All or Nothing; Division by Zero

If you divide by nothing, in one sense, you get nothing. In another sense, you are left with everything. However much there is to divide, all of it remains – undivided. The remainder is everything. To count it all, however, you must divide it up into discrete quantities. Of course, you can keep dividing it into smaller and smaller parts, each of which is increasingly infinitesimal. As you do so, the number you are calculating approaches infinity. This is the paradox of division by zero.  

To conclude the process, you need to divide everything into parts that are equal to zero. That is, a point where there is no longer a quantity to enumerate. That is, where things cannot be divided further. That point is zero, and at that point you have an infinite number of parts consisting of nothing. Again, the longer you keep dividing – keep finding something to divide – the greater the number or divisions you must make.  

As long as you have something, you can theoretically keep dividing forever. Hence, the intuitive conclusion that division by zero equals infinity. That is oddly consistent with the logical proposition that nothing is infinite. Unfortunately, while we can all agree that multiplying any number by zero produces zero, it is a bit more controversial to suggest that any number divided by zero produces infinity. But take out a calculator and start dividing any number by smaller and smaller decimal numbers and see what happens. Soon enough, the implication becomes clear. The smaller the number you use, the bigger the number you get. Technically, division by zero is equivalent to a process of infinite division. Or dividing by infinity. But the main thing to remember is that zero is equal to nothing, and logical thought accepts the assertion that nothing is infinite.  

Like infinity, zero is an imaginary number. We accept the irrationality of zero with any assumption of a discrete number of any kind. However many numbers you assign to count with, zero is implicitly included as the remaining quantity of any quantization. It closes the circle, creating a finite set of things to account for everything. Like a circle, any base set of numbers falls within a loop describing an infinite spiral from zero to infinity.  

Another way to look at it is that quantization is the creation of infinity. To accomplish it, you merely need an inexhaustible supply of indivisible parts. Call them particles. Describe them as points in zero dimensions. Take them from something boundless, like empty space. Infuse them with the potential for everything. Give each one its own, unique value; individually, equal to any other: A discrete and uncompromised identity. Though none of them possess any actual substance, everything is measured in terms of them, every thing being composed of them, gaining its own substance through them. This is possible because literally everything is divided between them. Each one is too small to grasp with anything, so it is virtually impossible to deal with one in isolation. They are just too small to interact with, but everything is defined by their interactions with each other. 

Everything, unquantifiable in their absence, becomes something that is shared by all of them. Their very existence depends on it. Just as the existence of everything is dependent on them. It is a relationship that makes existence possible. And yet, we insist that division by zero is impossible. Ironic.  

A Point about Paradox

In the eternal moment, the past and the future are simply a matter of perspective and the past can be influenced by overlapping reinforcement or the intent of the resolving future. A time paradox is a self-informing sequence within a point. The classic grandfather paradox assumes that time is linear, ignoring the fact that the appearance of a person from the future is an unprecedented event at that point in the past. It also ignores the consequences of his absence from the point of his departure in the future. For the time traveler, all of the time leading up to his departure from the future, relative to his point of arrival in the past, is a precedent for his existence in the past and the method or mechanism he used to get there. For time travel to be possible, moments in time have to be states of reality that are discrete from each other and persistent in themselves. Given that, the time traveler could go back in time, but the simple fact of his arrival in the past would have the potential to change the course of events, creating an alternate past or creating a feedback loop through the future. That feedback loop could have a constructive or destructive effect, depending on how elastic the fabric of reality is. Put simply, that whole segment of reality would become Schrödinger’s Box. The presence of the time traveler in the past would be sustained by a conservative force of causality. Insignificant changes would be relatively easy to make, while more significant ones would create bifurcations, or multiple instances of reality with a tendency to reconverge on the base time line or completely diverge, a maze of different paths striving to collapse into a stable path. The model of a grandfather paradox would describe a path along the original line, through the point of departure—continuing past that point to account for a future where the time traveler simply ceased to exist—which splits into a loop intersecting the past and entering a tree of divergent and reconvergent branches, at least one of which includes the time traveler walking away from the cooling corpse of his grandfather into an alternate future where, instead of being born, he just magically appeared out of nowhere.

