2. Basic Concepts: Position (Lage) and Direction (Gegend)

Before we can use Kant’s material absolutism from his works prior to 1768 to explain the argument of 1768, we need some sense of the fundamental notion of Gegend or direction and its contrast notion Lage or position. Position is clear: it means ‘the reference of one thing in space to another’ and ‘the positions of the parts of space in reference to each other’ (GUGR, 2: 377).Footnote ¹⁵ Lage also seems to be represented most primitively by ‘the concept of being next to each other’ which Kant mentions in the Inquiry (UDG, 2: 280). This symmetrical relation of ‘being next to’ relates to our most basic idea of extension and impenetrability, because two things are only next to each other (contiguous) insofar as they do not penetrate the same place. It is important to note that Lage encompasses the whole of Leibnizian analysis situs.Footnote ¹⁶ Analysis situs concerned both quantitative and qualitative aspects of space and only excluded the theory of measure.Footnote ¹⁷ Leibniz explains the content of analysis situs through the example of ‘a seeing mind concentrated at a point’ being shown two temples in which ‘nothing can be found in one which you do not also find in the other’ (Leibniz Reference Leibniz1956: 225). The two temples may differ in magnitude but ‘the proportions of walls, columns, and all the rest are the same; the angles are everywhere equal or in the same ratio to a right angle’. Neither a body, an external reference point, a measuring stick, nor even continuous experience allows the seeing mind-point to compare the temples. In effect, the mind-point only has its equations.

Analysis situs lacks not just a theory of measure but even the conditions for measuring: the two temples are effectively two different worlds, and the mind-point that inhabits both cannot be thought of as a continuous being that travels spatially from one to the other. But presumably, even without the shared world that allows for such measuring, the mind-point could still grasp the spatial features of each temple on its own, whereas Kant’s argument is that this is not so, and what is left out, at least in temples that are similar but incongruent, is Gegend or directionality. The notion of Gegend has been defined as the idea of ‘the direction of an orientation’ (Walford in Kant Reference Kant1992: 457), directionality of order, or, synonymously, a ‘directionality of sense’ (Walford Reference Walford2001: 423).Footnote ¹⁸ The idea is that there is a different sense to directional orientations given the asymmetrical nature of these sorts of relations. ‘X is to the right of Y’ is not equivalent to its converse nor to the symmetrical relation of being-next-to. In effect, Kant’s problem in the essay is to explain why there is such a difference of sense, and how it can hold of some relations or objects even considered in isolation by the mind-point and not in our shared world.

There are two notions of directionality at work in Kant’s essay, one of which is internal to the object and one of which concerns external spatial relations. The former will be called handedness, which is a non-metric, qualitative, asymmetric property of things. The latter concerns the locating of things in space relative to some set of directional coordinates, which is a derivative notion that results from applying the primitive, non-metrical directionality of sense to a theory of spatial coordination and measurement. This metric directionality of position can obviously change with movement (X is to the right, now it is to the left) whereas handedness cannot. Once we are occupying the shared world with its system of distances and measurements, then it is tempting to reduce directionality as a whole to metrical directionality and to see the latter as a description that can be eliminated for a purely mathematical or quantitative one. In other words, the directional system of above and below, right and left, and front and behind, may seem to be reducible to a non-directional co-ordinate system (GUGR, 2: 379). But such a reduction to a purely mathematical space also eliminates the sense of asymmetric directionality that allows there to be handedness, whereas there are, undeniably, such asymmetrical relations in nature, in particular, in incongruent counterparts.

This system of directional notions – up, down, left, right, front, back – is what Kant refers to in the title of his work as ‘directions in space’ (Gegenden in Raume) (GUGR, 2: 375).Footnote ¹⁹ Kant believes that other directional notions, such as north, south, east and west (compass points or Weltgegenden) are dependent on the more basic directional system (up, down, left, right, front, back). The availability of metrical directionality to us is due to the three-dimensionality of physical space, thought of as planes, intersecting at right angles at the point of an observer (GUGR, 2: 378–9). The directional system centred at the three-dimensional planes dividing the observer’s body is not reducible to a series of symmetrical relations. There remains a difference of sense between different directions, suggesting that there is an underlying asymmetry behind the notions. For example, down or below is distinguished from up or above by the traditional force of downward gravity. Front and back are obviously distinguished by the forward-facing nature of our senses and propensity to move in that direction. And left and right have physical differences, according to Kant, such as the fact that we favour one side over the other.

