Modernizing the philosophy of mathematics

The distinction between analytic and synthetic propositions, and with that the distinction between a priori and a posteriori truth, is being abandoned in much of analytic philosophy and the philosophy of most of the sciences. These distinctions should also be abandoned in the philosophy of mathematics. In particular, we must recognize the strong empirical component in our mathematical knowledge. The traditional distinction between logic and mathematics, on the one hand, and the natural sciences, on the other, should be dropped. Abstract mathematical objects, like transcendental numbers or Hilbert spaces, are theoretical entities on a par with electromagnetic fields or quarks. Mathematical theories are not primarily logical deductions from axioms obtained by reflection on concepts but, rather, are constructions chosen to solve some collection of problems while fitting smoothly into the other theoretical commitments of the mathematician who formulates them. In other words, a mathematical theory is a scientific theory like any other, no more certain but also no more devoid of content.     

Redrawing Kant's philosophy of mathematics

This essay offers a strategic reinterpretation of Kant's philosophy of mathemat- ics in Critique of Pure Reason via a broad, empirically based reconception of Kant's conception of drawing. It begins with a general overview of Kant's philosophy of mathematics, observing how he differentiates mathematics in the Critique from both the dynamical and the philosophical. Second, it examines how a recent wave of critical analyses of Kant's constructivism takes up these issues, largely inspired by Hintikka's unorthodox conception of Kantian intuition. Third, it offers further analyses of three Kantian concepts vitally linked to that of drawing. It concludes with an etymologically based exploration of the seven clusters of meanings of the word drawing to gesture toward new possibilities for interpreting a Kantian philosophy of mathematics.     

Duhem and history and philosophy of mathematics

The first part of this paper consists of an exposition of the views expressed by Pierre Duhem in his Aim and Structure of Physical Theory concerning the philosophy and historiography of mathematics. The second part provides a critique of these views, pointing to the conclusion that they are in need of reformulation. In the concluding third part, it is suggested that a number of the most important claims made by Duhem concerning physical theory, e.g., those relating to the Newtonian method, the limited falsifiability of theories, and the restricted role of logic, can be meaningfully applied to mathematics.I am indebted to Professors Douglas Jesseph and Philip Quinn for helpful comments on this paper.     

The architecture of modern mathematics: Essays in history and philosophy

It is a little understood fact that the system of formal logic presented in Wittgenstein’s Tractatusprovides the basis for an alternative general semantics for a predicate calculus that is consistent and coherent, essentially independent of the metaphysics of logical atomism, and philosophically illuminating in its own right. The purpose of this paper is threefold: to describe the general characteristics of a Tractarian-style semantics, to defend the Tractatus system against the charge of expressive incompleteness as levelled by Robert Fogelin, and to give a semantics for a formal language that is the Tractarian equivalent of a first-order predicate calculus. Of note in regard to the latter is the fact that a Tractatusstyle truth-definition makes no appeal to the technical trick of defining truth in terms of the satisfaction of predicates by infinite sequences of objects, yet is materially equivalent to the usual Tarski-style truth-definitions     

"不可或缺性论证"与反实在论数学哲学

The indispensability argument for abstract mathematical entities has been an important issue in the philosophy of mathematics. The argument relies on several assumptions. Some objections have been made against these assumptions, but there are several serious defects in these objections. Ameliorating these defects leads to a new anti-realistic philosophy of mathematics, mainly: first, in mathematical applications, what really exist and can be used as tools are not abstract mathematical entities, but our inner representations that we create in imagining abstract mathematical entities; second, the thoughts that we create in imagining infinite mathematical entities are bounded by external conditions. __________ Translated from Zhexue Yanjiu 哲学研究 (Philosophical Researches), 2006, (8): 74–83     

Ideas and processes in mathematics: A course on history and philosophy of mathematics

This paper describes an attempt to develop a program for teaching history and philosophy of mathematics to inservice mathematics teachers. I argue briefly for the view that philosophical positions and epistemological accounts related to mathematics have a significant influence and a powerful impact on the way mathematics is taught. But since philosophy of mathematics without history of mathematics does not exist, both philosophy and history of mathematics are necessary components of programs for the training of preservice as well as inservice mathematics teachers.     

The influence of silence upon clinet-perceived rapport

The effect of frequency of instances, length and overall amount of silence was investigated during nine trainee counsellors' interviews with a Standardized Client. Data on client-perceived rapport were collected each minute, allowing for comparison of overall interviews as well as during all stages of the inteviews. Results indicated that there were significanly more instances of silence and significantly greater overall amounts of silence in interviews rated as having higher levels of client-perceived rapport. In addition, there were clearly different pattersns of usage of silence during moderate versus very high rapport minutes during interviews. Suggestions are made for the integration of these findings into counsellor training courses where trainees often report anxiety regarding the occurence of silence. Issues for further research are discussed.     

The influence of theological attitudes upon evangelism

Pastoral Psychology -     

The influence of interpolated recall upon recognition

An experiment was designed to show how immediate recall may affect recognition. A number of subjects were shown a picture. Some were asked to recall it and were then given a recognition test. Others were given only the recognition test after the same interval. Only 4 of the 16 subjects who had recalled it identified it; whereas 14 of the 16 others did so. A second experiment gave similar results.

Recall was constructed round dominant items of the picture. This distribution of emphasis together with the acceptance of an invented detail as genuine were the common causes of errors in subsequent recognition. Both the dominant and invented items in recall were those which became most obviously merged into an organization of related experiences and in consequence those which militated against subsequent recognition.

The recognition test was applied in two further groups of 16 subjects with a change in one of the dominant details (i.e. the wording). The number of subjects who now correctly identified the remainder of the material was 9 when there was no intermediate recall, and nil when immediate recall of the original material was interposed.     

The influence of Platonism on mathematics research and theological beliefs

An age-old debate in the philosophy of mathematics is whether mathematics is discovered or invented. There are four popular viewpoints in this debate, namely Platonism, formalism, intuitionism, and logicism. A natural question that arises is whether belief in one of these viewpoints affects the mathematician’s research? In particular, does subscribing to a Platonist or a formalist viewpoint influence how a mathematician conducts research? Does the area of research influence a mathematician’s beliefs on the nature of mathematics? How are the beliefs regarding the nature of mathematics connected to theological beliefs? In order to investigate these questions, five professional research mathematicians were interviewed. The mathematicians worked in diverse areas within analysis, algebra, and within applied mathematics, and had a combined 160 years of research experience. Although none of the mathematicians wanted to be pigeonholed into any one category of beliefs, the study revealed that four of the mathematicians leaned towards Platonism, which runs contrary to the popular notion that Platonism is an exception today. This study revealed that beliefs regarding the nature of mathematics influenced how mathematicians’ conducted research and were deeply connected to their theological beliefs. The findings are presented in the form of vignettes that give an insight into the mathematical and theological belief structures of the mathematicians.     

Ontology and logic: remarks on hartry field's anti-platonist philosophy of mathematics

In Science without numbers Hartry Field attempted to formulate a nominalist version of Newtonian physics—one free of ontic commitment to numbers, functions or sets—sufficiently strong to have the standard platonist version as a conservative extension. However, when uses for abstract entities kept popping up like hydra heads, Field enriched his logic to avoid them. This paper reviews some of Field's attempts to deflate his ontology by inflating his logic.