A Mathematical Universe

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.

A Reintroduction to Truth

(Note: This is a rehash of a much earlier blog post.)

I discovered physics at a relatively young age and fondly remember reading every book I could find on the subject at the local library; quantum physics, higher dimensions, multiple worlds: this stuff fascinated me to an extent I still can’t fully communicate. I will admit that I didn’t understand the vast majority of what I read; although, at the time understanding wasn’t the point. Instead, I was simply enamoured with the idea of a deeper truth to the universe. The very idea that the world was understandable and that some people held access to it was appealing. I wished more than anything else to be one of those people someday.

I entered my undergraduate degree with the goal of becoming a physicist, but I didn’t make it very far. I got to third year before finally failing a course and giving up. I can see now that that I didn’t really fit into the physics world for two primary reasons. The first was that I misunderstood what physics was about. I thought it would be an opportunity to explore and understand the deeper realities of our world, but instead I found myself involved with a community interested only in creating rigorous mathematical models, concocting experiments to test those models, and modifying the models based on the experimental feedback. The entire undergraduate degree being nothing more than a desperate attempt to catch the student up on on hundreds of years of mathematical models. No explanation about why these models are important beyond ‘it accurately predict the outcome of experiments’ was offered or required.

This is not a point of criticism. I accept that teaching is hard and I have no interest in starting a conversation on how to do it better as it really was the second problem that prevented me from working through the first. The year was 2008 and a man by the name of Christopher Hitchens had only the previous year published an influential book by the name of “God is not great”. That book, along with others published around that time, created a movement I would come to know as ‘new atheism’ that was founded on the idea that humans must evolve beyond the need for religion in order to progress. These ideas circulated wildly among the undergraduate physics society during my degree, and created a problem for me specifically because I was still working through my own beliefs as a person who identified as an Evangelical Christian.

My father would later describe the church I attended growing up as a, “church people came to when they were fed up with their other churches”. Because of this it is difficult to categorize the exact theology I was taught. I was exposed to a wide variety of theologies and they all competed equally for my attention. I remember honest conversation about the nature of God, the question of evil, and actual struggles to understand the tragedies that the bible chronicles. Likewise I also remember being brought into a dark room solemnly taught the ‘truth’ of revelation while having the entire timeline of the apocalypse laid out before me. I had conversations about determinism, creation, eschatology and the many different ways Christians around the world express their belief. Even though I was introduced to much fundamentalist doctrine, I never became a fundamentalist, and indeed never viewed my religion as set in stone, unchanging, or inerrant in any way. If anything, this variety of Christian religious experiences only reinforced in me the idea that God is mysterious; that humans are flawed beings trying in vain to express something that they can’t fully comprehend. I wasn’t blind to the evils in my religion; there were plenty of false prophets, hypocrites, and manipulators. Yet, those too only demonstrated what I now believe to be the bible’s strongest and most consistent message: that whenever humans believed themselves to be closest to God they were instead farthest from him. I was well aware of the crimes of the church, and had already spent most of my life passively taking in conversations about the relationship between these crimes, humanity, and religion as a whole. So it was a little jarring being thrust into a social scene who, having had none of these conversations, viewed religion as at best ridiculous and at worst an intellectual disease from which humanity needed saving.

So what was the replacement supposed to be? Well it was science of course, and in the physics world science is just another word for mathematics. Over the course of three years in an undergraduate physics degree I took five courses in calculus, two courses in linear algebra, two courses in complex numbers, and several others I don’t care to list out. The physics courses made even less sense because they were also mathematical courses; they just didn’t begin with a list of axioms and were therefore more confused about what transformations were valid and which were not. We had one token experimental course where we actually ran some of the experiments that physics claimed as its source of truth, but the labs we used were so underfunded and the technicians, us, were so poorly trained that our data never aligned with accepted theory. The reports were always a desperate attempt derive a plausible sounding narrative out of the random data our experiments generated. All in an attempt to impress whoever marked our work.

