When responding to Mathematician in

*Triangular Circles*(a little play on words, I know circles can't actually be triangular), I wrote this:
On parameterisation, I did some thinking about this along the lines
of saying that if you have a 1/3 answer, then it seems (to me) that your
selection method must simply have missed some of the chords. In my way of thinking (standard caveat about
the possibility of being wrong), if we are asked to select a chord "at
random" then it follows that we would be selecting from a set of ALL
chords, rather than from a specific subset, unless advised otherwise. Thought from this perspective, our first
concern is making sure that we have ALL chords available to select from. The question then is how to express this
properly. I'm probably going to mess
this up in some obscure way, but if you can at least try to understand what I
am saying (and criticise the best formulation of my argument, rather than the
worst), it would be appreciated.

I suggest that an expression for ALL chords in a circle defined by x

^{2}+y^{2}=1 (in units of R where R is the radius of the circle) goes something like this:
The infinite set S
of all unique sets S

*of points that fulfil the following criteria:*_{i}
S:

-1 > c > 1 (defining the y axis intercept of the chord)

0 > θ > 2π
(defining the gradient of the chord)

S

*:*_{i}
-√((-cosθ)

^{2}+(c-sinθ)^{2}) > r > √((cosθ)^{2}+(c+sinθ)^{2})
(x,y) = (r.cosθ,r.sinθ+c)

Note: the combined
effect of these two conditions is (or is intended) to include all and only
points between intercepts of the line defined by (x,y) = (r.cosθ,r.sinθ+c) and
the circle defined by x

^{2}+ y^{2}= 1, thus defining a chord. In other words a unique set S*is intended to define a unique chord.*_{i}
When corrected in
terms of mathematical terminology, etc, is this a parameterisation and, if so,
does it establish or define a structure (per

*u/Vietoris*) for which there is a defined probability measure (per*u/Vietoris*) or probability distribution (per*u/overconvergent*)? And, if so, what Bertrand Paradox related answer would be expected from this parameterisation and associated probability measure/distribution?
---

Well, I was certainly right. I
did mess it up.

First, I've doubled the number of chords by using the
intervals c:[-1,1] and θ:[0,2π] (hopefully this terminology is clear, it's slightly
more convenient than using the -1 > c > 1 and 0 > θ > 2π structure.

I should have used either c:[0,1] and θ:[0,2π]

**c:[-1,1] and θ:[0,π]. Mathematician, in his response, went with the latter, so I'll use that to explain the second, more egregious stuff up.***OR*
Note that I said that "our first concern is making sure
that we have ALL chords available to select from". The whole purpose my sets was to achieve this
and they don't.

For any value of θ<>0 (assuming that θ=0 is standard and
aligns with the positive x-axis, the notional horizontal axis and that c is the
point at which the resultant chord intersects the y-axis or notional vertical
axis (I did use the word "intercept" before, which is apparently
right in some cases but there may be some subtlety that I am missing - or perhaps
my wording was just clumsy), there are chords are missed in my schema.

We've agreed that the interval [-1,1] is (or can be)
uniform, so imagine 11 equally spaced points on the y-axis in that range and
say we look at θ=π/4:

This cannot produce the set of ALL chords. Additionally, there is a "skewing"
of chords towards those that are longer, so it should come as no surprise that
(as Mathematician intuited) there would be substantially more chords of length greater
than √3.R. For this reason, I don't think the following comment
was nearly as silly as Mathematician later thought it was:

Ok, actually I'm not
sure that I am computing the correct probability here. Tell me if this is your
idea :

First you pick a
number between -1 and 1, uniformly on the interval [-1,1]. And then you pick an
angle between 0 and pi, uniformly on the interval [0,pi]. The chord corresponding
to the couple (r,θ) is the unique chord that has slope θ and that cuts the
horizontal axis at r. Is that okay ?

So this defines a
probability on the set of chords. And with this, the probability that a random
chord is longer than sqrt(3) is given by the following formula :

P= 1/3 +
ln(7+4*sqrt(3))/2pi = 0.7525...

