My Problems with Mathematician's Circular Argument

In the comments to Rectangular Circles - Yet Another Responseto Mathematician, I wrote what I thought (with some hubris) was a great argument, a killer argument:

I get the point that you are making here (or at least I think I do). It's why I've talked about the "set of ALL chords".

When you say (paraphrased) "there could be far more chords with c in [R/2,R] than in [0,R/2]" you are, I presume, assuming a "proper" mathematical circle/disc - we are talking about Euclidean space and not talking about curved space, or anything tricky like that. If so, I'd have to ask, on what basis, other than your selection process for problems like this, can you suggest that we might have more possible chords (and thus more possible lines defined by extending those chords out to infinity) passing through the interval [R/2,R] than any other interval of the same length? The circle/disc under consideration is essentially undefined as far as location, size and rotation go, so we should (reasonably) be able to change the locus and not have our answer change on us - but what you are suggesting is that if we shift the locus up by R, and rotate the circle/disc by π, then we'll change the number of lines passing through the intervals [0,R/2] and [R/2,R]. Ditto if we expand our circle/disc by a factor of 2 while retaining the locus at the notional (0,0).

We could even have two overlapping circles/discs, both of radius R, one with a locus at (0,0), the other with a locus at (0,R). This would mean that you'd simultaneously have more lines passing through [R/2,R] (as defined by chords in the first circle/disc) and more lines passing through [0,R/2] (as defined by chords in the second).

This seems odd to me. Does it not seem odd to you?

Mathematician replied (with my current responses interspersed):

> "set of ALL chords"

Wow, maybe I understand what you mean, but it would be odd.  We have a given circle, right?  When you say the "set of ALL chords", are you including the chords that are NOT inside the given circle (but inside another circle somewhere else ...)?

Was that your point all along for repeating "ALL chords" all the time? It would make sense with the rest of the argument:

I first talked about "ALL chords" in a response to a comment on Triangular Circles.  In Mea Culpa, I put some effort in to explain what I meant by "ALL chords" – there is hopefully no indication whatsoever that I had any thought about considering all chords in all circles, and thus including chords that are not in the circle being considered.  No, I meant "ALL chords" in the circle being considered.

My apologies for not making that sufficiently clear.

Sadly, the banks rarely have this sort of confusion, so when I go and tell them that I want to take out "all the money", they don’t pop out the back and give me every single dollar from everyone’s account … they just give me the money that that was in my account.

> on what basis can you suggest that we might have more possible chords (and thus more possible lines defined by extending those chords out to infinity) passing through the interval [R/2,R] than any other interval of the same length?

Since the beginning, you are thinking of chords as the intersection of a straight line with the disc. That's a great characterization and a good way to get chords. (but not the only one, as we both know)

So if I'm not mistaken, for your point of view, there is an existing set of all straight lines on the entire plane (like an infinite net), and you are just taking the intersection of this existing set of straight lines with a given disc. And you say that if you move the disc around, it will not cross the same straight lines, but the answer to Bertrand question should remain the same. Am I correct to assume that this is more or less your reasoning?

With this interpretation you are absolutely correct and agree with Jaynes argument. This is a mathematically correct argument.

At least I have that right!

But that's not the only natural point of view on this problem.

See, I'm taking another characterization for chords. For me a chord is a segment between two points on the circle. So there is nothing "outside" my circle. I have no reason to extend a chord out to infinity. The chords are not intersection of lines with the disc, they are segments inside the disc! There is no reason to consider objects (lines) that are not chords on the given circle, don't you think?

So, If I change the locus of the circle in the plane, the chords are moving with it. If I double the size of my circle, then the chords inside it will double their size. If I rotate the circle, the chords will rotate with it. So the final answer to Bertrand question will not change at all.
And with that point of view, it's perfectly natural to have more chords close to the rim than close to the locus. You only think it's odd because you are thinking of an existing "net" of straight lines on the plane, and you place your circle on that existing net of lines. But from my point of view, there is no "net" of existing straight lines.

I note that you write here "The chords are not (the) intersection of lines with the disc".  In one of your comments at Triangular Circles, you wrote (my emphasis):

A circle is a 1-dimensional curve in the plane. A disc is the 2-dimensional surface that is enclosed by the circle. So it's important to make the distinction, whether you talk about the endpoints (which are on the circle) or the midpoint (which is in the disc) of a chord (which is the intersection of a line and a disc).

Perhaps you can see why someone might get confused.

So my problem with this approach, and I’ve hinted pretty strongly at it (that is by writing it), is that as you have said yourself a chord is the intersection of a line and a disc – irrespective of how you select that chord – and a chord is thus also a segment of a line – again irrespective of how you select the chord (it just happens to be the segment of the line that intersects with the disc).

If you could have a circle and nothing else, then I guess I would have to agree with you, at least on the basis of my ignorance with respect to the implications and also in recognition of your authority as a Doctor of Mathematics.  However, immediately after invoking this free-floating circle you refer to something outside the circle – specifically when talking about changing the locus, altering the size and rotating it.  You have thus invoked an external reference plane on which the circle/disc rests.  Would you not agree that it is meaningless to talk about translational, rotational and scalar invariance if the circle is all there is?

It seems, therefore, to me, that you are attempting to have your cake and eat it too.