The worst part for me was that everything seemed so familiar. A preacher in front of the congregation giving long lectures justifying a conclusion I didn’t understand out of a data source I couldn’t understand. The only difference being that lecturers had whiteboards and preachers had pulpits. These people seemed just as certain in the inerrancy of mathematics to speak truth about the universe, as my religious friends were in the inerrancy of the bible to do the same. So when I saw my peers talking about the ‘obviousness’ of the nonexistence of God, I couldn’t help but compare them to the other side who talked about the ‘obviousness’ of his existence. It was a debate between people who were so certain that they themselves were correct that they couldn’t possible see the world through each others eyes. Indeed, their certainty required that they never try.

Reading Hitchens today only reinforced my suspicions back then. ‘God is not great’ is a damning catalogue of religions many crimes, but its argument against religion relies heavily on the readers predisposition to hate religion. He describes in great deal how religion has been, and continues to be, a contributor to warfare, a tool of political control, and a shield protecting histories most disgusting criminals. Yet, the conclusion implied in the books subtitle “how religion ruins everything” that we would be better off without religion isn’t really argued so much as assumed. For example when describing the barbaric practice of female circumcision Hitchens points out that, “No society would tolerate such an insult to its womanhood and therefore to its survival if the foul practice was not holy and sanctified.” Here the subtext is obvious, if religion couldn’t be used as a justification for this horrific attack on women, then the act wouldn’t have happened. He doesn’t go into detail, but the whole statement hinges on a deeply evolutionary argument. Women are necessary for our species to reproduce, so therefore an attack on women is in essence an attack on our ability to reproduce. This behaviour cannot come from an evolutionary standpoint and is therefore not natural. So such an attack can only be possible if something else, something evil, was overriding our fundamentally good nature.

But is this really true? Does removing the justification for a horrific act suddenly prevent the act itself? Unfortunately, Hitchens makes an assumption here that is prevalent in western philosophy; that humans are rational animals. In this context rationality means that we are always acting in such a way as to maximize some internal good. We have an internal model of how we think the world works, we use that model to weigh actions, and then we act on the results. If humans worked this way then yes it would be logical to conclude that getting rid of an incorrect model would force us to seek out a better model, and by extension act better. However, what if the opposite is true? What if our nature is not rational and we instead act first and only search for justification later. If this were the case then getting rid of religion accomplishes nothing. The act would still happen and the culprit would simply corrupt something else to act as justification for the action. This is a point that Hichens all but concedes when he tries to explain away the horrors of the ‘secular’ totalitarian government in Soviet Russia, “Communist absolutists did not so much negate religion, in societies that they well understood were saturated with faith and superstition, as seek to replace it.”

What about culture? The line between religion, culture, and ethnicity is something Hitchens never even bothers to address. If we accept that religion is evil then how do we excise it without stripping away a peoples’ cultural identity? While discussing the ethnic and religious violence in Yugoslavia he comments that, “Elsewhere in Bosnia-Herzegovina, especially along the river Drina, whole towns were pillaged and massacred in what the Serbs themselves termed “ethnic cleansing.” In point of fact, “religious cleansing” would have been nearer the mark.” The distinction between ethnic and religious is of fundamental importance to his argument and yet he fails to elaborate beyond this snide remark. Yet, just as easily as Hitchen’s can turn the word ethnic into religious the opposite is also true. When discussing Martin Luther King he says that, “the examples King gave from the books of Moses were, fortunately for all of us, metaphors and allegories. His most imperative preaching was that of nonviolence. In his version of the story, there are no savage punishments and genocidal bloodlettings. Nor are there cruel commandments about the stoning of children and the burning of witches…. If the population had been raised from its mother’s knee to hear the story of Xenophon’s Anabasis, and the long wearying dangerous journey of the Greeks to their triumphant view of the sea, that allegory might have done just as well. As it was, though, the “Good Book” was the only point of reference that everybody had in common.”

