You’ve probably heard of Zeno’s Paradox – the famous one about Achilles and the tortoise. It’s a 2000+ year old puzzle about the nature of infinity. An equivalent formulation is roughly this:
- Imagine someone running from point A to Z. At some time t, the person will be half way between A and Z, let’s call it B.
- The person will run from point B to Z. After time t/2, the person will be half way between B and Z, let’s call it C.
- The person will run from point C to Z. After time t/4, the person will be half way between C and Z, let’s call it D.
- One can continue this pattern of subdivision infinitely.
- Therefore, the person will never reach Z.
It’s hard to believe that this little puzzle was taken too seriously by those clever Greeks. Formally modeling it in math is easy – arithmetic of infinite convergent series is taught in high schools, so it’s clear that at time 2t, the runner will reach Z. But the infinity is bothersome enough that even 2000 years later we take the problem seriously. Some even bring up silly stuff like quantum mechanics and uncertainty principles to try to work around it.
But I came across another way to approach the problem – to sever the Gordian Knot, so to speak. That is to recognize an implication of the basic fact that argumentation about a situation is not the same thing as the situation itself.
In this case, the argumentation can indeed go on infinitely, as one talks about shorter and shorter distances & time intervals. But the error in logic is the last step of the list above. The “therefore” doesn’t hold, because the only thing that’s infinite is all this argumentation. The situation is quite simple and evolves independently of how a goofy observer might want to talk about it – or to imagine breaking it up.
In other words, just because someone chooses a degenerate, infinite, useless way to talk about a situation, the situation itself can be perfectly finite, reasonable, intuitive. There is no paradox.
In other words, the map (argumentation) is not the same thing as the territory (subject of the argument).