I might be wrong
here, but it seems reasonable.

To Mathematician, in answer to the embedded question " …
Is that okay?" Yes, I am reasonably
happy with that, once I get over my confusion about the use of r (which I
normally think of as the length of a vector from (0,0) to some other point). If forced to pick something similar, I'd have
gone with (c,θ) since we already have c defined - this would be the unique
chord with slope θ that is offset from the x-axis by c when x=0. But I get what you mean,

I'll try to define a set of ALL chords again (this requires
more than just a minor shuffle, I suspect).

An expression for ALL chords on a disc defined by x

^{2}+y^{2}=1 (in units of R where R is the radius of the disc) goes something like this:
The infinite set S of all unique
sets S

*of all unique sets S*_{i}*of points that fulfil the following criteria:*_{j}
S:

0 > θ > π (defining the
gradient of the chord)

locus defined as (0,0)

S

*:*_{i}
-1/cosθ > c > 1/cosθ
(defining the y-intercept of the chord)

S

*:*_{j}
-√((-cosθ)2+(c-sinθ)2) > r
> √((cosθ)2+(c+sinθ)2)

(x,y) = (r.cosθ,r.sinθ+c)

Note: the combined effect of
these two conditions is (or is intended) to include all and only points on the
intersection of the lines (x,y) = (r.cosθ,r.sinθ+c) and the disc defined by x

^{2}+y^{2}=1, thus defining a chord.
Defining the locus as (0,0) removes
some complications to the equations that would otherwise be required to achieve
invariance in terms of translation (by which I mean movement of the circle to
another location). Setting the radius of
the disc to R and making R the units of length in all considerations addresses
the question of invariance in terms of scale.
Defining the set of y-intercepts such that c:[-1/cosθ,1/cosθ] goes only
part of the way to addressing invariance in terms of rotation.

If we revisit the image above but
extend out the range of c, we will get:

The gap has gone, but we've now
got more chords at θ=π/2 than we had at θ=0, so we no longer have rotational
invariance. To get it back, we need to
introduce a concept that probably has another proper term to it, but I call
"granularity".

Say we select an arbitrarily
large number (N+1) of evenly spaced samples over the interval from which we
take c. If c:[-1,1] because θ=0, then
there would be N/2 samples above the locus and N/2 below the locus and one on
the locus. If we generalise this, for
c:[-c

*,c*_{i}*], then there would be still N/2 samples above the locus and N/2 below the locus, but with a different separation - rather than the samples being 2/N apart, they would be 2/N/c*_{i}*apart. I refer to this figure, 2/N/c*_{i}*, as the "granularity".*_{i}
In order to maintain invariance
in terms of rotation, we need to set the granularity of the sets to 2.cosθ/N with
N->∞. If there is a better way to word this, please
let me know.

If there is an iron-clad rule
that says that I cannot parameterise my chord selection with anything akin to
this concept of granularity, then I guess I have to graciously concede defeat,
albeit with the residue of the itchy feeling that

*maths shouldn't be like this*. But if it**possible, without necessarily being conventional, then I think my selection of chords makes sense, is invariant in terms of scale, translation and rotation and results in the 1/2 answer. And while it does not seem quite as elegant as my first (incorrect) version, it is more general and I don’t know that an attempt to do something similar with the 1/3 and 1/4 methods can be done***is***elegantly. Perhaps it can be done, perhaps there are even more elegant ways to do it, I'm in absolutely no way certain of this.***as*
> If there is an iron-clad rule that says that I cannot parameterise my chord selection with anything akin to this concept of granularity, then I guess I have to graciously concede defeat, albeit with the residue of the itchy feeling that maths shouldn't be like this.

ReplyDeleteIt seems that you still don't understand what I'm saying.

You can parametrize your chords as you want. You can ask that the probability distribution have nice properties of invariance. And you can compute the answer using this probability distribution. Yes, your concept of granularity makes sense, even if you explained with a very poor choice of words. That's not the problem at all (or at least that's not what I'm trying to explain).

