What is the relationship between the world and logic, between intuition and language, between objects and their quantitative determinations? Rationalists, on the one hand, hold that the world is structured in a rational way. Representationalists, on the other hand, assume that language, logic, and mathematics are only the means to order and describe the intuitively given world. In World and Logic, Jens Lemanski takes up three surprising arguments from Arthur Schopenhauer’s hitherto undiscovered Berlin Lectures, which concern the philosophy of language, (...) logic, and mathematics. Based on these arguments, Lemanski develops a new position entitled ‘rational representationalism’: the world is always structured by human beings according to linguistic, logical, and mathematical principles, but the basic vocabulary of these structural descriptions already contains metaphors taken from the world around us. (shrink)

Kant often seems to suggest that a cognition – whether an everyday cognition or a scientific cognition – cannot be beautiful. In the Critique of Judgment and the Lectures on Logic, he writes: ‘a science which, as such, is supposed to be beautiful, is absurd.’ (CJ 184 (5:305)) ‘The expression "beautiful cognition" is not fitting at all’ (LL 446 (24:708)). These claims are usually understood rather straightforwardly. On the one hand, cognition cannot be beautiful since on Kant’s account, it is (...) all about concepts whilst beauty is defined by its non-conceptual nature. On the other hand, beauty cannot contribute to cognition since the former is grounded on subjective feelings whilst cognition is all about objective knowledge. However, I will argue that Kant’s view of the relationship between cognition and beauty is not as straightforward as it may seem, and that both of these claims are in fact false. As I will show, cognition can be beautiful, and the feeling of beauty is cognitively valuable. Yet it is not because beauty is a sign of the truthfulness of a theory. Nor is it because the process that gives rise to the feeling of beauty, the free play, furthers scientific progress. Rather, it is because the experience of beauty stimulates our cognitive powers and thereby enhances our cognitive activity. On this basis, contrary to what is usually thought, cognition can, and in fact should, be beautiful for Kant. (shrink)

It is a common thought that mathematics can be not only true but also beautiful, and many of the greatest mathematicians have attached central importance to the aesthetic merit of their theorems, proofs and theories. But how, exactly, should we conceive of the character of beauty in mathematics? In this paper I suggest that Kant's philosophy provides the resources for a compelling answer to this question. Focusing on §62 of the ‘Critique of Aesthetic Judgment’, I argue against the common view (...) that Kant's aesthetics leaves no room for beauty in mathematics. More specifically, I show that on the Kantian account beauty in mathematics is a non-conceptual response felt in light of our own creative activities involved in the process of mathematical reasoning. The Kantian proposal I thus develop provides a promising alternative to Platonist accounts of beauty widespread among mathematicians. While on the Platonist conception the experience of mathematical beauty consists in an intellectual insight into the fundamental structures of the universe, according to the Kantian proposal the experience of beauty in mathematics is grounded in our felt awareness of the imaginative processes that lead to mathematical knowledge. The Kantian account I develop thus offers to elucidate the connection between aesthetic reflection, creative imagination and mathematical cognition. (shrink)