On the bankruptcy of the synthetic a priori

1.

For Kant, an analytic proposition ‘contains within the predicate the identity of the subject’ (cf. ‘Transcendental Aesthetic’). Quine had something to say about the vagueness of a predicate's being able to 'contain' a definition within itself, but this point of his 'Two Dogmas' was already attacked by the time Wittgenstein invented analytic philosophy.

2.

For Wittgenstein, an analytic proposition is something which is defined by its 'results' (i.e., its application) and the way in which these results co-determine the definition of the subject of the predicate. For example, the proposition 'the straightest line between two points is always at the same time the shortest distance' is for Wittgenstein an analytic one. Like Kant, an analytic proposition is for Wittgenstein not determined by experience. Unlike Kant, however, Wittgenstein thought that the same applied to synthetic a priori knowledge. For him we define a shortest line to be the straightest within two points. This is not something which we test against our representations (not a synthesis of regulative principles in the mind and the intuitions of our experience); nor is it something which we recognise as having no criterion against which to test it (as Hume said in book one of his Treatise on Human Nature). Empirical information simply doesn't come into it. The shortest distance between two lines can't be 'shown' to be analytic; it is simply defined as such. The shortest distance is in all cases the straightest line because the definition is co-determined by these two constituents. So, analytic propositions cannot divine the consequences of their rules because their meanings are contained in their results (e.g., the angles of a triangle add up to 180°; if they did not, we’d be talking about a different shape). In spite of this, such propositions must count as analytic because they are valid by definition, because if their validity is affected, then so is their meaning. To those geometries which show the actual shortest distance between two points to be a curved line (through the assumption that space-time itself is curved) we attribute a different meaning than the geometry in which the rule concerning shortest lines applies. We recognise that the rule does not hold in this case, but we also recognise that this is a special case of the rule, which exists alongside the ordinary rule. Of course, one could argue that the ordinary rule about the shortest distance between two points is determined by the limits of our powers of mental representation, even if it has no absolute ontological validity.

3.

Kant thought that Euclidian geometry held in all cases and determined the possibility of mental-visual representations. This seems to be true for the majority of our actual visual representations (our 'intuitions' as he put it in the 'Transcendental Aesthetic') – with the crucial exception of the parallel postulate. This is disproved on so simple an object as a globe bisected with lines of latitude and longitude. The globe shows that parallel lines - which are defined as the sum of two right angles each - converge at the apex of the globe. The fact that we can actually represent this to ourselves in our field of vision sounded the death-knell for Kant’s theory of the transcendental limits of representation. Modern speculation about ghost particles, etc, too, shows that our knowledge of the world needn’t be predetermined by the limits of what is in some sense directly representable to us. In other words, the theory of transcendental idealism has no ultimate bearing on our knowledge of the world, and cannot provide a foundation to a fruitful and realistic ontology.

The problem is that Kant’s doctrine of Transcendental Idealism led him to treat propositions as if they were the tools of a static thing – the mind – projecting its results onto the world, as if a synthetic priori proposition belonged to a storehouse in the mind, which ensures its correctness in virtue of its transcending space and time, which form the basic constituents of empirical experience. Wittgenstein demolished this transcendental ontology of the separation of mind and world by suggesting that the binding force of an a priori proposition is the formulation of the sentence and the use of the constituent terms in relation to what already exists in space and time. The analytic proposition, therefore, is not a ‘regulative principle’ or doctrinal postulate that determines the possibility of existing and future objective knowledge. This point is made forcefully in Wittgenstein’s Philosophical Grammar, and certainly more succinctly than this. Wittgenstein thereby avoided the ontological problems to which transcendental idealism succumbed.