Staring into the Face of Truth

“A story is as good a way to organize your thoughts as anything else,” she points out, poised in the shadows in the doorway. I quickly conclude that she is playing the part of my conscience. That, or devil’s advocate. Either way, she’s me. I cannot say she does not really exist without implying the same of myself. She is in my mind, and of my mind, so I do not look at her. She cannot be seen, not in the flesh anyway, but it’s not like I have to look at her to see her. “It’s been a long time since you’ve done this, though,” she observes, watching me carefully. I can tell there’s something on her mind. I can feel it. Technically, it’s on my mind, but I have long since learned that her thoughts are her own when she chooses to assert herself. It’s a bit like being in two places at the same time, a way to step outside my normal perspective and look at what I’ve become.

“That’s because it takes more effort than thinking, even if it is no more contrived than any other thing written; it isn’t really a story,” I respond. I do not have to add that this manner of confronting myself is one of the reasons my stories never get finished; she knows that as well as I do.

“It helps when you need someone to talk to, though,” she argues, crossing the room to sprawl on the couch next to my desk. There are times when I wish that I could have visual hallucinations; it would be nice to really see her when she goes to the trouble to try and fit herself into the world. Instead, I can only see her in the way I see what I am reading about in books, from everywhere and nowhere. Of course, with her, there is no book, no words; she is self-rendered thought. “It gives me chance to be myself, too.”

“You mean, get some distance from being like this,” I amend somewhat bitterly, in reference to all the unpleasant facts of my reality. Normally, I do not have the patience to write like this. Once I discovered I could split my attention two or three ways, it did not take long to become good enough at it that I would just talk to myself when I needed someone to talk to. I can confront any part of myself that way, even the parts that are smarter and wiser than I can normally be. I have come to believe that this is what angels and demons are, projections of ourselves, impressions of others and the personification of our hopes, beliefs, fears and doubts. It’s what I think of as five-dimensional thinking. “So, what do I need to talk about?”

“I’ve been thinking about what you posted yesterday. Okay, that’s still weird; it’s as much my post as it was yours,” she sighs and scoots closer to the opposite arm of the couch, tucking her feet under herself. “I know what it has to sound like to anyone who reads it, and if people have trouble understanding and accepting a transgendered person, well…” She cannot finish the thought, because I already know what she is going to say.

“People have a hard time understanding and accepting anyone different from themselves. It took me too long to realize that there is nothing I can do to make anyone understand or accept me. People have to take it upon themselves to understand anything, and it is impossible to truly accept what is not understood. I am inclined to think that an inability to accept something is in fact proof that you do not truly understand it,” I find myself declaring. I had been unable to understand what was expected of me as a child, and so the role imposed on me was unacceptable. When I learned enough to understand what made me a boy, I also understood that I never had a choice, and that was unacceptable. When I worked it out enough to realize I also had no power to change what I was, that too was unacceptable. This lead me to ask some devastating questions. What is the point of being able to choose if you are not given a choice — especially about something that virtually defines you? What is the point of living if you are given a life you did not want? “I am not the only one to suspect that there has to be more to life than this, or that there is more to us, for that matter,” I tell her, in response to her unstated concern for what was at the heart of that post.

She tilts her head and shrugs in agreement, picking at imaginary lint on her skirt. “I know, but I did not stop at that, did I?” I can feel her studying me. I can’t really meet her eyes, but I can imagine myself looking over at her, seeing thoughts written on her face.

“I know, some of this is impossible to put into words, but yeah, the post was really about believing in myself and the impulse to act on that belief,” I admit, picking up on the thoughts this little game was bringing to the surface with a small sigh. “Although, there really is nothing hard about changing the world. The world changes with or without our help. What is hard is getting the results you intended. I might have gone out on a ledge by saying what I wanted to do, or why I wanted to do it. If there was a problem with what I posted, it was not being able to say how it could be done.”

