Einhorn is Finkle. Finkle is Einhorn.
- Ace Ventura (1994)
Albert Einstein and Kurt Gödel were an unlikely pair. They could be seen walking together to and from the Institute for Advanced Study in Princeton, where they were each residents. Cherished as the public’s genius, Einstein generated awe whenever he entered a room. Although little known to the public, Kurt Gödel would meet similar veneration to a room full of scholars. Freeman Dyson once remarked of Gödel that he was “the only one of our colleagues who walked and talked on equal terms with Einstein.” In short, both men were revered at Princeton. And both men could not have appeared more different from each other. Gödel’s pressed suits, cropped hair, paranoia and general unease in the world was in stark contrast to Einstein’s unmistakable and celebrated foibles. In their own ways, both men were misfits. But dear friends they were. And Einstein at one time remarked that he only ventured to his office “just to have the privilege of walking home with Kurt Gödel.” Clearly the differences between the two men were superficial.
It is interesting to contemplate the deep conversations the two might have enjoyed. Did they spend much time discussing the creative insights that had led them each to infamy? Did the two men realise that Godel’s 1931 seminal proof – On formally undecidable propositions of Principia Mathematica and related systems – was a manifestation of the very same principle that underpins the bold leaps in Einstein’s formulation of his two Theories of Relativity?
Along with the constancy of the speed of light, Einstein elevated the Principle of Relativity to a postulate when formulating the Special Theory. With the formulation of the General Theory, he introduced a third postulate: The Equivalence Principle. Upon a detailed examination and a tour of several thought experiments, we will find that these two principles are one and the same. Moreover, we will find that Godel discovered the very same principle within mathematics when he first proved the Incompleteness and later the Inconsistency of arithmetic. This principle is fundamental to all knowledge – The Principle of Dimensionality.
Dimensionality
Dimensionality can be illustrated through a simple thought experiment. In this experiment, I want you to imagine that you are a pirate. That’s right, you’re a pirate. Who doesn’t want to be a pirate? Recently, you’ve started to notice a pattern: Whenever you circumnavigate your ship on a long voyage in one direction, you eventually arrive at the place at which you left. It’s the sixteenth century and you expect to find an edge to the Earth. But you never find it. Putting your pirating life to the side, you decide to make a few tests. You mark an X-marking-the-spot at your island home. And then you take long journeys, going north, south, east and west. At the end of each journey you arrive at the place you once left; the X-marking-the-spot.
We can see that no matter how many times you traverse the seas, you will never explain your predicament. Something else is needed: A creative leap.
One night you fall asleep, thinking of the many journeys to your X-marking-the-spot. You find yourself lucid dreaming. You are at the helm of your ship, deep naval, surrounded by the ocean. You find yourself floating into the sky, surrounded by the ocean. Floating higher and higher, surrounded by the ocean. Then, as you see your X-marking-the-spot peek up and over the horizon, it hits you. You see the curvature. You wake up, jump out of bed, and yell “the Earth is a ball!”
What just happened is simple yet profound. Your leap of creativity reveals something fundamental about all knowledge. Notice that were you to continue traversing the seas, not only would you not explain your predicament, you would be making circles. These circles are apparent constants. I say apparent as you cannot be confident in their constancy until you explain it. In this instance, the apparent constants occupy the surface of the Earth. Notice that the surface of the Earth is 2-dimensional. Your creative leap was a literal jump into the 3rd dimension where you explained the surface of the Earth as being wrapped around a sphere.
We can convert the components of the thought experiment into epistemological machinery. The apparent circles around the surface of the Earth may be recast as problems occupying a problem-space.[1] Notice that the problems are never solved: The problem-space is explained and the apparent constants transform into accepted constants. Notice also that the problem-space cannot be explained using itself. To do so would be circular; paradoxical. Just like the constants. This brings us to the Principle of Dimensionality: A problem-space cannot be explained using itself; its explanation resides in another dimension. This conceptual situation is depicted in Figure 1.
