¹ For details on Kant's understanding of Euclidean geometry and the fundamental principles of Newtonian mechanics see Friedman (1992a).

² I am especially indebted to Charles Parsons for raising this problem of historical contingency and stimulating me to take it very seriously.

³ Kant often makes such claims to explanatory uniqueness, for example, in the Transcendental Exposition of the Concept of Space added to the second [B] edition (B41): ‘Therefore, only our explanation makes the possibility of geometry as an a priori synthetic cognition comprehensible. Any mode of explanation that does not achieve this, even if it appeared to be similar to ours, can be most securely distinguished from ours by this criterion’ (translation by MF). I am indebted to Dagfinn Føllesdal for emphasizing to me the importance of the problem of uniqueness in this connection.

⁴ Thus, in considering the questions ‘how is pure mathematics possible?’ and ‘how is pure natural science possible?’ in § VI of the Introduction to the second edition of the Critique, Kant simply takes it for granted that the actual existence of these sciences puts the existence of synthetic a priori knowledge entirely beyond all doubt. In particular, in considering Hume's skepticism concerning the necessity of the causal relation—which then leads to skepticism about the possibility of any a priori metaphysics—Kant blames this result on Hume's insufficiently general understanding of the problem (B20): ‘[H]ume would never have arrived at this assertion, which destroys all pure philosophy, if he had kept our problem before his eyes in its [full] generality; for he would then have seen that, according to his argument, there could also be no pure mathematics (for it certainly contains synthetic a priori propositions), and his good sense would then surely have saved him from this assertion’ (translation by MF). Similarly, while considering (in § 14 of the second edition) the circumstance that neither Locke nor Hume posed the problem of the transcendental deduction, and instead attempted a psychological or empirical derivation of the pure concepts of the understanding, Kant concludes (B127–8): ‘But the empirical derivation which both fell upon cannot be reconciled with the actuality of the a priori scientific cognition that we have—namely of pure mathematics and universal natural science—and is thus refuted by this fact [Faktum]’.

⁵ Again, see Friedman (1992a), chapters 1 and 2 for details.

⁶ For Helmholtz's characteristic combination of empirical (or ‘naturalistic’) and transcendental (or ‘normative’) elements, see Hatfield (1990). For my reading of Helmholtz's conception of space and geometry, see Friedman (1997); (2000).

⁷ Bernhard Riemann's general theory of manifolds includes spaces of variable curvature not satisfying the condition of free mobility, and it is for precisely this reason that Hermann Weyl later attempted to generalize Helmholtz's approach to comprehend the (four-dimensional) (semi-)Riemannian geometries of variable curvature used in Einstein's general theory of relativity. Moreover, as I explain in Friedman (2000), pp. 209–211, Weyl, too, conceived his work as a generalization of Kant's original theory of space as an (a priori) ‘form of experience’. The important point here, however, is that Helmholtz is ‘closer’ to Kant's original theory (in so far as his generalization preserves the possibility of geometrical constructions analogous to Euclid's), whereas Weyl's work arises only as a further generalization, in turn, of Helmholtz's.

⁸ This hierarchical conception is developed especially clearly in Poincaré (1902). For details see Friedman (1999), chapter 4; (2000).

⁹ The ‘law of relativity’ is first introduced in Chapter V, ‘Experience and Geometry’, of Poincaré (1902), p. 96, English translation (1913b), p. 83: ‘The laws of the phenomena which will happen [in a material system of bodies] will depend on the state of these bodies and their mutual distances; but, because of the relativity and passivity of space, they will not depend on the absolute position and orientation of this system. In other words, the state of the bodies and their mutual distances will depend only on the state of the same bodies and their mutual distances at the initial instant, but they will not depend at all on the absolute initial position of the system and its absolute initial orientation. This is what I shall call, for the sake of brevity, the law of relativity’. Moreover, ‘in order fully to satisfy the mind’, Poincaré continues, the phenomena in question should also be entirely independent of ‘the velocities of translation and rotation of the system, that is to say, the velocities with which its absolute position and orientation vary’ (1902), p. 98, English translation (1913b), p. 85. Thus, because of ‘the relativity and passivity of space’, the absolute position or orientation of a system of bodies in space can have no physical effect whatsoever, and neither can any change (velocity) of such absolute position or orientation. In emphasizing that Poincaré's treatment of the relativity of motion rests squarely on his philosophy of space and geometry, I am in very substantial agreement with the excellent discussion in DiSalle (2006), § 3.7.

