Abstract

Is space absolute, relational, or is the distinction a conceptual confusion? This is the question at issue in Kant's critique of Leibniz. On first blush, it would seem that Kant and Leibniz share some common ground, as neither sees space as an entity with completely independent existence. Yet they differ significantly as to its role in cognition. For Leibniz, space is merely a relation between objects, one which does not necessarily depend on the mind. This made him oppositional to absolutist views of space (such as those held by Isaac Newton) that treated space as an object-independent entity that objects lie within, which Leibniz considered contrary to a priori principles of inference. Ever the contrarian, Kant considered both views to be confused; he did not see space as being a thing unto itself or a relation between objects, contending that both views go beyond what our reason experience can tell us. To Kant, both views make space something which we cannot know, either by making it a thing in itself (as in the absolutist view) or a relationship between things in themselves (as in Leibniz’s relational view). For knowledge of space to be possible it cannot relate to things in themselves, but must be a form of our intuition, meaning that which allows us to have cognition at all. In this post, I will explain the underlying justification for each view in greater detail and, further, elaborate on some of the other questions which might turn on these differing understandings (in other words, why it matters that they disagree on this point and what different conclusions it might imply). In particular, I will explore what each view says about the possibility of knowledge of space and of the objects therein; to Kant, Leibniz’s view leaves our cognition of objects uncertain, and thus unable to ground the physical sciences and protect them from skepticism. Evaluating this charge from Kant, I claim that his arguments are ultimately successful in countering Leibniz, but limit the sciences to a point that is hard to reconcile with developments that would come after his time.

Background to the Debate and Leibniz’s Ontology

The quarrels of Newton and Leibniz are often framed with respect to which one invented calculus, but they were fiercely opposed in many other respects. Included here are their different conceptions of space, either as absolute in the case of Newton, or relative in the case of Leibniz. As Leibniz is the subject of this essay, I will devote considerably more time to his ontology than to Newton’s, and indeed, Leibniz’s most forceful statements on the subject were usually in dialogue with Newtonians rather than Newton himself. It suffices that to say space is absolute means that “[a]bsolute space, in its own nature, without regard to anything external, remains always similar and immovable.”¹ To explain, this means that space exists whether or not there are objects placed in it, that each part of space is identical to every other, and that space cannot be affected by the objects placed in it. With this in mind, I will give a compressed overview of Liebniz’s ontology, to show why the absolutist conception is so anathema to it.

The first important thing to understand about Leibniz’s ontology is its fundamental unit: the monad. Monads are simple substances, in that they have “no parts, though [they] can be a part of something composite.”² These monads are unextended, but nonetheless comprise all composite objects, including “portions of matter” which, presumably, do have extension.³ Further, each monad “must be qualitatively unlike every other” monad, and in such a way that these differences are internal to the monad, i.e they cannot merely differ in their relation to one another.⁴ The second important principle of Leibniz’s ontology is that for every existing thing (and thus for every true statement) there must be a reason for that thing’s existence (or truth). This is the principle of sufficient reason (hereafter the PSR), and it is what grounds all contingent truths⁵ “even if in most cases we can’t know what the reason is.”⁶ This may seem, on first glance, to be an epistemological rather than an ontological principle. However, Leibniz takes the PSR to constrain not just our reason, but in fact what God will create, as we shall later see.

The contours of Leibniz’s ontology are fascinating and interesting in their own right, but what I have given thus far is enough to explain why Leibniz takes a relative view of space. He puts his view very simply in his correspondence with Samuel Clarke, who represents the absolutist perspective:

As for my own opinion, I have said more than once that I hold space to be something purely relative… that I hold it to be an order of coexistences… For space denotes, in terms of possibility, an order of things that exist at the same time, considered as existing together, without entering into their particular manners of existing. And when many things are seen together, one consciously perceives this order of things among themselves.⁷

Even on a cursory examination, it is easy to see why the absolutist view would be incompatible with Leibniz’s monadic ontology. Were space absolute, what would it be to Leibniz? It could not be a monad, as it consists of divisible parts and thus is not a simple substance. Moreover, these divisible parts are supposed to be “always similar” in that each part of space is identical with every other.⁸ But this cannot be the case for Leibniz, as he holds all monads to be internally distinct, which as I explained, means they cannot merely differ in their location in space.

So absolute space would pose a major problem for the rest of Leibniz’s ontology, but this may not itself form an independent counterargument to independent space. For that, Leibniz turns to the PSR. He poses a question: if space really exists and is absolutely uniform, then why should god have placed objects into space as he did? Why for instance, would not God have made “east into west,” or placed each object 7 units away from where he placed them in fact?⁹ The answer is that there would be none: “it is impossible there should be a reason why God, preserving the same situations of bodies among themselves, should have placed them in space after one certain particular manner and not otherwise.”¹⁰ A relational view completely sidesteps this issue. Were God to have made east as west, “those two states… would not at all differ from one another” on the relational view, and thus there is no violation of the PSR.

