The Greek philosopher Zeno of Elea (c.490-c.420 BCE) put forward a paradox that in a foot race, Achilles would never be able to overtake a tortoise if the tortoise was given a small head start. Today the Achilles paradox is an example of the summation of an infinite series, which always approaches, but never reaches, unity. One variation of this paradox occurs when teaching students about intermediate forms (or “missing links”) between any two nominal taxa in an evolutionary lineage. Theoretically, the discovery of a single transitional form creates the need for two more transitional forms (for a total of three intermediaries), which then requires the discovery of four more intermediaries (for a total of seven) and so on, so that every iteration (n) requires 2ⁿ-1 intermediaries. Similar to the paradox, it appears that an infinite number of intermediaries must be found to bridge the gap between any two taxa.

While the mathematical solution to the Achilles paradox is simple enough, the concept of the paradox plays a role in comparing gradual and punctuated modes of evolution to students. From a purely gradualistic approach, the need for an infinite number of intermediaries is self-defeating as there can never be an infinite number of organisms in the time frame between the two ends of the lineage. The punctuated model deals with the problem by allowing new taxa to evolve in the geological equivalent of an instant, thus eliminating the need for infinite intermediaries. Beyond evolutionary models, the Achilles paradox has applications in historical geology such as radiometric decay and gravity. Showing how the paradox relates to several aspects of science may encourage better critical thinking on the part of the student when faced with similar problem dealing with infinite series.