In Michio Kaku’s book “hyperspace: a scientific odyssey through parallel universes, time warps, and the 10th dimension” Kaku describes a moment that inspired his intellectual journey. “When I was 8 years old, I heard a story that would stay with me for the rest of my life. I remember my schoolteachers telling the class about a great scientist who had just died. They talked about him with great reverence, calling him one of the greatest scientists in all history…. I didn’t understand much of what they were trying to tell us, but what most intrigued me about this man was that he died before he could complete his greatest discovery. They said he spent years on this theory, but he died with his unfinished papers still sitting on his desk.” Kaku credits this mystery as contributing to his desire to pursue physics and a deeper understanding of the world. The man in the story was Albert Einstein and the theory was a unified theory of physics.
Einstein is a household name in physics for good reason. Through the simple act asking questions, and exploring the logical ramifications of those questions no matter how unusual, Einstein was able to reason his way into a new theory of gravity: relativity. The problem was that relativity, regardless of how successful it was as a theory, only explained gravity; the other fundamental forces, electromagnetism and the nuclear forces, were not addressed. Einsteins final task, which he never completed, was to unify gravity with these other fundamental forces. To create a theory that accounted for all of the fundamental forces in physics. However, Einstein did not succeed and the search for such a unifying “theory of everything” inspired a generation of physicists, Kaku among them.
We humans exist in a three dimensional world. Objects have height, width and breadth, and to identify an objects location on our planet we would need to identify three number: a latitude, a longitude, and an altitude. We could think of time, duration, existing as a forth dimension, but we can only do that if we accept that it is a different type of dimension that we humans experience separately from the other three. I can rotate an object in three spatial dimension, but time seems to be constant and unchanging. The dominant theory of time prior to Einstein’s relativity Newton’s mechanics. Newton viewed time as an immutable quantity. Time moved forward at the same pace regardless of who, or what, was measuring it. Time was a universal constant and, unlike space, cannot be changed. Relativity changed this. In relativity time isn’t fixed but instead can bend. Einstein’s theory of special relativity postulates that the experience of time is ‘relative’ to how fast an observers is moving. As my speed compared to an observer increases both our experiences of time and space also change. Time, for fast moving travellers, will be observed to be passing slower to their stationary friend; this is known as time dilation. As well, the perceived size of a fast moving traveller will also appear to contract: length contraction. General relativity takes this idea one step further by recognizing that our experience of acceleration and our experience of gravity are fundamentally the same thing. Large masses bend both space and time in a similar fashion. Relativity insists that space and time are not two separate entities that follow two separate sets of rules. Instead, they are a single object, “space-time”, following a single set of rules. Thus Einstein simplified physics by adding a higher dimension.
Kaku’s book introduces this idea from Einstein and follows it through a number of logical expansions. If adding a forth dimension allows us to explain gravity through geometry then maybe we can add even more dimensions to help us to explain the other fundamental forces. Hyperspace is ultimately a book about string theory and all of the various false starts and dead ends physicists took in order to expand Einstein’s four dimensions into the ten that the theory requires. Fundamentally, Kaku is arguing that the laws of physics simplify when viewed from higher dimensions.
Hyperspace was an important early inspiration into my own intellectual journey. The book was my first introduction to physical theory and Kaku’s main argument has stuck with me to this day. Of course, I was a child at the time and my underdeveloped brain didn’t understand anything about the physics or mathematics Kaku was arguing for, instead I connected to the simpler explanations of how higher dimensions can make possible the seemingly impossible.
Imagine a creature whose entire world is a single piece of paper. From the perspective of this creature a we humans can do the impossible. We could bend the paper in on itself causing two ends to touch and allowing the creature to instantly ‘warp’ from one edge of their universe to another. We could also remove this creature from their paper world, ‘turn them over’, and magically transform right into left. Through a simple act of geometry we can permanently disfigure the creature because it is unable to flip itself back due to that act requiring a third dimension. Likewise if two such creatures saw each other, they would only be able to describe the exterior shell or outline of each other, but we can easily describe their internals. This knowledge is trivially gained by us but is functionally unknowable to the two dimensional creatures.
It seems almost trivial now, but the simple idea of a higher dimension beyond the three that we live in changed everything for me. Just by existing it opened up the possibility of two simple truths. The first is that there are things in this world that I am physically incapable of experiencing, like extra dimensions, that exists, and effects me. The second is that even though such a reality is physically beyond me, I can still come to understand it. In a single chapter Kaku taught me a single unknowable truth, that it is possible to visualize a four dimensional cube, and in doing so convinced the younger me that with sufficient education there is no reason why this three dimensional being couldn’t be made to understand the unknowable truth of the universe, even if I can’t experience it for myself.
