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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 as in mathematics, crucially depend upon their fundamental synthetic a priori judgments.There is no question, therefore, but that we are actually in possession of such judgments in natural science. What is needed is an explanation of how we can possess them.