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Date: Mon, 30 Jun 1997 11:52:22 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: conception of "modern..."
Bill,
I think I understand your point now!
May I challenge you and anyone else on this list with the problem I mentioned
last week? I believe that some VERY important issues relating to continuity
can be displayed in considering answers to this question: You are given a
circle -- what are the odds that a randomly drawn chord will be longer than the
radius? Please give a numerical answer, e.g., p = .5, and explain your
method.
Tom Anderson
> From: Everdell[…]aol.com, on 6/27/97 10:09 AM:
> Tom Anderson asks: "I had thought that a function either was or wasn't
> continuous -- in general or at a given point -- according to specific tests.
> I'm failing to imagine how by shifting the analytical approach one could
> view the same function as continuous that under another approach appeared
> discontinuous."
>
> Quite right. Some functions are continuous everywhere, some are continuous
> in some interval or intervals but not all, and some (often called
> "pathological") are discontinuous everywhere.
>
> But the problem I was dealing with was the basis of analysis/calculus itself.
> You can get a pretty powerful differential calculus if your premises talk
> about "approaching" a "limit," as Euler did, but that calculus will be
> hornswoggled when discontinuous functions are found, or if continuous
> functions are found that are not differentiable at some points. This was the
> point made by Weierstrass in the 1870s (and earlier by Bolzano and Cauchy)
> against the elegant analysis/calculus of Euler and Lagrange. His move was to
> confine the limit concept to statements about sets of numbers, and to speak
> rigorously of numbers smaller by a given numerical amount than a given
> discrete number, all of these very small numbers, as small as necessary, but
> not infinitely small, because infinitely small numbers -- infinitesimals --
> entail continuity land you right back in the "approach to a limit" mode.
>
> This was a major change in mathematics and was much talked of at the time by
> mathematicians and others who could follow mathematical arguments.
>
> In 1961, Abraham Robinson showed how infinitesimals could be used in an
> analysis/calculus without loss of rigor and provability. This rehabilitated
> continuity in the foundations of analysis/calculus and in other branches of
> math. Once again, the foundational premises of these branches of math cannot
> assume both continuity and discontinuity. They must choose one or the other,
> because the two exclude each other.
>
> -Bill Everdell, Brooklyn
>
>
>
------------------------------
Date: Mon, 30 Jun 1997 17:15:23 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Challenge: Random chord
I posted a challenge, and I'd like to say just a little more about my
motivation in doing so. I got this problem from a very interesting Web page
about mathematical pedagogy that urges exploration using fundamentals as a way
of learning math.
The problem is one that is accessible for novices and challenging to experts.
What you gain from exploring this problem is some development of your
intuitions about probability and distributions -- or that's what the author of
the Web page wants you to gain -- and you do gain that. But I also believe you
gain some insight into some aspects of continuity, and that the exercise is a
good one to stretch some Peirce related muscles.
One needs only minimal mathematical knowledge to work on the problem -- little
more than knowing the definition of a chord of a circle: a line segment drawn
between any two points on the perimeter of a circle, with a 'one-point chord'
allowed.
The problem is just: What are the odds that a randomly drawn chord will be
longer than the radius of the circle?
It's a relatively old problem, known as Bertrand's paradox. Despite the
simplicity of stating and the ease of access, experts don't agree on how to
approach it!
Tom Anderson
------------------------------
Date: Mon, 30 Jun 1997 21:38:44 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord
Below is a proposed solution to Tom's challenge: What are the odds that a
randomly drawn chord will be longer than the radius of the circle?
Imagine a large circle with radius R. Using the center of the circle as
the vertex construct an equilateral triangle whose sides equal R. Note
that the base of this triangle forms a chord of length R. Now, within
this large circle construct a concentric smaller circle whose
circumference is tangent to the chord constructed above. The radius of
this smaller circle equals the height of the equilateral triangle and can
be computed using the pythagorean theorem by Tom.
Now, any chord of the larger circle greater than R must pass through a
point within the smaller concentric circle. So the probability of
randomly doing so is proportional to the area of the smaller circle
verses the area of the larger circle minus the area of the smaller
circle. N'est pas?
Jim Piat
------------------------------
Date: Mon, 30 Jun 1997 22:56:45 -0500 (CDT)
From: dkawecki[…]ix.netcom.com (David Kawecki )
Subject: Re: Challenge: Random chord
longest chord = 2r
hence 1/2 of possible chords > r
50/50 odds
David
------------------------------
Date: Tue, 1 Jul 1997 14:39:00 +1000 (EST)
From: Cathy Legg cathy[…]coombs.anu.edu.au
Subject: Re: Challenge: Random chord
Using Jim Piat's idea that the chord of the circle is equal to R when it
forms an equilateral triangle with two radii....
In fact one can think of the chord as always "subtending" two radii
separated by a particular angle between 0 and 180 degrees. Where the
angle is less than 60 degrees, the chord is less than R. Where the angle
lies between 60 and 180 degrees, the chord is greater than R.
Think of the chord as being "randomly drawn" by a process of choosing
two radii randomly from somewhere in the circle, and then connecting
them up at the top. We can see that wherever the first radius falls, the
second one can either fall:
- within 60 degrees either side, in which case the chord is less than R
- between 60 and 180 degrees on either side, in which case the chord is
greater than R.
The first option describes 1/3 of the circle while the second describes 2/3.
The answer to Tom's question then should be p = 1/3, it seems to me.
Cathy.
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
Cathy Legg, School of Philosophy,
A14, University of Sydney,
Sydney, 2006.
We had fed the heart on fantasies,
The heart's grown brutal from the fare;
More Substance in our enmities
Than in our love; O honey-bees,
Come build in the empty house of the stare.
http://coombs.anu.edu.au/Depts/RSSS/Philosophy/People/Cathy/Cathy.html
}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
------------------------------
Date: Tue, 1 Jul 1997 14:41:15 +1000 (EST)
From: Cathy Legg cathy[…]coombs.anu.edu.au
Subject: Correction: Random chord
The q. was actually what are the odds that the chord is *larger* than the
radius, in which case my answer needs to be: p= 2/3.