Clarke, for his part, attempts to show that a relational view of space would run into the same exact problem. He imagines “three equal particles… placed or ranged in the order a, b, c” and asks why, on a relational view, God would not have arranged them as ‘b, c, a’ or ‘a, c, b’. Just as in the case of space, there could be no sufficient reason to arrange the particles in any specific way, and thus says Clarke, the absolutist and relative conceptions rise and fall on the same grounds. However, I find Clarke’s response to misunderstand Leibniz’s point. As Leibniz himself states in reply, one can simply say that “no such [particles] will be produced by [God] at all, and consequently there are no such things in nature.”¹¹ One can argue that this response seems ad hoc, but in fact it offers an independent argument for a principle which Leibniz already accepts, i.e that no two things are absolutely identical. If God were to have made two identical things, there would be no sufficient reason to have placed them in the order “a, b” or “b, a”, and he would not have created two identical things. What’s more, even if it is ad hoc, it nonetheless successfully differentiates the two views. On a relational view, you get the odd result that there cannot be identical particles; meanwhile, on the absolutist view, you have the result that God would never have reason to place objects in space at all, which is manifestly false as things do indeed exist. So long as one accepts the PSR and a divine creator (which were, at the time, considered quite indispensable), I believe Leibniz’s arguments against the absolutist view to be more or less watertight.

Kantian Space and the Possibility of Natural Science

Thus far, I have operated as if the absolutist and relativist views are opposites. Leibniz seemed to operate on this assumption implicitly, as his argument for his own views are more easily construed as arguments against absolutism. If they are true logical opposites then this would be sufficient. But what if, rather than one being the negation of the other, they were merely a false dichotomy that precluded another set of options? Such was the contention of Immanuel Kant; in his mature philosophical works, Kant articulated a view which repudiated both the absolutist and relative views of space as not only being wrong, but as dangerously skeptical.

Kant famously attributed the inspiration for his most mature philosophical works to David Hume, not because he agreed with Hume’s conclusions, but because his skeptical challenges “broke into my dogmatic slumber, and pointed my work in speculative philosophy in a completely new direction.”¹² It is important to view Kant’s philosophical explanation of space in precisely this context; that he was attempting to ground our knowledge of space and of physics more generally against possible skeptical challenges by dissolving the ground on which those challenges stood. To that end, he erected a wall between two worlds, that of phenomena or “sensible entities,” and noumena, or “a thing that isn’t to be thought as an object of the senses but is to be thought (solely through a pure understanding) as a thing in itself.”¹³ There is a sense in which objects of sense are “correlated” with sensible entities, but it is notable that any knowledge we have of noumena must be that of pure understanding and cannot be empirical knowledge. When we cognize an object, we are not seeing it as it really is but merely a representation conditioned by our “pure intuition.”¹⁴ This is where space enters the picture for Kant, as it is one of the “pure forms of sensible intuition, serving as principles of a priori knowledge” (emphasis in the original).¹⁵ In other words, space is not something that we represent, but is the way that our mind represents sensible objects.

This move is extremely significant, both in the context of Kant’s broader project and in our question of his ontology of space. What we see here is that, in effect, Kant has no ontology of space, as it is not a thing at all but instead the way that we represent things in the first place. Moreover, because it is the precondition of representation and empirical knowledge rather than something we acquire knowledge of through investigation, knowledge of space is a priori. This is how Kant overcomes skepticism. By making space something which we have knowledge of a priori and does not really exist, he paradoxically secures it against skeptical doubt based on the unreliability of our senses and makes empirical knowledge possible. This may seem confusing at first, but consider it with respect to the concept of cause. A skeptical challenge against causation may say that all we experience are constant conjunctions, and thus it may only be, as Kant puts it, “a mere fantasy of the brain.”¹⁶ But if cause is instead a law which governs our intuition that can be understood a priori, it once again becomes legitimate to speak of causality and inferences we make from it as being real knowledge. Far from “turning the sensible world into illusion,” it actually secures knowledge against illusion, for otherwise “it would be quite impossible to decide whether the intuitions of space… aren’t mere phantoms thrown up by our brain, with nothing adequately corresponding to them.”¹⁷

This concept of space thus recontextualizes the relational/absolute binary as being built on a faulty premise, one which would make knowledge of space absolutely impossible. It is easy to see how this applies to the absolutist view; if space exists absolutely prior to experience then this would make it a thing in itself, and thus knowledge of it would be impossible. The incompatibility is not as immediately obvious on the relational view, but it comes into focus with the understanding that on the relational view, objects are still said to in some sense exist independently. In Leibniz’s case of the monads, these simple substances are not objects of sense, and must be noumena. But one of the most important things we cannot know about noumena is how they relate to other things, or as Kant puts it “we must… strip them of any relations to other things.”¹⁸ If nothing can be said of the relations of noumena, then this will invariably include its spatial relationships. And indeed, this distinction is not without purpose. Leibniz was, like Kant, concerned with grounding scientific knowledge as real, and believed “[t]hat science becomes real and demonstrative” through the PSR.¹⁹ Yet he simultaneously maintained that “most of the perceptions in human souls are but confused,” and so even if we are to say, as Leibniz does, that “the succession of these perceptions is regulated by the particular nature… which always expresses all universal nature,” what ground would allow us to accept this assertion?²⁰ It is this notion of “confused” representation that Kant expertly avoids. By allowing that space is a form of our intuition of which we have a priori knowledge, he does what Leibniz is unable to do by grounding scientific knowledge in something which we know with absolute necessity.