Hitchens has already concluded that religion is evil, and so the very fact that King ignored the problematic parts of the passage somehow saves him from being ‘religious’. Instead, any goodness that originated from King must have came from something else. In this case that something else is language and folktales, an important component of what we would call ‘ethnicity’. The people King was talking to were ethnically Christian. The bible is something they all knew, and when King attached his ideas to something his audience understood he had a better chance of getting those ideas across. In essence, King’s message wasn’t important because of its religious affiliation, it was important because of its ethnic affiliation.

This type of slipper definition is precisely what stood out to me in undergrad, even though I didn’t have the vocabulary to express it at the time. These definitions begin with an absolute statement, “Religion ruins everything”, and when faced with a situation where religion is not ruining everything they must immediately explain why the religion is in fact not a religion. Once again, this is all too familiar because this style of argument is the very glue that holds fundamentalist Christianity together. This is the logic that creates what Hitchens is so desperately trying to destroy.

Evangelicalism specifically focuses on the gospel of the ‘good news’. It is important that a Christian spread the good news of Christ because we are actively making the lives of those who hear it better. Books like, “Run Baby Run” by Nicky Cruz reinforce this message by painting the secular world as dark and grim. That world is full of gangs, drugs, unfulfilling sex, and the most extreme and grotesque forms of violence. The way out of this world is through the message of Jesus Christ. Likewise, by definition none of these things can exist within the Evangelical church as Christians leading better lives is core to the doctrine. We are then stuck in a situation where there is a fundamental disagreement about who gets to be religious. Hitchens arguing that good Christians are actually humanists, and Evangelicals arguing that bad Christians aren’t actually Christian. Of course, nobody agrees on what is good and what is bad and so nothing is ever decided. Both sides are in effect the same. The argument is simply western thought fighting over its own details. Concepts important to this discussion, such as the distinction between religion and ethnicity or whether God exists in a literal sense, just don’t mean as much in any other context.

All of this leads one way or another to a kind of negative morality. A position where we are focused entirely on the eradication of evil in order to allow good to flourish. If religion ruins everything then by getting rid of it we allow ourselves to return to the rational state it removed us from. If God is good and has rescued us from our sins, then we must destroy these sins so that it can’t capture us again. In both cases any violence that erupts is a necessary evil that transitions us into a better world. Another important idea in western thought is the inevitable triumph of good over evil. The trope of ‘saving the world’ in one heroic act of justified violence powers a majority of our popular media. The good guy always defeats the villain. The problem though is that the fight between ‘good and evil’ never ends. Once the evil bad is destroyed there is always an eviler bad to follow. Sooner or later the ‘war to end all wars’ simply becomes the ‘previous war to end all wars’. There has never been any scientific evidence that goodness is inevitable.

Atheism was never an alternative to my own religion. At the time it was obvious to me that jumping from one to the other only replaced one idol with another. I was taught that Christ and his word are truth, and Hitchens believes that science is truth. Yet, what truth was to both didn’t differ, it was a system founded on a single inerrant principle that denied the existence of anything not demonstrable through that system. Yet, here I was seeing both systems and finding both to be equally fascinating and equally flawed. I do not disagree with those who question religion. God doesn’t have to exist, science is a better explanation as to how we got here, and holding a thousands year old document as a source of inerrant truth is hard to defend. Yet, it wasn’t any triumph of human rationality that got me to doubt my own religious convictions, instead it was the idea of negative morality. Why was it that a religion founded on the ‘good news’ of Christ rescuing us from our own corruption was so focused on categorizing said corruption? If God is so powerful, why are we so afraid of evil? Why must the fear of hell power more of my decisions than the love of God? If someone is happy and content with their lives, why must I conclude that they are faking it if they aren’t ashamed of what I am personally labelling as their sin? And the same argument works against people like Hitchens. Why is he so focused on destroying something that he argues does not exist? Why must he continue to believe that religion holds no value when there are clearly billions of people worldwide who are continually attracted to it?

