to which intentional action must be causally produced “in the right way” by an agent’s proposition-involving reasons, has long held sway, we draw on Elizabeth Anscombe’s insights offer a path toward an alternative theory of action. In so doing, we reject the implicitly Cartesian conception of knowledge at the core of the intentional action argument, while hanging on to the idea that mental states are representations of a certain kind. Our argument provides theoretical support for anti-intellectualism by equipping philosophers with an account of non-propositional, practical content. (shrink)

Ordinary language treats reports of perceptual episodes as canonical justifications of beliefs. The challenge for empirically oriented epistemologists is to explain one's right to give credence to one's perceptual judgments. Traditionally, many empiricists have assumed that an epistemic subject is entitled only to some primitive judgments, so that judgments about kinds, modal properties, and dispositions are parasitic upon and less certain than those about the particulars given in perception. This paper contributes to an understanding of C.S. Peirce's alternative perceptual epistemology. (...) According to Peirce's model, perceptual experience must be conceived as including previously encountered content and... (shrink)

This paper examines a parallel between Aristotle’s account of phantasia and contemporary psychological models of working memory, a capacity that enables the temporary maintenance and manipulation of information used in many behaviors. These two capacities, though developed within two distinct scientific paradigms, share a common strategy of psychological explanation, Aristotelian Faculty Psychology. This strategy individuates psychological components by their target-domains and functional roles. Working memory and phantasia result from an attempt to individuate the psychological components responsible for flexible thought and (...) are thus implicated in most of our robust cognitive processes, from reading comprehension to problem solving. We then present two novel objections which suggest that these capacities cannot explain our ability to engage in flexible thought. To escape the resultant impasse, we survey alternatives and argue that most promising strategies depend on identifying the behaviors attributed to intelligent thought and action. (shrink)

In Metaphysics M, Aristotle aims to refute the Platonic view that mathematical objects are substantially prior to sensible things. For Aristotle, mathematical objects are the abstracted attributes of sensible substances required for geometrical analysis and proof. Yet, despite this derivative status of the objects of mathematics, Aristotle insists that they are logically prior to individual substances. This paper examines the distinction between logical and substantial priority, arguing that it underwrites Aristotle’s conception of mathematical necessity and explanation.

Some philosophers hold that mathematics depends on idealising assumptions. While these thinkers typically emphasise the role of idealisation in set theory, Edmund Husserl argues that idealisation is constitutive of the early Greek geometry that is codified by Euclid. This paper takes up Husserl's idea by investigating three major developments of Greek geometry: Thalean analogical idealisation, Hippocratean dynamic idealisation, and Archimedean mechanical idealisation. I argue that these idealisations are not, as Husserl held, primarily a matter of ‘smoothing out’ sensory reality to (...) produce ideal, ‘perfect’ figures. Rather, Greek geometry depends on assuming some falsehoods – equidistance from a source of illumination, a perfectly unwavering hand, or a machine that weights abstract objects – in order to make complex problems tractable. Although these idealisations were rarely discussed explicitly in antiquity, they can be systematically reconstructed from our extant sources. (shrink)

Descartes holds that, insofar as nature is a purposeless, unthinking, extended substance, there could be no final causes in physics. Descartes’ derivation of his three laws of motion from the perfections of God thus underwrites a rejection of Aristotle’s conception of natural self-motion and teleology. Aristotle derived his conception of the purposeful action of sublunar creatures from his notion that superlunar bodies are perfect, eternal, living beings, via the thesis that circular motion is more complete or perfect than rectilinear motion. (...) Descartes’ reduction of circular motion to rectilinear motion, achieved through his theological foundation of the laws of motion, thus marks a crucial break from Aristotle’s philosophy of nature. This paper argues that the shift from the Aristotelian conception of nature as self-moving and teleological to the Cartesian conception of nature as purposeless and inert, is not an empirical discovery but is rooted in differing conceptions of where perfection lies in nature. (shrink)

Aristotle Aristotle is a towering figure in ancient Greek philosophy, who made important contributions to logic, metaphysics, mathematics, physics, biology, botany, ethics, politics, agriculture, medicine, dance and theater. He was a student of Plato who in turn studied under Socrates. He was more empirically-minded than Plato or Socrates and is famous for … Continue reading Aristotle →.

Aristotelian imagination is widely understood as a psychological power by which retained perceptual states recur in consciousness. According to this view, imagination is decaying sense, a part of the psyche that is parasitic on perceptual acts for its content. This paper disputes this reading and provides an alternative account of Aristotle’s concept of imagination. I argue that Aristotelian imagination is a power of the psyche that is both productive like intellect, and presentational like perception. Unlike perception and intellect, however, imagination (...) does not correctly discriminate among beings, and thus cannot be relied upon to give one knowledge of the world. When one accepts this alternative conception of Aristotelian imagination, it becomes clear how it can take on the peculiar epistemic function of allowing a particular serve as the vehicle of a universal thought. This paper argues that Aristotle’s explanation of valid judgments in geometry depends on the imagination to allow the perception of a particular diagram to give rise to the intellectual grasp of a general proposition. (shrink)

Why is nature amenable to mathematical description? This question has received attention in the philosophy of science but rarely from a phenomenological perspective. Nevertheless Husserl’s late essay “The Origin of Geometry,” which has received some critical scholarly attention in recent years, contains the beginning of a striking answer. This answer proceeds from Husserl’s main claim in that essay, which he also makes in the Crisis of the European Sciences, that the original meaning of science has been covered over or “sedimented” (...) by concepts that obscure the true intentional core of scientific meaning. In the first three sections of this paper I develop Husserl’s central insights about mathematics in light of two contemporary critiques of his project of “reactivation” of the original, sedimented meaning of science. In the latter two sections, I argue that accepting Husserl’s account of the original meaning and development of science offers a promising explanation of why nature is amenable to mathematics. This explanation hinges on a conception of the objects and methods of mathematics and the mathematized physical sciences as accomplishments, that is, as constituted contents of consciousness. (shrink)