Page:Astounding Science Fiction v54n06 (1955-02).djvu/150

make considerable usage of mathematics as, for example, in the analysis of transient impulses and shock waves in mechanical systems by means of Heavisides Operational Method or the so-called Classical Method, whichever is preferred. I have often felt that the importance of mathematics as such is overrated. There is always the possibility of some mathematical thinker, like Zeno for example, who puts his tongue in his cheek and comes up with a logically inescapable result which can not possibly conform to reality. It seems fairly evident to me that a large part of the difficulty lies in the assumptions which are made.

Mr. Gunther, in his article in the August issue, mentions that physics is coming to recognize the necessity of a quantized structure of space. He then does an about-face and proves to his own satisfaction that such quantization can not exist. Both of these ideas are in the category of assumptions or postulates. I think in all fairness we should carry our thinking along one line or the other. As a matter of fact, after many years of devotion to this problem, I am personally satisfied that the idea of a quantized universe is completely compatible with reality but will also admit quickly that there are many cases where a quantized space structure would appear to be incompatible. But so far such difficulties have been successfully overcome by reorientation of thinking.

To get down to cases regarding Mr. Gunther's article, he states that "Zeno's thesis that Achilles must occupy as many positions as the tortoise is, and remains, unassailable." In my view this is by no means unassailable, particularly if you postulate a quantized structure of space.

Assume for the sake of argument that space is constructed of discrete "space quants" and that motion takes place in discrete "motion quants." Physically this is equivalent to a wagon wheel moving by small leaps and bounds over a cobblestone street. In the race between Achilles and the tortoise, Zeno's problem would follow his mathematical solution until the distance between Achilles and the tortoise was reduced to one space quant. At this point the succeeding pulse or motion quant would put Achilles at the same position or beyond the tortoise. Subsequent motion would put Achilles ahead which is, of course, obvious in the first place.

The point I am trying to make with this little physical example is that the whole dilemma of Zeno disappears when you look at physical reality. The paradox can only exist in a mathematical sense and even there it only exists by virtue of the fact that the primary assumption of the mathematical method is that number is infinitely divisible. Physical measurements of our universe quickly demonstrate that energy, motion, and I would say even time itself, comes in bunches or quanta.