This dissertation is centered around a set of apparently conflicting intuitions that we may have about mathematics. On the one hand, we are inclined to believe that the theorems of mathematics are true. Since many of these theorems are existence assertions, it seems that if we accept them as true, we also commit ourselves to the existence of mathematical objects. On the other hand, mathematical objects are usually thought of as abstract objects that are non-spatiotemporal and causally inert. This makes (...) it difficult to understand how we can have knowledge of them and how they can have any relevance for our mathematical theories. I begin by characterizing a realist position in the philosophy of mathematics and discussing two of the most influential arguments for that kind of view. Next, after highlighting some of the difficulties that realism faces, I look at a few alternative approaches that attempt to account for our mathematical practice without making the assumption that there exist abstract mathematical entities. More specifically, I examine the fictionalist views developed by Hartry Field, Mark Balaguer, and Stephen Yablo, respectively. A common feature of these views is that they accept that mathematics interpreted at face value is committed to the existence of abstract objects. In order to avoid this commitment, they claim that mathematics, when taken at face value, is false. I argue that the fictionalist idea of mathematics as consisting of falsehoods is counter-intuitive and that we should aim for an account that can accommodate both the intuition that mathematics is true and the intuition that the causal inertness of abstract mathematical objects makes them irrelevant to mathematical practice and mathematical knowledge. The solution that I propose is based on Rudolf Carnap's distinction between an internal and an external perspective on existence. I argue that the most reasonable interpretation of the notions of mathematical truth and existence is that they are internal to mathematics and, hence, that mathematical truth cannot be used to draw the conclusion that mathematical objects exist in an external/ontological sense. (shrink)

According to Roy Sorensen, when one looks at the Moon, during a solar eclipse, what she sees is its inner part of the farther, reflective one, and not the always-facing-Earth side of our natural satellite. To make his point clearer, he put forward the famous example of a double eclipse involving the fictional planets Far and Near. From the observer’s vantage point, the two planets have the same apparent diameter and overlap. What the agent sees is a dark disk, but (...) believes that what she is seeing is Near, because Far is behind it. Sorensen claims that what she actually sees is planet Far and that the causal theory of perception explains why this is the case. Of course, this position stands against common sense. Sorensen shows that it counters Alvin Goldman’s renowned observation criteria too. Nonetheless, he maintains, since Near is causally idle and the agent does see something, the only possible conclusion is that she sees Far, pace Goldman – and common sense. In this paper, I try to demonstrate that Sorensen is wrong and that the correct solution to the eclipse riddle is that the observer sees Near. As a matter of fact, besides meeting common sense and Goldman’s observability criteria, Near can be legitimately be considered the object of a successful perceptual discrimination even in the light of the causal theory of perception. (shrink)