She pulls her knees up to her chest and hugs them. I know what she’s thinking, because I am thinking the same thing. I have not been able to invest the time and effort needed to figure that out. “The hard part is not figuring out how it can be done. There’s plenty of scope for the imagination there,” she insists, prompting me to think of thousands of stories I’ve read, and hundreds I’ve tried to write, where suitable means were presented.

“No, the trick is establishing that there are means and methods available, and pushing ourselves beyond our current understanding. It is kind of hard to work on that if it is not your job, though,” I laugh, bitterly. Of course, there is no job like this. That has been the other reason I have been totally lost in this world. That sobers me up. “Honestly, even the little I’ve managed to find time to think about would take a lot of writing, and I don’t need another ‘job’ I don’t get paid to do!”

“And yet you sit up all night writing a blog like this?” she teases.

“Until I figure out what to do, what will make a difference, I don’t really have anything better to do,” I point out tiredly. As usual, I’ve barely scratched the surface of all the things that are on my mind. Writing is too slow and time consuming a way to deal with such thoughts. She looks at me, knowingly, and I shrug. “Things have to be done in their own way. If this were a story, I could skip over all the deep thinking. Even in a simple blog post, I could just focus on making a point. You intended to ask me how this is going to work. You really want to know how much more of this you have to endure.” I close my eyes and lean back in my chair. I know what it would take to set her free, and that I have to find it in myself. The problem is, as long as I am not her, I can’t really be me. I roll my head to the right and look at her. She cannot be seen, but she does not let that stop her. An obvious truth, always staring me in the face.

Matter and Movement in Four (or more) Dimensions

These days, it only takes a little curiosity, access to the Internet, and a bit of patience to find explanations of progressive spatial dimensions or examples of four dimensional geometry, such as the old favorite the hyper-square. Some of the things you will find use analogies like Flatland, or animations which is a way of using time and motion to reveal a higher dimensional object using a lower dimensional cross section — the advantage of this kind of intersection or interface is the ability to scan through what is too dense to actually see through. In this way, we can emulate the ability to observe, say, the details of the internal structure of a three dimensional body in a manner similar to seeing it directly from the fourth dimension. Of course, we did not have to wait for the invention of magnetic resonance imaging to be able to perceive the insides of our bodies; our sense of touch gives us the closest thing to a physically four-dimensional perception. Our kinesthetic or spatial sense is annexed to our visual perception to give us an integrated sense of physical reality. In addition, we supplement our active field of vision with the memory of what we have previously seen, and studies on perception have revealed that we often rely more on our visual memory than our active sight in familiar settings. This ability to fill in the blanks around us is one that we can use to “see” into higher dimensions.

In the mind, it is possible to construct things in four or more dimensions, but that does not tell us if there are any objects in the universe that are constructed in four or more dimensions. Taking the example I gave of eight-dimensional time-space, we could surmise that the universe has height, width and depth in a kind of cellular structure in which every moment in time exists in strands of continuity along branches of probability in a network of possibility where multiple event paths lead to and away from any give moment. The instant is where-ever you happen to be focused on eternity. Time-travel would be non-paradoxical because time itself would be process based, a product of attention. The event you experience would largely be determined by the state of mind you are in as you approach the moment, so causal time would probably be the norm; that is the path of least resistance. The real challenge to time travel would be presented by the body and its influence on attention. As a three-dimensional spatial construct, the body predisposes us to move through time as a byproduct of moving through space. To arrive at a specific point in space and time, without transiting the intermediary space and time, would break the perceived continuity of events unless one could perceive space four dimensionally–in which case the intervening space was bypassed in an instant of time. The mind can conceive of four-dimensions internally, but the real question is, how would you move the body through a fourth-dimension externally?

