Online research in philosophy

In this article, I will discuss the relationship between mathematical intuition and mathematical visualization. I will argue that in order to investigate this relationship, it is necessary to consider mathematical activity as a complex phenomenon, which involves many different cognitive resources. I will focus on two kinds of danger in recurring to visualization and I will show that they are not a good reason to conclude that visualization is not reliable, if we consider its use in mathematical practice. Then, I will give an example of mathematical reasoning with a figure, and show that both visualization and intuition are involved. I claim that mathematical intuition depends on background knowledge and expertise, and that it allows to see the generality of the conclusions obtained by means of visualization.

Foundations and Applications depend ultimately for their existence on each other. The main links between them are education and the axiomatic method. Those links can be strengthened with the help of a categorical method which was concentrated forty years ago by Cartier, Grothendieck, Isbell, Kan, and Yoneda. I extended that method to extract some essential features of the category of categories in 1965, and I apply it here in section 3 to sketch a similar foundation within the smooth categories which provide the setting for the mathematics of change. The possibility that other methods may be needed to clarify a contradiction introduced by Cantor, now embedded in mathematical practice, is discussed in section 5.

I argue against a formidable interpretation of Plato’s Divided Line image according to which dianoetic correctly applies the same method as dialectic. The difference between the dianoetic and dialectic sections of the Line is not methodological, but ontological. I maintain that while this interpretation correctly identifies the mathematical method with dialectic, ( i.e. , the method of philosophy), it incorrectly identifies the mathematical method with dianoetic. Rather, Plato takes dianoetic to be a misapplication of the mathematical method by a subset of practicing mathematicians. Thus, Plato’s critique of dianoetic is a not a critique of mathematics, as such, but of mathematicians.

The author examines He Lin’s interpretation of Zhu Xi’s method of intuition from a phenomenological-hermeneutical perspective and by exposing Zhu’s philosophical presuppositions. In contrast with Lu Xiangshan’s intuitive method, Zhu Xi’s method of reading classics advocates “emptying your heart and flowing with the text” and, in this spirit, explains the celebrated “exhaustive investigation on the principles of things (ge wu qiong li).” “Text,” according to Zhu, is therefore not an object in ordinary sense but a “contextual region” or “sensible pattern” that, when merged with the reader, generates meanings. Furthermore, by discussing the related doctrines of Lao Zi, Zhuang Zi, Hua-Yan Buddhism, Zhou Dunyi, and Zhu Xi’s own “One principle with many manifestations (li yi fen shu),” the author identifies the philosophical preconditions of Zhu’s method. Based on this analysis, the author goes on to illustrate Zhu’s understanding of “observing potential yet unapparent pleasure, anger, sorrow and happiness” and “maintaining a serious attitude (zhu jing).”.

Only recently, David Hilbert's program to axiomatize the sciences according to the pattern of geometry has left the shade of his formalist program in the foundations of mathematics. This relative neglect - which is surprising in view of the enormous efforts Hilbert had himself devoted to it - was certainly influenced by Logical Empiricists' almost exclusively focusing on his contributions to the foundational debates. The present paper investigates the stand of two core members of the Vienna Circle who had studied with Hilbert at Göttingen, the mathematician Hans Hahn and the theoretical physicist Philipp Frank. At bottom of their neglect of Hilbert's axiomatic method stands their conviction that reconciling Ernst Mach's empiricist heritage with modern mathematics required to draw a rigid boundary between mathematics and physics and to subscribe to logicism, according to which mathematics consisted in tautologous logical transformations. In this way, they missed the substantial difference between the logical structure of a particular axiom system and the axiomatic method as a critical study of arbitrary axiom systems. If this distinction is not properly observed - and admittedly Hilbert himself did deliberately obscure it at places - a core concept of the axiomatic method, "deepening the foundations" (Tieferlegung), becomes metaphysical because it might appear as an ontological reduction of basic physical concepts to mathematical ones rather than - as Hilbert intended - an epistemological reduction availing itself of the unity of mathematical knowledge. To be sure, Logical Empiricists considered the goal of axiomatizing the sciences as an important task, but in the way how they set it up axiomatization became much closer tied to a success of the foundationalist program for all mathematics than Hilbert's axiomatic method ever was.

While Gödel's (first) incompleteness theorem has been used to refute the main contentions of Hilbert's program, it does not seem to have been generally used to stress that a basic ingredient of that program, the concept of formal system as a closed system - as well as the underlying view, embodied in the axiomatic method, that mathematical theories are deductions from first principles must be abandoned. Indeed the logical community has generally failed to learn Gödel's lesson that Hilbert's concept of formal system as a closed system is inadequate and continues to use it as if there were no incompleteness theorem. In this paper I will stress the role of Gödel's incompleteness theorem in showing the inadequacy of such a concept of formal system and the need for a more articulated view of mathematical theories. More generally I will argue that Gödel's result entails that, as an alternative to mathematical logic, a new concept of logic is required: logic as the theory of communicating inference processes.

Those inquiring into the nature of mind have long been interested in the foundations of mathematics, and conversely this branch of knowledge is distinctive in that our access to it is purely through thought. A better understanding of mathematical thought should clarify the conceptual foundations of mathematics, and a deeper grasp of the latter should in turn illuminate the powers of mind through which mathematics is made available to us. The link between conceptions of mind and of mathematics has been a central theme running through the great competing philosophies of mathematics of the twentieth century, though each has refashioned the connection and its import in distinctive ways. The present collection will be of interest to students of both mathematics and of mind. Contents include: "Introduction" by Alexander George; "What is Mathematics About?" by Michael Dummett; "The Advantages of Honest Toil over Theft" by George Boolos; "The Law of Excluded Middle and the Axiom of Choice" by W.W. Tait; "Mechanical Procedures and Mathematical Experience" by Wilfried Sieg; "Mathematical Intuition and Objectivity" by Daniel Isaacson; "Intuition and Number" by Charles Parsons; and "Hilbert's Axiomatic Method and the Laws of Thought" by Michael Hallett.

By way of these investigations, we hope to understand better the rationale behind Kant's theory of intuition, as well as to grasp many facets of the relations ...

Although in the past three decades interest in mathematical explanation revived, recent literature on the subject seems to neglect the strict connection between explanation and discovery. In this paper I sketch an alternative approach that takes such connection into account. My approach is a revised version of one originally considered by Descartes. The main difference is that my approach is in terms of the analytic method, which is a method of discovery prior to axiomatized mathematics, whereas Descartes’s approach is in terms of the analytic-synthetic method, which is a heuristic pattern in already axiomatized mathematics.