Zeno's motion paradoxes and evolutionary theory

In the comments to my History of life entry I made reference to Zeno's motion paradox(es).

Zeno, a 5th century BCE Greek philosopher, believed that motion (change) was an illusion, and he proposed several similar thought experiments to demonstrate this view. For the sake of brevity, we'll look at just one: Achilles and the tortoise.

In the paradox of Achilles and the Tortoise, we imagine the Greek hero Achilles in a footrace with the plodding reptile. Because he is so fast a runner, Achilles graciously allows the tortoise a head start of a hundred feet. 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 a hundred feet, bringing him to the tortoise's starting point; during this time, the tortoise has "run" a (much shorter) distance, say one foot. It will then take Achilles some further period of time to run that distance, during which the tortoise will advance farther; and then another period of time 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, Zeno says, swift Achilles can never overtake the tortoise. Thus, while common sense and common experience would hold that one runner can catch another, according to the above argument, he cannot; this is the paradox.

Zeno is obviously wrong, as anyone can see that Achilles will in fact overtake the tortoise at some point if he is running faster, but it resisted mathematical refutation until Cantor came up with his theory of tranfinites in the late 19th century. What people had a difficult time accepting was that an infinite number of distances could converge on a finite sum (1.1, 1.11, 1.111 .... 2).

So what does this have to do with evolution? Hopefully, the point will have already presented itself, because I find myself stumbling to articulate it, but I'll try anyway.

In The Ancestor's Tale, Dawkins called our difficulty to conceptualize non-discrete categories as the 'Tyranny of a Discontinuous Mind'. For example, its easy for us to recognize the difference between reptiles and mammals, but hard for us to categorize species that were transitional. If we were to formulate an evolutionary version of Zeno's paradox we might ask: "at what point did a reptile give birth to a mammal?"

The answer is: never. This may seem counter-intutive, but only if you look at it from a discontinuous perspective. The chain of individuals that connect a mammal to a reptile ancestor consists of a continuous stretch of lives where each succesive generation is only slightly different than the previous one, so that if you look at any one point in the chain you will not be able to discern much of a difference between two generations. But the effects of these changes are cumulative, so if you compare far ends of the chain you can easily see the difference.

And while we're speaking about evolutionary change, one should check out this post at Carl Zimmer's blog about the tree of life that has been constructed as a result of genome sequencing. The tree is amazing in that it shows just how genetically similar we animals are relative to exhibited biological diversity on the planet.