In an article that deserves to be much better known [1], Arthur Lovejoy pointed out that the philosophy of Henri Bergson (1859–1941) depended on two Cartesian premises that also “serve as more or less tacit presuppositions of the most influential French philosophy of the twentieth century”, namely: (1) the assumption of a fundamental antithesis between “extended substance” and “consciousness”; and (2) the assumption of “the superior certainty of the existence of consciousness”. From these premises Bergson derives his familiar thesis “that the categories applicable to space and spatial things are essentially alien to that existence which we call our inner life, and vice versa” (467), while the latter is known with greater certainty.
These premises are then combined with a premise familiar from Immanuel Kant’s philosophy, that (3) time is the form of inner intuition, leading Bergson to formulate his task as follows: to determine what are the attributes of time as an inner intuition, firmly distinguished from any contamination by spatial concepts (466). I won’t repeat all Lovejoy’s subtle argument here, but the gist of it is this: Bergson follows his mentor Ravaisson in drawing a sharp contrast between the way concepts are represented in logic, mathematics and physics as extensive quantities, and the way they are presented to intuition, as intensive and devoid of extensive quantity (469–471). To represent a quantity as made up of distinct elements subject to numbering, or divisible into parts, is to spatialize it: its constituents can only be thought of as co-existent, and therefore spatial. Given all this, the time of inner sense —Bergson’s durée réelle— cannot consist in a succession of distinct moments or a continuum divisible into parts. But in that case, Lovejoy argues, “succession is inconceivable”, and yet “a time without succession is no time” (471). He takes that to be a reductio ad absurdum of Bergson’s position.
This article was written over a hundred years ago, so why should it be of interest now? One reason could be that these aspects of Bergson’s philosophy —the prioritizing of intuition, the conception of quantities given in intuition as intensive, the idea that consciousness is prior to and productive of the external world —were adopted by Gilles Deleuze, who continues to be revered for his original and iconoclastic metaphysics of “differential philosophy”.[2] But another reason —and this is what I will primarily be concerned with here— is that more or less the same three premises can be seen at play not just in French philosophy of the previous and present centuries, but also in the philosophical underpinnings of relativity theory, at least in how the theory was understood by those most responsible for its creation.
Let me begin with the case of Hermann Weyl (1885-1955), the philosophically astute mathematician and polymath who was a senior colleague and sometime mentor of Albert Einstein.[3] Weyl was much taken in his youth with Kant’s thesis that space and time are forms of intuition, and especially with the idealism that Fichte proposed as its natural extension, where Kant’s “thing in themselves” —independent noumena underlying our perceptions of them— are dispensed with.[4] Although Weyl came to reject much of the detail of Kant‘s philosophy, he clove fast to the doctrine of the primacy of intuition, and also to the idealist thesis that “the external world exists, in the final analysis, only as an object of consciousness” (Bell 2000, 1). Of course, as a mathematical physicist, Weyl was far from adopting a philosophy of time like Bergson’s, where time could not be extensively quantified.[5] Although he shared with Bergson the belief that reality is what is revealed to intuition, and that this could not be fully captured in mathematical models, [6] his appeal to intuition aligned rather with that of Brouwer, where the mathematical account of continuity given by Cantor and Dedekind failed to capture the “flowing” character of the continuum revealed in intuition. Weyl writes:
.. the continuity given to us immediately by intuition (in the flow of time and of motion) has yet to be grasped mathematically as a totality of discrete “stages” in accordance with that part of its content which can be conceptualized in an exact way. [ (Weyl 1987, 24; quoted from Bell 2000, 4).]
In conceiving the continuous flow of time as a succession given immediately by intuition, Weyl is clearly at odds with Bergson about what intuition delivers. But he still assumes a fundamental antithesis between “that constant form of my experiences of consciousness by virtue of which they appear to me to flow by successively” and what can be represented in mathematics.(Weyl 1987, 16; quoted from Bell 2000, 4). And the basis for reality is still that which appears to consciousness, even though “our consciousness does not directly grasp a transcendental real world which is as it appears”.[7]. Our belief in the existence of our own self, other people, and an external world, is just that: a belief. So actually it is an error of idealists like Fichte to think that “the phenomena of consciousness guarantee the reality of the ego in an essentially different and somehow more certain way than the reality of the external world.” But that concerns the existence of the subject, the Ego, not the existence of consciousness, which Weyl takes as indubitable. In “the transition from consciousness to reality,” Weyl maintains, ”the ego, the thou and the world rise into existence indissolubly connected and, as it were, at one stroke.”
