CHAPTER VIII.
GENERAL OBSERVATIONS AND RESULTS.
§ 46. The Systematic Order.
The order of succession in which I have stated the various forms of the Principle of Sufficient Reason in this treatise, is not systematic; it has been chosen for the sake of greater clearness, in order first to present what is better known and least presupposes the rest. In this I have followed Aristotle's rule: καὶ μαθήσεως οὐκ ἀπὸ τοῦ πρώτου, καὶ τῆς τοῦ πράγματος ἀρχῆς ἐνίοτε ἀρκτέον, ἀλλ' ὅθεν ῥᾷστ' ἂν μάθοι (et doctrina non a primo, ac rei principio aliquando inchoanda est, sed unde quis facilius discat).[156] But the systematic order in which the different classes of reasons ought to follow one another is the following. First of all should come The Principle of Sufficient Reason of Being; and in this again first its application to Time, as being the simple schema containing only what is essential in all the other forms of the Principle of Sufficient Reason, nay, as being the prototype of all finitude. The Reason of Being in Space having next been stated, the Law of Causality would then follow; after which would come the Law of Motives, and last of all the Principle of Sufficient Reason of Knowing; for the other classes of reasons refer to immediate representations, whereas this last class refers to representations derived from other representations.
The truth expressed above, that Time is the simple schema which merely contains the essential part of all the forms of the Principle of Sufficient Reason, explains the absolutely perfect clearness and precision of Arithmetic, a point in which no other science can compete with it. For all sciences, being throughout combinations of reasons and consequences, are based upon the Principle of Sufficient Reason. Now, the series of numbers is the simple and only series of reasons and consequences of Being in Time; on account of this perfect simplicity—nothing being omitted, no indefinite relations left—this series leaves nothing to be desired as regards accuracy, apodeictic certainty and clearness. All the other sciences yield precedence in this respect to Arithmetic; even Geometry: because so many relations arise out of the three dimensions of Space, that a comprehensive synopsis of them becomes too difficult, not only for pure, but even for empirical intuition; complicated geometrical problems are therefore only solved by calculation; that is, Geometry is quick to resolve itself into Arithmetic. It is not necessary to point out the existence of sundry elements of obscurity in the other sciences.
§ 47. Relation in Time between Reason and Consequence.
According to the laws of causality and of motivation, a reason must precede its consequence in Time. That this is absolutely essential, I have shown in my chief work, to which I here refer my readers[157] in order to avoid repeating myself. Therefore, if we only bear in mind that it is not one thing which is the cause of another thing, but one state which is the cause of another state, we shall not allow ourselves to be misled by examples like that given by Kant,[158] that the stove, which is the cause of the warmth of the room, is simultaneous with its effect. The state of the stove: that is, its being warmer than its surrounding medium, must precede the communication of its surplus caloric to that medium; now, as each layer of air on becoming warm makes way for a cooler layer rushing in, the first state, the cause, and consequently also the second, the effect, are renewed until at last the temperature of stove and room become equalized. Here therefore we have no permanent cause (the stove) and permanent effect (the warmth of the room) as simultaneous things, but a chain of changes; that is, a constant renewing of two states, one of which is the effect of the other. From this example, however, it is obvious that even Kant's conception of Causality was far from clear.
On the other hand, the Principle of Sufficient Reason of Knowing conveys with it no relation in Time, but merely a relation for our Reason: here therefore, before and after have no meaning.
In the Principle of Sufficient Reason of Being, so far as it is valid in Geometry, there is likewise no relation in Time, but only a relation in Space, of which we might say that all things were co-existent, if here the words co-existence and succession had any meaning. In Arithmetic, on the contrary, the Reason of Being is nothing else but precisely the relation of Time itself.
§ 48. Reciprocity of Reasons.
Hypothetical judgments may be founded upon the Principle of Sufficient Reason in each of its significations, as indeed every hypothetical judgment is ultimately based upon that principle, and here the laws of hypothetical conclusions always hold good: that is to say, it is right to infer the existence of the consequence from the existence of the reason, and the non-existence of the reason from the non-existence of the consequence; but it is wrong to infer the non-existence of the consequence from the non-existence of the reason, and the existence of the reason from the existence of the consequence. Now it is singular that in Geometry we are nevertheless nearly always able to infer the existence of the reason from the existence of the consequence, and the non-existence of the consequence from the non-existence of the reason. This proceeds, as I have shown in § 37, from the fact that, as each line determines the position of the rest, it is quite indifferent which we begin at: that is, which we consider as the reason, and which as the consequence. We may easily convince ourselves of this by going through the whole of the geometrical theorems. It is only where we have to do not only with figures, i.e., with the positions of lines, but with planes independently of figures, that we find it in most cases impossible to infer the existence of the reason from the existence of the consequence, or, in other words, to convert the propositions by making the condition the conditioned. The following theorem gives an instance of this: Triangles whose lengths and bases are equal, include equal areas. This cannot be converted as follows: Triangles whose areas are equal, have likewise equal bases and lengths; for the lengths may stand in inverse proportion to the bases.