Perhaps there was something in Bertrand's original phrasing that leads you to think that we can talk about a free floating circle, rather than one embedded in a plane.  I just don't know.  Unfortunately, my French is that of a rather forgetful schoolboy, I remember bits and pieces, I get the general gist of a menu or a wine list, but I never got to the stage at which I might have deciphered Bertrand's original text.  If only there were some French speaking mathematician I could call on to interpret …

This paper, interesting, explicitly refers to a plane, or rather the plane: "Consider a disk on the plane with an inscribed equilateral triangle."  In the afterword, the author writes (rather pleasingly from my point of view):

In his pointedly titled paper The Well-Posed Problem, (Jaynes) applies this principle to the Paradox of the Chord with success, uniquely identifying the uniform distribution over the distance between the midpoint of the chord and the center of the disk as the correct choice of measure, which he then proceeds to verify experimentally.

The use of the term "correct" is particularly satisfying, although I'm sure that there's some reason why my understanding of the term "correct" is limited and that that lack of understanding will be swiftly addressed.

> We could even have two overlapping circles/discs, both of radius R, one with a locus at (0,0), the other with a locus at (0,R). This would mean that you'd simultaneously have more lines passing through [R/2,R] (as defined by chords in the first circle/disc) and more lines passing through [0,R/2] (as defined by chords in the second).

If I understand correctly, with your point of view, if you have two circles in the position you gave, there is as much chords in the interval [0,R] than in the interval [R,2R], right? With your point of view, the fact that we have one, two or seven circles, do not change the "density" of chords at all, is that correct? This seems odd to me.

With my point of view, each circle has its own set of chords, so the "density" of chords will be higher in the intersection of both disc. So there will be "twice as much" chords in the interval [0,R] than in the interval [R, 2R], because there are two set of distinct chords.

If we permit a single mathematical line to "carry" multiple, overlapping, distinct chords, then I don’t see a major problem with the density of chords changing as you overlay circles (ie you could conceivably have ten identical circles on top of each other, with an infinite number of chords, each of which is replicated ten times).  But in the context that you quoted, I was not talking of chords per se, nor was I really thinking about the chords in multiple circles (except obliquely).  Perhaps I should not have even mentioned overlapping circles at all, because this has only served to confuse.

You seemed to understand what I was saying earlier, when you talked of a "net of straight lines on the plane" (my recollection is that all lines are straight, but perhaps you were just clarifying this for my benefit).

There will be no line that passes through the circle (across the disc) which is not also coincident with a chord, right?  (By this I mean that there will be a segment of the line which directly corresponds with a chord, the ends of which lie on the circumference of the circle/disc - the segment shares the same length, gradient and endpoints as the chord such that one could almost consider them to have the same identity.  When I said "carry" above, I mean to imply that such a chord is, in a sense, lying on the line with which it shares a segment.  It is possible, with multiple circles, for there to be multiple segments of the line, perhaps overlapping, lying on top of each other … perhaps entirely identical and overlapping. This might not be standard phrasing, but I hope that you can understand my intention.)

There is an infinite array of lines that pass through the circle, and a(n infinite) subset of those will intersect with a notional y-axis.  With your preferred method of selecting (or identifying) chords, via their endpoints, it seems to me (and even to you, apparently) that there will be fewer lines that intersect the y-axis at the locus of the circle than at the circumference – this is what does not appear to be justified.  Spiriting the circle out of this universe does not appear to be justified either, but perhaps the problem as originally phrased does demand it and I am simply unaware of that aspect of it.

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Would it help to move away from circles for a moment?  I understand that there are certain things about circles that might lead you to favour endpoint generation of chords.

I was thinking of a similar problem, but involving a square (squared circles!)  What is the probability that, on selecting at random an s-chord (my term for the equivalent of a chord within a square) the endpoints of which do not share the same side of a square, the selected s-chord is greater than the length of the longer sides of an isosceles triangle which has one of the sides of the square as its base (√5L/2 where L is the length of the sides of the square)?

This does, to me, seem a more complex question to answer.  I'm tempted to say that the endpoint approach will give, once again, a result of 1/3.  We don't seem have an equivalent of the other two approaches, because s-chords are not as constrained as chords - but perhaps there is a way of thinking about them using one side of the square as a reference.

But that's not what I was thinking about specifically.  What I was thinking about is the density of s-chords.  Would you expect to see them clustered around the edges of the square (or sides), or at the corners (or vertices), and more sparsely represented in the middle of the square?

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Why did I call your argument "circular"?  I was going to weave that into my response above, but there was no obvious or natural place for it.  So my explanation will have to stand on its own.

My argument is that a circle/disc doesn't exist outside of the plane on which it rests.  This in turn means that the "net" of mathematical lines of which the plane (conceptually) consists cannot be naturally separated from the disc, and hence from the chords (each of which is the intersection of a line and a disc).  If you ignore this, and consider chords as merely what you get if you connect two points on the circumference of a circle, a circle which (in my view) is strangely divorced from mathematical reality, then sure, you can float this circle around, squeeze or stretch it and spin it around.  With such an independent circle there is no need for invariance of any kind (translational, scalar or rotational) - because the circle is the circle is the circle.  If this is not circular …

(Please do note that this is, at least in part, a joke - a poor excuse for a pun.  My more serious efforts lie above this section.)