Unlike explanation in science, explanation in mathematics has received relatively scant attention from philosophers. Whereas there are canonical examples of scientific explanations, there are few examples that have become widely accepted as exhibiting the distinction between mathematical proofs that explain why some mathematical theorem holds and proofs that merely prove that the theorem holds without revealing the reason why it holds. This essay offers some examples of proofs that mathematicians have considered explanatory, and it argues that these examples suggest a (...) particular account of explanation in mathematics. The essay compares its account to Steiner's and Kitcher's. Among the topics that arise are proofs that exploit symmetries, mathematical coincidences, brute-force proofs, simplicity in mathematics, merely clever proofs, and proofs that unify what other proofs treat as separate cases. (shrink)

No agreed-upon account of mirror reversal is currently available although it has been discussed for more than two thousand years since Plato. Mirror reversal usually refers to recognized left-right reversal of a mirror image. Depending on the nature and layout of a reflected object, however, top-bottom reversal may be recognized instead of left-right reversal; no reversal at all may be recognized; and the presence or absence of reversal may not be decidable. Takano (1998) proposed a psycho-optic theory to explain all (...) these cases of mirror image recognition in a consistent manner. The proposed theory assumes different causes of mirror reversal depending on whether an object's left-right axis is parallel or perpendicular to a mirror's surface. This theory was later criticized in that it provided no explanation when the left-right axis is neither parallel nor perpendicular but at an intermediate angle between zero and 90 degrees with a mirror. This article completes the theory by presenting psycho-optic anal.. (shrink)

This paper develops a novel interpretation of Kant's argument from incongruent counterparts to the effect that the representations of space and time are intuitions rather than concepts. When properly understood, the argument anticipates the contemporary position whereby the meaning of indexicals cannot be captured by descriptive contents.

Philosophers and psychologists often assume that mirror reflections are optical illusions. According to many authors, what we see in a mirror appears to be behind it. I discuss two strategies to resist this piece of dogma. As I will show, the conviction that mirror reflections are illusions is rooted in a confused conception of the relations between location, direction, and visibility. This conception is unacceptable to those who take seriously the way in which mirrors contribute to our experience of the (...) world. My argument may be read as an advertisement of the neglected field of philosophical catoptrics, the philosophical study of the optical properties of mirrors. It enables us to recast familiar issues in the philosophy of perception. (shrink)

I examine some accounts that articulate the content of perception that occurs by means of a mirror. The defended account entails that a right hand seen in the mirror does not "become" a left hand.