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I recently watched Numberphile's excellent videos about Bertrand's paradox.

https://www.youtube.com/watch?v=mZBwsm6B280&t=0s

https://www.youtube.com/watch?v=pJyKM-7IgAU&t=0s

See also https://en.wikipedia.org/wiki/Bertrand_paradox_(probability) for a description.

At first, please excuse my ignorance. This is my second post about this subject, the first one was flawed and has since been removed. I don’t speak the math language but I do have a sincere wish to understand what I can learn from this “paradox”. Should it be that this post is flawed too, please explain why (as pedagogically as possible), close it and there will not be a third :).

Background

To me, the “choose a random xxx” obscures what I believe to be the core of Bertrand’s paradox, namely different spaces and functions to generate new spaces from existing ones. My understanding is that the act of “choosing a random xxx” first requires a well defined space to “choose an xxx” from, and that we hence can disregard the randomising aspects and instead focus on analysing the underlying spaces and functions.

Question 1

Unless it is deemed to be a completely pointless exercise, could someone please help me identify and express the involved spaces and functions in a rigorous form? My hope is that this would help me to better understand what I believe is the core of the “paradox”. Below are my own naive attempts.

Definitions (For a given circle and simplified by assuming a fixed rotation.)

Space A: The space of all arc lengths*. All real numbers from 0 to the circumference/2. PDF assumed (or defined?) to be uniform. This is the “starting” space for Space C below.

Space B: The space of all distances from the origin along a radius. All real numbers from 0 to the radius. PDF assumed (or defined?) to be uniform. This is the “starting” space for Space D below.

Space C: The space of all chords that can be created from Space A by applying the function “chord length as a function of arc length” to each element in Space A. The PDF is well defined. One property of the PDF is that P(chord length > sqrt(3))=1/3.

Space D: The space of all chords that can be created from Space B by applying the function “chord length as a function of distance from the origin” to each element in Space B. The PDF is well defined. One property of the PDF is that P(chord length > sqrt(3))=1/2.

Question 2

Can Space C and Space D be considered to be “equals”/"peers" under the axioms and rules of set/space theory or are we comparing apples and oranges?

Question 3

Is there a reason to believe that a "Space E" that contains "all chords of a circle" can be created from another Space?

Question 4

Are we "allowed" to define a "Space E" to contain "all chords of a circle"?

*Assuming that “choose two random points on the circle” is equivalent to “choose a random arc length”.

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Your background understanding is fine. The problem is to identify a space that represents the desired objects naturally and to define measure on that space, also in a natural way. “Pick a point from (geometric figure)” allows us to take said figure and use the volume/area/length as natural measure; of course this makes sense only if the total volume/area/length of the figure is finite so that we can normalise it to 1. Same goes for “Pick $n$ points from (figure)”. Note that there is no such natural way to “pick a natural number”, and while we can “pick a random real between 0 and 1” we cannot “pick a random rational between 0 and 1”.

The set of chords does form a topological space in a natural way, but there is no natural way to define a measure on that space. Note that the points (i.e., elements) of this space are not points in your standard intuition, but they nevertheless are called points in this context.

We can map the space of chords to the space of pairs of points on the circle, namely by mapping each chord to its endpoints. Or we can map our space to the interior of the circle by mapping each chord to its midpoint. Or we can map each chord to a pair of an angular direction and a length up to $2r$. All these are injective maps to nice spaces with natural measure (actually, the way I formulated it, the map to the midpoint is not injective, but this fails only for the unlikely case of diameter chords) and they all are continuous and respect the topology on our space (i.e., chords with “near” endpoints also have “near” midpoints as well as “near” angular directions and lengths). Perhaps one of these maps may occur more straightforward than the others, but neither is a really natural choice. Accordingly, the measures induced by these three maps on the space of chords are different and define (paradoxically) different probabilities for chord lengths.

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  • $\begingroup$ I am not an expert but ... (a) It seems to me that the fact those spaces can be considered topological spaces may not be very relevant here - what matters is that they are measurable spaces; (b) The transfer of measure "against" a function (from codomain to domain) - "pullback" for measurable spaces - seems to be problematic. Cf. mathoverflow.net/questions/122704/pullback-measures ; however this seems exactly the way the functions here play the role. $\endgroup$ Dec 28, 2021 at 21:39
  • $\begingroup$ Thank you for your reply. Let's get physical for a while. Imagine five empty hats marked "A" to "E". Hats A-D are supposed to eventually contain the elements from Space A to D as defined in the question, hat E is supposed to eventually contain the elements from the topological space that was defined in the answer. As far as I can understand, the only hats that we can fill from first principles are hats A and B. We can never fill hats C to E from first principles, the content of these three hats must be generated from the content of hats A and B. $\endgroup$ Dec 30, 2021 at 20:59
  • $\begingroup$ Before starting to filling up the hats, we are of course allowed to discuss hat E as containing something like "a not yet materialised concept of a topological space that represent the desired object naturally" but I can't wrap my head around what this would give us. If we allow ourselves to have that kind of discussion about hat E, we need to allow ourselves to have the same discussion about hats C and D and that seems equally pointless to me. $\endgroup$ Dec 30, 2021 at 20:59
  • $\begingroup$ My conclusions 1. When expressed/formulated in terms of set/space theory, it becomes obvious that Bertrand's paradox is an artificial problem that doesn't exist. No ambiguity, no "paradox" to be found, no lessons to be learned, and there are no known reasons for spending time trying to find out whether Space C or Space D is "the most correct space from which to create Space E". 2. To me the idea of a pre-existing Space E implies that we must axiomatically define Space E to "exist". We are of course free to create our own axioms, but then we must be clear about it and tell that... $\endgroup$ Dec 30, 2021 at 20:59
  • $\begingroup$ ... this is what we are doing. Please don't hesitate to point out any fallacies or gaps in my reasoning. I have been led astray by my intuition many times before and I appreciate a good lecturing :). I'm annoyed by this "paradox" being a thing and I would really like to understand what I might have missed. $\endgroup$ Dec 30, 2021 at 21:00

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