James R. O'Shea

Bounds on the levels of composition algebras

Certain families of quaternion and octonion algebras are conjectured to be of level and sublevel n. A proof of this conjecture is offered in the case where n is a power of two. Hoffmann’s proof of the existence of infinitely many new values for the level of a quaternion algebra is generalised and adapted. Alternative constructions of quaternion and octonion algebras are introduced and justified in the case where n is a multiple of a two power.

The needs of understanding: Kant on empirical laws and regulative ideals

Abstract This article examines the relationship in Kant between transcendental laws and empirical laws (focusing on causal laws), and then brings a particular interpretation of that issue to bear on familiar puzzles concerning the status of the regulative maxims of reason and reflective judgment. It is argued that the ‘indeterminate objective validity’ possessed by the regulative maxims derives ultimately from strictly constitutive demands of understanding.