The title of Kant’s essay tells us that there is an ‘ultimate ground’ of this system of directions, not to be confused with an epistemic ground.Footnote ²⁰ That ultimate ground is called direction (Gegend rather than Gegenden) ‘in the most abstract sense of the term’ and is described as ‘the relation of the system of these positions [of parts of space in reference to each other] to the absolute space of the universe’ (GUGR, 2: 377). Besides calling this a ‘relation’ to absolute space, which tends to be mistakenly thought of as an external spatial relation, Kant also says that the direction of order or sense ‘refers’ to ‘universal space as a unity’ (GUGR, 2: 378). How can absolute space be the ultimate ground of the primitive system of directional notions? That is the fundamental question of Kant’s essay. Absolute space, on the classical Newtonian model, is the directionally neutral, three-dimensional, continuous, unified, immaterial, immovable system of positions (Lagen).Footnote ²¹ This has meant that all that absolute space seemed to offer by way of a ‘relation’ is the relation of position (Lage). For example, Earman argues that Kant has ‘no coherent argument at all’, because appealing to absolute space to understand direction and asymmetry in an object can only make uses of ‘points and lines of the space external to’ the object, which merely reproduces the directional, asymmetrical order that was to be explained (1991: 137). Kant rejects considering this ‘relation to absolute space’ merely in terms of a relation to a place or ‘places in the space – for that would be the same thing as regarding the position of the parts of the thing in question in an external relation’ (GUGR, 2: 378). This, I believe, rules out the possibility of thinking that absolute space explains direction by virtue of the fact that it indicates absolute place. Kant’s solution stems from the fact that the directional system is irreducible to a non-directional co-ordinate system, but this solution is only available if we can explain how the sense of directional asymmetry is a part of nature rather than merely our perceptions.

3. Kant’s Argument for Absolute Space Based on Incongruent Counterparts

Kant’s argument against relational theories of space is based on the fact that the Leibnizian mind-point cannot distinguish between two temples that are identical in every way except that one has an asymmetry that is mirror-reflected in the other. Of two such objects, Kant says that they would be ‘exactly equal and similar, and yet still be so different in themselves that the limits of the one cannot also be the limits of the other’ (GUGR, 2: 381). These left-right reversed temples should be congruent on Leibniz’s account, but in actuality they are incongruent.Footnote ²² ‘The surface which limits the physical space of the one … cannot serve as a boundary to limit the other, no matter how that surface by twisted and turned’ (GUGR, 2: 382). Kant’s primary argument asks us to ‘imagine that the first created thing was a human hand’, which we clearly see would ‘have to be either a right or a left hand’ (GUGR, 2: 383). If we were relational theorists about space, then we would believe that there is nothing other than body and ‘all actual space would simply be the space occupied by this hand’ (GUGR, 2: 383).Footnote ²³ Because, it seems, directionality is ‘found in the constitution of bodies’, then the relational theorist will either admit that the solitary hand is either left or right and that this difference can be found in the geometric nature of the parts of the hand, or else the relationist must deny that the solitary hand would be either left or right (GUGR, 2: 383).Footnote ²⁴ The latter denies that there is any intrinsic property of things that determines left or right orientation.Footnote ²⁵ The former says that difference lies in the shape of a thing or how its parts are combined, although this is precisely what the Leibnizian analysis situs describes and, in so doing, leaves out incongruence.Footnote ²⁶

Kant believes that he has shown that no geometry that concerns only positions and magnitude can account for this intrinsic asymmetry of order, in particular, the sense of the difference between similar and equal but incongruent orders. The geometry of position will examine ‘the proportion and the position of its parts to each other’ and ‘scrutinize the magnitude of the whole’ to arrive a complete description, both qualitative and quantitative (GUGR, 2: 381). But it will not find the ‘inner’ difference that distinguishes a handed object from its counterpart, because that difference simply cannot be analysed in terms of ‘the relation and position of its parts to each other’ (GUGR, 2: 381).Footnote ²⁷ Kant does seem to say that this property does belong to a thing’s ‘physical’ or ‘corporeal form’ (GUGR, 2: 381). I take Kant’s point to be that it belongs to the materiality of the thing in contrast to its mathematical form. It is tempting to think that what he means is that this property is a feature of the way that the material extension that constitutes a thing is embedded within ‘universal absolute space’ (GUGR, 2: 381). But that is an entirely relational, positional description. I suggest that the ‘reference’ of this thing to ‘universal absolute space’ consists not in its positionality relative to the whole but in how its qualitative feature of handedness pairs with the asymmetry of space itself.