Sorry!
Cathy.
------------------------------
Date: Tue, 1 Jul 1997 07:41:19 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord
Cathy,
With respect to Tom's challenge. It seems to me that although 2/3 of the
angles are greater than 60 degrees this does not mean that the proportion
of chords are also greater than 2/3. Instead, I think the proportion of
chords is equal to the ratio of the area of the inner circle to the area
of the outer circle (.75 in the unlikely case my algebra is not wrong).
Granted all chords larger than R must subtend an angle greater than 60
degrees but your ratio does not (I think) reflect the right way to count
the chords. The "right " way to count the chords is to recognize that all
chords greater than R must have a point in common with the area of the
inner circle. This approach will result in "different" chords subtending
the "same" angle so that the proportion of chords is greater than the
proportion of angles. It also seems to me that one could argue that the
relevant ratio is between the two circumferences which I believe would
yield still another number...but I've got a feeling I'm about to be
embarrassed so I'll stop.
Jim Piat
------------------------------
Date: Tue, 1 Jul 1997 10:15:08 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord
Jim,
By now you have read three answers.
Jim, your answer is 3/4 -- that is, the odds of a random chord being longer
than a radius is 3/4. Verifying this calculation is an exercise for people
from Georgia.
Cathy Legg calculates 2/3.
David Kawecki calculates 1/2.
We now have some new reflections from Jim Piat that explain his perspective and
give some criticism of Cathy's.
Let me take a stab at recapitulating the logic.
Cathy essentially starts from a point on the perimeter, and considers the
distribution in question to be chords resulting from the movement of a line
sharing that point, such that any given range of angles gives an equal
opportunity for random selection. That is, the odds of getting a chord that
results from lines between 0 and 20 degrees is the same as from lines between
20 and 40 degrees.
Jim starts from a point on the perimeter (in my recasting of his case) and
slowly moves inwards to the origin, drawing chords perpendicular to the radius
traversed. He looks at the circle, and sees that once you get to a point that
crosses the equilateral triangle that Cathy drew, chords will be longer than
one radius. That point happens to be (by the pythagorean theorem, if Jim's and
my calculations are right) half the square root of three times the radius from
the origin. Jim then thinks: The distribution of chords has to be
proportional to the areas of the circle described by the larger circle and the
circle with radius half the square root of three, therefore 3/4.
David Kawecki offers the simplest solution. His space of distribution is the
possible range of values for a chord, all of which he suggests are equally
likely. The largest chord is the diameter equal to 2 times the radius, and the
smallest is one point with zero length, so half the time, a random chord ought
to be larger than a radius.
Now, the next step: What do all the answers have in common, and where do they
differ? And what does this have to do with Peirce & continuity? And can
anyone think of a computer experiment using random numbers to test one or all
of the three theories -- or any other kind of experiment??
Tom Anderson
> From: piat[…]juno.com (Jim L Piat), on 6/30/97 8:59 PM:
> Below is a proposed solution to Tom's challenge: What are the odds that a
> randomly drawn chord will be longer than the radius of the circle?
>
> Imagine a large circle with radius R. Using the center of the circle as
> the vertex construct an equilateral triangle whose sides equal R. Note
> that the base of this triangle forms a chord of length R. Now, within
> this large circle construct a concentric smaller circle whose
> circumference is tangent to the chord constructed above. The radius of
> this smaller circle equals the height of the equilateral triangle and can
> be computed using the pythagorean theorem by Tom.
>
> Now, any chord of the larger circle greater than R must pass through a
> point within the smaller concentric circle. So the probability of
> randomly doing so is proportional to the area of the smaller circle
> verses the area of the larger circle minus the area of the smaller
> circle. N'est pas?
> Jim Piat
>
>
>
------------------------------
Date: Tue, 01 Jul 1997 15:48:34 +0200
From: Hugo Fjelsted Alroe alroe[…]vip.cybercity.dk
Subject: Re: Challenge: Random chord
The tricky part in your challenge, Tom, is in the meaning of 'a randomly
drawn chord'.
My first solution went in the same way as Cathy's, taking a cord to be
chosen by picking an arbitrary point on the circle, and then chosing another
point on the perimeter of the circle randomly, the two points forming the
endpoints of the chord. This way , using the fact that an equilateral
triangle has angles of 60 degrees, the probability of a randomly drawn chord
being longer than the radius of the circle is 120/180 = 2/3.
But it is possible to take 'randomly drawn' as meaning something else.
Imagine you are making chords with a ruler, chosing any random direction,
you can slide the ruler across the circle and drawing a line randomly inside
the area delimited by the perimeter of the circle, across a possible sliding
distance of 2r. (Here 'randomly' refers to the diameter of the circle and
not the perimeter as above.) The proportion of chords longer than r can be
inferred again by using an equilateral triangle, with two radii as sides and
the non-radii side placed parallel to the chosen direction of your ruler.
(It is sufficient to look at one half of the circle.) Any chord made within
this triangle will be longer than r, thus the proportion we are looking for
is h/r, where h is the height of the triangle. h can be calculated using
pythagoras rule: sqr(h) + sqr(r/2) = sqr(r) <=> h = squareroot(3)*r/2 (or
simply as sin(60)*r), and the wanted probability is squareroot(3)/2 ~= 0.866
Perhaps there are other possible meanings of 'randomly drawn'?
Regards
Hugo Fjelsted Alroe
------------------------------
Date: Tue, 01 Jul 1997 17:16:47 +0300
From: Antti Laato antti.laato[…]abo.fi
Subject: Re: conception of "modern..."
Infinitesimals put end the Cartesian understanding of continuity.
Infinitesimals provide us a new way to understand the mathematical point.
The point is like an infinite microspace and continuity from one point to
another like a "trip" from one space to another. The problem is that we
have our Cartesian models to understand continuity and therefore it may be
difficult to understand Peirce's ideas about continuity.