The thought experiment reveals a relationship between inductive inference, infinite regress and self-referential paradox – The Trinity (Figure 2). Inductive inference is the misconception that knowledge can be acquired through repeated observations. We saw that this cannot be so. Without the creative leap, you would be stuck circling the Earth, in an infinite regress. The Principle of Dimensionality allows us to recast inductive inference as any attempt to use a problem-space to explain itself, a paradox. Inductive inference, infinite regress and paradox are one and the same. We will find evidence of this relationship at the heart of both Einstein’s and Godel’s work.
Relativity
Einstein conceived of the Special Theory of Relativity during his miracle year in 1905. However, the journey began when he was a teenager, at the age of 16, as he imagined what it would be like to ride alongside a beam of light. Disconcerted by the image of a frozen light beam, Einstein settled on a primary postulate that the speed of light be constant in all inertial reference frames combined with adherence to The Principle of Relativity. Originally conceived by Galileo Galilei in 1632 using a thought experiment involving a ship, we can explore The Principle of Relativity here using your pirate’s ship.
You board your pirate ship once again. Descend to the ship’s cabin and observe your surroundings. Being a pirate, you have a physical distaste for cages, so the first thing you notice is your pet parrot, Mr Jingle, flying freely around the cabin. Mr Jingle’s droppings, which he also drops freely around the cabin, have attracted a coterie of buzzing flies. Your experiment with cold brew coffee drips one drop at a time in one of the cabin’s corners and your pet fish, who shall remain nameless, swim in a fishbowl on your desk in another corner.
The cabin has no windows. You walk into the centre of the cabin, stop, and observe. Mr Jingle flies around the cabin, not influenced in one direction or another; the flies buzz in all directions around his droppings; the cold brew drips one drop at a time, with the drop always landing in the centre of its receptacle; the fish swim in all directions in their bowl. You yell out to your first mate: He leaves the dock, brings the ship to a uniform speed and reports back.
With the ship showing no signs of lurching forward or back, or rocking side to side, you observe the surroundings of the cabin. Mr Jingle continues to fly around the cabin unabated, the flies buzz in all directions, the cold brew drops continue to land in the centre of the receptacle, the fish swim in all directions. There is no apparent change. You jump up and down in the centre of the cabin and observe that you always land in the place where you jumped from: the ship’s velocity has no measurable effect on the inner goings on of the cabin.
Without a window to observe your motion, there is no way to discern the ship’s velocity from within the ship. The situation improves only slightly, even with a window. Looking out the window, you may be able to see the ship moving away from the dock, but there is no way to differentiate between the ship moving forward or the dock moving backward; or a combination of the two. We may only state that, relative to the dock, the ship is moving away. Conversely, relative to the ship, the dock is moving away. This is The Principle of Relativity in all its simplicity. A physicist might proclaim that the ship occupies an inertial reference frame and the dock an entirely separate inertial reference frame. We may only explain the motion of a given inertial reference frame in terms of another inertial reference frame. We cannot explain the motion of an inertial reference frame using itself. This should sound familiar because it is: The Principle of Relativity is The Principle of Dimensionality. It is The Principle of Dimensionality applied to inertial reference frames; and inertial reference frames are a variety of problem-space.
Einstein’s great insight was to inadvertently postulate The Principle of Dimensionality. A constant speed for light then becomes a necessity, which we can appreciate with another thought experiment.