The point is that YOU made choices about parametrization, distributions, properties, invariance, ... .Not the question, YOU ! The question could not be answered without some of these choices, and the question certainly did not imply any particular choice. So there is no mathematical reasoning that could justify these choices.

I think there is a very fundamental problem about all your reasonings. Mathematical statements are not ambiguous. They can not be ambiguous. However, english language can be ambiguous. So when you write a sentence in english to express a mathematical statement, you have to make sure that all the words have one single unambiguous meaning. When you ask a question using english language, it is possible that the question is mathematically ambiguous, even if the question makes perfect sense in english.

For example, a common question that was asked on reddit many many times is : "are there more whole numbers than even numbers ?". This seems to be a very natural question, and someone who never studied math probably thinks that a mathematician can give an answer to that question. The problem is that this question is ambiguous because the word "more" does not have a unique meaning in mathematics (especially when referring to infinite sets). So with this formulation, the question is meaningless.

Correct formulations of questions could be :

"is the cardinality of the whole numbers strictly greater than the cardinality of the even numbers ?" (answer in this case, NO)

"Is the density of whole numbers strictly greater than the density of even numbers ?" (YES)

"Is the Lebesgue measure of whole numbers strictly greater than the Lebesgue measure of even numbers ?" (NO)

"Is the set of whole number greater (for inclusion) than the set of even number" (YES)

Arguing about how to interpret an ambiguous question is something that is not in the realm of mathematics, but more in the realm of linguistics, psychology, philosophy, whatever ... You could argue for hours that the word "more" in previous question was obviously used by the guy asking the question to refer to cardinality/density/whatever ... The point remains that the word "more" is ambiguous.

Now, another point to understand is that sometimes, mathematician are lazy. They don't want to write everything every time when they talk about mathematical objects or ask mathematical questions.

DeleteFor example, if I ask the question : "does 1+1=0 ?" without any reference to anything, the common convention is to consider that 1 and 0 are integers, and that the "+" sign refers to addition and the "=" sign refers to usual equality of numbers, so that the answer is NO. But if I ask the very same question specifying a different way to interpret the symbols, for example : "does 1+1=0 in Z/2Z ?", then other conventions need to be applied (0 and 1 are now equivalence classes, and so on ...) and the answer is YES.

In the domain of probabilities, there are certain common conventions to avoid writing too many things when writing statements.

For example, the sentence "pick a whole number at random between 1 and 5", is a priori meaningless. The expression "at random" is not a well-defined mathematical term. But there is a CONVENTION, used by almost everyone around the world, that this is meant to be understood as "take a random variable X from the set {1,2,3,4,5}, with the uniform measure". But it's a convention that was chosen because it corresponded to the colloquial use of "at random" in the english language (the language that can be ambiguous). It's just to avoid writing this long sentence all the time.

In the same way, the sentence "pick a real number at random between 0 and 1", is BY CONVENTION meant to be understood as "take a random variable X from the set [0,1], with the uniform measure". But again, it's just a convention.

Now, there is no such convention for a general sentence of the form :

"pick an element at random in the set X"

especially when X is an infinite set. So the sentence is meaningless, because "at random" is not a well-defined mathematical term. For example, the sentence "pick a whole number at random" is meaningless. You cannot even say that the convention should be to take the uniform measure, because such a thing does not exist on the set of whole numbers. So clearly, the classical convention that works for finite sets and finite intervals, cannot be used here.

So the same apply for the set of chords. There is no EXISTING convention for the set of chords on a circle. So the question is meaningless, that's all. There is NO answer to the question.

So you might argue for hours about what the convention should be in this precise case. You might want to say that it's reasonable to take the "uniform" measure on the set of chord, but even the word "uniform" is meaningless for the set of chords, because there is no definition of what a uniform measure on this set should be. Again, you might argue that in this precise context, there is a particular probability measure that could be called "uniform" due to its resemblance with the usual uniform distribution on intervals. But as I said, this is a debate about mathematical conventions, it does not constitute a mathematical answer to the question. Because in fact, there was no question to begin with.

