Online research in philosophy

The practice of conceptual analysis has undergone a revival in recent years. Although the extent of its role in philosophy is controversial, many now accept that conceptual analysis has at least some role to play. Granting this, I consider the relevance of empirical investigation to conceptual analysis. I do so by contrasting an extreme position (anti-empirical conceptual analysis) with a more moderate position (non-empirical conceptual analysis). I argue that anti-empirical conceptual analysis is not a viable position because it has no means for resolving conceptual disputes that arise between seemingly competent speakers of the language. This is illustrated by considering one such dispute that has been pressed by a prominent advocate of anti-empirical conceptual analysis: Bennett and Hacker ( 2003 ) assert that psychological predicates only logically apply to whole living animals, but many scientists and philosophers use the terms more broadly. I argue that to resolve such disputes we need to empirically investigate the common understanding of the terms at issue. I then show how this can be done by presenting the results of three studies concerning the application of “calculates” to computers.

To answer the question of whether mathematics needs new axioms, it seems necessary to say what role axioms actually play in mathematics. A first guess is that they are inherently obvious statements that are used to guarantee the truth of theorems proved from them. However, this may neither be possible nor necessary, and it doesn’t seem to fit the historical facts. Instead, I argue that the role of axioms is to systematize uncontroversial facts that mathematicians can accept from a wide variety of philosophical positions. Once the axioms are generally accepted, mathematicians can expend their energies on proving theorems instead of arguing philosophy. Given this account of the role of axioms, I give four criteria that axioms must meet in order to be accepted. Penelope Maddy has proposed a similar view in Naturalism in Mathematics, but she suggests that the philosophical questions bracketed by adopting the axioms can in fact be ignored forever. I contend that these philosophical arguments are in fact important, and should ideally be resolved at some point, but I concede that their resolution is unlikely to affect the ordinary practice of mathematics. However, they may have effects in the margins of mathematics, including with regards to the controversial “large cardinal axioms” Maddy would like to support.

The practice of conceptual analysis has undergone a revival in recent years. Although the extent of its role in philosophy is controversial, many now accept that conceptual analysis has at least some role to play. Granting this, I consider the relevance of empirical investigation to conceptual analysis. I do so by contrasting an extreme position (anti-empirical conceptual analysis) with a more moderate position (non-empirical conceptual analysis). I argue that anti-empirical conceptual analysis is not a viable position because it has no means for resolving conceptual disputes that arise between seemingly competent speakers of the language. This is illustrated by considering one such dispute that has been pressed by a prominent advocate of anti-empirical conceptual analysis: Bennett and Hacker (2003) assert that psychological predicates only logically apply to whole living animals, but many scientists and philosophers use the terms more broadly. I argue that to resolve such disputes we need to empirically investigate the common understanding of the terms at issue. I then show how this can be done by presenting the results of three studies concerning the application of “calculates” to computers.

Most contemporary philosophy of mathematics focuses on a small segment of mathematics, mainly the natural numbers and foundational disciplines like set theory. While there are good reasons for this approach, in this paper I will examine the philosophical problems associated with the area of mathematics known as applied mathematics. Here mathematicians pursue mathematical theories that are closely connected to the use of mathematics in the sciences and engineering. This area of mathematics seems to proceed using different methods and standards when compared to much of mathematics. I argue that applied mathematics can contribute to the philosophy of mathematics and our understanding of mathematics as a whole.

Proof and Knowledge in Mathematics tackles the main problem that arises when considering an epistemology for mathematics, the nature and sources of mathematical justification. Focusing both on particular and general issues, these essays from leading philosophers of mathematics raise important issues for our current understanding of mathematics. Is mathematical justification a priori or a posteriori? What role, if any, does logic play in mathematical reasoning or inference? And how epistemologically important is the formalizability of proof? Michael Detlefsen has brought together an outstanding collection of essays in a volume which will be essential for philosophers and historians of mathematics who are interested in the nature of reasoning and justification. A companion volume, Proof, Knowledge and Formalization is also available from Routledge.