Let’s say you setup a horizontal light contraption on your ship. It consists of a lamp at the bow of the ship and a beam off the bow of the ship with a mirror at the end of the beam. The lamp directs light to the mirror and back again. Now, for arguments sake, let’s say that the speed of light is no longer constant; its speed is a function of the speed of light emitting from a body at rest plus the emitting body’s forward velocity in the direction of the light beam. You now accelerate your ship and bring it to a constant velocity. Then, at the bow of the ship, you shine a light beam along the horizontal contraption to the mirror and clock the time it takes for the light beam to return to the bow. Knowing how long the light should take to return to the bow were your ship to be at rest, you now have a device that can measure the velocity of your inertial reference frame using itself. A paradox! But is there also an infinite regress? Well, yes. Let’s say you slide in another mirror at the bow of the ship. You shine the light, it bounces from the mirror at the end of the beam back to the mirror at the bow and repeats. What happens to the speed of your light beam as it bounces between the two mirrors? Well, if light is additive to the emitting body then each time the light bounces from the mirror at the bow it should get a little push – we would see the speed of your light beam perpetually increase as it bounces between the mirrors. An infinite regress!
Given the thought experiment, it is clear that the speed of light must be a constant in all inertial reference frames. Were it not, then it would be possible to create the paradoxical situation of an inertial reference frame explaining itself. The Principle of Dimensionality as manifested in the Principle of Relativity demands constancy for the speed of light in all inertial reference frames to avoid paradox, infinite regress and attempted inductive inference.
Incompleteness
In the early 20th century a few operating at the edges of philosophy and mathematics concerned themselves with foundations. To a subset of mathematicians, the formalists, the failure to find foundations without paradox was deeply concerning. The famed German mathematician, David Hilbert, led the charge with his Program, detailing the requirements for a proposed solution to what was dubbed the ‘foundational crisis.’ Hilbert’s desperation was captured by his battle cry – “we must know, we will know” – concluding the retirement address he presented to the Society of German Scientists and Physicians in 1930. The words remain emblazoned on his grave in Gottingen.
The formalists wanted to reduce all systems of mathematics to be like games of chess. They required that it be possible to take a statement written in a mathematical system and work backwards to arrive at the axioms of that system. Such a system is said to be recursively enumerable. The game of chess is recursively enumerable. The axioms of a chessboard include the board, the pieces, the starting place for all the pieces and the legal moves of each piece. Given any chess game, it is possible to work backwards through each player’s legal moves to the starting point of the game. In effect, there are no surprises in a game of chess. Provided we restrict ourselves to legal moves, there are no situations that cannot be explained by the starting position of each piece. Consistent with Hilbert’s Program, we might say that the game of chess is complete: all true (legal) games of chess are provable from within the system of chess. We might also say that the game of chess is consistent: no true games of chess are also false (illegal); chess is free from contradiction. Lastly, we might say that the game of chess is decidable: an algorithm could analyse each game of chess and decide on its truth or falsity.
In 1931, Kurt Godel dashed the hopes of the formalists and took a blow to Hilbert’s Program. Godel proved that for any formal system of arithmetic of sufficient complexity, that both completeness and consistency could not be proved from within the system. The system could be shown to be complete, but then it would be inconsistent; likewise it could be shown to be consistent, but then it would be incomplete. Alan Turing took a final blow to the Program in 1936, when he showed that a decidability algorithm would also be untenable.
With a method of turning mathematics back upon itself, Godel constructed a mathematical statement that was the equivalent of ‘The Liars Paradox:’
This statement is false.
The statement is a self-referential paradox: If the statement is true then it evaluates as false; if it is false then it evaluates as true. A contradiction. The statement also contains an infinite regress: if the statement is true, then it is false, which makes it true, but then it is false; ad infinitum.
In his first theorem, Godel’s ingenuity was to reshape ‘The Liars Paradox’ into a statement about mathematical provability. In effect, he created the following mathematical statement:
This statement is not provable.
This slight modification allows the statement to evaluate as false or ‘true but unprovable.’ With his first Incompleteness Theorem, Gödel illustrated that the statement was ‘true but unprovable’ from within the system; the system was incomplete. One could get around this by going outside the system to formalise the statement as a new axiom, but then there would be another self-referential statement about the new system that would be ‘true but unprovable,’ and so on, ad infinitum – an infinite regress.