This brings me, surprisingly, to the topic of reference frames. For in a lecture delivered in May 1954, Weyl presents an analogy between coordinate systems and egos (or “subjects”), which could be taken as showcasing such an indissoluble connection between the subject and the world. In this analogy objects are compared to points on a plane, subjects to (barycentric) coordinate systems in the plane, and the appearance of an object to a subject is correlated with the coordinates of a point with respect to a such a coordinate system. Here a “barycentric coordinate system” S is one centred on a point p that is the centre of gravity of a (non-degenerate) triangle in the plane with equal masses at its three vertices; the coordinates of the three vertices, which are assumed as summing to 1, define the coordinate system. As Bell explains in his article, Weyl held that this put subjects and objects in the same “sphere of reality.” He continues:
On the other hand, the appearances of an object to a subject, i.e., triples of numbers, lie, Weyl asserts, in a different sphere, that of numbers. These number-appearances, as Weyl calls them, correspond to the experiences of a subject, or of pure consciousness. (Bell 2000, 11)
The idea is that the coordinates of p relative to some other coordinate system S′ —which can be determined by a straightforward algebraic transformation of the coordinates of S— “represent the appearance of the object corresponding to p to the subject represented by S′” (Bell 2000, 11). Of course, for the coordinate system S of pitself, comprised by the three triples (1,0,0) (0,1,0) and (0,0,1) —and for this coordinate system alone— the number-appearances will correspond to the numbers themselves, so that the “object” and the “appearance to a subject or pure consciousness” of the object will coincide, prompting Weyl to call it the ‘Absolute I’ in honour of Fichte.
I will not comment further here on the idealism of Weyl’s position; and it may be thought that his philosophical predilections are in any case idiosyncratic, and have not much to do with relativity as conceived by its first creators, Albert Einstein and Hermann Minkowski.
But as I have argued elsewhere [8], this is not such a far cry from Einstein’s conception of an inertial reference frame as giving a measure of time to an observer conceived as being at rest in it (or moving inertially with respect to it), a measure he dubbed “local time”. Where Minkowski had regarded space and time as mere projections of a fundamental four-dimensional reality that he called the absolute world, for Einstein “local time”, while mathematically a projection of four-dimensional reality, was epistemologically more fundamental, in that it is defined in terms of the measures available to the observer in that frame. Arthur Eddington, who possessed aknowledge of the mathematical physics of relativity theory that was unrivalled at the time, understood this, and presented Einstein’s perspective eloquently in his Space, Time, and Gravitation of 1920:
Physical space and time are found to be closely bound up with this motion of the observer; and only an amorphous combination of the two is left inherent in the external world. When space and time are relegated to their proper source —the observer— the world of nature which remains appears strangely unfamiliar; but it is in reality simplified, and the underlying unity of the principal phenomena is now clearly revealed. [9]
It is not a stretch to see in this privileging of “the observer” another instance of the assumption of “the superior certainty of the existence of consciousness”. It is not that Einstein and Eddington are denying the existence of the external world, and they certainly would not follow Bergson in regarding the mathematization of spacetime as constituting its spatialization. It is just that, like him, they try to found the reality of time on the experiences of the subject. Let me explain.
Minkowski had shown that, although the temporal intervals between events would vary according to the inertial frame adopted to coordinatize them, there are certain “absolute relations of before and after” in spacetime that are invariant under change of frame. A possible event that is not reachable by any process emanating from O at a velocity less than or equal to the speed of light is in the absolute future of O. The trajectory of any such process connecting the event with O is a timelike curve in spacetime, and is parametrized by the proper time. The latter gives a measure of the time elapsed along such a curve, and, as Minkowski showed, it is invariant: proper time gives the same value for the time elapsed along that curve in any inertial frame you might adopt. It is specific to the path taken, not to any reference frame.
