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Extra info for Lectures on an introduction to Grothendieck's theory of the fundamental group
The text of The Fourfold Root we usually read today is this later version. Schopenhauer begins The Fourfold Root with the single principle of sufficient reason which was the stock-in-trade of the eighteenthcentury academic tradition associated with Leibniz and Christian Wolff. The principle states simply: Nothing is without a ground or reason why it is (R, 6). Nothing is self-standing; everything is in relation to something else which is the reason for its being, or the explanation of it. However, there are, according to Schopenhauer, four distinct ways in which something may relate to a ground or reason, associated with four 19 different kinds of explanation, which, he claims, none of his predecessors has clearly distinguished.
We observe the way a state of affairs is in the world of objects, judge that it should be altered or preserved, and form the intention to act. Schopenhauer's chief point is that none of this is yet willing. The operations of perception, thought, and intention are quite separate preparatory events which may trigger the will the body, that is into action. He plays down the gap between willing and the movements one carries out with one's body, concentrating instead on the gulf between 42 representing the world of objects, and being in goal-seeking motion within it.
He makes two large claims: first, that nothing can be both object and subject; secondly that there can never be a subject without an object, or an object without a subject. It is the last point which he takes to establish idealism, and indeed to make it something obvious. Nothing can be an object for experience without there being a subject to experience it or think about it. But why must we think of material objects in space and time in this way? Schopenhauer would argue that the point of calling them objects is to indicate that they can and do fall within our experience.