Best regards, Antti
Antti Laato, Dr (theology), Phil.Cand. (mathematics)
Senior Research Fellow, The Academy of Finland
The Department of the Biblical Studies
Abo Academy, Biskopsg. 16 Turku FIN-20500 Finland
e-mail: alaato[…]abo.fi
------------------------------
Date: Tue, 1 Jul 1997 15:15:38 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord
Hugo,
Your second solution was my first solution! It's a one dimensional solution,
seeing the distribution of chords that are perpendicular to a diameter of the
circle as the universe, so that you have an equal chance of using any point on
the diameter to generate a chord. Therefore, chords generated from a point
closer than half the square root of three times the radius, from the origin,
would be longer than one radius. As you write, approximately 86% of such
chords would be longer than one radius.
So we have coherent presentations for probabilities equal to 50%, 67%, 75%, and
86%!!
AND each of these answers can present a probability distribution that results
in that answer.
Let me offer an absurd answer, and ask what might be wrong with it: Consider
Hugo's second solution looked at another way. Here is a picture of a diameter
of the circle:
------------------------------------------------------------------------------------------------
Now here's a picture of the diameter broken at the points Hugo suggests:
a b
c
---------x--------------------------------------------------------------------------x---------
(Not meant to be an exact picture!!)
Now we take the points on the line as our universe, and ask what proportion of
these points are on segments 'a' and 'c'. Well, since there are an infinite
number of points on any segment of a line, there are an equal number on each of
the three segments, and therefore the odds of picking a point on any one of
them is one-third. Two thirds would be on either 'a' or 'c', therefore two
thirds would generate chords shorter than one radius. I think we'd all agree
this argument is false, but where is the flaw?
And, what does this have to do with Peirce and continuity?
Tom Anderson
> From: Hugo Fjelsted Alroe alroe[…]vip.cybercity.dk, on 7/1/97 9:03 AM:
> The tricky part in your challenge, Tom, is in the meaning of 'a randomly
> drawn chord'.
>
> My first solution went in the same way as Cathy's, taking a cord to be
> chosen by picking an arbitrary point on the circle, and then chosing another
> point on the perimeter of the circle randomly, the two points forming the
> endpoints of the chord. This way , using the fact that an equilateral
> triangle has angles of 60 degrees, the probability of a randomly drawn chord
> being longer than the radius of the circle is 120/180 = 2/3.
>
> But it is possible to take 'randomly drawn' as meaning something else.
> Imagine you are making chords with a ruler, chosing any random direction,
> you can slide the ruler across the circle and drawing a line randomly inside
> the area delimited by the perimeter of the circle, across a possible sliding
> distance of 2r. (Here 'randomly' refers to the diameter of the circle and
> not the perimeter as above.) The proportion of chords longer than r can be
> inferred again by using an equilateral triangle, with two radii as sides and
> the non-radii side placed parallel to the chosen direction of your ruler.
> (It is sufficient to look at one half of the circle.) Any chord made within
> this triangle will be longer than r, thus the proportion we are looking for
> is h/r, where h is the height of the triangle. h can be calculated using
> pythagoras rule: sqr(h) + sqr(r/2) = sqr(r) <=> h = squareroot(3)*r/2 (or
> simply as sin(60)*r), and the wanted probability is squareroot(3)/2 ~= 0.866
>
> Perhaps there are other possible meanings of 'randomly drawn'?
>
> Regards
> Hugo Fjelsted Alroe
>
>
>
>
------------------------------
Date: Tue, 1 Jul 1997 15:38:08 -0400 (EDT)
From: Gerald McCollam gerald[…]cns.nyu.edu
Subject: Re: Challenge: Random chord
Pythagorean Peirceans:
I have a solution, though I'm not sure it is the *right* solution.
Nevertheless, I propose that we take all proposed solutions and
seek their asymptote as the *best* solution, as any law-abiding
pragmaticist would do!
Here goes: Choose any of the infinitely many lines that run
perfectly tangent to the circle's perimeter. From this draw two more
lines, one at 30 degrees, the other at 150 degrees counter-clockwise into
the circle. From the origin draw a 4th and 5th line, each bisecting the
lines at -30 and -150 degrees.
If you've followed along you'll have something like the following:
\ |
/ \30|
/ \ |
/ \|
origin->.___r120|--> point of perfect perpendicularity to r.
\ /|
\ / |
\ / |
/ |the tangent line (imagine the circle to its left)
Thus, any point to the left of the top and bottom of the two triangles
along the perimeter of the circle is equally likely to be crossed roughly
2/3rds of the time, where this line will be greater than the radius (r).
However, since we are referencing ourselves against a 180 degree tangent
line, where 0 degrees and 180 degrees are uniformly a single line -- an
infinitely small trajectory line -- the actual probability will be closer
to 120/(180-1) or .67039.
signing off,
gerald
=====+======+======+======+======+======+======+======+======+======+=====
Gerald Mc Collam (212) 696-1476 home
Center for Neural Science (212) 998-3928 work
6 Washington Place, RM 809
New York, New York 10003 e-mail: gerald[…]cns.nyu.edu
=====+======+======+======+======+======+======+======+======+======+=====
------------------------------
Date: Tue, 1 Jul 1997 14:21:07 -0700
From: Joe Wheeler jwheeler[…]islandnet.com
Subject: Re: Challenge: Random chord
>I posted a challenge, and I'd like to say just a little more about my
>motivation in doing so. I got this problem from a very interesting Web page
>about mathematical pedagogy that urges exploration using fundamentals as a way
>of learning math.
Tom, where is this Web site? What is the URL?
Joe Wheeler, Victoria, B.C.
"Great spirits have always encountered violent opposition from mediocre
minds."
--Albert Einstein
------------------------------
Date: Wed, 2 Jul 97 04:01:44 UT
From: "Andrew Loewy" AndyLoewy[…]msn.com
Subject: RE: Challenge: Random chord
Tom,
I believe Cathy's solution to be the correct one...not because I came to the same
conclusion the same way (an Australian thing?) but because of the following...
