Cartesian epistemology and infallible justification

In this paper I examine contemporary accounts of noninferential justification in light of what I take to be the Cartesian project of building epistemology on foundations made secure by the impossibility of error. I argue that familiar abstract arguments for foundationalism, by themselves, don’t seem to motivate Cartesianism. But I further argue that there is one version of foundationalism that is more closely linked to the way in which Descartes sought ideal knowledge.     

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.     

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.     

"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     

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 role of philosophy in the contemporary abortion debate.

Inspired by Patrick Lee's "A Christian Philosopher's View of Recent Directions in the Abortion Debate," this essay raises the question of how effective philosophical arguments can be in determining the moral status of legalized abortion. On one hand, Christian philosophers have been successful in explaining both the humanity and the personhood of the unborn child, as well as exposing the incoherence of those who would deny the unborn child's humanity or personhood. Nevertheless, in order to confront the pro-abortion position in its most radical form, a much more complex philosophical argument must be given. Following thinkers such as Alasdaire MacIntyre, Christian philosophers must articulate and promote a philosophical position according to which morality is conceived in richer terms than the mere respecting of individual rights. The social dimension of human nature must be rediscovered in order that the happiness and welfare of others becomes a desirable goal in and of itself. According to a morality where individual rights is the bottom line (for example, that of Judith Jarvis Thompson), women very well may have the right to "extricate" themselves from their pregnancy even when doing so will result in the death of their child. What must be explained, therefore, is the more profound insight that social morality is equally concerned with obligations to others, including those who are most helpless and unable to speak for themselves.     

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 history of philosophy in contemporary philosophy: The view from Germany

Conclusion I have frequently mentioned objective problems and topics in the preceding sections. But what exactly is the force of objective here? As my remarks should have made clear I have been using objective to contrast with purely historical. A purely historical approach never gets beyond reproduction, commentary, and interpretation. I call an approach objective when it involves a philosopher who advances his own theses and claims.This minimal understanding of objectivity (in the context of my remarks in this paper) by no means implies that there are problems and topics, systems of concepts, methods, and similar factors that are eternal, completely independent of the contingencies of history (of philosophy, of the sciences), that are not relative to a language, to a logic, to a model, etc. Indeed whether there are problems, etc., in just this absolute, atemporal sense is itself a question for systematic philosophy. It seems clear that the formulation of a problem can only take place against a cognitive background of some sort and within some conceptual scheme.³⁴ Such an assumption is made by most if not all analytic philosophers. But the fact that a philosophical tradition recognizes conceptual schemes does not make it a purely historical, non-objective philosophy, in the sense already introduced and described. A philosopher who explicitly accepts a certain conceptual scheme proceeds in an entirely objective and systematic (and not purely historical) manner when, within this framework, he formulates his own theses.This paper is the text of a talk. the title is due to Barry Smith.