Why Spheres?

Gravity is weak.  This is well known and endlessly recited, but the explanations for why it might be weak are often too esoteric to delve into.  One proposal  comes out of string theory research (now several decades old), and revolves around the properties of the graviton.  Although presently undetected, the graviton is a proposed member of the complete standard model, and serves as the carrier of the force of gravity.  These particles are theoretically exchanged by massive objects and thereby guide their attraction to one another.

But why is the strength of gravity so far out of whack from the other forces?  Employing an example of this weakness from Brian Green, think about a common refrigerator magnet.  Here we have a competition — a battle of strength against a few ounces of iron on the magnet, and the gravitation pull of the entire planet Earth.  Were gravity and electromagnetism on equal footing, the magnet would fall straight to the ground (and weight much more than a few ounces for that matter).  But that is not what we actually see.  In fact, the few ounces of iron have no trouble at all resisting the pull of the Earth’s gravity, and the magnet sticks resolutely to your refrigerator.

In string theory, there are more than three dimensions in real space, but those extra degrees of freedom are only accessible to certain types of string configurations, one of which is the graviton, and only over very small distance scales.  In normal 3-space, a force emanating from a point will expand as a spherical wave, thereby experiencing a flux density that drops in proportion to R squared.  Why squared?  Because the total flux emitted by this point must spread itself over the surface of a sphere growing  with time, therefore the density is the total flux divided by the surface area of a sphere.  This can be clearly seen built into the common equations of force.  Take for instance Gravitation:

‘G’ is the gravitation constant, and being a constant it can accurately be rewritten as any combination of constants, leading to the unusual but entirely accurate formulation:

And there we have it, a flux (numerator), divided over the surface area of a sphere.  This is visible again in Coulomb’s Law, and here the constant is often defined in terms of 4π:

Again the flux (numerator) is divided by the surface of a three-dimensional sphere.  So if you are dealing with these forces, you are dealing with spheres, and if you are dealing with three dimensions, you are dealing with ordinary looking “2-spheres” described by their usual properties.

Moving to N-Sphere

Without going into too much detail, the string theory explanation is pretty simple.  If string theory is correct about the true makeup of space-time, then over very small distances the force of gravity is much stronger.  As gravitons emit from a source, many of them drift off into extra dimensions.  The pieces that stay in our coordinates then move along and obey normal 2-sphere propagation attenuation, but appear as if they were very weak to begin with, as so many gravitons have already disappeared.  What I sought to accomplish was to determine what the force of gravity should actually look like in small scales, if space-time has more than three dimensions.  Clearly a 2-Sphere no longer describes the real “surface” that the net flux is distributed over.  The theoretical test, once a n-sphere formulation of gravity is identified, would be to find a means of balancing gravity against electromagnetism (a force that cannot see the extra dimensions, and is therefore 2-spherical all they way down) and see where they balance.

So one last time, the gravitational FLUX is described by the numerator in the equation above, and we are searching for the new surface to distribute it over.  Let psi-g represent the flux, and the constant k-g will be defined as follows (or merely the gravitational constant divided by 4π):

Spheres

When trying to imagine N-Spheres, like many before me, I work on analogy starting from lower dimensions.  Lets bear in mind the definition of a circle (a 1-Sphere) to help extrapolate: the set of all points equal distance from one center point.  In the case of a 0-Sphere, (a “sphere” that exists on a single line), the “set of all points” would refer to only two points.  The point R away from me along the line in one direction, and the point R away from me along the line in the other direction.  The “volume” of this sphere can be calculated by doing a “shadow integral” over an identity function running between R and -R.  As with any sphere, the surface area is then the derivative of the volume.