A More Specific Argument against Leibniz, and Kant’s Limitations

Of course, that Kant’s formulation secures knowledge of space against doubt does not necessarily show that Leibniz is wrong, only that there is some other way. To that end, Kant offers at least one independent argument against the relational view, which endeavors to show that it must invoke intuition after all. However, Kant’s use of this argument betrays that, despite his success in grounding physics, he may have limited it too much to account for what it would become.

Kant’s argument is quite hard to follow, and as explained by him it initially seems manifestly wrong. He first says of a relational view of space that “[i]f two things are completely the same in every respect of quantity and quality… you would expect it to follow that each can be replaced by the other in all cases and in all respects, without the exchange causing any recognizable difference.”²¹ This seems to follow fairly straightforwardly from Leibniz’s view, namely from his view that any two things which are internally identical would not simultaneously exist. But Kant says that this is not the case, and his counterexample is the hand in the mirror. Were we to hold our right hand up to a mirror, the mirror image of that hand would be internally identical to that right hand, and yet one could never “put such a hand as is seen in the mirror in the place of its original: for if the original was a right hand, the hand in the mirror is a left hand.”²² This is much easier to understand with his next example of a glove: no matter how hard one tries, they will be unable to make their right glove fit comfortably onto their left hand, because the right hand could never be translated or rotated such as to be able to occupy the same space as the left hand. So, then, space cannot be a relation between things, as one of the things which it would imply (the substitutability of internally similar things) is false.

As previously stated, it’s hard to understand the force of this argument. Namely, it’s hard to understand how the intrinsic rotation of the object would not be one of the internal qualities of that object. This confusion is mostly a result of inadequate explanation by Kant, and I hope to explicate it in such a way that it becomes intelligible. The point I believe Kant to be making is that, if you were to make intrinsic orientation part of the internal quality or quantity of an object, it would already be presupposing notions of direction that already involve a specific notion of space. Remember that one of Leibniz’s own examples for establishing the relational nature of space was to imagine what would happen if God turned “east into west,” and how supposedly nothing whatsoever would change.²³ Well, what Kant’s example shows is that despite all the relations between objects being the same in our world as in the west/east inversion world, it would nonetheless be impossible to translate or rotate one world such that it could lay on top of the other. They would be equivalent, but not congruent. That is really the point here; the example is meant to show how, whatever description you may give of something in the abstract, the way that our mind actually represents it will always be spatially oriented, and that this representation is the only way to account for the intrinsic differences between the objects. In other words, “it must be done by showing, and can’t be done by telling.”²⁴

And yet, in accepting the force of this argument, have we limited the project of physics too much? While Kant’s goal was to rescue the natural sciences, an interesting consequence of his argument is that it makes future developments in the physical sciences more or less unintelligible. This is well demonstrated by a seeming error in his argument. While he says that inner agreement does not entail congruence, he says that this only applies to 3 dimensional figures. In stark contrast, he says it “is the case with two-dimensional figures in geometry” that inner agreement would allow for seamless substitution.²⁵ But this just seems obviously false; the problem of intrinsic rotation is just as present in 2D as it is in 3D. This is quite easily demonstrated with the letters “p” and “q”, which like the right and left hand, are 2D mirrors of one another. There is no rotation or translation in 2D space that would allow one to substitute a “p” and “q”, and so the same arguments should apply to 2D figures. There is, on the other hand, a way to interpret this statement such that it is not inconsistent. If we allow that “p” and “q” can be rotated in 3D space despite being 2D figures, we would be able to overlap them after all by rotating them through the third dimension.

The point may seem trivial, but it has the interesting implication that if this move is available to 2D objects and not to 3D ones (and we are aiming to be consistent), it could only mean that there is no higher dimension through which we could rotate the 3D object, i.e the 4th dimension is not real. This implication is supported by the fact that, in the world of representation, we only have 3 dimensions, and so to Kant, a 4th would seemingly not be part of our a priori understanding of space. And here is where the walls of Kantian space begin to contract, for if we disallow a 4th dimension, we become unable to account for higher-dimensional geometry and even our most well accepted theories of physics, which do suppose higher dimensions. I won’t speculate on what Kant would have thought of these endeavors, but it seems fundamentally at odds with the form of our intuition and thus cannot be shielded from skepticism, as was Kant’s entire aim. Perhaps, then, a relational view of space is not so implausible after all. For whatever benefits there are in grounding space a priori, it seems much less amenable to our most empirically well-grounded scientific theories. In an ironic twist, it is a naive form of Liebniz’s relational view that is far more compatible with modern scientific theories, despite its lack of grounding.