“The Indefatigable Frog” was first published in Fantasy Story Magazine in July 1953. Page numbers come from Paycheck and Other Classic Stories by Philip K. Dick (New York: Citadel Press), pp. 221–229.
Professor Hardy introduces to his students Zeno and his paradox, defending it in a Physics class. Professor Grote, a philosopher, publicly challenges Hardy’s insistence on the truth of Zeno’s formulation that a frog will never reach its destination. Later, Hardy confesses that it is true no one ever performed an experiment that could test the truth of Zeno’s claim. Hardy and Grote agree to build a machine that will perform the experiment using a frog.
The “Frog Chamber” is built with support from the university. The way it works at replicating Zeno’s paradox is that one end of the tube is heated, forcing the frog to jump forward. With each hop, the frog is reduced in size by one half. From the frog’s perspective, the tube will get longer and longer. Hardy asserts that the experiment will prove that the frog can never get to the end of the tube. Grote who is suspicious of the machine and the experiment is urged by Hardy to investigate it in more detail. Grote is eventually caught in the same experiment as the frog. With relish Hardy urges his “dear frog” to hop.
Grote, although within the tube and getting progressively smaller is certain that Zeno is wrong and that he will reach the end of the tube in nine hours and thirty minutes, although at a vastly smaller size.
Meanwhile, Hardy is self-assured at his victory over his intellectual enemy and brags of this before his class. Grote continues to shrink eventually reaching subatomic size.
The original frog and later Grote disrupt another of Hardy’s classes. Unfortunately, the Zeno’s paradox cannot be tested using the tube because once the subject reached subatomic size, it escaped the tube and regained normal size. Another method is required to test the paradox.
This story comes to me as a bit of comic relief after Dick’s early masterpiece “The Variable Man.” The story centers on an intellectual dual between a philosopher and a scientist. The scientist, a bit strangely, believes Zeno’s paradox is true and can be tested experimentally. The philosopher, deduces that Zeno is wrong and the frog (in the original I always thought it was Achilles and a turtle) will reach its destination using only reason. To resolve the debate, the scientist builds a machine that will test Zeno’s paradox in a rather clever way. By shrinking the frog with each step, the device in effect doubles the distance to the destination with each step. As with so many academic debates, no matter how elaborate the argument and the experiment questions remain at the end. A solution eludes both sides. This reminds us of many academic debates that are presented as serious life-or-death struggles, but seem to have little meaning in the end. Of course, we have gotten along fine since the time of Zeno without the resolution of his paradox. (Well, the summing of infinite series resolves it, but Dick apparently was not aware of this or ignored this for the purpose of this story.) If ridiculous, it is nonetheless real enough as a reflection of the academic infighting and departmental struggles for supremacy. I doubt Dick could have foretold the current struggle of humanities to defend their very existence in the university from the assault of STEM fields. (I do not think all STEM and business faculty feel this way, but I have talked to enough who seriously asked what historians actually do.)
The wonderful part of the story is the end, when we learn that the massive (and we assume expensive) experiment did not answer the question because the experiment broke down at the subatomic level. In this way, the story seems to work as a defense of the humanities. The philosophy professor is the much more practical of the two, relying on reason and what can be seen and observed. It was the scientists who got lost in the math and felt the need to build a big device to solve what is really a silly question.
Solution to Zeno’s Paradox using the convergence of infinite series?