These answers make some sense intuitively.  If you have two points, separated by 2r then the “volume” of your 1-d enclosure is simply the length of line between them, or 2r.  The “surface” is really the sum of two points, which is harder to visualize, but expressed in the math.  Lets move ahead to 1-Spheres… commonly called circles.  We already know what to expect from these results, but lets test it.  Now I am going to do a “shadow” integral over an identity function, first describing a circle, and then from -r to r once again:

Lets do the last familiar case, so the pattern in the equation can emerge.  Here is the 2-Sphere:

Whenever you want to determine the “volume” of the next dimensional sphere, you need to first integrate over a circle that contains all of the degrees of freedom in question, and then treat that as your “shadow” and integrate up each dimension in the chain.  As expected, each dimension adds an extra power to the radius, which each time extends into a new degree of freedom.  Here are the results of the next several N-Spheres, arriving ultimately at a 9-Sphere whose surface can describe the flux distribution in an 11-dimensional string theory (special thanks to the TI-89 Titanium, and several sets of batteries!):

Here are the final order ones that matter for String theory.  These took hours to verify on my TI-89, although I originally determined these using an expansion I derived from the previous 5.

And there we have it.  String theory calls for at least 11 dimensions, meaning one time dimension and 10-spatial dimensions.  A ten dimensional sphere, or a 9-Sphere (9 because the numbering indicates the number of dimensions that makeup the bounding surface), therefore describes the needed volume and surface.  Of course I make no claim of having inventing the concept of shadow integrals, but I can take credit for deciding to use them to solve for N-Spheres in this way.  Just to cap this off, here is my formula (fully original, deduced only from my previous calculations) for finding the volume of an N-Sphere.  I have not proven this formula mathematically, but I have verified its predictions up to 10-Spheres:

New equivalence

If string theory is correct and gravity is very strong on a small scale, the relative strength of gravity compared to electromagnetism should change drastically on those scales.  I surmised that a test could be undertaken, attempting to find an equilibrium between the attractive force an electron’s gravity, and its repulsive force of its electromagnetism.  Let see on what scale such an equilibrium could be found:

Now we can solve this equation with the known values for an electron, and we find:

So when two electrons are brought to a half about one thousandth of a millimeter from one another, we should be able to get them to balance IF a) string theory’s proposition of 11-dimensions is correct, and 2) if the wrapped up dimensions described in the theory are on the order of 1000ths-of-millimeters or more.

In other formulations of string theory that require more dimensions, this equilibrium shrinks.  Unfortunately, current experiments have only been able to probe the strength of gravity on scales two orders of magnitude greater (tenths of millimeters), which suggests an answer will not be forthcoming for at least a couple years.  But in any case, lack of any finding is insufficient to falsify string theory because either of the two premises leading to the above equivalence may be wrong without string theory itself necessarily being wrong.

Avoiding the slippery slope that is my personal dislike of string theory, let me leave this entry alone as only a commentary on N-Spheres.  Well, now I can’t say my undergraduate degree in physics was a complete waste!  At least I got one Blog entry out of it.

The common portrayal of such a paradigm is never positive, exemplified most vividly with the Star Trek The Next Generation antagonists: the Borg.

The Brog are a cybernetic species that specialized in the indiscriminate assimilation of foreign biology and technology. The Borg are also pivotally characterized by a collective mind… the members of the Borg are merely drones without any personal awareness or sense of individuality. Indeed the horror of assimilation, and the compulsive replacement of your individuality with the collective, are recurring themes in Star Trek, as well as other scifi stories that touch the concept.

I take issue with several of these portrayals, and ultimately assume the unpopular perspective that a collective mind would be a huge opportunity and sign of maturity for humanity. It would also represent a fundamental paradigm shift of unprecedented proportions to the “human experience”.

Nodes in the Network

The key to keep in mind is that joining a “collective” does not alter the way individual brains process, it simple interconnects the brain with others. What a connection to a collective is supposed to entail is an instant and unfiltered exchange of all thoughts and experiences between all members of the hive. Each human connected (or node) remains an individual processing center, meaning they continue to have their own consciousness and their own interface with experiences. The difference is that after the instant of initial experience, the event becomes public and known to all, and free for everyone to individually react to.