The problem with such an revelation is that instead of offering up any concrete answers, it merely pointed out a direction through which further inquiry could be made. The unification Kaku sought was clearly a mathematical one. Even though Einstein unified the three dimensions into four dimensions, time still stands alone. If we imagine the second hand on the face of a clock. When the hand is at the twelve all it’s length is along the vertical dimension. As time passes and the hand rotates the length becomes less vertical and more horizontal. As we rotate the second hand its experience of the vertical direction shrinks until it becomes zero once the second hand is fully horizontal. Thus rotation in space can shrink sizes from it’s true size all the way to zero. Time works opposite to this. As an object speeds up, other observers will notice that time appears to slow down and thus expand for the traveller. How limit much time expands is only limited by the speed of light where time becomes infinite. However, the shortest possible measurement of time will always come from the person who is experiencing it. Thus rotation in time can expand sizes from it’s true size all the way to infinity. Time works backwards to space and is therefore still special.
The equations of relativity are written in mathematical language called Riemann geometry. Riemann developed a system for expressing a multitude of these ‘exotic’ or non-spacial geometries using the same common language. Both space and time fit within this paradigm and thus equally expressible as a four dimensional Riemann manifold. However, ‘dimension’ in this context suddenly becomes a distinctly mathematical word that loses a lot of its meaning when taken outside the context within which it was defined. So when Kaku says that the laws of physics simplify at higher dimensions, he is referring to a distinctly mathematical definition of the term that doesn’t have much meaning outside that context. Outside of mathematics time and space are different quantities following different rules. Inside mathematics both space and time are similar objects with slightly different, but still expressible, properties. Unfortunately, it is all to easy to jump to the conclusion that just because some things can be expressed within the same framework, that all things eventually will be as well. Kaku never goes so far as to make any wide sweeping epistemological claims, he simply seeks to unify the fundamental forces of physics and sees dimension as a way to do it. However, others have.
Pythagorus was an ancient Greek philosopher whose name will be familiar to anyone who has studied dimensions. The very formula that describes rotation in a spatial dimension carries his name. The Pythagorean theorem states that the three sides of a right angle triangle are related. That the square of the hypotenuse, the side opposite the right angle, is equal to the sum of the squares on both additional sides: A^2 + B^2 = c^2. As the second hand rotates its experience of the vertical and horizontal dimensions changes according to that formula. If the vertical dimension says that the hand is three centimetres long and the horizontal says it is four then the Pythagorean theorem tells us that its true length is five centimetres. Unfortunately, apart from this theorem very little in modern mathematics is attributed to that man. In fact, very little about Pythagoras’s actual beliefs and teachings are known, or even can be known, for certain today. This is because he wrote nothing down and everything we do know comes to us through other sources; most of which written hundreds or even thousands of years after his death. Worse, even his theorem likely did not come from him. Modern archaeology supplies plenty of evidence that the Pythagorean theorem was known in some form or another in ancient Egypt, a place where folklore states Pythagoras studied. At best he only rediscovered or popularized the theorem. At worst had nothing to do with it. However, what we do know is that Pythagoras, or the pythagoreans, believed that deep down at its core the universe was made out of numbers.
Legends has it that Pythagoras also had a Eureka moment that formed the foundation of his view of the universe. Kitty Furgeson in her book, “Life of Pythagorous” describes this moment as such. While experimenting with the strings of a lyre Pythagorus, “(or someone inspired by pythagorus) discovered that the connections between lyre string length and the human ears are not arbitrary or accidental. The ratios that underlie musical harmony make sense in a remarkably simple way. In a flash of extraordinary clarity, the Pythagorean found that there is pattern and order hidden behind the apparent variety and confusion of nature, and that it is possible to understand it through numbers.” The details of this statement are of course up for debate. Iamblicus, an important early biographer of Pythagoras, claims that Pythagoras came to this understanding while listening to the sounds of a hammer strike an anvil not while playing with a lyre. As well, the modern reader won’t find much enlightenment in what fragments we do have of Pythagoras’ metaphysics. Most of it sounds like numerology. The Pythagoreans believed in a theory of music that governed the heavens. The strings on a Lyre could be tuned in an unlimited number of ways, but it was only when they were tuned to the integer ratios that it could generate harmony, the same is true for the heavens. Pythagoras believed that each celestial body orbited a “central fire” and produced a sound as it travelled. Each heavenly body produced a different sound and together produced a harmony. Ten was an important number to Pythagoras because it represented perfection in his system. One can create a perfect triangle by starting with a base of four round balls and stacking three, then two, and finally a single ball on top of it for a total of ten balls. Likewise, Pythagoras needed for there to be exactly ten celestial objects orbiting the ‘inner fire’: the sun, the moon, the earth, five planets, the stars, and a mysterious counter earth that was never visible because it was always hidden on the other side of the inner fire. This counter earth existed not because he had observed it, but because his model wouldn’t make sense without it.
