Talk:Aristotle's wheel paradox

Example
The example given (of the wheels of a car rubbing on a curb) does not seem relevant; in that case the circumference of the wheels IS the same, and the outer wheels are rotating through a longer track than the inner wheels strictly because of frictional interference with the curb.

Bijections, really?
I'm glad that this article exists, but its statement of the problem is troublesome. Surely "sets of points" and "bijections" are anachronisms when recounting a problem posed by Aristotle? There is a paradox if one naively assumes that equal cardinalities have equal measures, but I fear that our modern eagerness to dispel that notion has caused us to project it backwards in time. Wolfram does the same, but it's a well-known hotbed of original research and shouldn't be trusted.

From this book and the included external link, it appears that the original paradox is a matter of geometry and mechanics, arising from the erroneous assumption that the small wheel traces out its own circumference. And this assumption is not necessarily made by corresponding points between the path and the circumference. It seems to be made without any explanation at all, which shouldn't be too surprising; paradoxes arise by sleight of hand and gaps in reasoning at least as often as explicit falsehoods. Melchoir (talk) 09:30, 2 March 2008 (UTC)


 * Since the external link is now broken, here's an alternative link. Will Orrick (talk) 16:57, 29 April 2023 (UTC)

The above comment is in reference to this version. I've since replaced the material in question, with a citation to Bunch, whose treatment is very similar although obviously more extended. Melchoir (talk) 10:01, 2 March 2008 (UTC)

One wheel
The paradox isn't hand waved away (slipping indeed, feh), as you folks are doing, if you consider the two circumferences to be the outside curve of the wheel and the inside curve of the same wheel.

In this event, the paradox holds and is a mathematical result of (our inability with respect to) mapping Reals to Reals. Lizard1959 (talk) —Preceding undated comment added 00:09, 29 December 2009 (UTC).

I agree with the first sentence. I believe the image could be improved in two ways. Lengthen the blue horizontal line so that it equals the circumference of the blue circle. Same for the red horizontal line and make it dashed to give some indication that is imaginary.

65.25.72.28 (talk) 17:40, 20 September 2017 (UTC)

I think it would be helpful to add to the description that a roll of duct tape, black electric tape, etc. are all examples for the case of the inner circle being fixed in place.

18:25, 20 September 2017 (UTC)~ — Preceding unsigned comment added by 65.25.72.28 (talk)

All the examples you give (tapes) are when the inner circle is dependent on the outer. The distance travelled is entirely dependent on the outer wheel if the inner is fixed. This is mentioned in the second paragraph of the Analysis section — Preceding unsigned comment added by 137.222.95.102 (talk) 16:55, 12 June 2018 (UTC)

this is a ridiculous treatment of the problem.
The point does NOT travel in a straight line as the wheel rotates (unless it is the center point of the wheel). It follows a SIN, COS, whatever... map the _actual_ location of each point in question and you will see two similar curves. However, the curve for the inner point will actually be shorter (in a 2D sense, not this "2D when convenient, 1D when it isn't" sense as currently presented) than the curve for the outer point. Is this seriously considered a useful entry? Shouldn't the "sleight of hand" be explained? — Preceding unsigned comment added by 134.134.139.78 (talk) 19:52, 21 August 2014 (UTC)

Wrong problem entirely
Aristotle's wheel paradox deals with the trail paths of the inner and outer rims of a wheel (which are necessarily of different radii and circumference). It has nothing to do with two wheels of different sizes. The problem has been stated incorrectly and discussed incorrectly, and is — as a consequence — entirely useless and misleading. Tagged as disputed. This may require an expert in philosophy or maths to address. 2602:306:3429:6050:45E9:6878:7646:58BE (talk) 17:31, 8 February 2017 (UTC)

Cycloid and Trochoid
Hi,

the movement of the wheel is not a translation but a rotation. Therefore a given point on the perimeter of the wheel performs in an x-y coordinate system a cycloid, beeing well described in standard mathbooks (see e.g. Bronstein-Semendjajew, "Taschenbuch der Mathematik"), an given point in a inner ring performes a shortened Cycloid oder Trochoid. The Lengths of the bows can be described by the formulas given there. This is no philosopic problem, rather a mathematical matter.

If you like me to improve this article, please contact me on my German WP-Account, --Fachwart (talk) 22:35, 22 March 2017 (UTC)

The movement of the wheel is rotation plus translation = rolling. https://en.wikipedia.org/wiki/Rolling 65.25.72.28 (talk) 12:46, 23 September 2017 (UTC)

Analysis and solutions
The paradox is that the smaller inner circle moves 2*pi*R, the circumference of the larger outer circle with radius R, rather than its own circumference. If the inner circle were rolled separately, it would move 2*pi*r, its own circumference with radius r. The inner circle is not separate but rigidly connected to the larger. So 2*pi*r is a red herring.

First solution: Let Pb be a point on the bigger circle and Ps be a point on the smaller circle, both on the same radius. For convenience, assume they are both directly below the center, analogous to both hands of a clock pointing towards six. Both Pb and Ps travel a cycloid path as they roll together one revolution. The two paths are pictured here: http://mathworld.wolfram.com/Cycloid.html and http://mathworld.wolfram.com/CurtateCycloid.html

While each travels 2*pi*R horizontally from start to end, Ps's cycloid path is shorter and more efficient than Pb's. Pb travels farther above and farther below the center's path -- the only straight one -- than does Ps. The closer Ps is to the center, the shorter and less curved its cycloid path.

If Pb and Ps are on any other radius, e.g. analogous to both hands of a clock pointing in any other direction, the same analysis applies. So every pair of points, one on the larger circle and one on the smaller circle, has been considered.

Second solution: The larger circle and the smaller circle have the same center. If said center is moved, both circles move the same distance, which is a necessary property of translation (geometry) and equals 2*pi*R in the experiment. QED. Also, every non-center point on both circles has the same position relative to the center before and after rolling one revolution (or any other integer count of revolutions).

Merjet (talk) 11:07, 25 August 2018 (UTC)


 * Everypoint inside the big circle travels the distance 2pi.R.2A00:23C5:C13C:9F00:7159:C602:A90E:100C (talk) 21:22, 20 March 2022 (UTC)

Add some comments about continuum and mensurability
The reader comes to understand, through Galileo's paradoxical vision, the difference between discrete and continuous mathematics... And it has also a modern vision on linear continuum.

See didactic explanation at https://www.youtube.com/watch?v=mrVg9GM5h7Q, at 5:15. Krauss (talk) 11:18, 17 April 2023 (UTC)


 * I watched the linked video, and it is problematic in a number of ways, particularly in its switching back and forth between using "size" to mean measure and to mean cardinality, without warning the viewer that these are different things. It also gives a misleading explanation of why the continuum is uncountable. Furthermore it seems to make the false statement that the property of being in one-to-one correspondence with a subset of oneself is only possible for continuous sets and not discrete ones. In fact, this is possible for all infinite sets, whether discrete or continuous.


 * There may be something to say about discreteness in connection with Galileo's wheel, but I'm not sure what that would be, as the issues seem rather subtle. If you can find something in the relevant philosophy literature, it could perhaps be added. Will Orrick (talk) 13:51, 19 April 2023 (UTC)