This is where the idea of losing one’s self enters the picture. Of course it is a matter of speculation, but I don’t subscribe to this model. It seems reasonable that people in a collective might arrive at interpretations or beliefs that they would not have held individually. From this deviation, we might deduce that the node is no longer an individual as it was unable to hold its own opinion. In other words, it may seem the individual’s opinion was forcibly overwritten by the collective. To the contrary, however, I would expect this sort of deviation. The change in a node’s “personal” opinion is not because the individual is unable to hold their own thoughts, but because their own thoughts mingle with every other person’s thoughts and a massive averaging takes place whereby every node individually aggregates the diversity of opinions and knowledge and arrives inexorably at the same conclusion. The key to remember is that the nodes share everything, so any differences of perspective or personality of individual nodes are subjected to every opposing opinion and perspective, allowing each node to personally agree with the “collective” personality and perspective.

This difference may seem subtle, but I insist it is not. Consider the elements that prevent people from agreeing on fundamental principals — take for instance an Evangelist and an Atheist. These two groups have entirely incompatible world views, and no amount of arguing could ever get them to agree. If they were connected to a collective, however, they would suddenly be able to exchange feeling associated with experiences, inherent instincts that cannot be explained, and they would be exposed to each other’s actual belief. As stated, they cannot have both beliefs, points would come into conflict and all internal reasoning would be shared and inclusive.

With the extra information and understanding, they would each likely arrive at some middle ground based on the various points one group or the other was unable or unwilling to internalize previously — In essence they are each so well informed and have such common experiences (personal or learned) that they nearly inevitable arrive at similar conclusions. The end result is that their opinions may have changed, but not because they had to… only because each individual grew beyond their original perspective and actually choose to agree with the collective. If disconnected from the collective, I would expect each individual to truly continue to believe whatever middle ground that had previously discovered

Averaging Knowledge

The ability to exchange information on the level of our “inner voice” opens up the door to this idea of true knowledge averaging. When we all have the same pieces and the same feedback on the best and worst way to use those pieces, then our interpretation of information is likely to average out to the “most-globally-reasonable” interpretation. This is not a loss by any means, it is a huge gain. It enables the enhancement of human understanding and influence to extravagantly unthinkable levels. It also does not require us to lose anything that we value in our current method of individual contributions, those contributions simple become lower level. For instance, an individual whose perspective is very innovative and new can still redirects the whole collective. But in a collective mind, that innovation can be leveraged to a greater capacity because as soon as it is discovered by a single node, it becomes available to all nodes to leverage.

Because the processing of information is still within the brain of the nodes, it makes sense that certain nodes would have certain values — some more likely to innovate and some more likely to make abstract connections, much like in our world. Again with the key difference that all nodes instantly understand how and why that innovation was realized, and can hopefully simulate the thought process.

This dispersal also allows humans to optimize themselves in ways previously unimaginable. Technology as is stands now — wikipedia, social networking, televised entertainment, music — none of it would be required anymore when culture and enrichment is available on demand. We would not lose these facets of our culture, we would simply be able to experience them without the technology middleman. I imagin a collective culture relying very little on technology or surroundings for happiness or entertainment.

Portability of Consciousness

I will close with a curious afterthought on this subject. If the individual consciousnesses in a collective were so interconnected that they could actually distribute their existence over multiple brains, a very sci-fi opportunity appears. Up until now, I describe a node as its own person who is fully connected to each other person. In many ways, this allows the group to control the group, because every decision (where to walk, what to say) is influenced and planned by the whole collective. However to execute the actual action, the host of that particular body must agree with the collective, and their brain must control their body. In this new sense of shared consciousness, individuals could actually move their consciousness between particular nodes, or even share control of nodes living primarily in the cloud. For physical tasks, a strong body might be occupied by an individual, and then for solving a problem, a node better suited to mental work might be occupied. In addition, several individuals might share control of multiple nodes at once.