The solution that I propose is drawn from Kant’s early natural science writings. Kant believes that there is a basic matter or Urstoff, made entirely of an active, attractive force. This primary matter is incapable of being thought of as impenetrable extension or as consisting of molar bodies in space, for such bodies and the repulsive forces that create them only emerge after the initial, infinite distribution of this ‘original material of all things’ (NTH, 1: 228, 264). In the proto-universe, this field of attractive force is divided into regions of different intensities and these regions ‘put each other into motion’ and are ‘a source of life for themselves’ (NTH, 1: 264). He calls the original matter of the universe a ‘subtle though universally active matter which, in the products of nature, constitutes the active principle, and, as a true Proteus, is able to assume all shapes and forms’ (FEV, 1: 211). From the original dispersal of this prime matter, ordinary matter and the forces described by Newtonian physics emerge. Most importantly, the Urstoff or ‘elements of self-forming nature’ all ‘acquire a general motion from west to east’ (NTH, 1: 266, 285). The Urstoff is originally just ‘attraction points’ – ‘nothing had formed yet’ (NTH, 1: 340, 263). This ‘elementary basic material at the beginning of all things … occupied the entire space of the universe’ and was the ‘simplest [state of nature] that could follow upon nothingness’ (NTH, 1: 263). Yet it is misleading to say this material fills space as if space is something other than it. Kant is clear that, after matter emerged and the field of force became less chaotic, our apparently empty space consists of an omnipresent ‘substance of infinitely small resistance’, which he believes is Newton’s view of absolute space (FEU, 1: 186). After the stronger attraction points coalesce into matter, what remains of the elementary material and what makes up what we call empty space is an ‘infinitely thin matter’ too weak to interact with ordinary matter but, presumably, still preserving the original rotation that all force initially had (NTH, 1: 338). In this way we have an answer as to how the solitary hand would be either left or right, without anyone being there to judge it so, because its asymmetry can be thought of as corresponding or not with the asymmetry of this basic material that makes up space itself.

4. Kant’s Solution and the Fall of Parity

Although there is nothing mysterious about asymmetrical objects, Kant’s puzzle is why there is a fundamentally different sense associated with incongruent counterparts that cannot be accounted for by the geometry of his day. This is not merely an epistemological puzzle but, like the classic Ozma Problem, it can be illustrated via the problem of knowing this difference of sense. The Ozma problem asks ‘how can we communicate … our meaning of left and right’ to some distant planet when ‘there is to be no asymmetric object or structure that we and they can observe in common’ (Gardner Reference Gardner2005: 167). Suppose you and I are each a mind reduced to a point in Leibniz’s temples, each of which has an asymmetrical order that is the mirror image of the other. The question is whether one mind-point could communicate to the other which temple it is in without any reference to a commonly shared exemplar of left-right asymmetry. The solution to the Ozma problem is to appeal to a fundamental law of nature that is invariably asymmetric across the entire shared universe. If the law of parity held and everything in nature fell into symmetrically divided classes, then there would be no such law. But the law of parity does not hold in nature. Gardner points out that ‘the south end of a cobalt 60 nucleus is most likely to fling out an electron’ (Gardner Reference Gardner2005: 216) or if one looks at neutrinos they all appear ‘like a bullet spinning from a rifle with left-handed rifling’ (McManus Reference McManus2002: 139). Rusnock and George note that ‘evidently, Kant was a firm disbeliever in parity’ but do not allow this as a solution to the ontological problem of directionality. They see this as nothing more than appealing to local asymmetries, and that this is ‘just as serviceable’ a solution to Kant’s problem, because he is only concerned with the epistemic problem of how we go about telling left from right (Rusnock and George 1995: 272).

But Kant’s examples in the 1768 essay are meant to show law-like deviations from the principle of parity (setting aside whether his examples are true or not): all human hair grows in a spiral left to right, all hops spiral from left to right, ‘whereas beans wind in the opposite direction’, ‘almost all snails, with the exception of perhaps, only three species’ have shells that spiral left to right, and these patterns do not depend on contingent circumstances or in location but are caused by ‘the seeds’ of the things themselves (GUGR, 2: 380). In the organization of our own body, ‘the right side … enjoys an indisputable advantage’ in terms of skill and power and ‘all peoples of the world are right-handed (apart from a few exceptions which … do not upset the universality of the regular natural order)’, whereas the left side is more sensitive (GUGR, 2: 380). My proposal is quite simple: Kant’s view is that there is some deep, natural and universal law that operates in an asymmetric and directional way, against which directional objects can be said to either match in their directionality or differ. That nomological foundation is the asymmetrical left-right directionality of the motion of space (and any part of space) itself.Footnote ²⁸ This differs from Rusnock’s and George’s claim that ‘if space has absolute direction, then hands are left and right because of their relations to these directions’, which is an appeal to ‘absolute north’ or ‘absolute west’ (1995: 271). That fails to explain the asymmetrical difference of sense that allows there to be a directional difference between east and west in the first place. For there to be absolute, irreducible metric directionality, there must be a non-metric directionality that does not rely in any way on spatial positionality and is instantiated in a way that is universal and lacking in parity.

