Space and time seem as fundamental as anything can get. Philosophers like Immanuel Kant thought that they were inescapable, that we could not conceive of the world without space and time. But increasingly, physicists suspect that space and time are not as fundamental as they appear. When they try to construct a theory of quantum gravity, physicists find puzzles, paradoxes that suggest that space and time may just be approximations to a more fundamental underlying reality.
One piece of evidence that quantum gravity researchers point to are dualities. These are pairs of theories that seem to describe different situations, including with different numbers of dimensions, but that are secretly indistinguishable, connected by a “dictionary” that lets you interpret any observation in one world in terms of an equivalent observation in the other world. By itself, duality doesn’t mean that space and time aren’t fundamental: as I explained in a blog post a few years ago, it could still be that one “side” of the duality is a true description of space and time, and the other is just a mathematical illusion. To show definitively that space and time are not fundamental, you would want to find a situation where they “break down”, where you can go from a theory that has space and time to a theory that doesn’t. Ideally, you’d want a physical means of going between them: some kind of quantum field that, as it shifts, changes the world between space-time and not space-time.
What I didn’t know when I wrote that post was that physicists already knew about such a situation in 1993.
Back when I was in pre-school, famous string theorist Edward Witten was trying to understand something that others had described as a duality, and realized there was something more going on.
In string theory, particles are described by lengths of vibrating string. In practice, string theorists like to think about what it’s like to live on the string itself, seeing it vibrate. In that world, there are two dimensions, one space dimension back and forth along the string and one time dimension going into the future. To describe the vibrations of the string in that world, string theorists use the same kind of theory that people use to describe physics in our world: a quantum field theory. In string theory, you have a two-dimensional quantum field theory stuck “inside” a theory with more dimensions describing our world. You see that this world exists by seeing the kinds of vibrations your two-dimensional world can have, through a type of quantum field called a scalar field. With ten scalar fields, ten different ways you can push energy into your stringy world, you can infer that the world around you is a space-time with ten dimensions.
String theory has “extra” dimensions beyond the three of space and one of time we’re used to, and these extra dimensions can be curled up in various ways to hide them from view, often using a type of shape called a Calabi-Yau manifold. In the late 80’s and early 90’s, string theorists had found a similarity between the two-dimensional quantum field theories you get folding string theory around some of these Calabi-Yau manifolds and another type of two-dimensional quantum field theory related to theories used to describe superconductors. People called the two types of theories dual, but Witten figured out there was something more going on.
Witten described the two types of theories in the same framework, and showed that they weren’t two equivalent descriptions of the same world. Rather, they were two different ways one theory could behave.
The two behaviors were connected by something physical: the value of a quantum field called a modulus field. This field can be described by a number, and that number can be positive or negative.
When the modulus field is a large positive number, then the theory behaves like string theory twisted around a Calabi-Yau manifold. In particular, the scalar fields have many different values they can take, values that are smoothly related to each other. These values are nothing more or less than the position of the string in space and time. Because the scalars can take many values, the string can sit in many different places, and because the values are smoothly related to each other, the string can smoothly move from one place to another.
When the modulus field is a large negative number, then the theory is very different. What people thought of as the other side of the duality, a theory like the theories used to describe superconductors, is the theory that describes what happens when the modulus field is large and negative. In this theory, the scalars can no longer take many values. Instead, they have one option, one stable solution. That means that instead of there being many different places the string could sit, describing space, there are no different places, and thus no space. The string lives nowhere.
These are two very different situations, one with space and one without. And they’re connected by something physical. You could imagine manipulating the modulus field, using other fields to funnel energy into it, pushing it back and forth from a world with space to a world of nowhere. Much more than the examples I was aware of, this is a super-clear example of a model where space is not fundamental, but where it can be manipulated, existing or not existing based on physical changes.
We don’t know whether a model like this describes the real world. But it’s gratifying to know that it can be written down, that there is a picture, in full mathematical detail, of how this kind of thing works. Hopefully, it makes the idea that space and time are not fundamental sound a bit more reasonable.
4 gravitons 
You ought to be careful when mentioning Kant. The “received view” of his work seems to misrepresent his work. In particular, a translation in Ewald’s anthology has him calling his peers to develop alternate geometries from Euclidean geometry long before Riemann or Lobachevsky.
What he wrote in “Critique of Pure Reason” had been a response to Humean skepticism. It presumes that space and time reflect aspects of our sensory experience, thereby grounding the meaningfulness of mathematical methods to the individual witnessing of objective events.
I am unaware of any point in “Critique of Pure Reason” or “Prolegomena to Any Future Metaphysics” in which he discusses Euclid’s parallel postulate. Given his earlier call for the development of alternate geometries, it would be charitable — presuming my recollection of his epistemological works to be correct — to interpret his use of the expression “geometry” as an ambiguous usage.
His contemporary critics, quite naturally, would only have thought in terms of Euclidean geometry, and, that this criticism of his work has been repeated for centuries reflects the sociological nature of academia more than critical assessment of his writings.
More significantly, perhaps, is that criticism of Kant also led critics to assert the unreality of mathematical forms. Fair enough. But, if all of the mathematics used in the sciences is “merely convention,” the exact nature of knowledge claims for science becomes rather murky.
Recently, I ran across the summary, “Kant saved science at the expense of scientific realism.” That seems quite accurate. Huxley is attributed with introducing the term “agnostic” in response to the plurality of metaphysical positions of his peers. Materialism had been among those positions which he could not accept in light of Hume and Kant.
Currently, the science community is overrun with materialists promoting scientific realism. Individuals more inclined to scientific agnosticism ought to be apalled at this advocacy. It feeds the anti-science constituency. And, it strengthens the defenders of theological beliefs. By doing such, it undermines the liberal hopes of Enlightenment philosophies.
Black’s paper on the identity of indiscernibles and the paper on the unity of opposites by McGill and Parry might be of interest to you. The problem of symmetry and the description of a definite model is discussed in Phillip Anderson’s paper, “More is different.” One might take this as a followup to Black’s paper. Similarly William Lawvere investigated the unity of opposites using category theory. Objections concerning the “intuitive becoming” of secant endpoints (twoness) transforming into a tangent point (oneness) go back to the nineteenth century; but this is central to applied mathematics. This is the topic to which Lawvere applies the unity of opposites.
If physics is headed toward denying the fundamental nature of space and time, then physicists are actually moving toward the Kantian interpretation of space and time as not subsumed as essential to an objective material reality.
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