A line "sweeps" around a point on the perimeter. When it sweeps from an angle
of 0 degs to 180 degs to the tangent at that point, it has described the complete set
of all chords at that point. That is, for every point on the perimeter, this sweep will
generate all possible chords that lie on this point. Let us ask ourselves, is the set
of chords forms by constructing lines perpendicular to an diameter equally
unconstrained (random). I think not. In doing the perpendicular-to-the-diameter
construction, the family of chords thus generated is constrained by its having to
be at exactly 90degs to the diameter. Now, here's the point....since it is the case
that for every point on the perimeter there is a unique diameter, we can in fact
compare in a one-to-one way the two families of chords generated in the two
ways just described. Since the "sweep" method is unconstrained and the
"diameter" method somewhat constrained, the sweep method gives the right
answer,
Andy.
------------------------------
Date: Wed, 2 Jul 1997 17:35:10 +1000 (EST)
From: Cathy Legg cathy[…]coombs.anu.edu.au
Subject: Re: Challenge: Random chord
On Tue, 1 Jul 1997, Tom Anderson wrote:
> And, what does this have to do with Peirce and continuity?
Continuity (as exemplified by the points on the perimeter of a
circle), has "room" for infinite multitudes in a number of different
proprtional relationships to each other...??? Or is that too vague?
I agree that this is intriguing...
Cathy.
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
Cathy Legg, School of Philosophy,
A14, University of Sydney,
Sydney, 2006.
We had fed the heart on fantasies,
The heart's grown brutal from the fare;
More Substance in our enmities
Than in our love; O honey-bees,
Come build in the empty house of the stare.
http://coombs.anu.edu.au/Depts/RSSS/Philosophy/People/Cathy/Cathy.html
}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
------------------------------
Date: Wed, 2 Jul 1997 17:36:21 +1000 (EST)
From: Cathy Legg cathy[…]coombs.anu.edu.au
Subject: Re: conception of "modern..."
On Tue, 1 Jul 1997, Antti Laato wrote:
> Infinitesimals put end the Cartesian understanding of continuity.
> Infinitesimals provide us a new way to understand the mathematical point.
> The point is like an infinite microspace and continuity from one point to
> another like a "trip" from one space to another. The problem is that we
> have our Cartesian models to understand continuity and therefore it may be
> difficult to understand Peirce's ideas about continuity.
Antti, how would you define "the Cartesian understanding of continuity"?
Cheers,
Cathy.
{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{{
Cathy Legg, School of Philosophy,
A14, University of Sydney,
Sydney, 2006.
We had fed the heart on fantasies,
The heart's grown brutal from the fare;
More Substance in our enmities
Than in our love; O honey-bees,
Come build in the empty house of the stare.
http://coombs.anu.edu.au/Depts/RSSS/Philosophy/People/Cathy/Cathy.html
}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}}
------------------------------
Date: Wed, 2 Jul 97 10:43:31 UT
From: "Andrew Loewy" AndyLoewy[…]msn.com
Subject: RE: Challenge: Random chord
>And can anyone think of a computer experiment using random numbers to test
one or all of the three theories -- or any other kind of experiment??
Tom,
It occurred to me that constructing a regular polygon with an even number (N)
of sides (and therefore vertices) allows one to calculate fairly easily the
number of "chords"...defined as lines joining vertices..that are obtained by
the 2 methods..sweep vs. diameter. The "winner"..ie the method generating more
"chords" will be apparent quickly, as will the proportion greater than the "radius".
This should work because a circle is just a regular polygon with an infinite
number of sides.
Andy.
------------------------------
Date: Wed, 2 Jul 97 12:02:10 UT
From: "Andrew Loewy" AndyLoewy[…]msn.com
Subject: Correction: chord challenge.
I'm wrong.
The number of "chords" drawable vertex-to-vertex in a regular even sided
polygon is the same when either method (sweep vs diameter) is used.
However its not hard to count the "chords" longer than the "radius"...these
approach 2/3 of the total as the number of sides of the polygon gets larger.
I will think about the difference in probabilty some more, but I gotta get
to work,
Andy.
------------------------------
Date: Wed, 2 Jul 1997 07:44:51 +0100
From: Tom Burke burke[…]sc.edu
Subject: RE: Challenge: Random chord
>And can anyone think of a computer experiment using random numbers to
>test one or all of the three theories -- or any other kind of experiment??
The fact that there seem to be several viable answers to the "random chord"
question reminds me of James's early example of "going around the squirrel"
as a good example of applying the pragmatic maxim (in the article, I
believe, where he first publicized "pragmatism" as a philosophical method).
In that example, each answer to the question is correct relative to how you
"practically" set up the situation to begin with. In the case of computing
probabilities of any kind, and more generally, of justifying probabalistic
inferences, what counts as a "correct" result depends essentially on what
kinds of "experimental" or "practical" methods you use to set up the
problem *to begin with*. This particular insight was stressed by Peirce
in several of the later articles in the 1877-78 -Popular Science- series
where he first introduced the pragmatic maxim. Cf. Houser and Kloesel,
_The Essential Peirce- vol.1, pages 169, 179, 193.
_______________________________________________________________________
Tom Burke URL: http://www.cla.sc.edu/phil/faculty/burket/
Department of Philosophy Phone: 803-777-3733
University of South Carolina Fax: 803-777-9178
------------------------------
Date: Tue, 1 Jul 1997 20:06:33 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord
>And what does this have to do with Peirce and continuity?
> " A true continuum is something whose possibilities of determination no
multitude of individuals can exhaust...Once you have embraced the
principle of continuity no kind of explanation of things will satisfy you
ecept that they grew."
> From _Philosophical writings of Peirce_ , edited by Justus Buchler
>"Is there no end to this michief?",
> anon.
Jim Piat
------------------------------
Date: Wed, 2 Jul 1997 12:33:03 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord
Joe,
This is the URL:
http://www.tufts.edu/%7Euwilensk/papers/paradox/lppp/msppp.html
You can find more information on the project Uri Wilensky reports on at this
URL:
http://el.www.media.mit.edu/groups/el/elprojects.html
Uri Wilensky discusses the use of computer program "StarLogo" to explore this
problem. If you follow the links from the "elprojects.html" you can download a
Macintosh version of that program. A PC version is in the works. You can also
find a PC version of LOGO. StarLogo is a LOGO that allows graphic manipulation
of a large population of 'turtles' -- Wilensky argues that this allows the
learner to develop intuitions about distributions in probability and statistics
in a nicely fundamental way.
