A Systems Theory of Relativity and Gravity, #1.

(Mattson, 2015)

TABLE OF CONTENTS

1. Introduction: Small Worlds

2. Every System Is Incomplete

3. The Speed of Gravity

4. The Three-Body Problem

5. The Singularity Isn’t Near

6. Conclusion: The Great Awakening

The essay that follows will be published in three installments; this, the first installment, contains sections 1 and 2.

But you can also download and read or share a .pdf of the complete text of the essay, including the REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.

A Systems Theory of Relativity and Gravity

1. Introduction: Small Worlds

We keep the wall between us as we go. (Frost, 1916)

I am, surely, not the first person to mourn the unsettling symmetry between the plot of Coraline—a novella by Neil Gaiman that was adapted into a film in 2009—and the modern predicament of researchers who study phenomena that refuse to scale. Quantum physicists predict the behavior of subatomic particles; systems theorists predict the course of life on Earth, including our climate and global economy. Astrophysicists armed with relativistic field equations measure the fallout of gigantic events occurring in the far reaches of observable space. All of them pursue knowledge inside an uncanny, claustrophobic version of the universe, hemmed in by theoretical constraints. When Gaiman’s Coraline finds herself swallowed up by a sinister otherworld, full of unctuous simulacra, she tries to escape by walking away. She ends up right where she started. “How can you walk away from something and then come towards it?” she cries. “Walk around the world,” her cat replies. (Cats can talk, as readers of Lewis Carroll have long known. However, unless the situation is very dire, they prefer not to.) “Small world!” Coraline snorts.

It is high time that scientists, mathematicians, and theorists took the full measure of our own enclosures, just like Coraline did. Moreover, her question is the right one. We have to come back, willy-nilly, to certain problems we actively chose, at some point, to walk away from. The three-body problem is one good example. The expansion of objects moving at tremendous velocities also qualifies. So does the speed of gravity. To solve them, we have to find our way back to approaches that most scientists and mathematicians currently won’t touch. At the same time, who can blame them? Applying quantum mechanics to macroscopic phenomena has led us on a wild goose chase for gravity “particles.” Looking for “emergent” phenomena like consciousness or homeostasis in the cosmos has led to panpsychism and the untenable proposal that the universe is, if not actually sentient, then at least “alive” in some way. Applying the laws of classical physics to living organisms has saddled us with questions like these: “if the brain’s entropy is greatest during periods of intense mental activity, why isn’t it fatal to think?” We have to abandon these lines of attack, acknowledging that not every model or metaphor can survive in every habitat. We must do this without abandoning the search for universal truths.

2. Every System Is Incomplete

It seems you cannot have one without the other. (Doctor Who, 2006)

Where, then, does one begin the search for such universals? They must be hiding in plain sight—and they are. The key to reimagining microscopic, anthropic, and cosmic phenomena as part of a single continuum begins with work already completed by Douglas Hofstadter in his oft-misinterpreted magnum opus, Gödel, Escher, Bach: An Eternal Golden Braid (Hoftstadter, 1979). Thanks to Hofstadter’s self-indulgent, overly whimsical presentation of his own ideas, his book is generally perceived as a vivid, playful exposition of well-established principles. In reality, it is a sleeping giant. Hofstadter’s logical framework is derived from the foundations of mathematics, which was revolutionized by Kurt Gödel’s Incompleteness Theorem. Gödel proved that Principia Mathematica-like logical systems are incapable of proving the truth of their own tautologies, which means they are incapable of demonstrating their own validity and soundness, no matter how seamless or encompassing their axioms may appear.

That is common knowledge; when Hofstadter begins applying Gödel, though, he stumbles onto something profound. His examples aren’t numerical; they’re systems. Consciousness, his overriding concern, is a system. Computers and artificial intelligence are systems. Languages are systems. Highly formalized artworks, meanwhile—I including the music of Johann Sebastian Bach and the mind-bending drawings of M.C. Escher—are not true systems, because they aren’t dynamic. They don’t evolve over time in response to internal or external changes in their environmental conditions. Hofstadter confuses his own argument by eliding the difference between, for example, a fugue and a living human brain. Still, when our eye travels across a canvas by Escher, or we spend time listening to Bach, we subjectively experience their creations as dynamic, evolving objects, precisely as if the artwork was undergoing cycles of pressure and change. So it is rigorously accurate to insist that artworks built upon perceptible, consistent rules function as a kind of symbol for systems. They symbolize what it is like for human beings to encounter systems.

Since systems are not identical with the artworks that simulate them, let’s set aside Hofstadter’s “complete liberal education in a single volume” and focus on the true systems he chooses as his testing-grounds for Gödel. We have to work backwards, starting from Hofstadter’s conclusions and arriving at his axioms. How is it possible to apply Gödel to consciousness or machine learning? To answer this, let’s do something systems theorists have failed to do for the better part of a century: formalize the patterns that systems exhibit. A system is more than a collection of interacting objects. It also includes a set of rules that govern those interactions. The law of gravity, the relationships between different animals in the same ecosystem, and the way long-term memories are consolidated in the brain are all consistent systemic phenomena expressible as rules—or, in mathematical terms, as operations. All such operations can be encoded as “Gödel numbers,” exactly like the logical relationships that constitute sets, and represented (as unique primes) on a horizontal, numerical continuum that also includes the system objects they act upon.

If it is true that it is possible to express the total extent of a system’s contents and interactions in Gödel’s code, then it follows that systems are also susceptible to the Incompleteness Theorem. From a systems perspective, the enormous import of Gödel’s theorem is what it says about systemic boundaries. If systems are finally not reducible to their operations and contents, because they aren’t capable of expressing or producing their own totalities, their boundaries must be both finite and permeable: more like membranes, in other words, than walls. They also cannot ever conclusively shield themselves from the possibility of systemic changes that dissolve their boundaries and, simultaneously, the internal web of relations that body them forth.

The same property that makes systems functional—their interrelatedness—also makes them finite in both the spatial and the temporal sense. The significance of this double finitude is immense. It gives rise to the familiar, explanatory, but inadequately explained tendency of all systems to decay and collapse over time, which we know as the Second Law of Thermodynamics. In fact, it is possible to speculate that our fundamental experience of time (the irreversible “Arrow of Time”) is produced by the entropy of finite systems; and, following Gödel, we must accept that no other kind of system exists anywhere in the universe, even at the maximal (Hubble) scale of the universe itself.

Arguably, the implications of spatial finitude for complex systems are even more complex and far-reaching. To begin with, this property of physical systems implies that every discrete system exists in an ongoing, dynamically evolving relation to an indeterminate “outside” produced by the system’s own boundaries. We are thus confronted with the same paradoxical structure described by Werner Heisenberg’s Uncertainty Principle, which demonstrated that it is impossible to simultaneously determine the location (i.e., the persistent boundary) of an object and that same object’s velocity. Velocity, in other words, is one way of expressing the dynamic relation between discrete systems and the larger “environments” that supersede them. Heisenberg’s principle is an elegant formulation of the insuperable, somewhat disappointing truth that we cannot view a system from both the inside (as a localized phenomenon) and the outside (as a body in motion) at the same time. If we could, we would have accomplished a complete, totalizing representation of an infinite reality. As Gödel demonstrated, such transcendent representations are impossible.

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