Abstract
Hegel’s Philosophy of Nature begins with the concepts of space and time, and all of the concepts and phenomena that follow in the text are spatio-temporal. But the actual content of Hegel’s theory of space and time has remained obscure despite two centuries of interpretation, and obscure for at least two reasons. First, it is unclear how Hegel’s theory relates to those of his immediate predecessors (Newton, Leibniz, Kant) as well as other theories in the history of philosophy. Second, the theoretical function to be played by this theory of space and time has remained unclear. In this paper, we clarify the first issue by interpreting Hegel’s theory as a theory of space-time, and we clarify the second issue by interpreting that theory as securing the theoretical possibility of motion. Of course, it will not be the same theory of space-time as the one that proceeds from modern relativistic physics; but designating Hegel’s theory as a theory of space-time helps to mark it out from its most prominent predecessors and helps to make sense of some of its most distinctive features—particularly Hegel’s insistence that time has three dimensions just as space does. This latter feature—the three-dimensionality of time—serves to distinguish Hegel’s view both from the classical mechanics of Newton and Kant (in which time does not interact with the spatial directions) and from contemporary relativistic physics (in which there is such interaction but time is only a single dimension added to the spatial dimensions). In Hegelian space-time, three-dimensional time is the flip side of three-dimensional space—it is this conception that makes Hegel’s view so distinctive.
In der Vorstellung ist Raum und Zeit weit auseinander, da haben wir Raum und dann auch Zeit; dieses “Auch” bekämpft die Philosophie.
Enc2 § 257 Z
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Notes
- 1.
In contrast with most of the literature on Hegel’s theory of space and time, we will have almost nothing to say about the status of space and time as subjective, objective, or both. Our reconstruction focuses on the structure and function of Hegelian space-time. For a good review and original contribution to the former debate, as well as a detailed analysis of the subjective side of space and time, see de Vries (2016).
- 2.
Technically this is a vector function f that assigns to each time t a position f(t) in three-space. In coordinates, this would be f(t) = (x(t),y(t),z(t)) where the three entries are the three spatial dimensions and the variable t is the one-dimensional parameter.
- 3.
Mathematically, if the motion is reasonably well behaved, this is indeed a curve, i.e., a locally one-dimensional continuous or even smooth subspace in three-dimensional space. It may have self-intersections like a figure 8.
- 4.
Another good visual image are the light paintings, e.g. “Walker Street” by Eric Staller.
- 5.
This is a general concept: a graph of any function f from a set X to a set Y is given by the pairs (x,y) in the Cartesian product of X and Y, that is x in X and y in Y, for which y = f(x). Hence formalizing the motion as a function f from time T to space S, the world-line or graph will be the points (t,p) where p = f(t), i.e., those time-position pairs (t,p) the where p is the position at time t. This is again a one-dimensional subspace, but now for the four-dimensional product of time and space. One gets back the curve C by the projection onto the space part, which maps (t,p) to p. The image of the world-line is the set of all the f(t), which is precisely C. Hegel’s insight is that the trace of C as a geometric object, as in the statement that the obits of planets are ellipses, is not sufficient to capture motion.
- 6.
For special relativity there is an additional step to a third conception of the relation between space and time precisely by saying that there is more to the relation of space and time than the Cartesian product. In special relativity, for example, the Lorentz boosts with the contraction of length and the dilatation of time are a way of adding this additional character to the relation. This surprising relation means that there is a possible interaction of space and time directions and dimensions.
- 7.
Of course, the mathematical versions of dimension all coincide for nice examples like the ones we are discussing here. For the mathematician: the dimension here is the dimension n of the affine space An which is the linear or vector space dimension of Rn. This coincides with the covering dimension of Rn in the standard topology and the algebraic Krull dimension of the ring R[x1,…,xn] which is an inclusion definition. We mention this to illustrate that these different concepts lead to conceptually different definitions of dimensions mathematically formalizing the particular aspect.
- 8.
For the sake or full disclosure, real n-dimensional space is the set of n-tuples of real numbers Rn. This has the same cardinality (size) as affine n-space An. And furthermore, the cardinality is the same for any n greater than 1. This cardinality is the cardinality of the continuum which is \( {2}^{\aleph_0} \). This type of algebra of sets is taken up by Hegel in his discussion of mathematics, see Kaufmann and Yeomans (2017). Here he moves beyond the logical/mathematical setting to a physical one, that of nature. Consequentially he utilizes a more sophisticated notion of dimensionality which is more that the granular, set-theoretic being-next-to-each-other (Nebeneinander).
