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By the end of the first quarter of the twentieth century, a consensus had grown up that Kant’s view of the synthetic a priori nature of mathematics is untenable. The basis for the consensus can be illustrated most easily in the case of geometry. Euclidean geometry – the criticism runs – should be considered as either an a priori or an a posteriori discipline. Let us assume, first, that it is an a priori discipline. As such, the whole Euclidean system is a set of uninterpreted formulae, like a logical calculus. Proofs, on this account, are entirely formal, and there is no need to draw any diagrams, whether in reality or in the imagination, to establish any axiom or theorem. Thus Euclid’s axiom ‘A straight line is the shortest distance between two points’ is taken as an uninterpreted definition of the term ‘straight line’; and so on for all the other axioms of the system.The theorems of the system are established by means of rules of inference from these axioms. So understood, Euclidean judgments can have no application to the world, since the terms in the proofs are all uninterpreted, being either undefined terms or defined by means of the undefined terms. Admittedly, because the theorems result logically from the axioms, i.e. by means of the system’s rules of inference, they do follow necessarily from the axioms. But since the basic terms are merely uninterpreted symbols, there can be no question here of any Euclidean judgment holding for the structure of space.

Second, let us assume that Euclid’s system is a posteriori. As such, the Euclidean axioms can, in principle, be true judgments about the structure of space. Further, since they are based on experience, they must be synthetic (to deny any of them is not self-contradictory). But although the axioms may be true synthetic judgments, they are not, pace Kant, necessarily true. They will be merely true empirical generalizations about the structure of space. Thus the straight line axiom should here be taken as an empirical claim to the effect that since it has always been found in fact that the shortest distance between any two points in space is a straight line – where ‘straight line’ is given some suitable empirical definition, e.g. ‘the path of a light wave in a vacuum’ – it is held, on inductive grounds, that this is always the case. Consequently, the theorems of the system, since they follow logically, by rules of inference from these axioms, will equally express empirical generalizations about the structure of geometrical figures in space. They cannot, therefore, state anything with necessity or strict universality about the structure of space. On the contrary, all the theorems as well as all the axioms will have (at best) a posteriori validity because the content of the axioms – and so the theorems derived from them – are dependent upon experience. On this way of looking at Euclidean geometry, its theorems as well as its axioms will, if true, express synthetic a posteriori judgments about the structure of space.

But, the criticism concludes, you must choose one or other of these ways of viewing (Euclidean) geometry: either as a system that embodies purely a priori formulae or as a system that has axioms that are based upon a posteriori evidence about the structure of space. If you choose the first, the theorems of geometry do indeed follow necessarily from the axioms, but they have no reference to the structure of space. If you choose the second, the theorems do refer to the structure of space, but they carry neither necessity nor strict universality. There is no via media between these two ways of viewing geometry. The main points of the objection were well summed up by Einstein: ‘As far as the laws of mathematics refer to reality, they are not certain [necessary or strictly universal]; and so far as they are certain, they do not refer to reality.’

The force of this objection is not strengthened merely by the discovery or invention of alternative pure ‘geometries’. Since these pure ‘geometries’ are uninterpreted marks on paper, they are not systems that make any claims about the structure of space, and it is doubtful whether Kant himself would even regard them as genuine geometries. (In themselves, they do not show that Euclidean geometry, when its basic terms are given a common-sense interpretation, cannot describe the structure of space.) What does appear to be a serious objection to the Kantian thesis about the status of Euclidean geometry is that some of these pure ‘geometries’, when given a physical interpretation, fit the spatial universe – the Einsteinian universe – despite contradicting Euclid’s system.That non-Euclidean figures hold for certain regions of space has been thought to enforce the objection that, to the extent that one conceives of a pure ‘geometry’ as having application to the spatial world (when its basic terms are given a physical interpretation), it is a question of fact, an a posteriori matter, whether it will do so. In other words, if one considers geometry to be a body of synthetic judgments holding for the structure of space, as Kant does, then there can be no necessity or strict universality about its holding for space. No geometry, so considered, can be a body of synthetic a priori judgments.

In fact, the contemporary view is that Euclidean geometry does not fit any region of the spatial world; it is only a close approximation over short distances and under our local conditions. Consequently, the Kantian thesis that Euclidean geometry holds for the structure of space is not even accurate if its judgments are taken to express synthetic a posteriori, let alone synthetic a priori, truths.

I should add that, more recently, some philosophers have attempted a reinterpretation of the Kantian thesis about mathematics. This reinterpretation is known as ‘the constructivist view’. It denies that the theorems in a system of geometry are already contained in the axioms independent of a certain type of construction. Rather, a proof has first to be given or constructed (in accordance with the axioms and rules of the system) before a theorem is true in that system, just as on the Kantian thesis it is the construction of a figure in intuition (in accordance with a priori geometrical concepts) that makes possible the holding of a geometrical judgment. Moreover, since, for the constructivists, it is this proof that alone determines the validity of the mathematical theorem, there is no question of its being falsified by recourse to experience, i.e. a posteriori.

Although the constructivist view of mathematics does, indeed, appear to give a sense to the thesis that the validity of mathematical judgments depends upon our carrying out a process of construction, it does not, so far as I can see, help Kant to prove or even to confirm the mind-dependence – the so-called ideality – of space and time (at least in the way in which he is seeking to prove it).Yet, as we shall find in the Transcendental Aesthetic, this is what he principally hopes to achieve with his thesis that the judgments of pure mathematics are synthetic a priori. So, even if it is accepted that a constructivist view does show that a Kantian-style thesis about mathematics is after all defensible – and the constructivist view itself remains a minority one compared with the position summed up by Einstein – it would not seem to be directly relevant to furthering Kant’s own Copernican revolution.