In Physics, Aristotle starts his positive account of the infinite by raising a problem: “[I]f one supposes it not to exist, many impossible things result, and equally if one supposes it to exist.” His views on time, extended magnitudes, and number imply that there must be some sense in which the infinite exists, for he holds that time has no beginning or end, magnitudes are infinitely divisible, and there is no highest number. In Aristotle's view, a plurality cannot escape having bounds if all of its members exist at once. Two interesting, and contrasting, interpretations of Aristotle's account can be found in the work of Jaako Hintikka and of Jonathan Lear. Hintikka tries to explain the sense in which the infinite is actually, and the sense in which its being is like the being of a day or a contest. Lear focuses on the sense in which the infinite is only potential, and emphasizes that an infinite, unlike a day or a contest, is always incomplete.
Aristotle created logic and developed it to a level of great sophistication. There was nothing there before; and it took more than two millennia for something better to come around. The astonishment experienced by readers of the Prior Analytics, the most important of Aristotle's works that present the discipline, is comparable to that of an explorer discovering a cathedral in a desert. This article explains and evaluates some of Aristotle's views about propositions and syllogisms. The most important omission is the difficult subject of syllogisms involving modalities. Aristotle distinguishes two relations of opposition that can obtain between propositions with the same subject- and predicate-expressions: contrariety and contradiction. In every canonical syllogism, one term is common to the two premises: it is called “middle term,” or simply “middle.” The remaining two terms of the premises are the only ones occurring in the conclusion: they are called “extreme terms,” or simply “extremes.”
Much of Aristotle's thought developed in reaction to Plato's views, and this is certainly true of his philosophy of mathematics. To judge from his dialogue, the Meno, the first thing that struck Plato as an interesting and important feature of mathematics was its epistemology: in this subject we can apparently just “draw knowledge out of ourselves.” Aristotle certainly thinks that Plato was wrong to “separate” the objects of mathematics from the familiar objects that we experience in this world. His main arguments on this point are in Chapter 2 of Book XIII of the Metaphysics. There are three distinct lines of argument: The first concerns the objects of geometry (that is, points, lines, planes, and solids); the second deals with the Platonist principles which are applied to arithmetic and geometry; the third is about substance as living things, especially animals, and perhaps man in particular. In addition to the above, this article also examines Aristotle's treatment of infinity.
Kentaro Fujimoto and Volker Halbach
This chapter sketches the motivations for treating truth as a primitive notion and developing axiomatic theories of truth. Then the main axiomatic systems of typed and type-free truth are surveyed.
James M. Joyce
This article is concerned with Bayesian epistemology. Bayesianism claims to provide a unified theory of epistemic and practical rationality based on the principle of mathematical expectation. In its epistemic guise, it requires believers to obey the laws of probability. In its practical guise, it asks agents to maximize their subjective expected utility. This article explains the five pillars of Bayesian epistemology, each of which claims and evaluates some of the justifications that have been offered for them. It also addresses some common objections to Bayesianism, in particular the “problem of old evidence” and the complaint that the view degenerates into an untenable subjectivism. It closes by painting a picture of Bayesianism as an “internalist” theory of reasons for action and belief that can be fruitfully augmented with “externalist” principles of practical and epistemic rationality.
The principle that every statement is bivalent (i.e. either true or false) has been a bone of philosophical contention for centuries, for an apparently powerful argument for it (due to Aristotle) sits alongside apparently convincing counterexamples to it. This chapter analyzes Aristotle’s argument, then, in the light of this analysis, examines three sorts of problem case for bivalence. Future contingents, it is contended, are bivalent. Certain statements of higher set theory, by contrast, are not. Pace the intuitionists, though, this is not because excluded middle does not apply to such statements, but because they are not determinate. Vague statements too are not bivalent, in this case because the law of proof by cases does not apply. The chapter goes on to show how this opens the way to a solution to the ancient paradox of the heap (or Sorites) that draws on quantum logic.
Matthew L. Jones
This chapter sketches the challenges Leibniz faced in building a calculating machine for arithmetic, especially his struggle to coordinate with skilled artisans, surveys his philosophical remarks about such machines and the practical knowledge needed to make them, and recounts the eighteenth-century legacy of his failure to produce a machine understood to be adequately functional.
This chapter discusses the history of Leibniz's work on infinitesimal calculus of which a considerable part is still unknown. His new method, emerging from studies in the summing of infinite number series and the quadrature of curves, combines two procedures with opposite orientation, differentiation and integration. These two procedures are united in a common formalism introducing in 1675 the symbols d and ∫ for differentiation and integration. Subsequently, Leibniz and his followers developed new rules and solution methods, and applied the calculus to physics. During Leibniz’s lifetime the public success of his calculus was overshadowed by discussions over the foundations of his methods and the priority dispute with Newton. While infinitesimals were eliminated from the calculus during the 19th century, non-standard analysis reinstated them again. The status of infinitesimals in Leibniz’s own philosophy of mathematics is still disputed.
The coherence theory holds that truth consists in coherence amongst our beliefs. It can thus rule out radical scepticism and avoid the problems of the correspondence theory. Considerations about meaning and verification have also pointed philosophers in the same direction. But if it holds all truth to consist in coherence it is untenable: there must be some truths that do not, truths about what people believe. This causes problems for traditional coherence theories, and also for verificationists and anti-realists. The admission of a grounding class of truths that do not consist in coherence also raises the question why there should be such systematic agreement between these. This cannot properly be explained by anything that is said within the theory whose truth is constituted by coherence with the grounding class. Kant saw this problem, and postulated “things as they are in themselves.” Others dismiss it; but that is not satisfactory.
Computational economics is a relatively new research technique in economics, but it is inexorably taking its place alongside the more traditional methods of general theory, abstract modeling, data analysis, and the more recent experimental economics. Perhaps because of its relative newness, the term computational economics currently has no determinate meaning. In contemporary use, it refers to a heterogeneous cluster of techniques implemented on concrete digital computers ranging from the numerical solution of the Black-Scholes partial differential equation for pricing options through automated trading strategies to agent-based computer simulations of the evolution of cooperation. Because of this heterogeneity, it is not possible to provide a comprehensive coverage of the topic in this article. Another reason for this restricted scope is that many of the methods used in computational economics have considerable technical interest but no particular philosophical relevance.