Tom Anderson
> From: Joe Wheeler jwheeler[…]islandnet.com on 7/1/97 4:27 PM:
> >I posted a challenge, and I'd like to say just a little more about my
> >motivation in doing so. I got this problem from a very interesting Web page
> >about mathematical pedagogy that urges exploration using fundamentals as a way
> >of learning math.
>
> Tom, where is this Web site? What is the URL?
>
> Joe Wheeler, Victoria, B.C.
>
> "Great spirits have always encountered violent opposition from mediocre
> minds."
> --Albert Einstein
>
>
>
>
>
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--BeyondBoundary_1_Wed_Jul_02_12:33:10_1997__29--
------------------------------
Date: Wed, 2 Jul 1997 12:40:52 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: RE: Challenge: Random chord
Andy,
Can I ask you to re-consider? Or rather, can I ask you a different question:
Can you find SOMETHING that the 2/3 answer and at least one answer have in
common? That is, can you rephrase the question so that at least one other
answer appears to be the correct answer?
The trick, to me, surrounds the issue of defining a probability distribution
over an infinite space. I really like this problem because it forces one
exploring it to examine fundamental issues about probability over a
distribution -- or put another way, you have to look at probability as
depending on a specific distribution. Once the distribution question is
settled, the probability question is trivial.
Peirce shows himself at his teaching best in his 1883 "Theory of Probable
Inference". In that paper, he stresses the importance of predesignating a
hypothesis and a method of measurement prior to collecting data to answer an
inductive question. He gives some really telling examples of violating this
rule, and follows up some ad hoc pattern discovery by Playfair as Playfair
develops some theory about the relations among chemical weights. There are
some other issues going on here, but predesignation is relevant.
Tom
> From: "Andrew Loewy" AndyLoewy[…]msn.com on 7/1/97 11:30 PM:
> Tom,
> I believe Cathy's solution to be the correct one...not because I came to the
> same conclusion the same way (an Australian thing?) but because of the
> following...
> A line "sweeps" around a point on the perimeter. When it sweeps from an
> angle of 0 degs to 180 degs to the tangent at that point, it has described the
> complete set of all chords at that point. That is, for every point on the
> perimeter, this sweep will generate all possible chords that lie on this
> point.
> Let us ask ourselves, is the set of chords forms by constructing lines
> perpendicular to an diameter equally unconstrained (random). I think not.
> In doing the perpendicular-to-the-diameter construction, the family of chords
> thus generated is constrained by its having to be at exactly 90 degs to the diameter.
> Now, here's the point....since it is the case that for every point on the
> perimeter there is a unique diameter, we can in fact compare in a one-to-one
> way the two families of chords generated in the two ways just described.
> Since the "sweep" method is unconstrained and the "diameter" method somewhat
> constrained, the sweep method gives the right answer,
> Andy.
>
------------------------------
Date: Wed, 2 Jul 1997 14:38:32 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: RE: Challenge: Random chord
Andy Loewy wrote:
> I believe Cathy's solution to be the correct one...not because I came to the
> same conclusion the same way (an Australian thing?) but because of the
> following...
> A line "sweeps" around a point on the perimeter. When it sweeps from an
> angle of 0 degs to 180 degs to the tangent at that point, it has described the
> complete set of all chords at that point. That is, for every point on the
> perimeter, this sweep will generate all possible chords that lie on this
> point.
Andy, I believe that ALL of the methods people have submitted ALSO generate all
possible chords!! That's a very important point, because here you really get
up against the problem of sampling from an infinite universe.
Let's take my method that gives half the square root of three as the answer, or
roughly 87% odds of getting a chord longer than a radius. This method takes a
line tangent to the circle, and moves it down across the circle. The chords
generated are the distribution, and the odds of selecting any one of them are
the odds of selecting the point where the line is perpendicular to a diameter
line drawn from the original tangent point. To make that more precise,
consider the odds of selecting a particular point as equal to the odds of
selecting a finite segment of the line containing the point is divided into
some number of equal such segments -- and we can make these as arbitrarily
small as we like. Granted, this only generates SOME of the chords, but it
generates all possible LENGTHS of chords.
Now Jim Piat's solution -- 3/4 -- is just a two dimensional version of the one
that gives half the square root of 3!!
What Jim is doing is the same thing, but he's arguing: Hey! The RIGHT way to
do this is as you said, but you need to rotate the diameter around 360 degrees,
and generate a chord from each point, perpendicular to each diameter, in each
diameter, rather than just from one diameter. That requires that we look at
each point within the circle as a generating point, and the ratio chords longer
than a radius will depend on the size of the inner circle from which any chord
must be longer than a radius. That inner circle covers 3/4 of the area of the
whole circle.
> Let us ask ourselves, is the set of chords forms by constructing lines
> perpendicular to an diameter equally unconstrained (random). I think not.
> In doing the perpendicular-to-the-diameter construction, the family of chords
> thus generated is constrained by its having to be at exactly 90degs to the
> diameter.
> Now, here's the point....since it is the case that for every point on the
> perimeter there is a unique diameter, we can in fact compare in a one-to-one
> way the two families of chords generated in the two ways just described.
> Since the "sweep" method is unconstrained and the "diameter" method somewhat
> constrained, the sweep method gives the right answer,
You are right, but still the "diameter" method does give all possible LENGTHS
of chord, and you can overcome your objection by going to Jim's method of
taking in the other dimension. Still, all three methods still give ALL
possible LENGTHS of chords -- it's just that the histogram of them is a little
different in each case!! See the point?
Tom Anderson
------------------------------
Date: Wed, 2 Jul 1997 15:35:33 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: RE: Challenge: Random chord
Andy,
That's a good experiment -- and the result is, I predict, 2/3. But it is only
one way of viewing chords and their distribution. It's a good way, and I think
it shows a finite method of approaching an answer. The 'trick' is that if you
choose a DIFFERENT finite method, you approach a different answer. If you take
an n-gon, with large enough n, you'll gradually approach getting the same
values for chord lengths by taking parallel cuts, with each cut a definite
distance from the previous cut, going top down as you do by taking all
connections of all vertices -- BUT you'll get a different distribution of them.