What does it mean for a belief to be true? For that matter what does it even mean for anything to be true? Looking back on all those physics books I read as a child I cannot deny that what drew me to them is the promise of objective reality. There was something out there, independent of me, it had structure, it had order, and it was beautiful. Back then, as today, I believed in objective reality, in an objective truth. I believed that truth wasn’t a personal matter, it wasn’t unique to me and didn’t change from person to person. I believed this because it had to be true in order for my own experiences to make sense. If I were somehow capable of making something true for myself then the world would be a fundamentally different place: it would be one where I understood why the people around me reacted to me the way that they did.

One thing both physics and religion had in common was that they both, at least in their teachings, actively encouraged me to seek truth on my own. The preachers implored me to read the bible and pray to God for wisdom, while the scientists encouraged experimentation as those were repeatable and not beholden to the whims of an individual. What does one do when their personal truth is suspect, but the alternatives are no better? How does one rectify a belief in an objective absolute truth with the realization that my own understanding of that objective reality is clouded by the things that I believe?

Well, I didn’t have an answer back then, and I won’t pretend to have one now. However, the journey I’ve been on since deciding I wanted to find out has been an adventure and I’d like to share it with anybody willing to listen.

Does Juri exist? Classifying Existence.

Defining existence, like many philosophical terms, is a notoriously difficult task. Intuitively it is extraordinarily simple concept, which is a problem. When asked if anything exists anyone can give a quick binary answer: either it exists or it doesn’t. Humans exist, Unicorns don’t exist, black holes exist, and nothing that happens in a dream exists. It should be this easy, but it isn’t. Intuitive obviousness, while to some is a concrete argument, is both relative to the speaker and subject to the poorly understood complexities of the human mind. Unfortunately, intuition is also sneaky. It works its way into many arguments unintentionally, and is often difficult to uproot. The most bizarre situations happen when intuitive arguments are used to oppose other intuitive arguments. Existence is one of those concepts. The purposes of this post are not to solve the problems associated with existence, but instead to complicate it enough to shut down some of the more ridiculous arguments against the existence of god, the soul, or other equally ethereal entities.

Existence, as it is commonly thought of, is closely related to physics. If an entity exists in physics then we can say the object has material or physical existence. Therefore, a table exists because the laws of physics act on that table. If I pick it up, gravity will want to pull it back down. If I exert pressure on it, those forces will move it, and eventually break it. My body has physical existence for the same reason; everything I do with my body is subject to the laws of physics: same with trees, buildings, clouds, and the wind. In all cases, the defining quality of a physical entity is that the laws of physics govern their interactions with other physical entities. Therefore, they are quantifiable, observable, and absolute knowledge is assumed achievable through the application of the scientific method. Disproving physical existence is impossible using this definition. Just because I haven’t seen a unicorn, doesn’t mean that they don’t exist. It is always possible that they do, I just haven’t come across it yet. This is the problem of induction, and is usually the argument that erupts when existence comes into question. I have no intention of discussing induction, but to merely mention that physical existence has far more worrisome problems then just induction.

How about Narnia, does Narnia exist physically? Most certainly not. Under our previous definition, Narnia cannot exist materially. It is not subject to the laws of physics, and instead is subject to the arbitrary laws put forward by the brain of C.S. Lewis. The interactions of elements inside Narnia are entirely predefined, and therefore untestable by the scientific method. For these reasons, it cannot exist materially. However, there is no reason why we can’t put forward a new definition that manages to classify this bizarre entity. While Narnia cannot exist materially, we can say that it exists as an idea. Ideas are complicated entities in and of themselves, and there is much discussion about what creates them. However, for the purposes of this post I’ll just assume that ideas are created by observing material existent objects. Therefore, C.S. Lewis created Narnia by observing things in the real world, and taking certain elements and putting them together in ways that are not always allowable by the laws of physics. Unicorns then are ideas also. Someone saw a horse and a narwhal then combined elements of both to create a unicorn. Ideas don’t have a material presence, but they do exist within our minds.

So now that we have solved all the problems associated with existence, let’s use it to answer a few questions. Obviously, everything I am capable of talking about exists as an idea so let’s just answer some questions about material existence. Do Dragons exist? Probably not. Does sound exist? Yes. Do computers exist? Yes. Do computer animations exist? No… Do Video’s exist? Yes.. Do movies exist if they were only distributed over the Internet? No… Maybe. Finally, does Juri exist?