¹⁰ For the nineteenth-century development of the concept of an inertial frame, see DiSalle (1988), (1991); for Mach's place in this development, see DiSalle (2002).

¹¹ Kant develops this interpretation of ‘absolute space’ in his Metaphysische Anfangsgründe der Naturwissenschaft (1786), published between the first (1781) and second (1787) editions of the Critique of Pure Reason. For details see Friedman (1992a), chapters 3 and 4, and also the Introduction to my (2004) translation of Kant's work.

¹² In particular, ‘absolute space’ for Kant is a regulative idea of reason, defined by the forever unreachable ‘center of gravity of all matter’ which we can only successively approximate but never actually attain.

¹³ Kant's construction of ‘absolute space’, from a modern point of view, yields better and better approximations to a cosmic inertial frame of reference defined by the ‘center of gravity of all matter’. Such a cosmic frame, in which the fixed stars are necessarily at rest, also counts as a surrogate for Newtonian ‘absolute space’ in Mach's treatment. For details, see again DiSalle (2002).

¹⁴ Poincaré formulates ‘the principle of relative motion’ in Chapter VII, ‘Relative Motion and Absolute Motion’, of La Science et l'Hypothèse (1902), p. 135, English translation (1913b), p. 107: ‘The motion of any system whatsoever must obey the same laws, whether it be referred to fixed axes, or to movable axes transported by a rectilinear and uniform motion. This is the principle of relative motion, which imposes itself upon us for two reasons: first, the most common experience confirms it, and second, the contrary hypothesis is singularly repugnant to the mind’. This, of course, is the principle of what we now call Galilean relativity, which was originally formulated by Newton as Corollary V to the Laws of Motion, and then played a central role in the recent literature on inertial frames of reference (see the references cited in note 10 above). However, as Poincaré is well aware, such Galilean relativity holds only for (uniform) rectilinear motions and does not extend, therefore, to the case of (uniform) rotational motion Poincaré also wishes to subsume under his ‘law of relativity’. Nevertheless, Poincaré says, ‘it seems that [the principle of relative motion] ought to impose itself upon us with the same force, if the motion is varied, or at least if it reduces to a uniform rotation’ (1902), pp. 136–7, English translation (1913b), p. 108. Thus, Poincaré's a priori commitment to the law of relativity, derived from the homogeneity and isotropy of space, stands in prima facie contradiction with the well-known experimental limitations of the principle of relative motion (Poincaré presents a sophisticated analysis of this apparent contradiction in the following discussion, which I shall have to pass over here).

¹⁵ In his 1912 lecture on ‘Space and Geometry’, appearing in Poincaré (1913a), Poincaré explicitly considers what we now call the four-dimensional geometry of Minkowski space-time, and he clearly states his preference for an alternative formulation of the Lorentzian type—where, in particular, both the Newtonian laws of mechanics and ‘the relativity and passivity of space’ retain a foundational role. Thus, from a modern point of view, while Poincaré's most fundamental ‘law of relativity’ is a purely geometrical principle, expressing the necessary symmetries of three-dimensional (homogeneous) space, Einstein's ‘principle of relativity’ expresses the symmetry or invariance properties of the laws of Maxwell-Lorentz electrodynamics—which we now take to be the symmetries of Minkowski space-time. The central problem with Poincaré's hierarchy, from this point of view, is that it makes the three-dimensional geometry of space prior to the four-dimensional geometry of space-time: compare again DiSalle (2006), § 3.7 for a similar diagnosis.

