Philosophy, Physics

Reframing the Substantivalism Debate

Abstract: This paper highlights and examines a critical assumption embedded within the framework of the substantivalism/relationalism debate. This assumption is the unacknowledged priority given to the traditional properties of spacetime-as-coordinate-system that have been handed down over the course of the twentieth century. In this paper, I attempt to substantiate my claim that such an unreflective assumption exists and is decisive, and evaluate the propriety and consequences of accepting that assumption. Finally, I will offer an alternative framework within which to move the debate forward on terms that prioritize spacetime-as-substance over spacetime-as-coordinate-system.

Traditional Spacetime

Even before the ancients, such as Aristotle and Euclid, started pondering the nature of space and time, before they created elaborate systems and defined the relationships between points in three dimensions, there was already a far older and more sophisticated neurological mapping system built-in to all human brains. The perception of spacetime as a matrix wherein objects and events are located, relative to one another and to one’s body, was already instinctive and familiar long before it was standardized by cartographers and mathematicians. The notion of spacetime as a coordinate system is old, instinctive, and unreflective.

What is new and radical is the idea that, over and above its role as a matrix that contains objects, spacetime is also a substance, a thing in its own right. And far from being instinctive and unreflective, only the most nuanced reasoning (Relativity Theory) and counterintuitive scenarios (Newton’s bucket argument) can bring to light even the mere possibility that spacetime might exist independently. With this stark contrast in mind I believe it is fair to say that:

  • Spacetime as non-existent coordinate system is intuitive, natural, ancient, genetically encoded, obvious
  • Spacetime as existing substance is radical, innovative, new, counterintuitive, contrary to experience, obscure.

Participants in the substantivalism/relationalism debate might well be nodding their heads in agreement, but also skeptically questioning my claim that there is anything hidden or surreptitious about this distinction. Of course the idea that spacetime is a substance is radical, counterintuitive, and unproven; that’s the whole crux of the debate. Fair enough, but I have something slightly more elusive in mind.

The Epistemological Priority of Spacetime as a Coordinate System

The first thing we need to note about the priority of spacetime-as-coordinate-system over spacetime-as-substance is that it is entirely epistemological. It is the ease with which we arrive at the respective conclusions that dictates their priority, not anything to do with their ontological natures. Spacetime-as-coordinate-system is an epistemological heuristic; it helps explain things. Spacetime-as-substance is ontological and remains to be explained. Indeed, spacetime-as-coordinate-system does not have any obvious ontological nature at all. By claiming it is non-existent, nothing but a geometric abstraction, we more or less explicitly admit that it is not so much a thing to be examined, as it is a mental overlay used for navigation, or a heuristic employed to simplify and categorize the events that are actually taking place. Spacetime-as-substance, by contrast, is proposed as the fundamental stuff of reality, the ultimate answer to the most basic ontological question: What is?

Herein lies the issue. Spacetime-as-coordinate-system is simple and has epistemological priority (exactly because of its historical precedence and great explanatory power) over spacetime-as-substance. On the other hand, spacetime-as-substance has ontological priority over spacetime-as-coordinate-system because, if it really exists, any coordinate system that results from spacetime’s genuine structure will be dictated entirely by that structure. That is, if spacetime exists, its role as coordinate system is entirely derivative of its ontological character. It is not the case that spacetime-as-substance must bend to accommodate whatever notions of reality are embedded within our currently accepted notions of spacetime-as-coordinate-system. And there are a huge number of such notions embedded there.

The priority of spacetime-as-coordinate-system is, fairly obviously, based on the long history and undeniable success of Western science, and specifically the great sophistication and accuracy (within many domains) of modern physics. That entire history of success is built into the latest epistemological spacetime-as-coordinate-system. Meanwhile, current notions of spacetime-as-substance are new, immature, enjoy no comparable history of success, and can boast no great explanatory power. It is entirely understandable, perhaps even justified, that we would prioritize spacetime-as-coordinate-system and unreflectively assume that it ought to form the starting point of our investigations even if we have, formally, made the shift to substantivalism. In effect, it appears reckless, disrespectful, and a bit daft to replace, wholesale and outright, the highly refined coordinate system we already have with an unproven substance that we cannot even define. Certainly at first, and perhaps for the foreseeable future, this newly postulated substance will lack anything approaching the explanatory power of our highly developed spacetime-as-coordinate-system. Trailblazers who venture into this uncharted territory should expect more than a little ridicule as they build their case. But if there is any merit to substantivalism, and a key success or two are forthcoming, it will eventually begin attracting converts.

