Infinite Infinitesimal...

Many of us might consider numbers the most sure-footed way to come within sight of infinity, even if the mathematical notion of infinity is something we'll never even remotely comprehend.

Mathematicians tell us that any infinite set?anything with an infinite number of things in it?is defined as something that we can add to without increasing its size. The same holds true for subtraction, multiplication, or division. Infinity minus 25 is still infinity; infinity times infinity is?you got it?infinity. And yet, there is always an even larger number: infinity plus 1 is not larger than infinity, but 2^infinity is.

When we're told that the decimals in certain significant numbers, like pi and the square root of two, go on forever, we can somehow accept that, especially when we learn that computers have calculated the value of pi, for one, to over a trillion places, with no final value for pi in sight. (For more on pi, see Approximating Pi.) When we're told that there are 43,252,003,274,489,856,000 possible ways to arrange the squares on the Rubik Cube's six sides, we may feel intuitively (if not rationally) that we must be on our way to the base of that loftiest of all peaks, Mt. Infinity.

One reason we may feel this way is that such numbers are as intellectually unapproachable to the mathematically challenged as infinity itself. Take a Googol. A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West. The Buddhists have an even more robust number, 10140, which they know as asankhyeya. Just for fun, I'll name a larger number yet, 101000. I'll call it the "Olivian," after my daughter. Now, doesn't an Olivian get me a little closer to infinity than the Googolians or even the Buddhists can get? Nope. Infinity is just as far from an Olivian as it is from a Googol?or, for that matter, from 1.

Infinities do come in two sizes, of course?not only the infinitely large but also the infinitely small. As Jonathan Swift wrote, "So, naturalists observe, a flea/Has smaller fleas that on him prey/And these have smaller still to bite `em/And so proceed ad infinitum."

Of course, just when we think we have infinity in the palm of our hands, we watch it evaporate in the harsh light of another of those confounding paradoxes: the numerals 2 and 3 are separated by both a finite number (1) and an infinity of numbers. This conundrum spawned one of the great paradoxes of history, known as Zeno's paradox. Zeno was a Greek philosopher of the fourth century B.C. who "proved" that motion was impossible. For a runner to move from one point to another, Zeno asserted, he must first cover half the distance, then half the remaining distance, then half the remaining distance again, and so on and so on. Since this would require an infinite number of strides, he could never reach his destination, even if it lay just a few strides away.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

We believe Hamlet when he says "I could be bounded in a nutshell/And count myself a king of infinite space." The shortest length physicists speak of is the Planck length, 10^-33 centimeters. But might not there be an even shorter length, say, 10^-333 centimeters, or 10-an infinite number of 3's centimeters?

as one mathematician pointed out to me, infinity is an abstract concept, appearing only in our mental images of the universe. It is not actually in the universe.

The Greeks, in fact, invented apeirophobia, fear of the infinite. (The term comes from the Greek word for infinity, apeiron, which means "without boundary.") Aristotle would only admit that the natural numbers (1, 2, 28, etc.) could be potentially infinite, because they have no greatest member. But they could not be actually infinite, because no one, he believed, could imagine the entire set of natural numbers as a finished thing. The Romans felt just as uncomfortable, with the emperor Marcus Aurelius dismissing infinity as "a fathomless gulf, into which all things vanish."

The ancients' horror infiniti held sway through the Renaissance and right up to modern times. In 1600, the Inquisitors in Italy deemed the concept so heretical that when the philosopher Giordano Bruno insisted on promulgating his thoughts on infinity, they burned him at the stake for it. Later that century, the French mathematician Blaise Pascal deemed the concept truly disturbing: "When I consider the small span of my life absorbed in the eternity of all time, or the small part of space which I can touch or see engulfed by the infinite immensity of spaces that I know not and that know me not, I am frightened and astonished to see myself here instead of there ... now instead of then." Martin Buber, an Israeli philosopher who died in 1965, felt so undone by the concept of infinity that he "seriously thought of avoiding it by suicide."

...like that engendered by this gem from another anonymous sufferer of our common infirmity: "Infinity is a floorless room without walls or ceiling."

A Googol is 10100, or 1 followed by 100 zeroes, and is the largest named number in the West.

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

Wee, math trivia! We actually have a number called a googolplex which is a 1 followed by a googol zeroes.

It wasn't for 2,000 years that Zeno's paradox finally got "solved," for all intents and purposes, by the calculus. Its inventors, Isaac Newton and Gottfried Leibniz, showed us how an infinite sum can add up to a finite amount, that it can converge to a limit. Thus, even though we can't count all the numbers between 2 and 3, we know they converge to 1.

Fortunately physics has answered Zeno's paradox. Time and space are not continuous, they are quantized.

Physics has said no such thing.

No empirically verified theory exhibits any sort of geometric quantization. While some of the promising research topics do exhibit geometric quantiziation, they still don't exhibit quantization of position.

It is meaningless to speak in terms of absolute position.

So in any case, every particle that can possibly be associated with a position must necessarily take on a quantized position relative to other particles. In other words, it can only move in a way to take on, or give up, a quantum of energy.

P.S. don't forget that there is no quantum theory of gravity. Furthermore, TMK, there has only been one experiment, ever, that has demonstrated quantum effects in a gravitationally bound system. (Of course there has only been one experiment that has tested it)

If you bring Zeno up to date and imagine Achilles and the Tortoise to be two particles, each being one fundamental quanta in diameter, moving in a spacetime that is quantised into fundamental quanta of time and space (discrete moments and discrete positions), then relative motion between A and T makes no sense at all. That is, not unless you allow length contraction and time dilation.

Zeno's point was that if one takes spacetime to be quantized, or, equivalently, takes the number line to be a series of points, then motion is (mathematically) paradoxical.

If one takes spacetime as a continuum then the paradox disappears.

This is consistent with the effectiveness of the calculus, since a 'point' in a continuum is actually an infinitely divisible range. By that I mean, the only thing that is infinitely divisible is a point in a continuum. If spacetime (or the number line) is quantized then why do we need to use infinitesimals to calculate motion?