How can a line have an infinite number of dissections on that line and yet exist within finite, measurable realm? Parmenides's theory of the one can help solve this question.
Many people have claimed that either the line cannot have an infinite number of dissections possible or that there is some fundamental flaw with the logic. The logic is sound and the answer is valid; however, our understanding of that answer is invalid.
How can a person trace their finger along a line of infinite points? Isn't it infinite, meaning never ending, unlimited?
The theory of the one is stated that there is no such thing as motion as such that our understanding allows for. Parmenides believed that motion was impossible and stated the true nature of existence was bound in one eternal form. This similar theory can be found in many religions. It is the theory of oneness. It is best encapsulated by this thought from the Japanese poet Matsuo Basho, "There is nothing that you can see that is not a flower; there is nothing you can think which is not the moon."
The difference between Eastern and Western thought was that Western (Greek) thinkers were explicit in both their discoveries and their explanations. The eastern thinkers such as Lao Tzu, Basho, Chuang-tzu were at best, ambiguous. Many of their findings were similar to western thought but the problem was it did not contain the same level of specificity as the Greeks.
It is possible to understand reality based upon this system of one which Parmenides formed and apply other insights to random elements. For example, if all is oneness than how can their be numbers beyond one? Numbers, in our current sense are flawed. There is one repeating. There is not one additive to one equaling two because there are no like objects. Each contains within it its own oneness, and it's own inherent individuation. Yet math is our most useful tool.