- 9.
Note that we use the term here in the modern mathematical sense (as entailing a space with an orientation and a metric), rather than in the philosopher’s sense of a space in which the parallel postulate holds.
- 10.
We use the name ‘translation’ without a proper definition, which would make things a bit too formal.
- 11.
We note here in passing the correspondence with the Zariski topology of inclusion dimensions, since there is no space in the text to take it up. That correspondence is that there is a marked difference between the point and a line helps elucidate Hegel’s somewhat puzzling statement “That the line doesn’t consist of points, the plane doesn’t consist of lines [nicht aus Linien besteht], emerges from their concept” (Enc2 § 256R). Although one commonly represents the line as composed of points, this does not capture its one-dimensionality, which comes from the progression point-line. In terms of Krull dimension, the ideal (x,y) plays a different role than (x,y,z) and so on. The correct principle in algebraic geometry is that of specialization, in particular of the generic point to generic points of surfaces to generic points of curves to a geometric aka closed point.
- 12.
This is the four in the development of nature afforded by the doubling of the particular as discussed in Kaufmann et al. (2021).
- 13.
In Kaufmann et al. (2021), these progressions were tied to Hegel’s logical argument where the first progression corresponded to the level of progressing to four, made possible by two particulars as available in nature. For the enclosing surface a second distinction is necessary which allows one more step, that culminates in the ultimate sophistication of five in Hegel’s count.
- 14.
Cf. Carnap’s disctinction between logical, intuitive, and physical space in Carnap (1922).
- 15.
Inwood is certainly right to note that Greek philosophers are much more influential for Hegel’s theory of space and time than is Kant (Inwood (1987, 49)).
- 16.
In terms of the ideals, it is clear that the zero ideal is always an ideal and lies below any other ideal.
- 17.
The arithmetic starts with the “dead one [tote Eins]” (Enc2 § 295) obtained by paralyzing the principle of time. This one can be repeated hence added or multiplied but cannot not enter configurations as it does not have externality as space does. There can be three minutes or three intervals of time, but unlike three points which may define a triangle, they do not give a configuration. On the other hand, three times the same point is not a notion of Euclidean geometry. This type of multiplicity does appear in tangency conditions, which are implicit in Hegel’s unfolding of space.
- 18.
Actually, the progression continues into spirit, where we first get the actual experience of the past and the future, but that is too long a story for this paper. For the outlines, see Yeomans (2022).
- 19.
There are Aristotelian resonances here that we cannot explore in the current context. In any event, Hegel’s view ends up closer to Aristotle’s view than either Leibniz’s relational view or Newton’s container view. But we take this sophisticated story of dimensionality to be the way in which Hegel avoids the potential circularity of Aristotle’s view, on which it looks like spatial objects define the space that surrounds them. Cf. Zigliogi (2015).
- 20.
Hegel himself discusses the relationship of the two-dimensionality of the plane and theorems about Euclidean triangles in Enc2 § 265Z. One upshot of these consideration is that the dimension of the local space appears through a power count, as in the Pythagorean theorem, and coincides with the unfolding dimension of the global space. The tetrahedra, although not directly in Hegel, are part of the Pythagorean tetractys philosophy.
- 21.
There are actually two standard approaches to classical mechanics, the Euler-Lagrange version which uses coordinates and their time derivatives and the Hamiltonian, which uses positions and momenta as conjugate variables. Mathematically one then looks at a 2n dimensional manifold, with n space-like dimensions, with special features that is called a symplectic manifold which means woven together, that is the space and conjugate momentum (time).
- 22.
Strictly speaking, there is a difference in space and time directions for special relativity which is expressed by a sign in the metric, and in general relativity which is expressed through a signature. In technical terms space-time is a pseudo-Riemannian manifold, not a Riemannian one, like space alone. Nevertheless, the possible transformations are allowed to mix space and time directions, just like rotations in time mix the coordinates or length, width and height.
- 23.
In particular, speed is in m/s and indeed there is no naturally occurring phenomenon which has a physical unit of s/m. Although of course one can mathematically construct them, they do not fit into a logical exposition of nature like the one Hegel is undertaking.
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Kaufmann, R., Yeomans, C. (2023). Hegel’s Theory of Space-Time (No, Not That Space-Time). In: Corti, L., Schülein, JG. (eds) Life, Organisms, and Human Nature. Studies in German Idealism, vol 22. Springer, Cham. https://doi.org/10.1007/978-3-031-41558-6_6
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