Although traditionally neglected, mathematical diagrams have recently begun to attract attention from philosophers of mathematics. By now, the literature includes several case studies investigating the role of diagrams both in discovery and justification. Certain preliminary questions have, however, been mostly bypassed. What are diagrams exactly? Are there different types of diagrams? In the scholarly literature, the term “mathematical diagram” is used in diverse ways. I propose a working definition that carves out the phenomena that are of most importance for a (...) taxonomy of diagrams in the context of a practice-based philosophy of mathematics, privileging examples from contemporary mathematics. In doing so, I move away from vague, ordinary notions. I define mathematical diagrams as forming notational systems and as being geometric/topological representations or two-dimensional representations. I also examine the relationship between mathematical diagrams and spatiotemporal intuition. By proposing an explication of diagrams, I explain certain controversies in the existing literature. Moreover, I shed light on why mathematical diagrams are so effective in certain instances, and, at other times, dangerously misleading. (shrink)

Agents can think using concepts they do not fully understand. This paper investigates the question “Under what conditions does a thinker fully understand, or have mastery of, a concept?” I lay out a gauntlet of problems and desiderata with which any theory of concept mastery must cope. I use these considerations to argue against three views of concept mastery, according to which mastery is a matter of holding certain beliefs, being disposed to make certain inferences, or having certain intuitions. None (...) of these attitudes is either necessary or sufficient for mastery. I propose and respond to objections to my own recognition view of the conditions under which a thinker has mastery of a concept. (shrink)

Agustín Rayo’s The Construction of Logical Space offers an exciting and ambitious defense of a broadly Carnapian approach to metaphysics. This essay will focus on one of the main differences between Rayo’s and Carnap’s approaches. Carnap distinguished between analytic, a priori “meaning postulates”, and empirical claims, which were both synthetic and knowable only a posteriori. Like meaning postulates, they determine the boundaries of logical space. But Rayo is skeptical that the a priori/a posteriori or analytic/synthetic distinctions can do the work (...) Carnap wanted them to, so unlike meaning postulates, ‘just is’-statements aren’t assumed to be analytic or knowable a priori. This essay will concern the epistemology of ‘just is’-statements in Rayo’s picture. If not by a priori reflection, how can we determine which ones to accept? I’ll distinguish two competing strands in Rayo’s work. The less radical, Lewisian strand holds that the question of whether to accept a ‘just is’-statement can be addressed in a neutral, non-question-begging way, by a kind of cost-benefit analysis. The more radical, Kuhnian strand holds that there can be no ‘just is’-statement-independent, rational choice of which ‘just is’-statements to accept. I argue that Rayo faces strong internal pressure to adopt the Kuhnian picture. While it is possible for Rayo to resist these Kuhnian pressures, natural strategies for doing so leave his view more similar to Carnap’s than the above gloss suggested. (shrink)

The debate around the notions of a priori knowledge and a posteriori knowledge has proven crucial for the development of many fields in philosophy, such as metaphysics, epistemology, metametaphysics etc. We advocate that the recent debate on the two notions is also fruitful for man-made distributed computing systems and for the epistemic analysis thereof. Following a recently proposed modal and fallibilistic account of a priori knowledge, we elaborate the corresponding concept of a priori belief: We propose a rich taxonomy of (...) types of a priori beliefs and their role for the different agents that participate in the system engineering process, which match the existing view exceedingly well and are particularly promising for explaining and dealing with unexpected behaviors in fault-tolerant distributed systems. Developing such a philosophical foundation will provide a sound basis for eventually implementing our ideas in a suitable epistemic reasoning and analysis framework and, hence, constitutes a mandatory first step for developing methods and tools to cope with the various challenges that emerge in such systems. (shrink)

It is nowadays standard to distinguish between epistemic and metaphysical analyticity. Metaphysical analyticity has been widely rejected, while epistemic analyticity has been widely endorsed. I argue that we also have good reason to reject epistemic analyticity. I do so by considering all the plausible ways of characterizing epistemic analyticity and of drawing the epistemic analytic–synthetic distinction. I argue that on all of them, the distinction fails to carve at the semantic joints. I conclude that there is good reason to think (...) that there is no semantic distinction to be drawn between analytic and synthetic sentences. (shrink)

In this paper, we explore the traditional conception of a prioricity as epistemic independence of evidence from sense experience. We investigate the fortunes of the traditional conception in the light of recent challenges by Timothy Williamson. We contend that Williamson’s arguments can be resisted in various ways. En route, we argue that Williamson’s views are not as distant from tradition as they might seem at first glance.

Contemporary epistemologists typically define a priori justification as justification that is independent of sense experience. However, sense experience plays at least some role in the production of many paradigm cases of a priori justified belief. This raises the question of when experience is epistemically relevant to the justificatory status of the belief that is based on it. In this paper, I will outline the answers that can be given by the two currently dominant accounts of justification, i.e. evidentialism and reliabilism. (...) While for the evidentialist, experience is epistemically relevant only if it is used as evidence, the reliabilist requires that the reliability of the relevant process depends on the reliability of experiential processes. I will argue that the reliabilist account accommodates our pre-theoretic classifications much better. In the final part of my paper I will use the reliabilist criterion to defend the a priori—a posteriori distinction against recent challenges by Hawthorne and Williamson. (shrink)