The problem with Aristotle's wheel

There's an apparent mathematical paradox, of very long standing, that illustrates a fundamental problem with a lot of modern discourse.

It's called the Aristotle's wheel paradox, and it goes something like this.

Imagine you have a circular wheel, and attached firmly to it is a concentric smaller wheel.  You set the larger wheel on a flat surface, and allow it to roll one complete rotation without slipping.

The trouble comes in when you try to figure out how far each wheel has moved.  Let's call the radius of the outside wheel R.  So by the time it's turned once, it's traveled as far as its circumference, which we all know from high school geometry is a distance of 2πR.  So that should be the length of the horizontal blue dashed line in the above diagram.

But here's the snag; the same applies to the smaller wheel.  Let's say its (smaller) radius is r.  So by the same logic, after it's made one complete rotation, it's traveled a distance of 2πr, which is less than 2πR (because r < R).  That's the red dashed line in the diagram.

But... the two lines are obviously the same length!

While it's uncertain if Aristotle ever did puzzle over this seeming conundrum, it's definitely been known since antiquity.  The first written exposition of it was by Hero of Alexandria, who described it in his book Mechanics in the first century B.C.E.

The solution has to do with the fact that the way the question is posed is misleading.  Most people, reading the description of the paradox and (especially) looking at the diagram, would accept unquestioningly that this is a correct framing of the problem.  But in fact, by stating it that way I was engaging in deliberate sleight-of-hand -- giving you information that seems correct on first glance, but is disingenuous at best and an outright lie at worst.

[Nota bene: I'm not implying here that Aristotle and the other mathematicians who worked on it were lying; they seemed genuinely puzzled by it.  What I'm saying is that I was misleading you, because I know the answer and misdirected you anyhow, with complete malice aforethought.]

The truth is, the two circles haven't moved by the same amount, even if that's what the diagram plausibly leads you to believe.  In fact, the straight dashed lines in the diagram aren't the paths taken by a point on the circumference of either circle.  (Those lines' lengths are equal to the distance covered not by a point on the edge, but by the center of the wheel.)  If you trace the paths of actual points on the rims of the two wheels, here's what you get:

[Both this and the above diagram are licensed under the Creative Commons Merjet, AristotleWheel6, CC BY-SA 4.0]

Without even measuring it, you can see that a point on the outer wheel (the blue dashed curve) travels considerably farther than one on the inner wheel (the red dashed curve) -- and both, in fact, cover more distance than that traveled by the wheel's center (the green dashed line).

Just as you'd expect.

What strikes me about the Aristotle's wheel (non-)paradox is that this kind of thing underlies a great many of the problems with our current political situation.  How many of the hot-button topics in the news lately have come about because of a deliberate, disingenuous attempt to reframe the question in such a way that it ignores important facts or completely mischaracterizes the situation?  Examples include the Florida State Education Department's new standards for history requiring teachers to include information about how slaves benefitted from slavery, Richard Dawkins's statement to commentator Piers Morgan that biological sex is binary "and that's all there is to it," and Jason Aldean's defense of his controversial song and video "Try That in a Small Town," stating that "There is not a single lyric in the song that references race or points to it."

All three could be looked at with a shrug of the shoulders and a comment on the order of, "Okay, I guess that's true."  But in each case, that is to miss the deeper and far more critical truths those statements are deliberately overlooking.

This kind of thing is dangerous because it's so damned attractive.  We're taught to take things as given, especially when (1) they come from a trusted or respected source, and (2) they seem right.  This latter leads us onto the thin ice of confirmation bias, where we accept what someone says because it confirms what we already thought was true.  Here, though, the bias is more insidious, because the case is deliberately being presented to us so as to say nothing specifically false, and yet still to lead us to an erroneous conclusion.

So whenever you're reading the news, remember Aristotle's wheel -- and always keep in mind that what you're seeing may not be the whole story.  Like the two diagrams of the wheel's motion, sometimes all it takes is looking at things from another angle to realize you've been led down the garden path.

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