Imagining the "Zeroth" Dimension

Zero is powerful because it is infinity's twin. They are equal and opposite, yin and yang. They are equally paradoxical and troubling. The biggest questions in science and religion are about nothingness and eternity, the void and the infinite, zero and infinity.

- Science Writer Charles Seife, in his book Zero: the Biography of a Dangerous Idea

Last August I started a more in-depth series about the nature of each dimension, but I started with Imagining the Second Dimension. Some people have asked why didn't I go right back to the beginning, so let's try that now. Here's how I would start this discussion:

With Imagining the Tenth Dimension, we start from a zero, which some would call a "zeroth" (or "zero-th", if you prefer) dimension, and we move to the first, second, and beyond using a repeating logical structure to eventually end up at a timeless ultimate ensemble.  When you get right down to it, that's what every respectable TOE (Theory of Everything) needs to describe: some underlying "thing" that all else is derived from. Otherwise, you're back to the "turtles all the way down" joke that often comes up in these discussions. Reconciling this timeless everything (which with my project I'm calling the tenth dimension in its unobserved state) with the zero that we start from (a point of indeterminate size) is the mind-blowing concept we arrive at with my project once we have imagined all ten dimensions.

I have often insisted that zero is not a dimension, because after all we need two new directions to travel within to define a spatial dimension. I have also suggested that the zero and ten represented on my helix logo for this project are shown to be on a line within the other dimensions because they represent the two directions which take us towards the infinitely large one way, and the infinitesimally small the other. Connecting Zero to Ten is the most recent entry where we've talked about that idea. But consider this: even if we ignore everything but this zero, this point of indeterminate size, we should remember what that means: even though we tend to think of this point as having no size, the definition of "indeterminate" is really "all possible values are true". Incredibly, that means even this lowly point that we start from does have a way that we can think of it as being infinitely large, encompassing absolutely everything within the information that becomes reality! So with that in mind, and for the sake of discussion here, let's continue to pursue the idea this there could be a "zeroth" dimension.

We began this entry with a quote from Charles Seife's Zero: the Biography of a Dangerous Idea, a book which discusses the relatively recent origin of this most powerful of numbers. Why does Seife call zero a dangerous idea? As an example, his preamble chapter includes an anecdote about a billion dollar warship that was suddenly dead in the water when a software bug resulted in a "divide by zero" error which completely crashed the computers running the ship. Then, chapter one begins with these thoughts:

...as natural as zero seems to us today, for ancient people zero was a foreign -- and frightening -- idea. An Eastern concept, born in the Fertile Crescent a few centuries before the birth of Christ, zero not only evoked images of a primal void, it also had dangerous mathematical properties. With zero there is the power to shatter the framework of logic.

The beginnings of mathematical thought were found in the desire to count sheep and in the need to keep track of property, and of the passage of time. None of these tasks requires zero; civilizations functioned perfectly well for millennia before its discovery. Indeed, zero was so abhorrent to some cultures that they chose to live without it.

Gevin Giorbran, author of the brilliant Everything Forever, Learning to See Timelessness, liked to point out that some cosmologists say the accelerating expansion of the universe will eventually take us to an absolute zero of perfectly flat space, an empty and formless void which seems like the most grim future imaginable. Gevin's take on this idea was that this zero we're headed towards is not empty, but full of all the other possible states, and this can be supported by the commonly held viewpoint that our universe arises from the breaking of an underlying symmetry. This means that our universe is now headed back towards a natural return to the perfectly balanced whole that exists both "before" and "after" the existence of our universe. Here's a quote from chapter 20 of Gevin's book:

(1 + (-1)) + (2 + (-2)) + (3 + (-3)) +... = 0 + 0 + 0 + ... = 0

     The simplest most straightforward way of summing all numbers is to sum the equal but opposite numbers together as shown above. So for a moment we will imagine that the correct sum of all numbers does sum up to and equal zero. Except this means that we need to change the value of zero away from being "no" things. We need to treat zero as the largest value in the mathematical system which actually includes the two already vast infinities of positive and negative numbers. Suddenly zero has become an infinite whole that contains all other numbers. Every positive and every negative number on the real number plane is summing or combining together to form an ultimate number of absolute value. Obviously this is not math as we know it. This is a math without time, without process, a math of truly infinite values.

    So we have made a dramatic change and the next step is to see the effect that changing the value of zero has had on the value of other numbers. If we are going about this bravely, as if we are imaginatively exploring a series of ideas, and so the brain is actually working, we notice that the values of other numbers have also changed, transformed in the same shift that we have taken with zero. Ordinarily the nothing of zero is a foundational axiom. Our foundation has shifted dramatically. What now is the value of one or two?

    If zero is seen to contain all other numbers, then logically all other numbers must have a lesser value than that of zero. If zero is the largest value, the only way there can be lesser values is if we remove some measure of value from the whole of zero. For example, suppose that we take away a (-1) from zero. What remains in the absence of that (-1)? Zero is still very large but zero is no longer an absolute value containing all other numbers. Something has been removed from it. But what value does zero transform into to show that loss?

    The answer is simply that zero minus (-1) equals 1. The missing (-1) causes zero to transform into the value 1. If zero contains all numbers within it, and we take away a value, zero then contains all numbers except the removed value. If we remove a negative one from zero the value of zero records that loss by transforming into a positive one. It still contains all other numbers besides (-1). So it is still a very large number like zero. But it is no longer the complete whole of all numbers. It is one. A very large number one.

    So if we treat what just happened as the logical rule we can now discover the values of other numbers in this system. For example, one is the sum of all numbers, so it contains within it all numbers, except (-1) is removed. The number two is the sum of all numbers except (-2) is missing, so it is also near zero but its content is less than zero and less than one. The number three contains all other numbers except (-3) so it is very large but smaller than two, one, and zero. And so on, and so on. The transformation that has happened is not simply an inverse reversal of ordinary mathematics, rather in this mathematical system, the value of a number decreases as we count toward greater numbers, since more of the negative numbers are being removed and placed somewhere else.

    Now, I should point out, just for the sake of clarity, that switching to the negative, the number (-1) is a combination of all numbers except that a positive 1 is removed, which would otherwise create the balance of zero. And in removing a positive two the whole shows that loss by becoming the number (-2). Unlike ordinary math, where negative values are less than nothing at all, here the numbers (-1) and (-2) are very large. In fact the content of (-1) is equal but inverse to the content of (+1). In physics, matter and anti-matter particles are equally substantive yet inverse in form and structure.

Do you follow Gevin's logic here? Saying that zero minus negative one equals one really makes perfect sense in ordinary math, but framing this idea in terms of zero being "full" and any of the other numbers as being slightly less than that requires a powerful mental shift. This shift takes us to the understanding that the broken symmetry that creates our universe or any other is defined by what's "missing" from it. In the case of our own universe, we know there is much less anti-matter than would be expected if our universe is derived from an underlying symmetry state: so it is this absence of anti-matter which is one of the defining factors that resulted in our particular universe.

There's no question in my mind, Gevin Giorbran was a genius and I'm sad that he's no longer with us. And p.s., I really should remind everyone that Gevin's book is available in hard cover, soft cover or as a downloadable pdf from www.tenthdimension.com/store.

Enjoy the journey!

Rob Bryanton

Next: Imagining the First Dimension