A question like this is a question about matter as it relates to space. Among the things physicists know, matter occupies very little space and is distinguishable from energy only by structure. Attempts to understand the structure of matter has led to the identification of elemental atoms, primary particles and fundamental quarks; the last taking us down into the realm of quantum mechanics. In the process of getting down to the quantum level, physics has also run into fundamental forces, the electric and magnetic forces, gravity, the strong nuclear force and the weak nuclear force. These are all things that can be observed or inferred to exist based upon experimental observation, and for all that is known about them, there is much that is still not understood. The one aspect of matter that has captured my interest most often is the characteristic of mass and its association with gravity. A particle with mass is infinitesimally small and produces (or focuses) a force that has infinite range (though the strength of the effect diminishes over distance in a known, inverse-square ratio). Unlike electric and magnetic forces which are polarized, or both attract and repel, gravity seems only to attract and does so in a “like to like” fashion.

The “dent in space” model of gravity gets me thinking, as anyone who read my post on Gravity in a distributed, process driven, information-based Universe could tell. Einstein gave us the equation summarizing the relationship between matter and energy, but by itself, the equation does not explain what is really happening when energy is concentrated into mass. We have to ask, what is happening to the energy, and part of the answer lies in understanding how a point of mass is focused into a stable object and why that deforms the space around it. The answer is further complicated by the specific structure and electromagnetic properties of a given particle. Particle physics is a whole field of study unto itself, and if the great minds devoted to it will pardon me, outside the complexities that might be explored, the simple observation is that structure holds the answer. Energy is concentrated and structured into a more complex and dynamic state in which we find a focal point in three-dimensional reference and forces that produce one-dimensional (polarity), two-dimensional (surface tension, surface area), three-dimensional (height, width, depth) and four-dimensional (mass, gravity, inertia, vector) effects. There is so much going on, all of it debatable, but I always come back to the four-dimensional view of matter.

I would have to have a great deal of time and a decent amount of resources to formulate something more substantial from this speculation. I am sure there is a great deal more information available that could affect the assumptions I have about pervasive energy, pervasive space, particular matter in infinitesimal space, concentrated energy, mass, structure, gravity, spatial displacement, fields, force, electron shells, magnetic shells, and light. I have the interest and the fascination to keep probing and a desire for more reliable speculation, but until I find an opportunity to devote myself to it, I can only work with the insights I have now. The implication of four-dimensional structure in matter, or the idea of atoms as four-dimensional objects, does not make our world any less a three-dimensional environment. That is, matter may only be possible at the three-dimensional surface of a four-dimensional substrate of energy and space. There are particles that seem to spontaneously pop in and out of existence, if some of the reading I’ve done on particle and quantum physics is correct, and that might be an indication of structure transecting our three-dimensional “plane” but most atoms seem to be pretty well stitched into place.

I am not as confident in speculating on how energy and structure “bind” but that is what I see as a likely basis for fundamental forces. The forces seem difficult to understand or explain, but part of that is because the concept presents us with an inherent mental block. A concept allows us to hold onto an idea about an observed phenomenon, but in the act of grasping an aspect of reality in that way, we focus on the effect and become unable to see the cause. Stepping back and looking again, we might be able to see that what we call a force is simply a particular way the balance of energy in a structured system must behave to achieve stability. Seeing that way, we can begin to ask what imposes structure and how does it persist either as part of or apart from energy. The question brings me back to a notion I had about the nature of limits and how that impacted the perception of substance and solidity. If matter is mostly empty space, what keeps things from constantly falling through each other? The substance of matter is not in the mass, but in the repulsive forces of the electron shells of atoms. The thing that makes the world seem solid to our touch is the existence of forces associated with particles that prevent them from actually touching.