Contrast Between a Substance and the Coordinate System

When scientists overlay reality with a particular spacetime-as-coordinate-system, that structure comprises all of the mathematical, scientific, and philosophical underpinnings of the latest cosmological thinking. It is emphatically not simply a piece of blank graph paper onto which, or clear lens through which, reality is directly and objectively recorded. Indeed, it functions as nothing less than the horizon, the current limitations, and the conditions under which reality itself can appear at all. For example, there is currently no accepted explanation for the curious rotational characteristics of spiral galaxies, though dark matter is the leading candidate. But because dark matter is not understood, there is no aspect of the latest spacetime-as-coordinate-system, when overlaid onto reality, that will allow it to appear. Among other things, this failure announces to us that the current spacetime-as-coordinate-system, fortified with all the best ideas to-date, is incomplete in at least one major respect. Dark energy is another such example, and between them—dark energy and dark matter—we have two gaping holes right at the heart of the current theory, and therefore two gaping holes in the latest spacetime-as-coordinate-system.

Substantivalists believe that the acceptance of spacetime-as-substance and its subsequent elaboration will fundamentally alter our understanding of ontology in general and physics in particular, potentially filling the two holes mentioned above. It follows directly from that assumption that they also believe that the current state of the scientific art that is built into the latest spacetime-as-coordinate-system is wrong at the most basic level, and may be very far from the truth in most detailed respects as well. Yet the debate between substantivalists and relationalists usually unfolds from a curious effort on the part of substantivalists to conceive of a spacetime substance that faithfully explains the theories embedded in the latest spacetime-as-coordinate-system. Essentially, the standard of proof for a theory of spacetime-as-substance seems to be that it is able to replicate whatever success is accorded to the current spacetime-as-coordinate-system—even though substantivalists ought to recognize that any such standard would require a theory that they themselves have already at least implicitly rejected, and which demonstrably fails to explain both dark matter and dark energy, among many other things (e.g., black holes, cosmogony).

The Ontological Priority of Spacetime as a Substance

For substantivalists, this observation ought to be liberating and unnerving at the same time. Liberating, because there is no requirement to bend one’s notions of spacetime-as-substance to fit into the latest spacetime-as-coordinate-system. In fact, any such attempt will invariably lead to a characterization of spacetime-as-substance that is beset by all of the inadequacies of the current theories. So, for example, it ought not to be accepted as axiomatic that any coherent formulation of spacetime-as-substance must fully replicate quantum theory, with little or no alteration. And it is by reference to that radical degree of freedom that the trepidation ought to set in. Presumably, a fully developed and defended theory of spacetime-as-substance will be radically different from our current set of theories. And at least two things follow from that stark acknowledgement.

One, the relatively straightforward (not to say easy) project of shoehorning spacetime-as-substance into the latest spacetime-as-coordinate-system is almost certainly of very limited utility and may be wholly counterproductive. Nevertheless, and to be fair, it is unlikely that absolutely everything that is currently propounded by modern cosmologists is complete nonsense, and for that reason it will be, in most cases, extremely difficult to determine which elements need to be respected and which questioned or discarded—this is a perennial challenge of all scientific progress. But however it ultimately unfolds, a healthy skepticism of especially those aspects of the standard models that bear directly on spacetime, matter, and ontology will be indispensable.

Two, substantivalists need to fully embrace the ontological priority of spacetime-as-substance over spacetime-as-coordinate-system. As mentioned, this paradigm shift feels somehow unnatural, disrespectful, and premature, but in the end the latter must entirely conform to the former. On a more positive note, such a shift utterly transforms the landscape and horizon of discovery. Whereas previously the main focus of substantivalists has been the futile shoehorning exercise discussed below, going forward their investigations need to focus less on what it would take to make a physically real coordinate system, and more on what sort of substance is sufficient to explain everything in the physical universe. Whatever it turns out to be, the appropriate spacetime-as-coordinate-system will follow naturally and be entirely derivative thereof.

Geometrodynamics—Folding the Coordinate System

The most comprehensive and ambitious effort to-date to shoehorn spacetime-as-substance into a spacetime-as-coordinate-system was John Wheeler’s geometrodynamics. Wheeler’s program was to treat matter, in all its various manifestations, as so many complex geometric contortions of spacetime. He reasoned that if we can imagine a large, simple spacetime curve as a gravitational field, we can also imagine a very small and tightly curved spacetime region as a subatomic particle, what Wheeler called a geon. To slightly oversimplify it, he conceived all of reality as a kind of spacetime origami—surface-like spaces twisted and knotted such as to give rise to all the fundamental particles at one extreme of scale, and all of the large-scale astrophysical phenomena at the other extreme.

It is widely agreed that Wheeler’s efforts were not entirely adequate, but it is worth taking some time to appreciate his attempt and glean what knowledge we can from it. In particular, geometrodynamics points to just how confounding the relationship is between spacetime-as-substance and spacetime-as-coordinate-system. Wheeler, a classically trained physicist, naturally accepted the priority of spacetime-as-coordinate-system and attempted to fold and twist that coordinate system in such a way as to create spacetime-as-substance. The project hinged on the dubious notion that it is possible to contort a massless, geometric abstraction in such a way as to bring ontologically genuine matter into existence. It also points to a fundamental issue related to the distinction between properties and substances.

