Appendix 29. - (316)

Hume divided human cognitions into two classes: 1. those which consist in simple relationships of ideas, such as all reasoning in pure mathematics, and 2. those which deal with facts, such as the proposition: 'There is no effect without a cause.' He was concerned to eliminate this second type of a priori cognitions whilst leaving the former intact. However, even assuming such a distinction, one of these aspects of a priori knowledge could not subsist without the other. Hume's dialectic, based upon Locke's principle, is a kind of corrosive acid capable of dissolving everything; a priori knowledge vanishes completely and all a posteriori knowledge connected with it. My argument is, I think, utterly clear, and unanswerable. Hume's distinction between a priori knowledge which consists in simple relationships between ideas and a priori knowledge which deals with facts, would not affect the argument in any way even if it were sound; both would be equally doomed to destruction. What is more, the proposition, 'There is no effect without a cause', considered in all its universality, is a simple relationship between ideas like any proposition in pure mathematics, as, for example, 'Two things which are equal to a third are equal to one another.' That proposition, if applied to some particular effect or cause, involves the practical realm, just as the propositions of pure mathematics do when they are applied to bodies and thus become the source of applied mathematics. The proposition which is true in theory is also true in practice, provided care is taken to calculate all the practical elements in order to modify the result of the purely theoretical proposition. If I wish to calculate the thrust of a vault which I intend to build in order to discover the thickness of the supports which I have to provide, I start from theoretical propositions, from simple relationships regarding the nature of arches, gravity, movement, etc.; and even prior to this I begin from simple numerical calculations, in short, from the propositions of pure algebra and geometry. The certainty, therefore, of those universal, necessary propositions which are simple relationships of ideas and of those which refer to facts is intimately connected. If the first certainty exists, so does the second; they constitute a single certainty. Propositions which involve facts are merely applications of theoretical propositions which express a simple relationship of ideas. The theoretical propositions communicate their power to the factual propositions whose certainty cannot be shaken unless the certainty of theoretical propositions, communicated to the factual propositions, is shaken.

Assuming, therefore, that our a priori knowledge is divided into propositions which are only simple relationships of ideas and propositions which refer to facts, it is obvious that Hume cannot have examined closely enough the link between these two series of propositions. He assumes their mutual independence, although the second group are only derivations from the first; he assumes the presence of a priori propositions referring to facts without their being applications of antecedent propositions, that is, of meaningful, simple relationships of ideas, which is false. He was led astray by the outward form of the proposition 'There is no effect without a cause', which in referring to effects seems to refer to facts. But a careful examination shows that it refers to effects in general, to effects which are mere ideas. It does not refer to this or that real effect, in which case alone it would refer to facts. In short, it merely expresses a relationship between two ideas, that is, between the idea of cause and the idea of effect in exactly the same way as a similar relationship is expressed by this example: 'The number two is less than the number ten' or 'The angles of a triangle are equal to two right angles.' When these mathematical propositions are applied to a number of real things - for example, to a number of persons and to a particular triangle - they refer to reality in exactly the same way as the proposition, 'Every effect must have its cause', refers to facts when applied to a particular, actual effect.

Finally, Hume's distinction is false.

The universal principle from which these concrete propositions (that is, propositions dealing with facts) is derived, is mixed with the concrete propositions themselves. Consequently, the concrete propositions, too, have a certain a priori element. Nevertheless, a priori knowledge always lies in the principle itself. In other words, it lies in propositions which, although applicable to facts, express a simple relationship of ideas and as such are necessary and universal. Human cognitions, therefore, are certainly divided into a priori and a posteriori cognitions, although a priori cognitions cannot be divided, as Hume attempts to divide them, into 1. propositions which express a simple relationship of ideas and 2. propositions which deal with facts. The second kind of propositions are a posteriori, although they need the others if they are to be deduced on the occasion of external experience. Such experience provides particular facts to which general propositions can be applied and used to form a judgment about facts.

Finally experience, corresponding to the calculations of applied mathematics, bears witness to the truth and efficacy of the ideal propositions which surveyors use to rule nature and ensure its obedience.