If you adopted a different rule with the parallel cuts, and demanded that they
ONLY cut the vertices and NOT be an equal distance one from the other, then the
distribution, I predict, would be the same as yours.
Tom Anderson
> From: "Andrew Loewy" AndyLoewy[…]msn.com on 7/2/97 5:46 AM:
> >>And can anyone think of a computer experiment using random numbers to test
> one or all of the three theories -- or any other kind of experiment??
>
> Tom,
> It occurred to me that constructing a regular polygon with an even number
> (N) of sides (and therefore vertices) allows one to calculate fairly easily
> the number of "chords"...defined as lines joining vertices..that are obtained
> by the 2 methods..sweep vs. diameter. The "winner"..ie the method generating
> more "chords" will be apparent quickly, as will the proportion greater than
> the "radius". This should work because a circle is just a regular polygon
> with an infinite number of sides.
>
> Andy.
>
>
>
------------------------------
Date: Wed, 02 Jul 1997 22:55:51 +0200
From: Hugo Fjelsted Alroe alroe[…]vip.cybercity.dk
Subject: RE: Challenge: Random chord
Tom and Andy, just a minor point. I believe all the methods are capable of
generating not only all possible lengths of chords, but all possible chords.
Procedures that will do this can be these:
1/2: Pick a chord length at random on a 2r ruler. Then pick a direction on
the circle at random and draw the two chords paralel to this direction (or
one randomly).
2/3: Pick a point on the perimeter at random. Then pick another point at
random and draw the chord.
0.866: Pick a direction at random. Then pick a point at random on the
diameter paralel to this direction and draw the perpendicular chord.
3/4: Pick a point in the circle at random, then draw the chord that has this
point as midpoint (though this procedure is somewhat ambiguous as to the
chords through origon).
Hugo.
------------------------------
Date: Wed, 02 Jul 1997 22:31:45 +0200
From: Hugo Fjelsted Alroe alroe[…]vip.cybercity.dk
Subject: Re: Challenge: Random chord
Tom wrote:
>So we have coherent presentations for probabilities equal to 50%, 67%, 75%, and
>86%!!
>
>AND each of these answers can present a probability distribution that results
>in that answer.
>
>Let me offer an absurd answer, and ask what might be wrong with it: Consider
>Hugo's second solution looked at another way. Here is a picture of a diameter
>of the circle:
>
>---------------------------------------------------------------------------
---------------------
>
>Now here's a picture of the diameter broken at the points Hugo suggests:
>
> a b
c
>
>---------x-----------------------------------------------------------------
---------x---------
>
>(Not meant to be an exact picture!!)
>
>Now we take the points on the line as our universe, and ask what proportion of
>these points are on segments 'a' and 'c'. Well, since there are an infinite
>number of points on any segment of a line, there are an equal number on each of
>the three segments, and therefore the odds of picking a point on any one of
>them is one-third. Two thirds would be on either 'a' or 'c', therefore two
>thirds would generate chords shorter than one radius. I think we'd all agree
>this argument is false, but where is the flaw?
>
>And, what does this have to do with Peirce and continuity?
>
>Tom Anderson
>
Tom, here are my preliminary thoughts.
I think the problem in the argument above lies in the use of 'infinite'. The
inference from: 'there are an infinite number of points on each segment' to:
'there are an equal number of points on each segment' has several dubious
aspects. First, it seems to treat 'infinite' as a number, while I would take
infinite in: 'an infinite number of points on a line' to mean something like
inexhaustible. Formalistic mathematics might detest this interpretation
because of the implicit reference to reality, actions, time etc., a
reference which is natural in my own view of mathematics. But this
interpretation shows how the argument above, and in fact the whole of your
challenge, may be resolved.
The picking of points on the line segments being inexhaustible says nothing
on there being an equal number on each segment, in fact 'the number of
points on a line segment' has no meaning, given the continuous nature of the
line.
'Probability' and 'random' gain meaning by the same reference to reality,
actions, etc., 'probability' referring to the possibility of specified
actions or events, and 'random' specifying an even possibility.
We may use random both of continuous probability distributions and of
discrete, but it is the continuous case which may cause problems. In the
discrete case, the specification of 'random' comes naturally with the
description of the discrete outcomes; random meaning that each of the
(finite) outcomes are of equal probability. But the continuous case is more
tricky, as any point on a line can be said to have a possibility of zero.
The nature of continuity means that we cannot even talk of hitting a
predesignated point, there are no points on the line but the ones we mark
out. We are left with something like: 'dividing the line into equal
segments, no matter how small these segments are made, hitting them will be
equally probable'.
So, the argument on top asks a meaningless question in asking what
proportion of points are on segments 'a' and 'c', as there are no proportion
of points on the segments, points can only be picked out by some procedure
to be specified. And the inference it makes is wrong for the same reason.
As for the whole of your interesting challenge, there is nothing really
surprising in the number of different answers presented, given the view above.
Randomness and probability cannot be used in a meaningful way without
specifying the possible actions or events referred to. 'What is the
probability of Peter being at home?' we may ask, but the question has no
meaning unless we specify how the probing is to be done. If the probing is
done by checking the premises at noon every day (and Peter has a
conventional 9 to 5 job), we will get one probability, while checking at
midnight will give another. Though some might deny this, I find chords on a
circle to be quite the same; 'the probability of a chord' has no meaning if
not the becoming of chords are specified. And 'randomly drawn' is not a
sufficiently unambiguous specification, as chords can be drawn randomly in a
number of different ways.
As for a philosophical ground for the interpretation of probability and
randomnes given above, it can be found in the metaphysics of Aristotle and
Peirce. Probability and randomness are concepts describing 'potency' or 'the
possible', and we just have to use them accordingly.