To conclude, you seem to think that if a mathematical notion (such as the "uniformity of a measure") makes sense in a certain context (as the uniform measure on the set [0,1]), then there is a unique way to generalize the notion to more complicated objects (like the set of chords). This is simply wrong, and shows a misunderstanding of how mathematical definitions are chosen and generalized.

It may surprise you, but I have no difficulty with what you are saying. It seems (to me in my clumsy impure non-mathematical way of thinking of things) like a contractual difficulty - the client comes along with a poorly worded requirement, the contractor builds what the contractor builds (using the contractor's standard method) and then in court, the lawyers point out that client didn't stipulate requirements correct and the contractor didn't ascertain precisely what client wanted, but the contract left that open, so whatever is provided so long as it satisfies the contract is "correct".

DeleteI'm on the side of the client, and I understand that we are bogged down in points of law (or points of precise mathematics), but that there is a common understanding of certain terms and it's only because we have lawyers involved that we are fussing about these eroteric definitions - plus, to reach a solution that accords with the common understanding of the terms involved, not much effort is required (which the contractor's staff would have known right from the start if they had thought about it a bit and not just focussed on the way they do things).

So applying that to our situation, I understand that if we are going to be perfectly correct and mathematically rigorous, and there is a time and a place for that, then you are 100% correct - we have to carefully define our terms, even with as apparently simple things as 1+1=0.

It's almost like there is a "Bertrand's paradox" for you guys (who can arrive at an answer that says there is no definitive answer and pretty much any answer can be arrived at depending on how you select whatever it is you want to select in order to establish your set of chords) and there is another "Bertrand's paradox" for everyone else who lives in the real world. I am not saying here that you are off with the fairies, I am merely saying that if I wanted to establish the set of ALL chords, then I want to reach a usable answer (let's say I'm using chords of precise length with a certain granularity as the basis of raffle draw in a fundraising event for the "International Save the Globefish Fund".) Perhaps our function would get raided by mathematicians (see what I did there?), but I think I can come up with a common understanding of "pick chord at random from a circle of radius R" in this instance. Do you not agree?

I do recall many moons ago that I went through the process (in a university) of working through a physical understanding of differentiation. The area under a graph can be established by breaking it up into arbitrarily thin bars with heights determined by the average value of the graphed line across the width of the bar. This allowed us, the students, to grasp that differentiation isn't just some nonsense that a pointy-headed mathematician dreamed up, but that it has real world application. It occurs to me that pretty much the same sort of thing can be done within the disc, cram the disc with arbitrarily thin bars and you'll arrive at the area of the disc - with an increasing accuracy as you reduce the width of the bars - you'll never quite get there until the width is infinitesimal but there is never a requirement to change the arrangement of the bars. This (to me) implies that there is something more natural about chords arranged in the disc in such a fashion than there is associated with the 1/3 and 1/4 methods - because it has real-world application. Again, does this seem unreasonable? If there a real-world application that can only be achieved via the 1/3 and/or 1/4 methods?

Part 2 - got hit by the character limit:

Delete>This is simply wrong, and shows a misunderstanding of how mathematical definitions are chosen and generalized.

I don't think my issue was

about how mathematical definitions are chosen and generalised. This is a point at which we are talking past each other because we have different perspectives.everI do note that you haven't engaged on the notion of obtaining ALL the chords, rather than focussing on using a given method to select chords.

I think however, that I do understand the concerns of mathematicians on this question, and I think I agree that within their sphere of influence, these concerns are valid. I understand where you are coming from when you say, in terms of mathematics, there is no single clean answer to the question because of it's lack of definition, and I do understand that I'm bringing some (non-mathematical) definition with me from outside the original question (if thought of purely as a mathematical question). Can you comprehend that from outside your occasionally opaque crystal dome, out in the real world where things are often messy and imperfectly defined, my position is not completely crazy unreasonable?