In this paper I shall argue that to a very significant extent mathematics is concept analysis, and that though the analysis of mathematical concepts is in a number of ways different from the analysis of philosophic concepts, the similarities between these two types of concept analyses are as important and far reaching as the differences. I shall argue that because mathematics and philosophy are each concerned with the analysis of concepts, they are much more like one another epistemologically than is often recognized. In insisting upon fundamental similarities between mathematics and philosophy, I shall be agreeing with the classical rationalists, but on a very different conception of both philosophy and mathematics from that held by the rationalists. The rationalists wished to assimilate philosophy to mathematics as understood in their time; viz. as a body of necessary propositions, which followed from self-evident axioms and postulates, revealed to the natural light of reason. As against this rationalistic position, I wish to make a comparison in the reverse direction, in which I shall presuppose a certain conception of philosophy as something given, and then insist that mathematics is in many important respects similar to philosophy as so understood. In particular, I wish to insist that there is a significant comparison between mathematics on the one hand, and philosophy as understood by probably a majority of philosophers today on the other--viz., philosophy understood as concept analysis--, where it is conceded that the analysis of philosophic concepts is inherently a tentative matter, wherein it is impossible--at least in the usual case--to offer any one analysis of a given philosophic concept as absolutely certain and beyond all revision. I shall argue that by virtue of the fact that mathematics, like philosophy, is concerned with the analysis of concepts, many at least of the propositions advanced within it are inherently revisable, and do not possess the kind of certainty the rationalists ascribed to them.

The Conceptual Roots of Mathematics is a comprehensive study of the foundation of mathematics. Lucas, one of the most distinguished Oxford scholars, covers a vast amount of ground in the philosophy of mathematics, showing us that it is actually at the heart of the study of epistemology and metaphysics.

When the traditional distinction between a mathematical concept and a mathematical intuition is tested against examples taken from the real history of mathematics one can observe the following interesting phenomena. First, there are multiple examples where concepts and intuitions do not well fit together; some of these examples can be described as “poorly conceptualised intuitions” while some others can be described as “poorly intuited concepts”. Second, the historical development of mathematics involves two kinds of corresponding processes: poorly conceptualised intuitions are further conceptualised while poorly intuited concepts are further intuited. In this paper I study this latter process in mathematics during the twentieth century and, more specifically, show the role of set theory and category theory in this process. I use this material for defending the following claims: (1) mathematical intuitions are subject to historical development just like mathematical concepts; (2) mathematical intuitions continue to play their traditional role in today's mathematics and will plausibly do so in the foreseeable future. This second claim implies that the popular view, according to which modern mathematical concepts, unlike their more traditional predecessors, cannot be directly intuited, is not justified.

One recent trend in the philosophy of mathematics has been to approach the central epistemological and metaphysical issues concerning mathematics from the perspective of the applications of mathematics to describing the world, especially within the context of empirical science. A second area of activity is where philosophy of mathematics intersects with foundational issues in mathematics, including debates over the choice of set-theoretic axioms, and over whether category theory, for example, may provide an alternative foundation for mathematics. My central claim is that these latter issues are of direct relevance to philosophical arguments connected to the applicability of mathematics. In particular, the possibility of there being distinct alternative foundations for mathematics blocks the standard argument from the indispensable role of mathematics in science to the existence of specific mathematical objects.

Book description: This book contains groundbreaking contributions to the philosophical analysis of mathematical practice. Several philosophers of mathematics have recently called for an approach to philosophy of mathematics that pays more attention to mathematical practice. Questions concerning concept-formation, understanding, heuristics, changes in style of reasoning, the role of analogies and diagrams, etc. have become the subject of intense interest. The historians and philosophers in this book agree that there is more to understanding mathematics than a study of its logical structure. How are mathematical objects and concepts generated? How does the process tie up with justification? What role do visual images and diagrams play in mathematical activity? What are the different epistemic virtues (explanatoriness, understanding, visualizability, etc.) which are pursued and cherished by mathematicians in their work? The reader will find here systematic philosophical analyses as well as a wealth of philosophically informed case studies ranging from Babylonian, Greek, and Chinese mathematics to nineteenth century real and complex analysis.