Regards
Hugo Fjelsted Alroe
------------------------------
Date: Thu, 3 Jul 97 02:12:36 UT
From: "Andrew Loewy" AndyLoewy[…]msn.com
Subject: RE: Challenge: Random chord
Tom,
>Can I ask you to re-consider? Or rather, can I ask you a different question:
>Can you find SOMETHING that the 2/3 answer and at least one answer have in
>common? That is, can you rephrase the question so that at least one other
>answer appears to be the correct answer?
In thinking about the problem today i in fact HAD to do exactly what you
are asking. Specifically the diameter method of constructing chords kept
meeting every "objection" so that in fact the chance of choosing an
appropriate point on a line (diameter) was in fact a valid of framing the
question, "what is the probability that a randomly drawn chord is larger than
the diameter".
Now, of course, choosing an angle in the sweep method was also valid... but
more so...why?..because there are a lot more angles in an arc than real
numbers on a line segment. So more random chords are possible...and i take
random to mean, the most unconstrained.
Andy.
------------------------------
Date: Wed, 2 Jul 1997 21:53:47 -0500 (CDT)
From: tsander[…]ix.netcom.com (Thomas Anderson)
Subject: RE: Challenge: Random chord
Andy Loewy wrote:
> Tom,
>>>Can I ask you to re-consider? Or rather, can I ask you a different question:
>Can you find SOMETHING that the 2/3 answer and at least one answer have in
>common? That is, can you rephrase the question so that at least one other
>answer appears to be the correct answer?
> In thinking about the problem today i in fact HAD to do exactly what you
>are asking. Specifically the diameter method of constructing chords kept
>meeting every "objection" so that in fact the chance of choosing an
>appropriate point on a line (diameter) was in fact a valid of framing the
>question, "what is the probability that a randomly drawn chord is larger than
>the diameter".
> Now, of course, choosing an angle in the sweep method was also valid... but
>more so...why?..because there are a lot more angles in an arc than real
>numbers on a line segment. So more random chords are possible...and i take
>random to mean, the most unconstrained.
>
Well, part of the whole point is just "what is random?" and "how do you
define a distribution over an infinite set?"
I DO think that 2/3 and your method of finding it has many virtues --
but I disagree with the notion that there are more angles in an arc
than real numbers on a line segment. Cathy -- help us out here!! And
don't let Andy's national origin blind you!
What I wanted to say -- and I'll say it, but knowing that it's full of
holes -- was that you could map the angles in the arc to the reals on
the line segment. But I think part of what this problem displays is
that we DON'T have methods for such mapping, and if we want to map from
the one to the other, what we have to do is specify a method of
mapping, and our answer to what maps depends on our method of mapping.
Well, I guess we DO have methods, but the point is the question of
whether the mapping will be one to one or not cannot be answered
without specifying a particular method of mapping in advance.
Different methods are useful for different purposes.
Or, maybe, there are more angles in the arc -- but how do you count
them, Andy?
Tom Anderson
------------------------------
Date: Wed, 2 Jul 1997 22:06:38 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord
Tom,
I'll grant you that each method generates all possible lengths of chords
but I do not believe each method employs an unbiased sampling procedure
of the entire population of possible chords. When a chord is randomly
selected it must be selected from the entire population of chords. We
can't restrict the population to only those chords going through a point
on the perimeter or a point on a diameter even if these points were
randomly selected. For example we can't get an unbiased sample of the US
population by randomly selecting people from coastal towns or along major
highways. One must allow all possible chords to have an unrestricted
chance of being selected on any given selection so that the selection
will reflect the actual underlying distribution. It still seems to me
that the least restrictive sampling procedure which best approximates
this criterion is the one which yields the 3/4 solution. I think our
concern with the logic of partitioning the chords in each method has
obscured the fact that each of the proposed partitioning procedures is
also in fact a way of generating and selecting the chords. As you have
already pointed out the essential consideration is specifying the
population and underlying distributions to which I would emphasize
assuring that selecting the chords allows for unbiased sampling of the
full universe of chords. Consider the 2/3 solution. Granted its
specifies the underlying distribution of possible chords generated from a
particular point. Moreover it generates all possible lengths of chords.
And finally (and most misleading) it also generates sufficient chords to
cover every point covered by the circle. But it does NOT generate all
possible chords (of which there is an infinite number both greater and
less than R) because every point in the circle can have infinite chords
passing through it. So, the question becomes how do we best specify and
compare two infinite populations. I think some of the solutions have
rightly divided the wrong pie.
------------------------------
Date: Thu, 3 Jul 1997 10:04:29 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord
I'd like to offer a quote from Peirce about randomness, from REASONING AND THE
LOGIC OF THINGS, ed. Ketner, Cambridge, 1992: Harvard University Press,
starting on page 204:
". . . It is the operation of chance which produces the retardation of the
upper layer of air we were just considering; but surely it is no ignorance of
ours that has that effect. Chance, then, as an objective phenomenon, is a
property of a _distribution_. That is to say, there is a large collection
consisting say, of colored things and of white things. Chance is a particular
manner of distributions of color among all the things. But in order that this
phrase shall have any meaning, it must refer to some definite arrangement of
all the things.
"Let us begin by supposing that the multitude of colored things is _denumeral_,
and that that of the white things is likewise _denumeral_. The denumeral
multitude, as I explained in a former lecture, is that of all the whole
numbers. Every denumeral multitude may be numbered. That is, the number 1
may be affixed to one of its objects, 2 to another and so on in such a way
that every object of the collection receives a number. When that is done I
call the relation of an object receiving any number but 1 to the object
receiving the next lower number a _generating relation_ of the collection. It
is by no means indispensable to introduce any mention of numbers in defining a
generating relation. I do so for the sake of using ideas which which you are
familiar and thus save time and trouble. .
"Now in our collection of denumeral colored things and denumeral white things,
let F signify a particular generating relation, so that when the objects are
numbered according to that relation the object numbered n + 1 is F of the
object numbered n. Then, I say that a fortuitous distribution of color and
whiteness in the collection consists in this that any object of the collection
being colored or not is independent of it being an F of a colored thing, and is
also independent of its being an F of an F of colored things, and is also
independtend of its being at once an F of a colored thing and and F of an F of
a white thing; and in short that an object's being colored of not is
independent of its having or not having any character definable in terms of F,
color and whiteness. That satisfactorily defines a _fortuitous distribution_
when the colored things and white things are both denumeral.