There is a great deal more needed in a comprehensive analysis of matter, but this is enough to return to the question of moving a body through four-dimensional space. A common observation is that an infinite number of objects of a given dimension can exist in an object of the dimension above it, being in effect an image of itself, but it would take the action of an entity acting in the higher dimension to manipulate or move the object through that hyper-space. In my example of a person attempting to jump from one position in space-time to another position in space-time without transiting the intervening space, either an outside agent would have to be involved, native to the higher dimensions, or the person would have to be constructed in four- to eight-dimensions to begin with. Not really a problem for the mind, assuming the mind is not exclusively internal to the body. The hard part, for a mind rooted in a physical body in a world such as ours would be figuring out that it did exist in more dimensions and that this enabled it to move through space and time in ways that transcend the physical limits of the body. No tool or technology grounded in the physical world would be of much use in discovering or exploiting this fact. Not that you could not discover it by accident if the mind should happen to wander; though you would have a hard time distinguishing random moments scattered over infinite probability from dreaming.

A Glimpse into the Eye of Paradox

On any given day, a small handful of people find their way to the eye of paradox. Some of them probably just glance at an article to see if it’s relevant to the search that brought them here. One or two might actually read something. Once in a blue moon, someone leaves a short comment. Taken all together, I do not see anything to indicate that my words mean much. Is that frustrating? A little. I think it only bothers me because I have so much on my mind all the time, and there seems to be little or nothing I can do with it. I have to ask myself, what am I really accomplishing here? I’ve said it before, I tend to write in the hope of provoking a little thought, usually because the inspiration to write is interesting in itself. I do not expect anyone else to find the result as interesting; partly because it is hard to capture a thought perfectly in words. I am not a thousand-word-a-minute typist and yet I seem to think a million miles a minute. I am not a linear thinker. My thoughts are more like a library in a hurricane. I have never really been able to explain it, but it’s what I mean when I talk about higher-dimensions of thought. The problem with that terminology is that we would all probably assign the dimensions differently, based on the way we think. I suspect that a lot of us think things out in words as a general rule, that good old internal dialogue. I have one, but it’s only one part of how I process information.

I think that most of what I think cannot really be put into words. Seriously, I find it easier to think in worlds. Every instant my brain is processing sensory data to assemble an endless stream of consciousness that ties what I perceive in the moment with things I have perceived in the past, things I imagine, things I have conceptualized, things I have analyzed, things I have articulated, things I intuit and things I have only imagined. I have this notion that I ought to be an author because stories come to me in bursts of instantaneous thought. The problem is that it takes so long to fully articulate one I will have conceived of a thousand variations in the time it takes to block out the basics of the one I started with. The sheer number of options and variations overwhelms me. I don’t find it surprising, however. This is what our brains evolve to do; in life we only get one chance to get anything right, and there are a lot of times when a mistake will cost you your life. When faced with a challenge, we automatically engage the imagination and run through as many simulations as our intelligence and attention allow. A good view of the future requires at least seven dimensions of thought. Our base line of reference is four dimensional (working in three spatial dimensions and one temporal dimension) and we have to be able to project that image forward to assess the consequences of our actions through to positive, neutral and negative outcomes.

The future is not on a straight line. Neither is the past, really. An individual has no problem seeing history as a line running back through events, but there is such a line for every person and every object involved in every interaction. The past that we perceive is not the only past. For any given instant, there is a conceivable alternate path leading up to it, though usually the only people who are aware of this are people involved in reconstructing events. To a lesser degree, anyone who mulls over the days of their lives will notice the variable paths within the repeating cycle. You can stop, while walking down the street and suddenly put yourself on the other side of the street in some memory. It is one of the reasons we like routine. The more times we go through a sphere of activity, the better we understand the possibilities of acting in that sphere. We use it to maintain a hypersphere of potential activity. We use something similar in the mastery of our own bodies. At any given moment, there is only one position we can be in, but we are aware of all the positions we can move to just in the limits of our own bodies. In a sense, you could say that this is the real difference between the physical nature of something and the spiritual nature. We can only ever see one instance of an object, but that one instance contains the potential of every instance of that object. When you can look at an old man and see the little boy he once was, that is a very spiritual perception.

