Jeremy Chapman Reading Assignment
3726223 Kant & Geometry
The discussion of a priori and a posteriori knowledge is a complex and often argued subject. Kant's Critique of Pure Reason gives first definition, and then metaphysical treatment of these two types of knowledge. He defines a priori knowledge as knowledge that arises from intuition; that does not rely on experience for its justification. An example of a priori knowledge are mathematical propositions, for what rational person would attempt to deny that 2+2=4? A posteriori knowledge, on the other hand, is empirical and therefore is derived from direct experience or a secondary source such as reading or word-of-mouth. Two types of statements that are indispensable to this discussion are analytic and synthetic statements. Analytic statements are said to be true in virtue of definitions alone. Kant uses the example of a newspaper containing news as one of the analytic ilk, for without news in it, could one truthfully call it a newspaper? This class of statements are also termed explicative for they explicate definitions, clarifying or re-defining existing definitions. Synthetic statements are sentences that ascribe a new non-definitional property to their subject. These statements go beyond the original definition, amplifying them, and are therefore called ampliative. According to Kant's guidelines, all analytic sentences are a priori, whereas all a posteriori statements are synthetic. However, Kant does leave open the possibility of a priori synthetic knowledge, which cannot be discovered by empirical means (for it is a priori), and is a new concept (being synthetic). This knowledge springs from human reason alone, and Kant claims that our knowledge of space is exactly this: a priori synthetic knowledge.
This spate of definitions serve to allow for an understanding of Kantian geometry and Kant's views on the subject of space. The conception of space as a priori knowledge does solve some problems. This allows space to be the framework upon which all of our experiences (soon-to-be a posteriori knowledge) is arranged. It allows for spaciality, the placing of objects in relation to one another with the implicit knowledge that the medium in which they are to be placed is universal and rationally constructed. Thus we have the words above, below, beside and near and far. Without this intuitive understanding of space, one could never situate and object, or discover its relation to another object. Of course, to Kant, Euclid's axioms of geometry are knowledge of the same sort, and in fact must be a priori in order for Kant's views to be tenable. Also, since the properties of forms in Euclidean geometry are not furthering existing definitions, but are telling us something new about the world, they must be synthetic statements. This strange beast which arises not from experience, yet tells us something entirely new forms seems to run contrary to reality, yet Kant forms his conception of space upon it. Kant viewed space as intrinsically Euclidean, for all experience for him is performed in a Euclidean space that rationally "has to be". This of course was undermined shortly after his death by mathematical models of non-Euclidean space, which negated or showed inconsistency with Euclid's fifth axiom of parallel lines. This did not however hinder the acceptance and debate of Kant's definitions of knowledge or the firm footing that his Critique put the philosophy of science upon.
Questions