Juri most certainly exists as an idea. I see a picture of her, I press buttons, and my eyes and ears get the impression that she moves around. I press other buttons and those same sensors get the impression that she is kicking other idea people. However, does she exist materially? This is a question that different people will intuit different responses. On one hand, she is a visual representation and nothing else. She isn’t human, but she is representative of one. On the other hand, she is still a product of, and subject to, physics. She is a product of a computer. When I push a button, a series of physical interactions take place. My actions put electrons into motion, which interact with the computers processor in a very physical manner. Eventually these electrons will make it up to my computer monitor, which outputs lights and sounds all of which happen in a very physically testable way. How Juri moves is both observable, repeatable, and subject to the same process that would allow a scientist to link it back to the greater theory of physics. The laws of physics govern everything about her, from the initial button push, to the light signal sent from the monitor, to how my eyes receive the light. Yet even now, if I ask ‘Does Juri Exist?’ one might be tempted to answer in the negative. One could argue that Juri herself doesn’t exist; however, instead she is just an idea created by a separate physical entity. The computer creates the idea of the existence of Juri, but the computer, not Juri, comprise the material presence. The problem with this argument is that it I can easily rework it to argue something intuitively absurd. I claim that the computer itself doesn’t exist materially. Molecules contain large amounts of electrons and protons. The electrons and protons bump around each other, each subject to the laws of physics, causing physical processes that send signals to my brain. These molecules, through physical interactions, are creating the idea of a computer in my mind, but the molecules, not the computer, comprise the material presence. The computer then is just as much of an idea as Juri is. However, the argument works recursively. We have no reason to stop with molecules. We can continue breaking objects down into smaller and more theoretical elements until we arrive at the limit element of that series: physics itself. In this way, the only conclusion we can arrive at, if we allow this argument, is that the only thing we can consider physically real is physics itself. Continuing this train of thought, the next question to pop into my head is, ‘What is Physics?’ If you ask a physicist, they will probably give you a long lecture of sorts. This lecture will include an argument, some tables, some numbers, the scientific method, and plenty of math all of which fail our definition of material existence. I can’t touch arguments, interact with tables, and the mathematician inside me cringes at the thought of claiming math is anything more than an idea. Therefore, we have arrived at the conclusion that since physics itself is an idea everything else is as well.

As we can see, the semantic leap between arguing Juri has no physical existence and arguing nothing has physical existence is extraordinarily tiny. Yet the idea that everything is idea is just as intuitively absurd as arguing that everything is physical. Unfortunately, the symmetric argument holds. If we argue that Juri does exist physically then the leap to argue that unicorns, ferries, and even Narnia exist physically is just as small because each idea entered our mind through some physical process. The very fact that we can put ideas into physical objects and transmit them between ourselves is extremely troubling for anyone wanting to argue that only provably physical objects actually exist.

Existence then cannot be a binary between ideal and physical. It must be seen as a gradient with connections at both ends. Things can become so physically real that they exist solely as an idea: like most of theoretical physics, and ideas can become so physical that they actively interact with the world at large: like Juri. There is a give and take relationship between the two. Sometime physics creates ideas, and sometimes ideas create physics. Ideas have such a large effect on the world that any detractor would be foolish to argue against them. Ideas start wars, rule societies, and evidently change the path of world history.

This is why I love computers. They are a gateway between the physical and the ideal. Juri is real, because the computer acts as a gateway allowing her to interact with the physical world. However, the computer has not changed physics. It has only given us a portal to see the world as it always has been. Existence then is not about physics, it is about the universe. Everything that interacts within the universe must exist in one form or another. I do not want to detract from the problem of classifying existence, but attempting to organize them into any meaningful categories is an extremely non-trivial endeavor. Right now, I can accept that all ideas are not created equal. I just have no logical method for telling them apart, if one exists at all.