The details of what Pythagoras actually believed are eternally up for debate; we can’t really know anything for certain. But his impact was undoubtedly profound. Plato’s, inspired by his conversations with Pythagorean followers, included a detailed geometric view of the cosmos in his Timaeus. In this model the world was literally geometry and made up of atomic triangles. Each of the four basic elements, water, earth, fire, and air, gain their properties through the configuration of the triangles within them, and the world as a whole comes out of the interaction of these elements. Modern readers might see this construction as nonsense, but if we remember that the foundation of modern mathematics, Euclid’s elements, had not yet been written its much easier to see that the geometric view of the universe in Plato’s Timaeus is at least an attempt to explore the ramifications of a mathematical universe. As mathematics has developed so too have the mathematical models that describe our universe. There is no shortage of such models we could explore. Some are absurd, like Kepler’s early model of the solar system which used platonic solids to describe the orbits of the planets, some are useful, like Brahe’s geocentric model, and some would fundamentally alter how generations of scientists view the universe, like Newton’s laws of gravity. So the question remains, is there a correct fundamental model of the universe?
Physicist Max Tegmark takes this idea to its ultimate extreme in his book, “Our Mathematical Universe” where he argues that the universe isn’t just described by a mathematical model, it is a mathematical object. He calls this the Mathematical Universe Hypothesis. “If the Mathematical Universe Hypothesis is correct, then our Universe is a mathematical structure, and from its description, an infinity intelligent mathematician should be able to derive all these physical theories.”
Fundamentally, this is the question that stuck with me into adulthood. Does such a theory exist? I don’t mean does a unified theory of the fundamental physical forces exist? For all I know Tegmark’s confidence that the we will be printing t-shirts with the equations of a a unified theory of physical in our lifetime could be correct, but that question doesn’t interest me. Instead, what I’m asking is something more profound. Is there a unified theory of everything. Can a sufficiently powerful, infinite dimensional, creature derive our universe, everything inside it, and everything it is capable of becoming. Thinking back to my childhood both myself and Pythagoras had a similar moment. Reading Kaku gave me a brief moment of enlightenment where I realized that part of the world isn’t random. However, unlike Pythagoras my religious background prevented me from taking the next step. Just because something is understandable doesn’t imply that everything is.
The problem with the mathematical universe hypothesis, and the search for a “unified theory of everything” in general is that it is impossible to argue against. Sure Pythagorus’ model containing exactly ten celestial bodies is wrong, Newton’s theory describing a static and universal time is wrong, Einstein’s failure to account for electromagnetism and the nuclear forces implies that his theory is at best incomplete, and string theories inability to predict any experimental outcome implies that it probably isn’t the final theory either. Yet, none of these precludes the idea that a correct and perfect mathematical model of the universe doesn’t exist. There’s just is no way of proving a negative. Worse, the fact that each of these models improves on the previous model implies some sort of forward momentum. Each scientific breakthrough doesn’t invalidate the knowledge gained from the previous model, it merely casts that knowledge from a new perspective, a higher dimension, that let’s us see old knowledge along with new knowledge in a common framework. So how do we account for the success of the mathematical sciences without admitting that at some level the universe is mathematical?
What is mathematics if not the language of structure. If I say that 1 + 1 = 2, I am asserting a relationship between these two objects that offers clues as to the structure that these objects live in. If this statement is only true sometimes, or in some contexts, then that implies that the rules governing these objects still dictate that 1 + 1 = 2 in such contexts. Thus our knowledge might not be universal, but it still hints at a fundamental nature we still know something about. To even begin searching for knowledge we must first admit that such a thing, a stable foundation, exists at all. If the universe exists then it must be true to its own nature. If the universe changes it is not because the nature of the universe has changed, it is because the nature of the universe is to change. If we assume that the the universe is true to its own nature and that it follows its own rules then mathematics inevitably leaks out.
At this juncture we have not built a foundation strong enough to even explore that assumption in full. However, what we can do is explore what it means for something to follow its own rules and to be mathematical. This is where my own personal journey began. If mathematics can generate truth, then what does a mathematical universe look like? What, if anything, can math tell us about itself.