¹⁶ This idea is stated as a key part of Poincaré's ‘General Conclusions’ to his discussion of (classical) mechanics (1902), p. 165, English translation (1913b), p. 125: ‘[The fundamental principles of mechanics] are conventions or definitions in disguise. Yet they are drawn from experimental laws; these laws, so to speak, have been elevated [érigées] into principles to which our mind attributes an absolute value’. Later, in Geometrie und Erfahrung (1921), Einstein explicitly uses the language of ‘elevation’ [erheben] in connection with precisely Poincaré's conventionalism (1921), p. 8, English translation (1923), p. 35: ‘Geometry (G) [according to Poincaré’s standpoint] asserts nothing about the behavior of actual things, but only geometry together with the totality (P) of physical laws. We can say, symbolically, that only the sum (G) + (P) is subject to the control of experience. So (G) can be chosen arbitrarily, and also parts of (P); all of these laws are conventions. In order to avoid contradictions it is only necessary to choose the remainder of (P) in such a way that (G) and the total (P) together do justice to experience. On this conception axiomatic geometry and the part of the laws of nature that have been elevated [erhobene] to conventions appear as epistemologically of equal status' (I shall return to Geometrie und Erfahrung below). To the best of my knowledge, this striking language in Einstein's 1905 paper (together with its reappearance in 1921) has not been previously noted in the literature.

¹⁷ The crucial point, in this connection, is that Newton's third law—the equality of action and reaction—implicitly defines the relation of absolute simultaneity in a classical inertial frame, in so far as it allows us to coordinate action-reaction pairs related by the Newtonian law of (instantaneously propagated) gravitation. Einstein's two postulates take over precisely this role in the case of his new, relativized relation of simultaneity defined by (continuously propagated) electro-magnetic processes.

¹⁸ Euclidean geometry is singled out, for Poincaré, in that it is both mathematically simplest and very naturally corresponds—roughly and approximately—to our pre-scientific experience of bodily displacements. Just as Helmholtz's conception, as I have suggested, is the minimal extension of Kant's original conception consistent with the discovery of non-Euclidean geometries, Poincaré's conception is the minimal extension of Helmholtz's consistent with the more sophisticated group-theoretic version of the principle of free mobility due to Sophus Lie, the new perspective on the relativity of motion due to the modern concept of an inertial frame, and, most importantly, the apparently paradoxical new situation in electrodynamics arising in connection with precisely this relativity of motion—where, in particular, Poincaré's hierarchical conception of the mathematical sciences allows him to retain the foundational role of both Euclidean spatial geometry and the laws of Newtonian mechanics in the face of what we now call Lorentzian (as opposed to Galilean) relativity .

¹⁹ Friedman (2001a), pp. 86–91, develops more fully the parallel between these two cases of ‘elevating’ a mere empirical fact to the status of a (relativized) a priori principle by first examining the relationship between the invariance of the velocity of light (as recently verified in the Michelson-Morley experiment) and Einstein's new definition of simultaneity, and then the relationship between the equality of gravitational and inertial mass (as recently verified in the Eötvös experiments) and the principle of equivalence.

²⁰ See Norton (1985) for the details of Einstein's early applications of the principle of equivalence.

²¹ I discuss at length the crucial importance of the rotating disk example in the development of Einstein's thought—following Stachel (1980)—in Friedman (2001a), (2002).

²² For a detailed analysis of Geometrie und Erfahrung, against the background of both Helmholtz and Poincaré, see Friedman (2001a), (2002).

²³ Ryckman (2005), § 3.3, emphasizes the importance of this passage in relation to the earlier argument of Geometrie und Erfahrung.

²⁴ As I suggested, Einstein could not embrace Poincaré's philosophy of geometry even in 1905, since it privileges a priori the three-dimensional geometry of space over the de facto symmetries of the laws of motion (which, on our current understanding, express the four-dimensional geometrical symmetries of space-time). Einstein's divergence from Poincaré on this point is even stronger in general relativity; for, not only do we now use non-Euclidean geometries to describe both space and space-time, but we have also definitively given up (in both cases) the homogeneity and isotropy (constant curvature) of the underlying geometry. Einstein thereby ultimately arrived at a radically new conception of physical geometry envisioned by neither Helmholtz nor Poincaré. For details, see again Friedman (2002).