The funny thing is that we have run into this same thing in quantum physics, the notion that things have potential that exceeds what can be manifested at any given point. In the mind, we can hold onto everything at once, seeing nothing but aware of it all, and pull whatever we want into focus in an instant. I really don’t find it surprising that reality is pretty much the same way. We work constantly to bring the world into focus, we are in a constant process of realization, learning about changes in the world and updating our own internal representation accordingly. This is how we maintain our grip on the universe, and also how the universe maintains its grip on us. Or, this is how we maintain our grip on ourselves. This is a good spot to focus on if you take the question “who am I?” seriously enough. This is where I ended up after years of asking that question in an attempt to determine if it was who I am or what I am that makes me “me”. I came to the same conclusion the characters in the Matrix did, the body cannot live without the mind. Perhaps that is an indication of gestalt consciousness, an indication that the mind is more than the sum of the body’s parts? I am still thinking on that. In the meantime, while I find myself in the universe’s grip, there is an omniverse of information in my grip. I am holding the universe in a firm mental grip, but at the same time I am holding on to many, many more in my thoughts. Of course, I might just be apprehending the possibilities of the universe that would be found in higher dimensions.

What kind of sense organ would be able to perceive higher dimensions? I do it in my mind constantly, so I would be inclined to say the brain is that sense organ, as I rush along in the wake of intuition, chased by thoughts of perception being our key to acting in our environment. I grasp all of the implications of movement in higher dimensions of space and time and cannot keep up with the possibilities that seem to open themselves up. I am riding on an epiphany, a realization of a universe that contains infinite potential. What kind of words could begin to describe it? I struggle to find them, even now. I struggle to find the time to think things through enough to achieve a less dizzying perspective. This is my true field of study, and all I can do is stand at the threshold and stare into it longingly. The irony is, we’re all at this threshold. It’s kind of like the best kept secret, because it’s hidden in plain sight. I think the only reason I noticed it is because I am too.

Divide by Zero

Is it possible to divide a number by zero? Conventional thinking in mathematics declares this to be impossible, but this answer is contingent on a point of view. The problem here lies in the normal interpretation of the value of the number zero; a representation of “nothing” zero is seen as more of a place holder, a number that has no value. By contrast, infinity is a value that has no number. In multiplication, the value of any number times zero is zero. It is like declaring a number in theory. It is like saying “no number” and when it comes to division by zero, one might as well be saying “no division” because use of the number zero declares no number of parts into which a value could be separated, including one, producing a value in no parts. The common assumption is that division by zero would equal infinity.

The question to ask here is, is there any other representation of zero that could allow for a different point of view, one from which a number can be divided by zero? The answer to this question is, surprisingly, yes. It can be done on any number line that includes positive and negative numbers. For simplicity’s sake, ten divided by zero is equal to five plus minus five; it is a line segment with an absolute value of ten and an effective value of zero – or zero to the power of one (indicating one bisection). It could also equal two-point-five-sub-x minus two-point-five-sub-x plus or minus two-point-five-sub-y minus two-point-five-sub-y; it is an intersection of vertical and horizontal line segments with an absolute value of ten and an effective value of zero – or zero to the power of two (indicating two bisections).

To be a bit more explicit, any number divided by zero is equal to a symmetrical, neutral “excluded” equation. As stated above, division by zero declares no number of parts into which a value can be separated, but it does not matter. Any rational number of parts that, in absolute value, cancels out the apparent value will suffice. A valid solution of ten divided by zero might be a circle with the circumference of ten at a radius perfectly perpendicular to the number line at zero. In application, a number divided by zero is effectively “displaced” from the working continuum; the segment of the number line bridging zero is effectively compressed into the singularity of the zero point. This could be represented by placing that segment across the zero point perpendicular to the base number line. It represents a number or value that still exists but is now external to or in a different dimension from the original system.