[Excerpt Be and Become, © ProCreative, Sydney 2000]
The observation of anomalies is an excellent fillip for suspecting that our current view of reality is incomplete.
The high degree to which we have biased our perceptions in terms of our local physical senses (sight, hearing, touch, smell and taste) has blinded our consideration of some glaring anomalies in our thinking—anomalies (theoretical paradoxes) which have persisted for nearly two and a half thousand years.
Broken Arrows
Around 450 B.C. Greek philosopher Zeno of Elea introduced a number of paradoxes that revealed how motion (of any kind) was theoretically impossible. What he managed through straight-forward reasoning was to show that our theoretical perception of reality didn’t match our experience of it. And that mis-match between theory and practical experience has persisted ever since. In fact, it has become all the more entrenched in the last few hundreds years since the on-set of the industrial revolution.
The first paradox established that the commencement of movement is theoretically impossible.
Consider a runner, ready to run the 100 metres at the Olympics. She’s a top athlete, but, according to Zeno’s paradox, she can not only never run the race, she can’t ever start it. For the athlete to run the 100 metres, she must first get to the halfway (50 metre) mark. But before she gets there, she must get to the quarter-way (25 metre) mark. And before that, she has to get to the “one-eighth way” (12.5 metre) mark and so on, ad infinitum. And this is were the paradox arises, for in taking time (albeit ever smaller portions of it) to traverse an infinite number of increasingly smaller and smaller initial points in the race, the runner can never (as in ever) get started. Each halfway point is a finite number, requiring finite time to traverse and since there are infinite such traversals required to run a race, the runner requires infinite time to run the race.
Basically, the paradox is that before we move anywhere, we have to first get through or beyond infinity.
The second paradox involved Achilles’s attempt to overtake the tortoise in the classic hare and the tortoise race. Even though Achilles is much more fleet of foot than the tortoise, he can never catch up to and overtake the head start initially given to the tortoise. Essentially, the same problem as the Olympic runner is involved—taking infinite smaller and smaller bits of time to traverse infinite smaller bits of space. It would take anyone forever to traverse infinite small bits of time (no matter how small). The head start cannot ever be beaten now matter how fast Achilles runs or how slow the tortoise crawls (so long as the tortoise does not stop completely).
The third paradox is more helpful in understanding how the paradoxes can be resolved, in that it introduces ideas which are crucial to resolving all three paradoxes. The third paradox involves the motion of an arrow through the air. At any point in time, an arrow (or indeed any object) flying through space must be at rest in that space. The arrow, in real terms, must be completely at rest in some specified section of space at some specified time, for otherwise the arrow could not be said to be anywhere specifically in space. The arrow would not be real—it would instead be some ghost-like apparition, not real-enough or solid enough to pierce armor or its intended target. We can better visualize this by imagining that we have a very fast movie camera taking consecutive snapshots of the arrow in flight. In each frame, the arrow will appear stationary, i.e. it will appear entirely real, simply because it is real (it has a fixed positioned in space). As physicist Fred Alan Wolf explained,
... if it (the arrow) is occupying a place, it must be at rest there. The arrow must be at rest the instant we picture, and since the instant we have chosen is any instant, the arrow cannot be moving at any instant. Thus the arrow is always at rest and cannot fly.1
To get around this paradox, philosophers and scientists have imagined that reality could be broken down into infinite minutely-small time-frames, each with its own slightly different “snapshot” of reality. This is another way of treating time as being infinitely divisible. When all these frames are run consecutively, just like a movie-film through a projector, we get the sense and experience of motion. When we have infinite such frames, they end up blending together into the perfectly continuous flowing reality that we normally take for granted. However, this line of thinking does not resolve the paradox, it merely introduces another set of paradoxesi.
Wolf explains:
by assuming that the arrow’s motion was continuous, it was natural to imagine continuity as “made up” of an infinite number of still frames, even though we would never attempt to make such a motion picture. We just believed that “in principle” it was possible.2
The theorists believed that “in principle” we could span the infinite, even though in practice it would be impossible. The reason for the acceptance of such theories was that as the time-frame becomes infinitely short or small, we could “in principle” span or traverse them in finite time. As the frame became infinitely small, we could thus reasonably ignore specific consideration of it. In a sense, the mathematicians performed theoretical leaps of reason and logic by jumping over the intervening infinitely small spaces and times. In other words, it became convenient to ignore the spiritual (infinite, immeasurable and unknowable). And this is the crux of the dilemma that has persisted for the last 2.3 millennia.
Mathematicians invented the generic term “infinitesimals” to conveniently label quantities which were infinitely small, but which were still some value greater than zero. They were paradoxically real, in that they were numbers of some size greater than zero, but also, they were unreal, in the sense that no-one could measure or define them. They weren’t able to be defined as being some specific size.
These “infinitesimal” quantities, with their realness having evaporated in the unknowable mist of infinity, were nonetheless paradoxically still considered real, or at least real enough to be used to effectively explain reality. Consequently the scientists’ standard tool for mapping reality became the accepted confluence of real and unreal measures. These unreal measures were soon expanded to include imaginary quantities (based on the square root of -1) which were even less real, in that they were even less able to be meaningfully related to the real world.
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