This kind of collective allows humans to break the 1:1 connection that exists between a body and a mind — in fact it opens up the ability for n:m where n minds control m bodies, and n >= m. Now if a body is lost, it does not necessarily pull its host out of the collective — the host may exists redundantly across the network. Now any consciousness can actually control any body, a subtle difference from before where only one could control a body, even through its decision to do so was largely the decision of the collective.

Enlisting in the Borg

The technology to achieve the kind of interconnectivity a collective requires is no where near the horizon, and may be permanently delegated to the Sci-Fi realm. If it does every make it to reality, however, I think we stand to benefit greatly from its potential. The changes it represents to our way of life are so small compared to the amazing opportunity for peace, advancement, and growth as a species. If it ever comes to be, I envision plugging in will be a major point of contention, but inevitably everyone would seek its refuge and comfort, and be much happier they had done so.

Despite harboring these thoughts and questions for many years, it was only very recently that I began to consider the philosophical connotations of teleportation, in particular to the user of the hardware. I sought to answer the question, “What emerges on the other side of a transporter?” Of course I don’t have the answer, but I have an answer, and I wanted to write it all out.

Transporters

Before I can get into this too much, it is worth pursuing a quick tangent, and discussing how transporters work in the Star Trek series. The concept is fairly simple: leveraging Einstein’s E=M*C*C, a computer scans a user and dematerializes their matter into an energy stream along with data about their original configuration. Next ensues a handful of semi-relevant albeit esoteric techno-babble, including the likes of “pattern buffer”, “confinement beam”, and “Heisenberg compensator”. When it is all said and done, the computer delivers the energy stream up to 40,000 km and reverts it back into its original matter state… e.g, the person who was getting transported. In the Star Trek story line, the computer scanner is able to resolve the quantum uncertainty that should otherwise be present between the position and momentum (or other non-zero commutators in QM) of the particles. This stage is the only part of the transport process that is fundamentally resting on impossible science, so I will ignore it in my discussion. Here, I am curious with what might actually happen if one of these transporters were built, and under no circumstance could we build something with functional “Heisenberg compensators”.

Consciousness

Perhaps predictably, the real question at the gut of this whole thing is if human consciousness can be duplicated in the same manner that matter can. While arduously avoiding the word “soul”, I wish to following in the method of Rene Descarte and suggest a few fundamental truths about consciousness to serve as a starting point for subsequent deductions. While Descarte’s basic principle approach to philosophy only got so far as “cogito, ergo sum”, I propose granting the assumption that what applies to one individual must also apply to every individual, and thereby extending the foundation: you think, therefor you are. And thus combined we can agree that we each do exists, and we each are separately sentient.

In granting the supposition that we all exists, we have acknowledged that consciousness is something real and distinct from person to person. It seems obvious, but clearly my consciousness is not the same one as your. There is some mechanism that makes sure my consciousness stays with my body, and does not leak into others or else vanish altogether… in other words, it seems quite reasonable to conclude that a particular consciousness is mapped immutably to one instance of a human. Everything in our experience suggests that this is the case. I believe this conclusion still holds when we start to look at more unusual or even hypothetical situations, although it becomes less obvious and definitely arguable. Here are a couple cases I have thought about in an attempt to better define my own perception of the boundaries of a particular consciousness.

1. Monozygotic Twins

Okay, this one is not so hard. Identical twins (lets take two as an example) have nearly the same biological construction, but clearly there is “somebody home” inside each twin independently. At the point when consciousness is likely to have manifested (prior to sentience), variation between the two twins would be confined to errors during mitosis, and the minuscule differences in personal experiences while in the womb. Despite the differences being essentially immeasurable at first, each twin still gets assigned a separate consciousness.

2. Clones

We have to employ our imagination a little harder now. Suppose you go to the doctors office and you are cloned. While you watch, the scientists grow a copy of you at a rapid pace. It seems unlikely that when the clone reaches the point of being able to support consciousness, you would suddenly be affected. The idea that your awareness of self might suddenly span two bodies is unreasonable. Again, we are likely dealing with a new separate consciousness despite the mirrored biological construction. It seems to follow from these two examples that consciousness emerges independent of the particular brain construction. That is to say, the “person” who sits behind one’s eyes is not a function of biological construction.

