In the paradox of Achilles and the Tortoise, Achilles is in a footrace with the tortoise. Achilles allows the tortoise a head start of 100 metres. If we suppose that each racer starts running at some constant speed (one very fast and one very slow), then after some finite time, Achilles will have run 100 metres, bringing him to the tortoise’s starting point. During this time, the tortoise has run a much shorter distance, say, 10 metres. It will then take Achilles some further time to run that distance, by which time the tortoise will have advanced farther; and then more time still to reach this third point, while the tortoise moves ahead. Thus, whenever Achilles reaches somewhere the tortoise has been, he still has farther to go. Therefore, because there are an infinite number of points Achilles must reach where the tortoise has already been, he can never overtake the tortoise.
The Greeks had a fascination with paradoxes because they perceived that they illuminated deep truths about the true nature of reality which we may have been blind to via sense perception. But sometimes I think that a fixation on species as the taxonomic category to rule them all confuses and calcifies the understanding of evolutionary genetic processes in the eyes of the public. Just as it is difficult to communicate that science is not a collection of facts, that it is a process and a method, so many people seem to take species categorizations as reflecting the true order of the universe. Historically this goes back to the pre-evolutionary taxonomists whose aim was to catalog all of God’s creation. It persists today explicitly among Creationists, who bandy about terms like “kinds,” but are really talking about immutable ideal entities with particular essences. Species on steroids if you will.
A fixation on the species has also confused the public on the issue of microevolution vs. macroevolution. Creationists regularly accept the former and reject the latter, despite the fact that the majority of biologists would probably assert that evolutionary processes are a unified whole, with no difference of scale. The constant usage of the term “microevolution” by the enemies of evolution seems to have even cast that term into some disrepute, I’ll admit to be shocked when a reader was confused as to the non-Creationist usage when I recommended Alan Templeton’s Population Genetics and Microevolutionary Theory. One could argue that the subject of modern population genetics is fundamentally microevolution.
Species are obviously abstractions. But I think an analogy can be made between them and physical objects. At the end of the day we know that the solidity and boundedness of physical objects are perceptions and interpretations filtered through our brains. Fundamentally they’re a bundle of particles and forces, interacting with other particles and forces. We don’t need to deny this deep reality, all the while instrumentally acknowledging the usefulness of categories of physical objections.