For those unfamiliar, Zeno of Elea (ca. 490–430 BC) proposed a few mind puzzles very many years ago involving the basic logic of simple motion, generally referred to as "Zeno's Paradoxes" (or so history is written). Perhaps the most famous of these paradoxes (all of which are similar) is called the "Dichotomy Paradox".

The Dichotomy Paradox is presented thusly:

Dichotomy paradoxThat which is in locomotion must arrive at the half-way stage before it arrives at the goal.– as recounted by Aristotle, Physics VI:9, 239b10

Suppose Homer wishes to walk to the end of a path. Before he can get there, he must get halfway there. Before he can get halfway there, he must get a quarter of the way there. Before traveling a quarter, he must travel one-eighth; before an eighth, one-sixteenth; and so on.

The paradoxical conclusion then would be that travel over any finite distance can neither be completed nor begun, and so all motion must be an illusion. An alternative conclusion, proposed by Henri Bergson, is that motion (time and distance) is not actually divisible.

Aristotle answered these paradoxes by revealing that both time and distance are involved in motion and when the distance is cut in half, so is the time required to travel it. Thus no matter how infinitely one divides the distance required to travel, the time is divided as well yielding a constant speed. Simple solution. Motion flows on.

===============

But besides the issue of speed and motion, there is a confusing issue of the division of any distance. If one can infinitely divide a distance and one must add each divided segment together in order to complete the whole distance, how can the whole ever be assembled? If there are an infinity of numbers between 0 and 1, how can a number line between them exit at all? How could the number line ever get to 1?

Think about it for a moment. In order to get from 0 to 1, an infinity of fractions must be appended. But in reality, to get to any one of those fractions, an infinity of additional fractions must be appended. And, to make it even worse, to get to any of those additional fractions, an infinity of double-additional fractions must be appended. And that scenario has no end .. aka "is endless", aka "is infinite". In short, you can't get to 1 from 0. So what gives?

If you are like me and don't really like to read all that much, let me cut to the essential:

- The universe is not made of location points (numbers), but rather is made of the space between them.

The one thing that philosophers can contribute to (and would seriously improve) the world is to require physicists and mathematicians to take a course in ontology construction.

When constructing an understanding of the world/universe (an ontology), there are options (females tend to innately understand this for whatever reason you might like to believe -- males .. not so much). For sake of cognitive understanding and dealing with the world, a person can choose to allow a division between 0 and 1 meter (for example) up to a limit of 1000 (one millimeter segments). Perhaps that is fine for their need. 10,000 years ago, can you even imagine a need to be more accurate? And with that chosen limit, one can construct a "quantized" version of all physical reality. Later in human development, much greater precision was required. The point is that it is arbitrary as to what limit one chooses as the standard for their ontological construction.

People keep realizing that any distance can be divided, so no matter what standard is chosen, a more precise standard is conceived. Quantum Physicists have tried (once again) to declare that distance cannot be divided more than a particular chosen (too small to measure) amount (about 10^-38m), called "Plank's Constant". It's just another effort to get around the truth of the matter.

The distance 1 (of whatever) can be divided into 2 segments, or 4, or 10, or 1,000,000 or any number chosen. The thing to keep in mind is that it is a distance that was divided, not a number. And as long as one multiplies by the same number of segments, the original distance is reestablished. The distance can be divided infinitely, but as long as the infinitesimal segments are multiplied by the same degree of infinity (yes, infinity comes in degrees), the original distance is reestablished. And that resolves the issue of Zeno's paradox.

One must traverse half of every half of any distance in order to gain any distance. But all that is saying is that one must be accomplishing motion - the changing from one segment to the next. Any size segment may be used in the understanding. But every segment takes up space, else it isn't a segment nor a division of a distance, but merely a location point. And if something is proposed to take up absolutely zero space, then it doesn't physically exist (to exist is to have affect from one point to another - the affect between the points is the existence). You are made of the space between what isn't you on one side and what isn't you on the other (as is everything else). And if you are moving, every segment and affect within you is relocating to the next segment of space. What size you choose those segments to be is completely up to you and your ontology.

==================

This whole issue then relates to the recent debate concerning whether the value expressed as "0.999..." is really exactly equal to the value expressed as "1.0".

Carleas raised the question of how one could get from 0 to 1 if taking a 90% step and 90% of each remaining amount (aka "0.999...") did not add up to exactly 1.0. I contended that if you are required to only take 90% of each remaining amount with each step, then by definition, you could not ever get to 1.0.

- So how does the real number line ever get to 1.0?

The answer, as explained above, is simple. The real number "line" is made of line segments, not of location points. Any and every segment may be divided infinitely. As George Cantor pointed out long ago, the number of potential numbers between 0 and 1 (or actually between any two numbers) are uncountably infinite. The real number line can have an infinity of numbers between any pair of numbers and another infinity of numbers between each of those, and so on infinitely. But the number LINE doesn't care, because any and every "line" is made of segments, not numbers.