Now we can answer the real question I wanted to discuss. Does God exist? Well the atheist will say he is imaginary, ideal, and therefore not real. My answer to them is, ‘He is an idea, and therefore exists.’ To the theist who demands that God must be more than just an idea I say, ‘He is an idea, what more do you want?’

Suspended Disbelief: The Square Root of 2.

Several thousand years ago the Greeks made a mathematical discovery that rocked the intellectual world at the time. They believed firmly in logic, that any statement logically derived from a true statement must also be true. To doubt this would be to doubt their entire intellectual community, and in some places their entire culture. Pythagoras was a Greek mathematician, as well as a cultist. His religion was one of logic and reason. He and his followers developed a cult of numbers. They worshipped mathematical beauty, and strongly believed the universe was comprised completely of integer ratios. The patterns of the natural numbers defined the Universe and the world inherited it’s natural order from the natural order inherent to ratios. So it was all the more earth shattering when it was proven that the square root of two is not an integer ratio. Creating in mathematics a classification of numbers know as the ‘irrational numbers’. They were aptly named as they defied the logic of the time, and in some ways they still make little sense now.

Imagine a perfect right angle triangle. Also imagine that the artist of this triangle was absolutely exact when measuring it out. The two sides adjacent to the right angle in this triangle are exactly 1 unit long. The remaining side, the hypotenuse, it’s length can be determined by using the famous Pythagorean theorem which was known at the time.

\(x^2 = 1^2 + 1^2 = 2 \Leftrightarrow x = \sqrt{2}\)

The hypotenuse is exactly \(\sqrt{2}\) units long. So far so good.

Now Pythagoras didn’t know anything about \(\sqrt{2}\) or the irrational numbers, so he did the only thing his mathematical mind could do at the time; he tried to figure out what ratio \(\sqrt{2}\) was. Every ratio is in the form \(\frac{a}{b}\) so equating that to \(\sqrt{2}\) we get the formula

\(\frac{a}{b} = \sqrt{2} \Leftrightarrow \frac{a^2}{b^2} = 2 \Leftrightarrow a^2 =2b^2\)

(note: the symbol \(\Leftrightarrow\) means the two statements on either side are equivalent. In this case basic algebra can take you from one to the other.)

Now before we can properly analyze this equation we need to know something about measurement.

Measurement is a universal system for comparing objects. In this case we will be dealing entirely with length. In our society the meter is used as the primary unit of length. If someone told you they had a stick that was 2 meters long we can visualize that distance by joining two meter sticks end to end. Now if we wanted a distance of 1.5 meters we would then have to cut a meter stick in half and join it end to end with a full meter stick. Alternately, we can define a new unit the ‘half meter stick’ and join three of them together. The half meter stick is then \(\frac{1}{2}\) meters long. Notice the ratio. In fact we can define any unit this way. The centimeter is actually \(\frac{1}{100}\) meters and the millimeter is \(\frac{1}{1000}\) meters. If we joined 60 millimeters together we would get \(\frac{60}{1000}\) meters. Then given any ratio \(\frac{a}{b}\) the number b defines the unit we are using, and the number a defines how many of those units we have. Now \(\frac{1}{2}\) is bigger then \(\frac{1}{4}\), so a bigger b actually defines a smaller unit. If we cut b in half, our unit would end up twice as big as the original, and if we multiply b by two the resulting unit would be half the size. Perfect measurement in our system assumes that given any two sticks of arbitrary length, there exists some unit that can measure both sticks. That means that an integer number of those unit sticks can be joined end to end to form a stick exactly the same length as both the sticks being measured.

Now returning to our equation

\(a^2=2b^2\).

Since \(2b^2\) is an even number \(a^2\) must also be an even number. Now it was known at the time that a square number like \(a^2\) can only be even if a is even. I’m not going to prove it, but feel free to try it. Grab a calculator and try it out. If you square an even number the resulting number is always even, and if you square an odd number the resulting number is always odd. Since a is even, we can factor out the 2 resulting in \(a = 2c\). Our equation then becomes,

\((2c)^2 = 2 b^2 \Leftrightarrow 4c^2 = 2b^2\).