In a second theorem, Gödel demonstrated that a mathematical system could not prove its own consistency. By first assuming a system proved its own consistency, Gödel showed as per his first theorem that the system would contain a self-referential statement that is ‘true but unprovable.’ And since the statement is true, Gödel could assert that the statement would be provable within the system. Any complete mathematical system therefore contains true statements that are both provable and unprovable, which is a contradiction – a paradox.
These results upset the mathematical world. It was now possible to labour away at a proof unbeknownst to the fact that it was unprovable. Some even worried that Godel’s result implied that inconsistencies lay at the heart of mathematics.
We can take a less sombre view of Godel Incompleteness when we identify that Godel discovered The Principle of Dimensionality within mathematics. We can think of a system of mathematics as an abstract problem-space. The rules of this problem-space, the legal moves, are set by the axioms. Recast yourself to the original thought experiment. You are a pirate circling the Earth. Earth-circling is an axiom for the problem-space of the surface of the Earth. You can never prove that you circle the Earth from within the bounds of the problem-space. From this vantage point, Earth-circling is a ‘true but unprovable’ generalisation. You can, however, explain your Earth-circling axiom by transcending the problem-space to conjecture that the surface of the Earth is wrapped around a ball. Godel incompleteness is a celebration of the richness of the abstract mathematical problem-space. This problem-space consists of an infinity of dimensions and the game of mathematics an infinity of surprises. Each time a mathematician suspects a proof to be ‘true but unprovable,’ the mathematician may, with reasoned explanation, choose to conjecture a new axiom and add another dimension to the abstract problem-space. Like all scientific conjectures, each new axiom will be fallible.
Godel’s second theorem reveals that we cannot use the abstract mathematical problem-space to explain itself. It shows us that attempting to prove a ‘true but unprovable’ conjecture is akin to attempted inductive inference. Combined together Godel’s two incompleteness theorems are a manifestation of The Principle of Dimensionality.
Equivalence
Back in 1907, and still working as a clerk in the patent office, Einstein experienced what he would later refer to as the “happiest thought in [his] life.” The Special Theory of Relativity, as its name suggests, applies to only a subset of inertial reference frames – those that move at a uniform constant-velocity. The theory also remains silent on the impact of gravity. Einstein was troubled by these limitations. It was whilst sitting in a chair at the patent office that he came to the realisation that would stick in his mind and launch him on the path towards The General Theory of Relativity. Einstein recounts feeling “startled” when he realised that “if a person falls freely, he will not feel his own weight.” Over time, Einstein refined his realisation into a thought experiment involving a man in a free-falling chamber. We can explore this experiment once again using your pirate ship.
Let’s now say that you’re more than a pirate – you’re a space pirate! You traverse the galaxy looking to raid unsuspecting spaceships to steal their cargo. Now, I want you to travel to an empty part of the galaxy where there are no gravitational forces. Once again, descend to the ships cabin, close all the windows, and this time, make sure to turn off the ship’s gravity boosters. In a region of space where there is no gravity, you and your ship will be free-floating. With the gravity boosters disengaged, you start to float within the ship’s cabin. Now let’s assume that you previously screwed a hook into the ship’s deck and we organise another ship to free-float above yours with a hook affixed to the base of its hull. You order your first mate to tie a rope between the two ships.
All of a sudden a force forces you back towards the floor of the cabin. What just happened? It could be that the other ship has accelerated upwards, bringing with it the floor of your cabin towards you. It could also be that a magically appearing large mass below the ship created a gravitational field. In this case, you are gravitated towards the floor of the cabin and the floor of the cabin, supported by a rope to the ship above, remains stationary.
Einstein realised that from within the enclosed bounds of your cabin, there is no way to differentiate between the two situations. The floor of your ship accelerating towards you and you gravitating towards the floor of your ship are equivalent. Acceleration and gravitation are equivalent. Einstein coined this The Equivalence Principle. Put another way, we might say that accelerated and gravitated inertial reference frames obey The Principle of Relativity. We might also say that accelerated and gravitated inertial reference frames cannot be used to explain themselves: The Equivalence Principle is another manifestation of The Principle of Dimensionality.
