This paper explicates Descartes’ theory of intuition (intuitus). Departing from certain commentators, I argue that intuition, for Descartes, is a form of clear and distinct intellectual perception. Because it is clear and distinct, it is indubitable, infallible, and provides a grade of certain knowledge he calls ‘cognitio’. I pay special attention to why he treats intuition as a form of perception, and what he means when he says it is ‘clear and distinct’. Finally, I situate his view in relation to (...) those of his Scholastic predecessors on one hand and current theories on the other. His view anticipates the contemporary ‘perceptual model of intuition’, though it is much bolder with its promise of certainty. (shrink)

This paper explicates Descartes’ theory of intuition (intuitus). Departing from certain commentators, I argue that intuition, for Descartes, is a form of clear and distinct intellectual perception. Because it is clear and distinct, it is indubitable, infallible, and provides a grade of certain knowledge he calls ‘cognitio’. I pay special attention to why he treats intuition as a form of perception, and what he means when he says it is ‘clear and distinct’. Finally, I situate his view in relation to (...) those of his Scholastic predecessors on one hand and current theories on the other. His view anticipates the contemporary ‘perceptual model of intuition’, though it is much bolder with its promise of certainty. (shrink)

The article explores the evolution of the idea of correlating numbers and meanings, from ancient numerological systems to modern models of natural language processing based on vector representations and neural networks. The authors demonstrate that the aspiration to uncover hidden properties of objects by associating them with numbers and performing operations on these numbers has been a common thread across various cultures for millennia. The article traces the stages in the formation of the concept of mathesis universalis (universal mathematics), starting (...) from Aristotle and Proclus’s reflections on general mathematics and culminating in R. Descartes’s ideas about the existence of a universally applicable science of order and measure. Particular attention is devoted to G.W. Leibniz’s ambitious project of creating a universal characteristic – a formalized language in which all knowledge could be precisely expressed and logical inference could be carried out through calculations. It is noted that, despite inherent limitations, some of Leibniz’s intuitions have found embodiment in natural language processing technologies based on vector representations of words. However, the simple correlation of concepts with vectors is insufficient for the full realization of the idea of a calculus of reasoning, as it lacks a mechanism for deriving true statements from others. The development of the transformer architecture, which employs the mechanism of self-attention to construct context-dependent vector representations, has been a significant stride towards the automation of reasoning. Nevertheless, modern language models are based on the principle of maximizing likelihood rather than rigorous logical rules. The authors analyze conceptual solutions proposed for creating a comprehensive calculus of knowledge. In conclusion, it is emphasized that, while the idea of reducing reasoning to mathematical operations is undeniably attractive, one should be cautious about its maximalist interpretations. Cognition is constructive in nature, and even the most perfect model of formalization of reasoning cannot substitute for the practical wisdom and freedom of human thought. (shrink)

This article examines how deduction preserves certainty and how much certainty it can preserve according to Descartes’s Rules for the Direction of the Mind. I argue that the certainty of a deduction is a matter of four conditions for Descartes. First, certainty depends on whether the conjunction of simple propositions is composed with necessity or contingency. Second, a deduction approaches the certainty of an intuition depending on how many “acts of conceiving” it requires and—third—the complexity or difficulty of the acts (...) of thinking, which is determined by the content of the thoughts and on external factors. Fourth, certainty depends on the intellectual aptitude of the person using the deduction. A deduction lacks certainty when it relies on memory such that it is not apprehended with immediacy. However, the mental capacity and speed of a mind can be increased by training the special mental faculties of perspicacity and discernment. Increasing one’s intellectual aptitude allows for more steps of a deduction to be inferred in fewer acts of conceiving, thereby helping preserve the certainty of a deduction. (shrink)

Method may be second only to substance-dualism as the best-known among Descartes's enthusiasms. But knowing that Descartes wants to promote good method is one thing; knowing what exactly he wants to promote is another. Two views seem fairly widespread. The first rests on the claim that Descartes endorses a purely procedural picture of reason, so that right reasoning is a matter of proprieties of operation, rather than (say) respect for its objects. On this view, a method for regulating our reason (...) would offer general rules of procedure, abstracted as much as possible from the content of particular problems. Second is the view that Descartes maintains what we might call an ‘intellectualist’ approach to method, one that restricts right reasoning to operations internal to the mind, and allows the use of external bodily resources (such as sensory perceptions, activities in the corporeal imagination, or written notes) only as initial inputs or as helpful props— convenient, but marginal to the procedure and readily eliminated from it. (shrink)

The distinction between the mental operations of abstraction and exclusion is recognized as playing an important role in many of Descartes’ metaphysical arguments, at least after 1640. In this paper I first show that Descartes describes the distinction between abstraction and exclusion in the early Rules for the Direction of the Mind, in substantially the same way he does in the 1640s. Second, I show that Descartes makes the test for exclusion a major component of the method proposed in the (...) Rules. Third, I argue that in Rule 14 the exclusion-abstraction distinction is connected to a theory of distinctions, which includes a notion of real distinction as essentially tied to the imagination. This sheds light on Descartes’ development in and after the Rules. (shrink)

Descartes’s most extensive discussion of the law of refraction and the shape of the anaclastic lens is contained in Rule 8 of "Rules for the Direction of the Mind". Few reconstructions of Descartes’s discovery of the law of refraction take Rule 8 as their basis. In Rule 8, Descartes denies that the law of refraction can be discovered by purely mathematical means, and he requires that the law of refraction be deduced from physical principles about natural power or force, the (...) nature of the action of light, and the behavior of light rays in a variety of transparent media. For over a century, however, there has been broad agreement that Descartes discovered the law of refraction by purely mathematical means, and that he only later provided the relevant physical rationale (via comparisons or analogies) in Dioptrics II. I execute each step in Descartes’s proposed deduction of the law of refraction and the shape of the anaclastic lens in Rule 8 and concretely show how Descartes could have discovered the law of refraction and the shape of the anaclastic by its means. Rule 8, I argue, reflects Descartes’s actual path to the discovery of the law of refraction and the shape of the anaclastic lens. (shrink)

Method may be second only to substance-dualism as the best-known among Descartes's enthusiasms. But knowing that Descartes wants to promote good method is one thing; knowing what exactly he wants to promote is another. Two views seem fairly widespread. The first rests on the claim that Descartes endorses a purely procedural picture of reason, so that right reasoning is a matter of proprieties of operation, rather than respect for its objects. On this view, a method for regulating our reason would (...) offer general rules of procedure, abstracted as much as possible from the content of particular problems. Second is the view that Descartes maintains what we might call an ‘intellectualist’ approach to method, one that restricts right reasoning to operations internal to the mind, and allows the use of external bodily resources only as initial inputs or as helpful props— convenient, but marginal to the procedure and readily eliminated from it. (shrink)