Search Results

Given that natural science no longer dictates the course of mathematical development and that no curtailment of the free flowering of pure mathematics seems prudent, what are the appropriate methods to use? This chapter argues that those methods are not properly based in extra-mathematical metaphysics, that internal mathematical ends and values should carry the day. The various branches of mathematics aim at different goals, which explains why set theorists strive for a single unified theory of sets while geometers and algebraists embrace a variety of structures. Set theory's foundational role requires that its fundamental axioms form a unified list and that they be as generous as possible. These desiderata count against an axiom candidate like the Axiom of Constructibility (V=L).

This book explores social learning concepts and methods as well as new developments in the field. These methods include: experimental and statistical methods that allow researchers to categorize cases of social learning according to their underlying psychological processes and learning mechanisms; neuroscientific methods for identifying the brain structures, neural circuitry, and physiological processes underlying both social learning and social influences on decision making; and mathematical methods for predicting the pattern of diffusion of novel learned innovations, and for modeling cultural evolution and gene-culture coevolution. This introductory chapter presents some definitions such as “social learning,” “imitation,” “innovation,” and “social transmission” and explains why social learning is an important field of study. It also provides an overview of the chapters that follow.

This chapter discusses the reason for the amplification of mathematical and other formal methods used in natural philosophy and astronomy. It explains that Oxford humanists' attacks which were so frequently generated from the standpoint of neo-Platonism resulted in the concealment of very similar trends in scholastic mathematics and natural philosophy. It discusses that the commentaries clarifying Oxford's significance in the evolution of medieval physics was bounded in time. This chapter also evaluates the central idea in hylomorphism. It argues that one of the more regrettable aspects of the entire medieval debate is that the supposed solutions were so often internally inconsistent. It explains that there were different attitudes to natural philosophy.

Both in his pre-critical writings and in his critical works, Kant criticizes the Wolffian tradition for its use of the mathematical method in philosophy. The chapter argues that the apparent unambiguousness of this opposition between Kant and Wolff notwithstanding, the problem of ascertaining the relationship between Kant’s and Wolff’s methods in philosophy cannot be dismissed so quickly. Only a close consideration of Kant’s different remarks on Wolff’s approach and a comparison of the methods that Wolff and Kant actually used in philosophy can allow us to determine when Kant’s criticisms are justified and where the differences in their methodological proposals for philosophy actually lie. We see that Kant’s account of philosophical method in fact has some elements in common with the Wolffian paradigm, even though there are also relevant differences.

In December 1693, a little book caused a furor when the English Parliament ordered to have it burned and launched a search for the author, the printers, and the publishers. The controversial book, divided into two parts, was written by William Freke, an Antitrinitarian, and it was entitled A Dialogue By Way of Question and Answer, Concerning the Deity. All the Responses being taken verbatim out of the Scriptures and A Brief, but Clear Confutation of the Doctrine of the Trinity. Gottfried Wilhelm Leibniz obtained a copy of the pamphlet and made some reflections before sending it to Princess Electress Sophie. In his pamphlet, Freke argued that the Son and the Holy Spirit are angels, a departure from the Socinians who view the Holy Spirit as only one of God's virtues. In January 1694, Leibniz advised his nephew Friedrich Simon Löffler to confute Freke's pamphlet as the subject of his dissertation for his theological studies at the University of Leipzig in Germany. Löffler replied by telling Leibniz that he intended to conduct the confutation using a “mathematical method”.

Chapter 4 has three main parts. The first develops an account of Kant’s mathematical method, based partly on work by Philip Kitcher, called the KV account. This account allows for epistemically independent applications of geometrical method to objects in pure space—the mind-dependent space of imagination—as well as to objects in a putatively mind-independent empirical space. The a priori synthetic necessity of geometry is understood as counterfactual necessity. Objections from Friedman and Waxman are considered. In the former case the application constitutes pure geometry, in the latter, physical geometry. In the second part we see that Kant compares the theorems of both kinds of geometry, finds them identical, and seeks to explain this coincidence in a Second Geometrical Argument. The explanation is Transcendental Idealism. In the final part, I consider how Kant’s Euclidean theory of pure geometry fares in light of modern non-Euclidean theories of physical geometry.

This chapter argues for three claims concerning the relation between Christian Wolff’s philosophy and the methodological views of early modern experimental philosophers. First, Wolff’s system relies on experience at every step and his views on experiments, observations, hypotheses, and the a priori are in line with those of experimental philosophers. Second, the study of Wolff’s views demonstrates the influence of experimental philosophy in early eighteenth-century Germany, well before Tetens and Feder endorsed the experimental method in the last three decades of the century. Third, Wolff’s thought is shaped by two distinct, but compatible, methodological commitments: to develop a thoroughly experimental philosophy and to build a system according to a mathematical demonstrative method. References to Wolff’s alleged empiricism and rationalism are best identified with references to his endorsement of the tenets of experimental philosophy and of a mathematical demonstrative method.

This chapter recounts Lawrence R. Klein's story. Klien received the Nobel Prize in 1980. Klien was born in 1920, studied in Berkeley, and was a scholar at MIT. He worked with Paul Samuelson; and he pushed for the greater acceptance of macroeconomics in the academic world. He also worked on researching methodologies using the mathematical method. In Ottawa, he put together the first model of the Canadian economy. Back in the U.S, he worked on the Klein-Goldberger model using computers for estimation. His other work involved the Oxford model which was focused on numerical presentations and on the creation of a model of the American economy improving upon the earlier Klien-Goldberger model. Among the books he wrote or collaborated on include An Econometric Model of the United States and Econometric Models as Guides for Decision Making.

Bolzano’s best-known and arguably best mathematical work, the Rein analytischer Beweis of 1817, promises to deliver a ground-revealing proof of an important theorem from the theory of equations, which Bolzano shows to follow from a generalization of the Intermediate Value Theorem. This paper explains and assesses this promise against the background of Bolzano’s early account of mathematical method, in which the idea of grounding plays a central role. In addition, it shows how Bolzano’s early views on proof, though still in an early stage of development, provided the basis for decisive criticisms of previous attempts to prove the theorem. Finally, Bolzano’s early views on proof and grounding are briefly considered in light of the later development of his thought.

This chapter presents an overview of Bolzano’s work in mathematics and its philosophy, while presenting some interesting samples of his work. It begins with a discussion of his views on mathematical method in their historical context, followed by an exposition of some of his best work in real analysis. In particular, the chapter discusses his early work on infinite series and his analysis of continuity, beginning with the Purely Analytic Proof (1817), and extending to his construction of a continuous, nowhere differentiable function in the 1830s, called Bolzano’s function.

This chapter is an examination of Herbart’s psychology, especially his attempt to make psychology a science. To make psychology a science meant for Herbart the application of the method of mathematics to the mental realm. What exactly the application of mathematics meant to psychology is considered closely. This chapter also considers Herbart’s critique of faculty psychology and his dispute with Friedrich Beneke about the role of metaphysics in psychology.