Eddington interprets this situation as follows. For him Minkowski’s “absolute relations of before and after” are unchanging relations among events that are simply there, waiting to be apprehended: “Events do not happen; they are just there, and we come across them” (1920, 46). The temporal relations to which we have access are those given by the time coordinate t, and this is determined by the motion of the observer. Eddington writes:
It is in the external world that the four dimensions are united—not in the relations of the external world to the individual which constitute his direct acquaintance with space and time. Just in that process of relation to an individual, the order falls apart into the distinct manifestations of space and time. (1920, 51)
But the coordinatization of possible events according to a given inertial frame with time coordinate t, even though perfectly objective, is an abstraction: I have no “direct acquaintance” with possible events that are simultaneous with me at a given instant. My existence and motion through time is mappable in spacetime as a timelike trajectory, and proper time gives a measure of time along it. But it is not necessary for me to be undergoing inertial motion in order for that to be so. It is therefore a mistake, sometimes suggested by Einstein, and often by physicists since, to identify proper time with Einstein’s local time. And with respect to the epistemological priority claimed for the latter, it should be noted that the measurement of time is indirect, and not a matter of what the subject experiences; to verify our conceptions through practice and experiment is not to define time in terms of the experiences of a conscious observer. Our experience of time has nothing to do with our supposedly “inhabiting” a reference frame that gives us our temporal experience, as is so often supposed.
Time lapse is an objective feature of reality. But it is no more the time of an observer occupying an inertial reference frame than it is a time of inner sense devoid of any extensive quantity. It is not determined by consciousness, although of course our awareness of it is.
Endnotes:
[1] Arthur O. Lovejoy, “Some Antecedents of the Philosophy of Bergson,” Mind, 88, October 1913, pp. 465–483. I thank Steve Savitt for bringing this article to my attention.
[2] Cf. “The physical object and mathematical space both refer to a transcendental (differential and genetic) psychology of perception” (Deleuze 1993, 88); “the object itself ceases to be an empirical given in order to become the product of [differential] relations in conscious perception” (Deleuze 1993, 89). See also his own description of his debt to Bergson (Deleuze 1994, 331, n. 14).
[3] As John L. Bell writes (“Hermann Weyl on Intuition and the Continuum,” 2000), Weyl’s “work had a vast range, encompassing analysis, algebra, number theory, topology, differential geometry, relativity theory, quantum mechanics, and mathematical logic. He was also unusual among mathematicians in possessing acute literary and philosophical sensibilities — sensibilities to which he gave full expression in his writings.” (Bell 2000, 1).
[4] For an incisive analysis and supporting quotations from Weyl’s writings, see John Bell’s article on Weyl’s philosophy cited in the previous note.
[5] In fact, as Steve Savitt remarked to me, if Lovejoy’s analysis is correct, you don’t need to probe the details of Bergson’s critique of Einstein’s theory of relativity to see why he could not accept the theory. Spacetime, as a combination of a spatial quantity with a non-quantifiable form of intuition, would be an impossible entity for Bergson.
[6] Cf.: “… the conceptual world of mathematics is so foreign to what the intuitive continuum presents to us that the demand for coincidence between the two must be dismissed as absurd.” (Weyl, The Continuum: A Critical Examination of the Foundation of Analysis, tr. S. Pollard and T. Bole. Kirksville, Mo.: Thomas Jefferson University Press. (English translation of Das Kontinuum, Leipzig: Veit, 1918.) 1987, 108; quoted from Bell 2000, 4).
[7] This and the following quotes in this paragraph are from Weyl, The Open World, 1932, 24; (quoted from Bell 2000, 2).
[8] Richard T. W. Arthur, The Reality of Time Flow (Springer, 2019), ch. 6.
[9] Arthur S. Eddington, Space, Time, and Gravitation (Cambridge UP, 1920; Preface, vii)
© Richard T. W. Arthur, September 2026