3. Replacement Clone

Now lets say you are cloned through a process that necessarily kills you. The doctors take your blood, multiple samples, and the end result is your death. Then they use the materials they acquired and create your clone. Does the exit of your consciousness have any effect on “who” wakes up inside that clone? Being that your original consciousness is gone, could the new one actually be your original consciousness again, or is this case really the same as the one above? We are past the point where I can offer any certain answers, but my hunch tells me that there should be no relationship between the existence of one consciousness and that of another. If we agree that the particular mind to emerge is not a function of the biological construction, then I believe that the clone in this example, just like before, would be a brand new consciousness — albeit one that thinks they are you, that acts, talks, and behaves like you, but would actually be different. This case is very proximal to the central discussion about transportation, so I will hold off further thoughts until we get back to that.

Let me present one last thought about consciousness before moving on. The line between “you” and “your consciousness” is very vague. In general, those things that define who you are, are all bodily. You personality comes from your experiences, your sense of self accomplishment comes from your memories, your purpose, your self worth, all of the facets to your temperament… they are all the result of years of experiences, memories, thoughts, and interactions. Of course there is an innate component to many of these things, but I argue that those items that really define us — the people we love, the people who loved us, our proudest moments, our deepest understanding of life — these things that have shaped us, are entirely contained in our physically-stored memories. Experiments with animals, as well as studies of humans after accidents and with certain memory-related diseases, have well established that personality and memories can come and go with alterations to the brain. In other words, the common concept of who you are is not dependent on your consciousness. I propose that consciousness provides nothing more or less than the “self” who is able to experience what the brain processes.

This apparent tangent is very important. It means that “your consciousness” is not synonymous with “you”. Who you appear to be to others is defined by the makeup of your brain… two people with the same predispositions and the exact same experiences, would likely act as if they were the same person. Quite contrary to this, we have established here that consciousness is not related to the physical makeup (or else clones would be controlled by a single consciousness). When I talk about “you” in regard to transporters, I really mean the combination of your physical identity (memories, feelings) as well as you particular self-awareness. Either one without the other is not the entire you.

Teleporting Humans

Alright, so 1000 words later and still the question remains unanswered: what would happen if a human was transported? From the physical perspective, we know that the original human is decomposed into energy and a copy is created at a distance. Note that sans the Heisenberg Compensator, we cannot truthfully state that the same physical particles are moved to the new location… but nonetheless, we undoubtedly have a better copy than our “Replacement Clone” example above. Lets further clarify that a transport process need not kill the transportee (in practice this might make little sense, but the point is that the same relationship that existed between clones and replacement clones exists here). We found previously that a clone and a replacement clone were really the same phenomenon, each independent of the exit or entrant of other entities. In this case alike, I doubt that the same “self” that existed before the transport somehow moves or shares the new “structure” created by the transporter. It seems inevitable that we are dealing with a new consciousness.

“Beam Me Up”, or “Count Me Out”?

So if a human enters a transporter, they are not in fact transported, rather a duplicate is created elsewhere while they are killed. We are forced to wrap up on a final philosophical curiosity: would it really matter to society as a whole?

In every quantifiable respect, the copy would be the original person. We have already discussed how personality, memories, experiences, and even temperament are parts of the physical body, and would therefore operate in the copy precisely as they did in the original. The copy would walk out of the receiving end of the transporter with a perfect memory of getting in at the other end moments before. In fact nothing about them could give any indication that anything had changed (since we know nothing physical did change), so for all intents and purposes, it would be the same person.

But the “self” inside their head would actually be only moments old, and completely distinct from the original.

My guess is that if transporters are ever invented, many many people will use them without worry and apparently without cause for worry. Myself ? I’ll just call for a shuttle.