If we look at one of the most obvious influences on Kant’s early natural science writings, Descartes’s Principles of Philosophy, we find that there is an essential directionality to nature, and it is bound up with incongruent counterparts. Descartes argues that ‘secondary matter’ consists of spherical contiguous bodies that form ‘curved-line’ triangular gaps and that elements of some of the ‘first matter’ fill and pass through them (Descartes Reference Descartes1644: 3. 90). Although some elements of the first matter are extremely fast-moving and almost constantly changing shape, there are slower moving and more regularly shaped bits of first matter that have straight-line motions through these triangular gaps made by the contiguous spheres of the secondary matter (Descartes Reference Descartes1644: 1. 89). This bulkier primary matter has the following shape: they are ‘small fluted cylinders with three grooves or channels which are twisted like the shell of a snail’ which allows them ‘to corkscrew their way through the little triangular spaces’ between the secondary matter (Descartes Reference Descartes1644: 3. 90).Footnote ²⁹

So it is clear that the rotating centre of the heavens causes some portion of the material elements of the plenum to become, according to a universal and invariable law, incongruent counterparts.Footnote ³⁰ Because of the parity of these particles emanating from the poles, this may offer no solution to the Ozma problem. Nevertheless, it shows that a major influence on Kant’s own cosmology thought that the most basic, primary matter of the universe was not only asymmetrical in its rotation but actually had the corporeal nature of being incongruent counterparts.

5. Resolution of Two Interpretative Difficulties

A Newtonian conception of space that is entirely ‘positional’ has led to a failure to properly characterize both the property of handedness and its relation to absolute space. As Walford notes, Kant’s ‘central conclusion’ must be that directionality (such as being right-orientated or left-orientated) is a ‘primitive and non-metric spatial property’ that belongs to some objects (2001: 418). But not even Walford has recognized that Kant attributes that property to space itself, making his absolutism a relative of Aristotle’s, in which there was a natural above-below directionality (Aristotle 1984: 208b9–25; Jammer Reference Jammer1954: 19). The confusion can be seen in Rusnock’s and George’s claim that Kant cannot ‘make up his mind’ whether direction is an ‘inner characteristic’ or an ‘outer characteristic’. They decide that is an outer one, because those are given ‘through comparison with other bodies’ – the other outer characteristics being ‘magnitude and external place’ (Rusnock and George 1995: 265). Similarly, Van Cleve says Kant’s solution is a form of ‘extrinsicism’ because ‘leftness or rightness of a hand does depend on something outside of it’, whereas the ‘intrinsicist’ holds that the difference between left and right pertains to the parts of a thing (1991: 204). These interpretations fail to recognize anything other than positional relations either internal to the hand or to something ‘outside’ the hand. But the entire point of Kant’s essay is to challenge the completeness of such a view, by showing that non-metric directionality (handedness) is something belonging to the object, but only subjectively available to a knower who has a sense of the difference between left and right, and only objectively in the world insofar as there is a universal seat of the difference of sense between the asymmetrical relation left-right. The puzzle about how to interpret handedness rests on applying a division between inner and outer characteristics that is spelled out in entirely positional terms, which Kant is rejecting.Footnote ³¹ To use Van Cleve’s terms, Kant is a ‘holist’ – left and right is a ‘thoroughly nonrelational affair’ and these are ‘monadic properties’ – of which, Van Cleve says, ‘no one I know of has held this view’ (1991: 205). As it turns out, Kant did.

A second interpretative difficulty has been confusion over the difference between handedness (a primitive, non-metric property) and metric directionality by which we orientate things in spatial relations of distance via compass points or other Gegenden. In the latter relations, such as ‘X is 5 metres to the left of Y’, a change in the perspective of the observer seems to change the relation itself. Thus these relations seem reducible to a purely quantitative relation. Of course metrical, directional relations are important for Kant’s point, leading him to discuss them on several occasions, because, insofar as we are not eliminativists regarding them, they are dependent on a fundamental difference of sense that accompanies and distinguishes the right-left relation from, for example, the 5-metres-apart relation. But Kant’s fundamental concern is the non-metrical property of handedness, which no change of position or motion can modify. Kant’s conclusion, I maintain, is that there is a dependency ‘on the reference of that physical form to universal absolute space’ which is a ‘ground’ of this non-metric, monadic feature of things because space itself has this property (GUGR, 2: 381). It is not the mere unity nor tri-dimensionality nor even metric directionality of space that accounts for this unless, at least in the last case, one recognizes that metric directions of absolute space itself (absolute up, down, left, right) are only possible insofar as there is a universal asymmetrical relation that lacks parity in a nomological way, such as all space having one direction of rotation.

