March 7, 2009

Musings on the Nature of Time

What is eternity? Is it simply a never ending series of days?

If Einstein taught us anything, it’s that time possesses extraordinarily complex physical properties. For example, one of the questions Brian Green raises in his book, Fabric of the Cosmos, is: Why does time appear to have an arrow? Why, in other words, does it seem to progress only into the future, when nothing in Einstein’s equations prevent it from slipping into the past?

Interest in the nature of time isn’t a recent phenomenon, as people have been musing on its properties for centuries. A central theme in Jewish mysticism, for example, is the unknowable quality of the infinite and eternal. The Greeks, too, considered time’s many complexities, asking whether it could be subdivided to some atomic equivalent, or if it was continuous in a way that defied division. They had two words for it: chronos and kreinos—the former being man’s time and the latter God’s time—which suggests it comes in two versions.

The study of mathematics has had far-reaching effects on our ideas of time. Einstein’s general theory of relatively, for instance, relies upon a complex number line to define what has been labeled the fourth dimension or time space. Complex numbers—sometimes referred to as imaginary numbers—appear often during the course of solving certain mathematical equations (the term x2 = -1 is an example) that results in the square root of a negative number. Such a solution shouldn’t exist—at least not in any world we understand. Mathematicians have invented a sleight-of-hand to deal with them, but for a long time people didn’t know what to do when complex numbers appeared. Then Einstein came along and showed how they were essential to a fully developed theory that would replace much of Newtonian physics.

Perhaps the most interesting discovery that furthered our understanding of time is Georg Cantor’s studies on infinity (which, if you think about it, is the mathematical corollary of eternity). His proofs are elegant and simple, but rather than show them here, let me summarize the implications of his work with a single sentence.

Infinity comes in various levels of value and complexity!

It seems incredible, I know, since every infinite set has no end to it, but they can differ with respect to dimension. One of the least complex varieties is, for example, the set of all integers, which relates to our common reckoning of eternity as a never ending number of days. It can be thought of as a number line—a single dimension. However, there are higher—much more complex—versions of infinity that are not bound by anything resembling a series of integers.

Cantor showed, for instance, that all the numbers between 0 and 1 (let’s call this set A) is greater than the set of all integers (this we’ll call B). Again, I won’t show the proof, but suffice it to say that A contains dimensions not included in B, including an infinite set of transcendental numbers. Transcendental numbers, as you probably know, include pi (π), which in their various decimal representations never end or repeat. (Pi is approximately 3.14159, but the digits continue on and on without pattern). If you’ve ever thought about transcendental numbers, you’ll have considered how they relate to a number line. We can show approximately where they are and do so with increasing degrees of precision, but they defy being placed in a precise location.

Is it possible that eternity is more than an endless number of days, but something that is unbound by time in the way of a complex number? Can a particular moment in eternity defy placement in the way of a transcendental number? At the very least, these are interesting ideas to ponder.

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