We can divide both sides by two to get

\(2c^2 = b^2\).

Now it is easy to see that b is also even using the exact same argument that we used to determine that a is even. So

\(b=2d \Leftrightarrow 2c^2= (2d)^2 \Leftrightarrow 2c^2=4d^2 \Leftrightarrow c2=2d^2\).

Which is the exact same formula we started with. Solving for \(\sqrt{2}\) gives us

\(\sqrt{2} = \frac{a}{b} = \frac{c}{d}\).

But remember \(b=2d\) so the unit d is twice as big as the unit b. Which means if there exists a unit b that can measure the hypotenuse of our original triangle then there also exists a unit d that is twice as big that can do the same job. Now by the same logic if there is a unit d that can measure the hypotenuse of our triangle then there is another unit twice as big as that, and then another one twice as big as that. Eventually we come up to the completely ludicrous conclusion that I can use a meter stick \(2^{99999}\) times or more larger then our original unit. A number that is clearly larger then the length we are measuring.

People were killed trying to let this secret out, that’s how dangerous this knowledge was at the time. One thing the Greeks worshiped was logic, and here we have a completely logical basis followed by completely logic reasoning, resulting in a totally absurd conclusion. Today we know that \(\sqrt{2}\) is not an integer ratio. \(\sqrt{2} \neq \frac{a}{b}\) for any integer a or b. In mathematics we just accept it as fact and move on. However, for early empiricists this is a catastrophe. Imagine that when this triangle is drawn there is exactly 1 billion evenly spaced molecules inside the unit edge. How many molecules are in the hypotenuse? The answer is approximately 1.41 billion molecules, but the decimal trail never ends. Eventually one of the molecules will have to be split into parts, but we can’t measure how to split it. Even if we displaced the molecules so that they weren’t evenly spaces, we wouldn’t be able to measure their displacement. So we are forced to arrive at the unpleasant conclusion that one of two things cannot be possible. Either it is impossible to measure all lines in a right angle triangle, or the triangle was not really a right angle triangle to begin with. In both cases the conclusion is the same, either it’s impossible to measure some lines, or to draw some angles. In both cases our system of measurement is logically flawed.

Today quantum mechanics has given as a way around this problem, but it is still characteristic of a common problem that crops up again and again in every field I have studied. Before this proof came across, it would have been impossible to convince Pythagoras that portions of the universe were fundamentally unmeasurable.To him the universe had to be measurable, the world just wouldn’t make any sense otherwise. This was a statement that coheres so well with the remainder of his world belief system that it was not only true, but it’s truth was self evident and required no further explanation. However, his beliefs aside, the statement was still wrong.

The remainder of the history of mathematics is nothing more then a long series of extremely intelligent people creating hard and fast rules that were simply ignored by the next generation. The greatest leaps in the field come when old unquestionable systems of truths are questioned and then thrown out. This is most certainly not limited to mathematics. I can’t begin to describe how many conversations I’ve been in where someone has tried to convince me of an utterly wrong statement. Like Pythagoras they do not yet have the knowledge necessary to understand why they are wrong, and like Pythagoras whatever it is that they believed coheres so fundamentally to their worldview that absolutely nothing I say can ever change them of their mind. Likewise, I am aware of the same things withing myself. To then arrive at anything that can even remotely resemble the truth of the universe, it becomes necessary to individually pick out every single assumption, challenge it, then keep it if and only if it survives. Even then it must be acknowledged that it is still possible that the evidence of it’s falsehood simply does not yet exist. To do this, I willingly admit, that I may not have the knowledge necessary to challenge those things which I accept as truth. However, here is where the main difference between mathematics and philosophy aids me. Philosophy tries to understand the universe, without defining anything for fear that those definitions may be wrong. In math, we just define away, and deal with the consequences later. Does that make it true of the universe — absolutely not. However, it makes it true of the ideal universe defined by the assumptions I was forced to make. The question concerning which assumptions correspond to the actual universe we live in, is still an open question.