Fun With Numbers (II) – September 26, 2010

Last week I wrote that “mathematics is simply a language of numbers that incorporates a certain kind of logical thinking. But pure logic and mathematics cannot of themselves answer questions about origins, purpose, or the structure of reality. And it’s entirely possible to construct statements in either numbers or words that bear no relation to reality. Further, using either numbers or words requires the user to make certain underlying assumptions that neither mathematics nor logic can test.”

Here are some of the assumptions I see embedded in the use of math by physicists and cosmologists.

In most descriptions of reality, physics assumes that time (t) is just another dimension of space in the same way that the conventional Euclidian three (x, y, and z) are dimensions. You can make a lot of interesting equations from this assumption. And certainly time is an integral part of most physics equations, being deeply involved in matters of velocity, acceleration, and gravity.

But time is qualitatively different from the dimensions of side to side (x), up and down (y), or forward and back (z). For one thing, to move along any of these spatial dimensions, I have to apply a certain amount of initial force. Otherwise I sit in one place. And yet all of my component atoms are moving forward in time and always have.

Now, you might say that my atoms acquired their proper motion through the universe from the spin of the planet, the planet’s revolution around the sun, the sun’s revolution around the galaxy, and the galaxy’s outward acceleration stemming from the Big Bang. And so, in similar fashion, my atoms acquired their initial movement through time from the Big Bang as well. But the fact is, I can change the velocity and direction of my atom’s proper motion through applications of sufficient force (governed by the equation f=ma, or force equals mass times acceleration). And, through the equations of relativity, I can even slow my initial motion through time—at least relative to other observers—by speeding up my motion through one of the spatial dimensions. But no application of energy in any dimension can reverse my motion through time in the same way that I can turn around and travel backwards along the x, y, or zdimensions.

Time works in one dimension. I can write four dimensional equations using negative time (-t). But that doesn’t mean they define a reality in which I can move backwards in time.

Another such assumption is infinity. This, like zero, is a human concept in mathematics. Our minds invented it long before we had any notion of a universe beyond the spheres on which the stars were hung. In most cases in human history, however, “infinity” was just a number that we were too lazy to try to count. In such a way did the Greeks use “myriad” to represent a great number of things—usually more than 10,000, which seems to be the limit of reliable human counting in ancient times (consider the wildly inaccurate troop counts in ancient battles). But rational people are still not comfortable with the idea of infinity when applied to reality, such as the defining the limits of the universe. Mathematics may use concepts like infinity and zero, but they do not necessarily represent anything in our experience.

Similarly, mathematics also has its own deep pockets of illogic. Take the imaginary numbers. In the system of counting, mathematicians allow for there to be both positive numbers, which take you up the ladder, and negative numbers, which take you down the ladder. But in the real world, negative numbers cannot be displayed. Show me a negative three apples without first displaying at least three positive apples. Negatives only work if you are moving along a dimension (x, y, or z) for which a zero point and positive and negative directions have been arbitrarily assigned.

By the rules of mathematics, also, two numbers multiplied together are always positive. Negative two times positive three equals positive six. Negative two times negative three equals positive six. And yet, when dealing with the artificial construct of square roots (a number that multiplied by itself yields the target number), it is often useful to look for the root of a number that is negative. Of course, that root multiplied by itself will yield a positive number, not the original negative. So the square root of a negative number is considered imaginary. Useful but doesn’t exist.

My point is that a lot of mathematics is like that: useful for manipulating numbers in a consistent fashion but treacherous when applied to the real world.

Physicists, for example, can't reconcile the four main forces—electromagnetism, the strong and weak nuclear forces, and gravity—from Einstein’s relativity and the equations of quantum mechanics in a three-dimensional universe. But they can do it with four, or eight, or eleven dimensions.

Cosmologists confronting the dead end of the Big Bang (what came before that point of infinite temperature and infinite density?) have adopted a multiverse: many, perhaps infinite numbers of, universes all coexisting and trading matter and energy back and forth by squirting it through gaps in the boundaries.

If anyone has seen and visited a spatial dimension beyond the Euclidian x, y, and z, please contact me at the address below. We have a wonderful book to write together.