With reference to the ontological argument for the existence of God, Kant pointed out that existence is not a predicate. The exact same list of properties can be ascribed to the idea of God as can be ascribed to God Himself. All that distinguishes the two is existence. Kant explained, correctly I believe, that properties are predicated of a substance while existence is the substance itself. It is meaningless to predicate existence of a substance because, in effect, existence and substance are synonymous. Substance is nothing other than whatever we mean by existence. To say that substance exists is like saying heat is hot—the predicate adds nothing to the concept because it is the concept.

In a strangely analogous way, Wheeler was attempting to create existence by merely altering the description of the underlying coordinate system. In effect, he was trying to add the property of existence to an otherwise purely abstract concept, thereby bringing the previously non-existent idea into full reality.

Black Holesblackhole

All of these ideas come to a head in the most enigmatic entity in the universe—a black hole. The current (relationalist) theory postulates that there is a real thing (an infinitely dense singularity) in the center of a very intensely curved, but formally non-existent, spacetime region. In this model, the gravitational relationship between a black hole and objects within its influence is exhaustively determined by the singularity and the external object. The spacetime-as-coordinate-system surrounding the black hole is merely a mathematical overlay, fortified with the latest theories (including, for example, gravitons, Higgs fields, etc.), that functions entirely as a heuristic. But what is a substantivalist committed to in this case?

According to a substantivalist, spacetime is not merely a geometric abstraction but is the one true substance of the cosmos. So, here’s the big question: What, exactly, exists in the region one inch from the center of a black hole? If we were to demand of a relationalist that he discard temporarily his helpful heuristic, the spacetime-as-coordinate-system, and instead say what is really there, he might hypothesize that there is likely a very high graviton particle density and a Higgs field with an array of very extreme values, but no genuine “matter” in any standard sense of the term. All of the actual mass is concentrated into an infinitely dense, infinitesimally small, object in the center of the black hole. A substantivalist would say something completely different.

If we take seriously for a moment the central tenet of substantivalism, then the spacetime surrounding the center of a black hole is real, not a mere abstraction, not simply a coordinate system. From that single assumption, we are suddenly faced with a fascinating question: Is the real mass—viz. the spacetime surrounding the center—different from or the same as the mass traditionally attributed to the singularity in the center itself? In other words, are there two types of substance, one type in the center and another type surrounding it? Or, on the contrary, is it spacetime all the way down? If spacetime is the one and only substance, then it follows necessarily that this substance (regardless of how we ultimately characterize it) becomes increasing dense as we approach the center of a black hole, and the center itself is nothing more than the densest location, but it is not qualitatively different from any other point in the entity. And, most importantly, there is nothing special at the core; there is no singularity.

The Singularity as Absurdum

What does any of this have to do with the thesis of this paper, that spacetime-as-coordinate-system is mistakenly prioritized over spacetime-as-a-substance? The singularity hypothesized to exist at the center of a black hole is exactly where the rubber hits the road in this whole debate. And in nearly every respect, this is great news for the substantivalist. In the first place, a singularity is only necessary because spacetime is not currently thought to be a real substance. If spacetime is a substance, then there is no difficulty explaining the extreme mass of a black hole. Whatever else this implies, it means that the current (relationalist) theory has been forced to fabricate, ex nihilo, an utterly preposterous entity in order to maintain its stance that spacetime is a non-existent coordinate system. The singularity exists entirely outside of all other accepted science, has no coherent physical (or logical) description, and will most likely never enjoy a single shred of empirical evidence. It very clearly runs afoul of Karl Popper’s falsifiability test of scientific coherence. One could easily imagine the singularity being used by an opponent of the existing theory rather than by one of its proponents. “You say, my good man, that spacetime does not exist. But you realize, don’t you, that you must then assume the existence of a pure fantasy, a hole in reality, at the core of black hole? Are you sure you wouldn’t like to reconsider?”

Cosmologists know that the singularity, whether at the core of a black hole or at the origin of the universe, is an ominous threat to the standard models. The singularity is no less disconcerting than quantum theory’s abysmally inaccurate prediction of zero-point energy—the vacuum pressure—120 orders of magnitude greater than its measured value. It also draws uncomfortable attention to the intractable incompatibility of quantum and relativity theories, invalidating at least one if not both of them. Indeed, if we were to recast the entire standard model as an elaborate reductio ad absurdum proof, the singularity fits very nicely into the role of absurdum. Immediately upon demonstrating that this absurdity follows necessarily from the theory, we can say with perfect justification: therefore, the opposite must be true, spacetime is a substance. However comfortable we have become talking about singularities, until such time as they are given a physical description that is not utterly incoherent, their necessary existence ought to be regarded as strong counterevidence to the standard models.