Rotation
It is seemingly mysterious that rotation appears to contravene The Principle of Relativity. That is, there is a way to determine whether you are situated within a rotating inertial reference frame from within that reference frame. The situation is described by a famous thought experiment proposed by Isaac Newton – The Rotating Bucket.
Lets say you take the ships bucket and you screw a hook into the centre. You tie a rope to the hook and then to another hook in the ceiling of the ship’s cabin. Now you hang the bucket from the ceiling via the rope. Next you fill the bucket with seawater. If you twist up the rope tightly and then release it, then the following happens: The bucket rotates and the water begins to creep up the sides of the bucket as the centrifugal forces take effect.
If you were to find yourself within a large rotating bucket filled with water you would be able to conclude that you were in a rotating inertial reference frame just by observing the concavity of the surface of the water. It seems you do not need to compare your reference frame to that of another. Newton had concluded that rotation is absolute. The physicist and philosopher, Ernst Mach, disagreed. Mach argued that the bucket experiment is incomplete and that the centrifugal forces are produced by the bucket’s relative rotation with “respect to the mass of the Earth and other celestial bodies.” Einstein later coined the argument as Mach’s Principle and, although less fundamental than The Equivalence Principle, used it as a guiding light when he developed The General Theory of Relativity. Mach’s Principle can be stated simply as “mass out there influences inertia here.” It is a generic way to say that an inertial reference frame cannot be explained using itself. Mach’s Principle is another manifestation of The Principle of Dimensionality.
In 1949, Godel presented a new paper to Einstein for his 77th birthday. Within the paper, Godel had solved Einstein’s field equations of General Relativity for a rotating universe. This design of Godel’s universe specifically violates Mach’s Principle, as the stars rotate faster and faster as one moves further away from the centre. Godel’s universe is an example of absolute rotation and a contravention of The Principle of Dimensionality: One is left with the question of “rotating within what?” Like all contraventions of The Principle of Dimensionality, Godel’s universe produces infinite regresses and paradox in the form of closed timelike curves that return an observer back to his starting point. This opens the door to backward time travel and the Grandfather paradox.
A Universal Principle
Both Einstein and Godel were unwitting champions of a fundamental principle that underscores how the nature of reality must be explained – The Principle of Dimensionality: A problem-space cannot be explained using itself; its explanation resides in another dimension. Although compatriots, they each worked from different angles. Einstein’s genius lay in his ability to recognise new manifestations of The Principle. Godel’s genius lay in his ability to illustrate how infinite regress, paradox and induction (The Trinity) emerge when one contravenes The Principle.
References
Einstein, A. (1905). Zur Elektrodynamik bewegter Körper [On the Electrodynamics of Moving Bodies]. Annalen der Physik, 17(10), 891–921.
Isaacson, W. (2007). Einstein: His life and universe. Simon & Schuster.
Gödel, K. (1931). Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme I [On Formally Undecidable Propositions of Principia Mathematica and Related Systems I]. Monatshefte für Mathematik und Physik, 38(1), 173–198.
Gödel, K. (1949). An Example of a New Type of Cosmological Solutions of Einstein's Field Equations of Gravitation. Reviews of Modern Physics, 21(3), 447–450.
DiSalle, R. (2020). Space and Time. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy (Winter 2020 ed.). Metaphysics Research Lab, Stanford University. Section 2.1: Newton's Bucket Experiment. Retrieved from https://plato.stanford.edu/entries/spacetime-theories/#NewBucExp
Healy, J. (2022). Mach's Principle. In E. N. Zalta & U. Nodelman (Eds.), The Stanford Encyclopedia of Philosophy (Spring 2022 ed.). Metaphysics Research Lab, Stanford University. Retrieved from https://plato.stanford.edu/entries/mach/
[1] Problems come in two varieties. Apparent constants in need of explanation, or expected constants that fail to be observed.