‘That which distinguishes the third class of representations, in which Space and Time are pure intuitions, from the first class, in which they are sensuously (and moreover conjointly) perceived, is Matter, which I have therefore defined, on the one hand, as the perceptibility of Space and Time, on the other, as objectified Causality.’The first part of this is straightforward enough, but the ‘therefore’ was initially rather puzzling. Why should those two definitions of Matter follow from the representability of both pure and empirical intuitions? Well, after some discussion, we decided that the ‘therefore’ only pertains to the first definition of Matter, ‘as the perceptibility of Space and Time’. Perhaps this is Schopenhauer writing poorly, or a problem with the translation. In any case, having bracketed the second definition, I could understand the connection, namely that it is Matter which makes spatio-temporal perceptions possible, in the synthesis of Matter and Form which comprises an empirical intuition. Of course, space and time can be represented independently of matter, as pure intuitions, according to Schopenhauer (and Kant). I think I need to clarify the different senses which the word 'representation' seems to have in Schopenhauer.
Schopenhauer gives two brief arguments for the representability of space and time as pure intuitions. Regarding space, he says that complete representations (empirical intuitions) add the determination of being empty or filled. I presume the implication is that we must be able to represent the formal (infinite) container - namely space - first. Secondly, he states that ‘the infinite expansion and the infinite divisibility of Space and Time are exclusively objects of pure intuition and foreign to empirical perception.’ Presumably this is a reference to Kant’s Transcendental Aesthetic, and I suppose he would refer anyone who asks him to say more about the representability of pure intuitions to those pages in Kant (in fact, he rehashes Kantian arguments in §36).
In §35 paragraph 3, Schopenhauer states that the form of Causality is not a pure intuition, belonging as it does, to the understanding. He writes that we cannot have consciousness of it (i.e. represent it) ‘until it is connected with what is material in our knowledge’ (p.154). Notably, he does not argue further for this claim. Why does he feel the need to state this view, yet not to argue for it? It reads as though he’s reminding us of an obvious point, proved elsewhere. Perhaps he simply thinks that Kant’s arguments in the Aesthetic just obviously don’t apply to the form of Causality; it being, after all, merely a condition of thought, not a condition of experience.
The PSR of Being is the ‘law by which the divisions of Space and Time determine one another reciprocally’. That is to say, the law by which ‘all parts stand in mutual relation... [and] each... conditions and is conditioned by another.’ Dina asked why he called it the PSR of Being. I think he chose the term to connote two things, namely: (1) the eternal, unchanging nature of these forms and (2) their status as conditions of experience, and hence, of objective existence.
A consequence of all this, for Schopenhauer, is that axioms of geometry and mathmatics can be explained as founded upon pure intuitions. In §39 he distinguishes between a reason of Being - corresponding to a conviction that something is the case - and a reason of Knowing - which supports an understanding of why something is the case. The fundamental axioms are of the former sort. So, a reason of Being can underwrite a series of reasons of Knowing. Someone asked whether the opposite can happen too. I think not, because I take the distinction to be asymmetrical: a reason of Knowing must surely contain a reason of Being - knowledge why X is the case must presuppose that X is the case. So a reason of Knowing couldn’t support a reason of Being.
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