2.5.8  Mathematical Solution

Consequently, after the successful development of mathematics and science in recent centuries, many scientists, mathematicians, and philosophers rejected Aristotle’s treatment, replacing it with what is known as the Standard Solution that is based on the theory of sets that was developed in late 19th century. However, many other philosophers of the 20th century, such as Henri-Louis Bergson, Max Black, Franz Brentano, L. E. J. Brouwer, Solomon Feferman, William James, James Thomson, and Alfred North Whitehead argued in different ways that the standard mathematical account of continuity does not apply to physical processes, or is improper for describing those processes, mainly because it is based on actual infinity that cannot be encountered in experience, and thus is unreal.

While mathematics can calculate where and when the moving Achilles will overtake the Tortoise, it does not really address the central point in Zeno’s infinity argument that is indispensable if we want to solve his paradoxes. Yet Aristotle’s ingenious idea of Potential Infinity is also not sufficient, since it inevitably presupposes actual infinity.

There is no doubt that Zeno presented a profound problem associated with infinity which, despite centuries of philosophical and scientific efforts to resolve it, still seems to lack a truly satisfactory solution. As Frankel remarked:

“The human mind, when trying to give itself an accurate account of motion, finds itself confronted with two aspects of the phenomenon. Both are inevitable but at the same time they are mutually exclusive. Either we look at the continuous flow of motion; then it will be impossible for us to think of the object in any particular position. Or we think of the object as occupying any of the positions through which its course is leading it; and while fixing our thought on that particular position we can’t help fixing the object itself and putting it at rest for one short instant.” Frankel (1942)

Zeno, of course, was not trying to deny motion altogether, because it is clearly occurring after all; objects reach their destinations, Achilles does run faster than the Tortoise and the Arrows move in their paths. What Zeno was trying to say is that our common understanding of motion, and plurality, and hence space and time as containers of events, can not be correct or complete. Actually, it is this wrong or incomplete understanding of space and time that is now the main obstacle in modern physics which is preventing any reconciliation between relativity and quantum theories.

The human mind is accustomed to classifying quantities as either countable or uncountable, or either discrete or continuous; there is no other way. This is inevitable on the outward level of multiplicity. On the level of oneness, however, there would be no meaning for such terms. The problem is then how to link this apparent physical multiplicity with the real metaphysical oneness!

The Single Monad Model and the resulting Duality of Time hypothesis, and its dynamic creation of space and matter in the inner levels of time, that will be discussed further in chapter IV, offer the unique solution to this persisting problem, because it is not based on either discreteness or continuity. Rather, these two contrasting and conclusive views are now complementary aspects of the ultimately reality of oneness. Motion, change, and all associated phenomena of physical multiplicity, are temporal manifestations of this metaphysical oneness.

The Duality of Time postulate of the Single Monad Model provided the link between the ultimate metaphysical oneness and the apparent physical multiplicity. This led to the genuinely complex nature of time which is the only way to conceive of granular and self-contained space-time structure. This dynamic space-time structure can easily reconcile Relativity with Quantum Mechanics, because it does not require any background geometry or topology in which the space-time could be artificially quantized, as Loop Quantum Gravity is now trying to achieve. The new genuinely complex nature of time achieved this goal using a simple Euclidean space, getting rid of the singularities in the ill-defined Riemannian geometry, and eliminating the need for renormalization, in addition to other even more significant achievements far beyond what scientists ever aspired to achieve, because it can easily lead to full Quantum Field Theory of gravity, as well as explaining the other fundamental interactions in terms of its genuine complex-time geometry.