The failure to separate non-metric directionality from metric directionality derives in part from Kant’s ambition to show that all material extension (compounded matter) stands in metric directional relations in space. Kant begins the 1768 article with the claim that ‘the positions of the parts of space in reference to each other presuppose the direction in which they are ordered in such a relation’ (GUGR, 2: 377). We may certainly grant that our epistemic position, as self-moving beings who navigate the world and locate ourselves and objects via directions, is one for which all positional relations necessarily appear as directionally orientated. But Kant is interested in more than this claim about our directionally dependent cognition.

Kant holds that absolute space is the very foundation of the compound character of matter. This view is one articulated in his early natural scientific writings. The proto-universe, prior to the emergence of compound matter, was a chaotic field of material, attractive force that would have seemed empty to us given the complete lack of any aggregation, but it would not have allowed the sort of regular, uniform movement that we now see in space. The winnowing of the chaotic intensities of force gave rise to compound matter and left behind the weakest, most rarefied forces as the material foundation of the seemingly empty space in which we now experience free motion. Although this may show why extension belongs to ‘universal space as a unity’ and that absolute space is the ground of the actuality of compound matter, it does not show that it is the ground of the possibility of the compound character of matter – that is presumably a stronger claim about necessity (GUGR, 2: 378). That necessity is established by Kant’s argument from handedness, which tells us that materiality itself, which is capable of being formed into figures with handedness, must be a part of a space that contains a universal, non-regional, non-metric, directional asymmetry. And this only seems possible if we reject both the relationist view of space as well as an immaterialist space without qualities. Absolute space is certainly independent of all matter or bodies with a reality of its own, as Kant says in the passage just quoted, but it must be thought of as a material substrate with at least the dynamic, asymmetrical quality of directional rotation.

The door seems opened for Kant’s earlier force-based absolutism based on his argument from incongruence. In that story, cosmic, absolute metric directionality emerges. After all the regions of varying intensity aligned, produced aggregated matter and left behind a rarefied material space, what also happened was an organization of infinite space itself (and all compound matter) around a central, gravitational core. Around the rotating centre of the universe there were poles, and the early vertical fields of matter and force were condensed to a horizontal plane or disc of projected force around which all matter sank into regular orbits. Although there is no analogue to front-back directionality, we can say that the universe and space itself has north-south poles, a central plane indicating the middle and the above-below distinctions, as well as a left-right or east-west rotation. These directional notions seem to emerge at the level of simplest attraction-points of space itself, as well as at the level of the cosmic unity of the whole of space. If this is so, then it will be the case that all positionality does in fact have directionality relative to absolute directions of the universe. Thus it will make sense to say of, for example, an actual, material isosceles triangle that the angle made by its two equal sides is orientated absolutely left of its unequal side (and not just opposite it) based on its relation to the centre of the universe and that one equal angle is above (or to the north of) the other in absolute relation to the central plane of the universe.

The aforementioned problems have led scholars to argue that Kant has no way to connect handedness and absolute space. But because of the connection between the cognition of the handedness of objects and our own intrinsic sense of the difference between left and right orders, scholars have decided that Kant effectively gives up on the metaphysical claims of the essay and arrives at a view near to that of the Critical philosophy. For example, Walford claims that what Kant means by the primitiveness of certain concepts, and of Gegend in particular, is that ‘they can only be apprehended intuitively’ (2001: 415). Rusnock and George argue, with regard to appeal to absolute space as an explanation of direction, that Kant ‘takes the steam out of this line of argument’ because he also suggests a solution ‘independent of absolute directions’ – we can apprehend incongruence by comparing it to some norm or available standard (1995: 272). Nuzzo argues that Kant senses the difficulties of his own argument and ‘ends, as it were, with a proof of a different thesis’, which is that ‘the body’ and our own intuitive grasp of directionality ‘replaces absolute space’ (Nuzzo Reference Nuzzo2008: 35). This mixing of the epistemic issue of how we know left from right (in ourselves and in objects) with the ontological problem of the ground of directionality in nature and in space is something Kant would simply not allow in 1768. The obvious reason for this is that it would mean that, without some being to actually intuit the universe with a solitary hand, then it would be neither left nor right, which, he has argued, is absurd.