Zeno’s paradox and the infinitesimal

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In a recent post I wrote about the split among neuroscientists between the “determinists” and the “compatibilists.” The former see choice/free will/volition as an illusion created by our brains, while the latter see an active role for our “minds” in determining our future actions, although not necessarily as the “master decision-maker” that our personal “homunculus” often perceives.

In that post I hinted at two stories which, to me, demonstrate the primary weakness of the determinist argument. In short, the stories are about the perils of reductionism, digging so deep into a phenomenon that it seems to disappear. So it is with human choice.

Zeno’s Paradox

Zeno of Elea was a Greek philosopher who lived in the 400s BCE. In a classic version his “thought exercise” now called Zeno’s Paradox, the great Achilles is in a footrace with the Tortoise. Confident of his athletic prowess, Achilles gives the Tortoise a big head start of one Greek stadion (about 185 meters). But then, Achilles begins to think about the mathematics of catching up with Tortoise. In order to catch Tortoise, he realizes that he must first get to Tortoise’s starting point, by which time Tortoise will have moved a small distance farther toward the finish line, perhaps a tenth of a stadion.

But by the time Achilles would get to that new point, Tortoise would have moved a small distance yet further. Achilles realizes that, before he can catch Tortoise, he first has to get to where Tortoise had just been, but each time he gets to that point, Tortoise will no longer be at that point, having moved a small amount further. Because of his impeccable logic, Achilles convinces himself that he can never catch Tortoise, and so he gives up the race before ever starting out.

This is a classic case of reductionism leading into a logical dead-end. There are many alternative explanations for Zeno’s Paradox, and the best try to explain what happens to the very concepts of “motion” and “distance” when the increments get increasingly tiny, down to the subatomic level. An ancient story says that Diogenes the Cynic disproved the paradox simply by standing up and taking a step. This explanation still is lacking a critical “How?” explanation, but there is an important sense in which Diogenes was correct.

The word “footstep” really only has meaning at the level at which we, along with Achilles and Tortoise, interact with the world. We normally do not work at distances of tiny fractions of an inch without technology. So the “footstep” is an emergent property, which is saying that, although it has a root cause in the most elemental forces of nature at the atomic and subatomic level, it only has human meaning in higher-level interactions of those atoms. I can take a “footstep,” but an electron cannot. Likewise, I cannot behave like an electron.

So this is my answer to those who say that volition is an illusion because when we get deep into the brain, we can’t find any “yes/no” switch. Our perceptions of time, consciousness and free will, in this context, are all emergent realities. If you try to “drill down” too far to find their sources, they all disappear.


My second example comes from a fascinating story of a second, lesser-known battle between the Catholic Church and the emerging scientific revolution at the time of Galileo Galleli in 17th-century Italy. Everyone knows the story of Galileo being forced to recant his theory that the Earth orbits the Sun. In his book Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World, Amir Alexander recounts how, in that same century, the Church also successfully tamped down the mathematics relating to objects approaching the “infinitely small,” called infinitesimals. [1]

Zeno’s Paradox, with its description of infinitely small distances, was held in deep suspicion by Jesuit educators. They felt that, like Galileo’s theory, the “natural order” of God was being challenged.

Here is my example of a vexing infinitesimal mathematics problem that was once theologically dangerous, before the invention of calculus by Isaac Newton and Gottfried Leibniz around the turn of the 18th century. [2] The red line in the diagram below is a classic parabola, the path that a ball thrown into the air would take as it returns to the ground. The blue line intersects the parabola at two points, so it is easy to compute the slope of that line. We would just divide the change (Δ) in units on the Y axis by the change in units on the X axis. In this case, the slope would be approximately 11 divided by 10, or 1.1.


But what about the green line? It touches (is “tangent to”) the parabola at just one point. Since a point, by definition, has zero length and zero width, then my slope equation becomes zero divided by zero, which is an “undefined” value. By taking this green line to the infinitesimal tangent point, I now require a new kind of mathematics to calculate its slope, and the 17th-century Catholic Church was none too pleased with this math. The classic definition of “slope” since the days of the Greek mathematician Euclid had now been violated, and this violation of the natural order of “God’s world” could not be tolerated!

Amir Alexander recounts how the Church managed to stymie Italian mathematics for a century, but the Protestant Reformation and other forces freed the mathematicians of northern Europe, Newton and Leibniz among them, to invent “infinitesimal calculus” to deal with mathematics of parabolas and other geometric constructs plagued by infinitesimals. Ironically, the path of that ball thrown into the air is pulled into the shape of a parabola by the very “natural order” of gravity, and calculus can be used to calculate the point at which it will land. Thrown hard enough with the help of a rocket several centuries later, that same ball could orbit the Earth in a path predicted by that same calculus.

Getting back to free will, choice or volition, let me suggest that excessive reductionism takes us into “infinitesimal” territory at the biological cell level. There are ways to deal with this territory without insisting that a zero-length, zero-width point cannot exist.

Also ironically, infinitesimals, at least mathematically, no longer get theologians bent out of shape. No religion that I know of condemns the study of calculus anymore. Could it be that some other concepts long held as violating “God’s truth” by theologians are the infinitesimals of the future?


  1. Alexander, Amir R. Infinitesimal: How a Dangerous Mathematical Theory Shaped the Modern World. Scientific American/Farrar, Straus and Giroux, 2015.
  2. See my post on Newton and Leibniz here.

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