In his doctoral dissertation On the Principle of Sufficient Reason, Arthur Schopenhauer there outlines a critique of Euclidean geometry on the basis of the changing nature of mathematics, and hence of demonstration, as a result of Kantian idealism. According to Schopenhauer, Euclid treats geometry synthetically, proceeding from the simple to the complex, from the known to the unknown, “synthesizing” later proofs on the basis of earlier ones. Such a method, although proving the case logically, nevertheless fails to attain the raison (...) d’être of the entity. In order to obtain this, a separate method is required, which Schopenhauer refers to as “analysis,” thus echoing a method already in practice among the early Greek geometers, with however some significant differences. In this essay, I here discuss Schopenhauer’s criticism of synthesis in Euclid’s Elements, and the nature and relevance of his own method of analysis. (shrink)

Following the Kantian critique of metaphysics, the conscious subject is discovered to be an insurmountable obstacle with respect to knowledge of things themselves. For this reason, Kant concludes that metaphysics as the science of being as being is impossible. In this essay, the possibilities of metaphysics in light of the problem of subjectivity are reexamined. The nature and relationship between the conscious subject and the embodiment of the subject is first examined. Following this, the subject’s “encounter with being” within consciousness (...) is analyzed yielding two fundamental structures that are determined to hold with respect to being in relation to the conscious subject. Further examination of the act of reflection coupled with judgment reveals that these structures may in fact be transcended, from which an encounter with being as such follows. On the basis of reflection and judgment, metaphysics is thereby determined to be possible. (shrink)