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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.

      Natural science

      On the face of it, Kant does not provide any detailed reason for affirming that the first principles of physics are genuine instances of synthetic a priori judgments. (Physics is regarded as determining the behaviour, the dynamical relations, of matter in space.) Of course, there seems no great difficulty in comprehending why he should think that these principles claim this status. Thus the principle ‘Action and reaction must be equal in all communication of motion’ does claim necessity and universality, and it is also synthetic (B 17–18). The difficulty lies in understanding his grounds for affirming that the first principles of physics not only claim to hold but actually do hold as a body of synthetic a priori judgments.

In the Introduction to the First Critique, his assertion appears to rest solely on the near unanimity of opinion, among scientists, as to which group of synthetic a priori judgments forms the first principles of physics (or natural science). Undoubtedly, he does think of the first principles of physics as in a privileged position compared with those of transcendent metaphysics, where there is no unanimity about what judgments constitute its first principles. Undoubtedly, too, he is suspicious of the claims of transcendent metaphysics because of this lack of unanimity. But his ground for accepting a given group of synthetic a priori judgments as forming the genuine body of first principles of physics does not rest on the mere fact that this group commands unanimous assent among scientists. It is based primarily on the close parallel which he sees between the procedures in mathematics and physics.

      This parallel is argued for in the second edition Preface (B x–xiv), just before he turns to consider whether a procedure similar to that employed in mathematics and physics might be attempted in metaphysics. With regard to mathematics and physics, his first point is that both are plainly in the canon of the sciences: they both possess a set of first principles, and they both yield, partly by means of their first principles, a vast body of results that are everywhere agreed to hold with a priori certainty. In fact, if we study the procedures of these two sciences, he believes that we shall find that their results are based either wholly (in the case of mathematics) or substantially (in the case of physics) on non-empirical foundations.A proof in mathematics, e.g. concerning some property of an isosceles triangle, depends on axioms or principles like ‘A straight line is the shortest distance between two points’ (together with certain non-empirical constructions and observations); and a proof in physics, e.g. concerning some property of moving balls on an inclined plane, depends on first principles like ‘In all communication of motion, action and reaction must always be equal’ (together with certain empirical constructions and observations). These axioms or principles are, in both cases, synthetic as well as a priori – even though the first principles of physics, as opposed to those of pure natural science upon which they depend, are not entirely free from the addition of some very general empirical input.

So Kant’s claim that the first principles of physics are genuine instances of synthetic a priori judgments is by no means based solely on the unanimous assent concerning these fundamental judgments (as opposed to the palpable lack of such unanimity in transcendent metaphysics). Rather, it is based mainly on what he sees as a close parallel between the procedures in mathematics and physics, together with the extraordinary success of these procedures in yielding a huge number of results that are everywhere acknowledged to hold with necessity and universality. If the procedures in mathematics yield results, and a huge number of results, that have a priori certainty, and if closely analogous procedures in physics are similarly successful, then the first principles of physics (or natural science) as well as the axioms of mathematics must be genuine instances of synthetic a priori judgments. For the certainty of the results in physics, just

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