My rationale for bringing this to light here is to cast doubt on the wisdom of starting from spacetime-as-coordinate-system and attempting to build spacetime-as-substance up from there. That traditional prioritization is entirely epistemological, built partly on our innate neurobiological mapping skills and partly on the notions of space and time (absolute, non-relativistic) used throughout most of scientific history. But as black holes make clear, spacetime-as-substance has ontological priority and should be assumed as the starting point for the purposes of understanding substance itself. Among many other things, this reprioritization means that we are under no obligation to demonstrate how mass can be magically created, via some form of spacetime origami (e.g., geometrodynamics), out of an inherently massless and theoretically inadequate spacetime-as-coordinate-system.

Over and above the manner in which black holes expose the dubious singularity notion, is the fascinating way in which they reveal spacetime’s mass-like characteristics at extremely high densities. Nothing new or special or mathematically complex (geometrodynamic) had to be done to spacetime to make it into matter. All we had to do was compress it. Indeed, if we examine—armed with our new assumption that it is nothing but spacetime all the way down—the entire gravitational (spacetime) field of a black hole, we will find a smooth continuum of spacetime, distributed according to the inverse square law, approximating the vacuum pressure at the most distant extent and the density of atomic matter at the core, and nothing else. That is, once we discard the singularity, a black hole illustrates all by itself the manner in which spacetime behaves as both matter and gravity. And that is not a coincidence.

Rethinking the Third Dimension

Finally, another very questionable assumption sneaks into the debate when we prioritize spacetime-as-coordinate-system over spacetime-as-substance; we accept, without even realizing it, that it makes sense to talk about a three-dimensional substance that does not exist. That is, we embrace the bizarre notion that nothing fundamentally new results from adding a third dimension to the two-dimensional surface-like analogy that undergirds spacetime-as-coordinate-system. It is easy enough to imagine a three-dimensional coordinate system that is nothing but a heuristic, a geometric abstraction. But is not possible to imagine a three-dimensional substance that lacks any form of existence whatsoever. In a nutshell, the lack of depth, and therefore lack of ontological reality, of a two-dimensional surface cannot be translated into a three-dimensional system. Coupled with the assumption that it is a substance, the addition of the third dimension is enough to make spacetime exist. To suppose that some additional form of folding or knotting or twisting is also required is, at bottom, an incoherent idea that follows from taking the two-dimensional analogy too far—from embracing uncritically the priority of spacetime-as-coordinate-system. Consider that, if the non-existence, the pure geometrical abstractness, of the coordinate system persisted into its conception as a genuine substance, then no amount of folding or twisting would do any good. If we start folding or stacking two-dimensional planes, one atop the other, each one possessing exactly zero depth, we will never end up with a 3-D object with mass.

We can see this clearly if we go back to our new conception of a black hole. Whether we use spacetime-as-coordinate-system or spacetime-as-substance, the curvature is very simple and follows the inverse square law from the center of the black hole all the way out to the farthest reaches of its gravitational influence. What made the spacetime into a substance was not any elaborate folding or twisting but simply our elimination of the singularity at the center. Once we grant ontological priority to spacetime, the third dimension is enough to make it a real substance all by itself. Indeed, it is not even possible to imagine a massless, but nevertheless 3-dimensional, substance. If a substantivalist assumes that spacetime is a three-dimensional substance, he has already assumed it has mass, otherwise he has not really assumed it exists at all. There is no additional work to be done on that score. What needs to be done to prove that the substantivalist theory is correct is not to demonstrate how spacetime-as-coordinate-system can be creatively folded to create matter—that would be to, once again, prioritize the coordinate system—but rather to demonstrate the explanatory power of a fully developed spacetime-as-substance with mass as an already integral component. If such a demonstration is successful, as it appears to be with a black hole, then the burden of proof will fall onto the relationalist, and it will be abundantly clear that the ontological priority of spacetime-as-substance trumps the epistemological priority of spacetime-as-coordinate-system.


In summary, spacetime-as-coordinate-system and spacetime-as-substance serve two different purposes. The former is an epistemological heuristic, embedded with the sum total of our current knowledge of physics. The latter is immature, but is proposed as the fundamental ontological substance of the cosmos, and it is hoped that its full development will result in radical changes to our picture of reality—in other words, that it will supplant the current coordinate system. Any notion of substance that is derived from the current spacetime-as-coordinate-system will inevitably be plagued by exactly the same problems that substantivalism is chartered with solving. Therefore, if only as a formal matter, spacetime-as-substance must be recognized as ontologically primary, and all conflicting aspects of spacetime-as-coordinate-system should be regarded as highly suspect and subject to change in accordance with new ontological discoveries.

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