> I do note that you haven't engaged on the notion of obtaining ALL the chords, rather than focussing on using a given method to select chords.

DeleteYou keep coming back on that point, and I don't understand why. You can obtain ALL the chords with the 1/3 and the 1/4 method. So what's the point ?

> Can you comprehend that from outside your occasionally opaque crystal dome, out in the real world where things are often messy and imperfectly defined, my position is not completely crazy unreasonable?

Can you understand that out in the real world where things are often messy and imperfectly defined, the 1/3 method is not completely crazy unreasonable ?

As I said, what makes mathematics powerful is that it is never messy and imperfectly defined. If you ask a mathematical question (such as what is the probability of this ...) you should expect a mathematical answer with mathematical arguments. If you want to ask a physical question with application to a real-world problem, then ask it in terms of real world and physics. For example "if you throw many straws of length L from a certain height H on a circle drawn on the floor of radius R, what is the proportion of straws that will define a chord of length greater than R/2".

> If there a real-world application that can only be achieved via the 1/3 and/or 1/4 methods?

I thought you read the wikipedia page, where it's explicitly said that there are. I'm not going to write a long speech here, but I advise you to read ENTIRELY this interesting article :

http://arxiv.org/pdf/1503.09072.pdf

Perhaps it's a little bit too formal for you, but I hope that even a small mathematical background is enough to understand it.

There is also this seemingly interesting article by Tissier :

http://www.jstor.org/stable/3615385?origin=crossref&seq=1#page_scan_tab_contents

but I can not access the full text. However, the introduction explicitly says : "the article attempts to [...] offer experimental evidence for the different results should evidence be deemed necessary."

To finish :

> This (to me) implies that there is something more natural about chords arranged in the disc in such a fashion than there is associated with the 1/3 and 1/4 methods

You insist on the "more natural" part. I insist on the "to me".

To me, there is something more natural about chords arranged in the disc by the 1/3 method (and yes, I'm really honest here). I don't mind the "clustering" or "granularity" at all, because the disc have no reason to be "homogeneous". The points in the middle of the disc have no reason to be equivalent to points on the outer edge of the disc. There is NO reasonable symmetry that would send the points close to the boundary to points close to the middle. So I don't see why I should expect the "density" of chords to be the same around a point in the middle and around a point close to the boundary.

As I said, it's a reasonable thing to ask, and there are good reasons to ask for homogeneity. But there are many other reasonable things to ask for.

In the same way, in my example of "subintervals in interval", there is (to me) something completely intuitive about the fact that there is "more" subintervals close to the middle than close to the boundary. And the answer of 0 I arrived at, using your reasoning about "granularity" (or homogeneity), seems completely unrealistic to me.

But the difference between you and me is that I don't think that the answer based on my personal personal preference for this particular problem is the best/the only one/the correct/the real-world/the one true answer ...

> But the difference between you and me is that I don't think that the answer based on my personal personal preference for this particular problem is the best/the only one/the correct/the real-world/the one true answer ...

DeleteWell, that seems to be a bit judgmental. If my fiddling around with figures arrived at 1/3, then 1/3 would be "my personal personal preference".

As to the "if you throw many straws of length L from a certain height H on a circle drawn on the floor of radius R, what is the proportion of straws that will define a chord of length greater than R/2" - you're venturing into the real world now :) I'd really have to look at the mechanism by which they do this exercise. If it's set up in such a way as to emulate the method with Point 1 on the circumference, then sure, it'll end up with 1/3. If it's setup in such a way as to emulate the method with Point 1 at infinity (or 10R), then it'll end up with 1/2. Do you not agree? If not, why not and how can you assure me that experimental error isn't at play in this?

As to the 1/4 method, I have an idea about that which is so crazy that even

think it sounds crazy, so I am going to have to do some research to see if anyone else has come up with anything similar to it before I float it, if I ever float it at all.I