No groaning allowed for non-mathematicians: I'm not going to use more than a couple pieces of math, and none of these will be mind-blowing.
Complex math is a system of algebra that allows for roots of negative numbers. Remember that grade school class, where the teacher taught that the square root of -4 was not -2, that in fact, it didn't exist? Maybe your teacher even stated cryptically that it became 2i, an imaginary number. In complex math, this little "i" is the heart and soul of every problem. The definition of "i" is:
i^2 = -1 ( I squared = minus one )
After a lot of drilling in this form of math, I remembered that cryptic statement from Grade 4. These numbers are imaginary. They don't exist in real life. Yet, thanks to my training in physics, I understand they produce answers experiment and observation dictate. How is this possible? If these numbers don't exist in 3-D reality, then how can they be used to predict correct answers?
My solution to that is that they do exist in reality, but not in a visible manner. They exist in your mind, that little corner of your consciousness (or unconsciousness) reserved for holding these numbers and performing calculations so you can write the answer down on the paper in front of you. But since that is the only place they exist, and they seem to be able to predict outcomes for real events, therefore having a direct impact on reality, then your mind has a direct impact on reality.
I understand that psychologists have been stating this for a while now, and it's considered proper to think that your view of reality is completely fabricated by your unconscious mind based on input. But this implies that there is something that exists beyond the information that gives us the reality that we perceive. Just like a computer will only show the end product on the screen, and not the unbelievably long list of functions that were performed to get the product, your mind also leaves the useless details out. Unfortunately, these details include exactly where the information came from. But it's a fair assumption then, if we make any parallels between the mind and a computer that, just like typing on a keyboard or clicking on a mouse, you can interject your thoughts and expectations into reality and expect to get feedback. Now, I'm not implying that you can change reality completely just by thinking, any more than you can change the program you're in just by typing. But, just like the program, you can change the "program options" of reality to make your life easier. This is still only a subtle change, but a change nonetheless.
I've had a psychology professor tell me that if we were just heads on a lab shelf being fed computer generated impressions of life, or just programs on a machine, we'd never notice. There's no way to tell if what we see is real. The notion of complex math, and my previous argument, suggests that there's more than what's being seen, and that we can affect it. But the idea that we can affect it at all is a good reason to believe that there is no such "program" that controls what we see. In the context of the computer example I used, some options that we can change should have "administrator passwords" on them to lock us out, but they don't seem to.
Now, I guess I had better offer some backing for my statements, before my argument dies of criticism. For the first point, I will ask a question. What makes up reality? I'll bet your answer can be rephrased to say, "Whatever I see, hear, taste, feel, and smell." Your answer might include feelings not attributed to these senses, but such feelings are always defined in terms of the five regular senses. You "feel" that slight gust of wind you intuitively know means you're not alone in the room. You "see" an event that you shouldn't be able to, but can. You "hear" things that other people can't. Anything that we receive as personal information comes from these five sources, whether the phenomenon is physical or not. The senses aren't just physical functions; they're concepts, axioms from which we define everything else. Even the idea 1+1=2 is validated because you can see the result.
I conclude that, since the only information we get about our external life is from these five senses, they to all intents and purposes, define reality. Now, let's assume that you observe some event. It can be anything, but for this example preferably a mechanical event - a lever, or something moving past you at fast speed; something that has measurable quantities. Since you've seen this event, perhaps heard or felt it as well, it fits into the definition of reality.
Now, let's reintroduce the complex math. There are methods of applying the math to the problem (quantum mechanics for the moving object, advanced engineering methods for the lever) so we'll gather the information in two ways. First, we'll take measurements from the event itself by use of whatever lab equipment necessary. This also fits into reality for the same reason.
Let me make a note here: I don't think it a valid point to say that complex numbers fit into reality the same way that the moving object does. In fact, everything we see and anything we learn points to the fact that the roots of negative numbers are not only a ridiculous notion, but also completely unnecessary. After all, since there is an infinite amount of observable numbers, why shouldn't they be able to solve all of our problems? Since a complex number can't be associated with something that is seen, heard, felt, tasted, or smelled, then it doesn't exist as far as your senses tell you.
Let's now predict the outcome of the experiment with our knowledge of math and physics. I'm sure that you don't want to see actual calculations performed, especially if you're not a physicist, but assume the answers match up. In the spirit of good scientists, let's repeat the measurements many times, and check the calculations for errors. Let's assume they still match. Since the math properly predicts the outcome of the experiment beyond coincidence, those imaginary numbers with no bearing on reality are now associated with something that is real. So, from that association, the numbers can, in a manner, be accessed by the senses, in the form of the event being predicted. The numbers have become a part of reality.
For mathematicians who live and die by their proofs, I'm not arguing that the mathematical proofs of complex numbers aren't sound. But I'm looking at it from an observational point of view. In mathematics, it's easy to get so caught up in the math itself, that the question the math was created to solve becomes less important. Mathematics is more than a tool; it's a powerful way of thinking. But, it's only one way of thinking. And if a concept in math isn't describable in any way other than math, then that concept has a very limited usefulness.
Warning: physics philosophy ahead. I wrote this paper with the idea of showing that something so obviously imaginary as complex numbers could do something real. As I pointed out, complex numbers are involved whenever math becomes unthinkable any other way. My Quantum Mechanics professor tells me that we use complex notation in Lorentz Transforms to "cheat the system," making the last component of a 4-dimensional vector negative if you square it. It's used in crystallography because the structure of a crystal, when viewed by X-ray, is literally inside out and backwards. These numbers are involved with practically everything that you cannot see. They can be used for things you can, but so can real numbers. Generally speaking, complex numbers are used whenever you begin using more than three dimensions. In the context of four dimensions, it's the fourth dimension (i.e. time) that is generally complex. If we recognize more than three dimensions to the universe, then maybe complex numbers do exist physically in reality as a representation of what a higher dimension looks like in our 3-D perception. The existence of observable complex numbers doesn't go against my earlier argument that your thoughts affect reality. It actually fits very well into the concept of mental dimensions (i.e. astral planes or inner planes). If you recognize multiple dimensions, and those dimensions are expressible by complex numbers, then those dimensions quite possible could be imaginary in origin. This is not to say that you made them up on the fly; these dimensions might be where the individual mind draws its innate knowledge, where we send signals to and receive information.
A current theory in physics espouses eleven dimensions, four of which we live in (the three spatial dimensions, and time), and seven other dimensions which are said to have "curled up" to a size that makes an atom look large. We're not going to notice anything that small, which means that the imaginary, even though it may actually have a place in the universe, must still be "imaginary" forever. You'll never know the process of your imagination, only the results. This idea also lends itself to Jung's idea of a "collective unconscious." What he proposed in psychology might be explainable with modern physics.
The ideas in this paper are not really provable. But then neither is the idea of eleven dimensions, or most philosophical issues. The only things that can be proven are those that can be heard, felt, smelled, tasted, or, arguably, most importantly to humans, seen. And this is only true so long as you trust your senses. But the concepts seem to fit together very well, almost too well to attribute it to coincidence. I think that most, if not all, explanations of how the universe works will eventually converge into one idea, with pieces from all the other ideas it has replaced.
If we're lucky, it might even be the truth.
January 4, 2000