Story Background
“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.
Plot Summary
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.
Analysis
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.
Resources
Wikipedia article on “The Indefatigable Frog.”
Introduction to Zeno’s Paradox.
Essay on defending the humanities.
Helen Small’s book on the value of the humanities.
Solution to Zeno’s Paradox using the convergence of infinite series?
Philip K. Dick Review 
What scientific explainations are there for the mysteries revealed within his strange,fictional reality,unless they lie in political realms,which can also cause existential changes?There is technology of course,but even that sometimes has to be achieved through mysticism and
faith.Often though,it is plausible I suppose within pseudo-scientific reasoning,and forms the background to much of his stuff,but doesn’t offer any feasible answers to explain the unusual happenings.The ultimate truth though,as you say of the philosophy professor above,will be ascertained by the evidence of what can be observed,rather than scientific reasoning.
Spirituality is the most concrete reason for the unexplainable events that were his particular trademark.He could rationalize everything through cause and effect in his writings.I think the same reasoning can be applied to the short story above.However,I think this particular piece was just another one of his tidbits to sustain him until he could write more mature works though,and seems to be just another piece he churned out in a continual rush.