" . . .
"If of the two subcollections, the colored things and the white things, one is
denumeral and the other is more than denumeral, we may still speak, and
sometimes do speak, of a fortuitous distribution. It is true that for a
collection more than denumeral there can be no generating relation. But still,
unless the total collection is a continuum of more than one dimension, with or
without topical [Peirce's word for topological] singularities, all the objects
in it may be placed in a sequence, at any rate by means of a relatively
insignificant multitude of ruptures and junctions. It must be understood that
the fortuitousness refers to the particular way in which the objects are placed
in sequence. It must furthermore be understood that by a definite mode the
whole sequence is broken up into a denumerla collection of subcollections and
the fortuitousness is further relative to that mode of breaking up. [And]
moreover this mode of dissection must be capable of a particular mode all at
once inclusive of less and less without limit, and fortuitousness is still
further relative to that mode of shrinking. If, then, no matter how small
these subcollections are taken the character of the subcollection having any
character definable in terms of the generating relation of the denumeral
collection, of containing a blue thing, and of not containing a colored thing,
then the distribution is fortuitous. For example, we may say that certain
marked points are fortuitiously distributed upon an infinitely long line,
meaning that if the line is cut up into a denumeral series of lengths, no
matter how small, the lengths containing marked points will be fortuitously
distributed along the whole series of lengths.
"We might speak of a finite number of points being fortuitously distributed
upon the circumference of a circle, meaning an approximate fortuitous
distribution. When we say that a finite number of points are distributed _at
random_ on the circumference, that is quite another matter. We then have in
mind a fortuitous distribution, it is true, but it is a fortuitous distribution
of the denumeral cases in which a man might in the course of all time. throw
points down upon the circumference."
What do people think? Do these ideas apply to our problem? I think they do.
Tom Anderson
------------------------------
Date: Thu, 3 Jul 1997 13:03:35 -0500 (CDT)
From: dkawecki[…]ix.netcom.com (David Kawecki )
Subject: Re: Challenge: Random chord
We are involved here in a narrative deconstruction of the integer and
aporia congenital to the Euclidean abstraction.
David
------------------------------
Date: Thu, 3 Jul 1997 17:23:36 -0400 (EDT)
From: Jeff Kasser jleek[…]umich.edu
Cc: John Gerald Devlin jdevlin[…]umich.edu
Subject: Re: Challenge: Random chord (fwd)
I haven't had a chance to give Tom's challenge the attention it deserves
(in part because I suspect I'd have little to contribute), but I have
forwarded most of the recent postings to a colleague who shares
Tom's interest in mathematical pedagogy, and he thought that the following
post might prove helpful. I will forward any replies to John, of course,
or he can be cc'd if you wish.
Bertrand's Paradox has taken on a life in the philosophy of science
literature, by the way, in ways that will be unsurprising to those of
y'all who have wrestled with the problem. The Paradox is usually deployed
to show how difficult it is to defend particular initial probability
distributions in accordance with the principle of indifference. In
general, there are many reasonable-sounding but incompatible ways of
describing states to which we might assign equal initial probability.
This probably isn't much news to many of you, but the problem with which
you are wrestling is of pretty general philosophical and methodological
interest.
Best,
Jeff
---------- Forwarded message ----------
Date: Wed, 2 Jul 97 14:39:48 -0400
From: John Devlin john[…]ghostrider.philosophy.lsa.umich.edu
Reply-To: jdevlin[…]umich.edu
To: Jeff Kasser jleek[…]umich.edu
Subject: Re: Challenge: Random chord (fwd)
Hello Jeff Kasser. Thanks for your note dated Tue, 1 Jul 1997:
Thought a little bit about the problem. I don't have an answer; not even a guess.
But I think the only way to tell if anyone is making progress is to think about it
from first principles. Otherwise you just have elaborate hunches and no way to test them.
Suppose you have a circle with radius 1. There are two ways to think about points
on the circle. If you graph it on a plane and center it at the point (0,0), you can think
of the points on the circle as ordered pairs (a,b) where a^2 + b^2 = 1. Alternatively,
you can pick a point, say the point (1,0), and measure the length of segments going
clockwise round the circumference, in which case the points would range
from 0 to 2 * pi. It turns out, I think, that you need to keep both in mind.
So we have
y
|
| B
_ |_
/ | \
| | |
------------------------------- x
C | | | A
\ | /
-|-
| D
|
Where the points A, B, C, and D on the circle are
A = (1,0), or 0 * pi (or 2* pi)
B = (0,1), or 1/2 * pi
C = (-1,0), or pi
D = (0, -1), or 3/2 * pi
depending on your choice of co-ordinates.
A chord is just a pair of points on the circumference of the circle -- for example,
the chord (A,B), the chord (A,C), the chord (A,D), etc. And the simplest way to
represent ordered pairs is on a plane. But to do that we need to think of chords
as *points* on some plane -- ie as ordered pairs of real numbers. This requires
a coordinate system that represents *each* point on the circle as a *single* real
number, which is what our second system of co-ordinates does. So draw a new
graph and let both the x' and y' axis represent points on the circumference
from 0 to 2 * pi. Now we have a square with area = 4 * pi^2.
Here's how it would look, roughly, with some points labelled ...
y'
|
|
|------------------------------------
2* pi | (B,A) (C,A) (D,A) (A,A) |
| |
| |
| (B,D) (C,D) (D,D) (A,D) |
| |
| |
pi | (B,C) (C,C) (D,C) (A,C) |
| |
| |
| (B,B) (C,B) (D,B) (A,B) |
| |
| |
------------------------------------------ x'
(A,A) (B,A) (C,A) (D,A) (A,A)
0 pi 2 * pi
Notice that we just get symmetry along the line x' = y' ... since the chord (A,B) is
just the chord (B,A). But within the triangular region bounded by y' = 0, x' = 2 * pi
and x' = y', every point on the plane represents a unique chord and every chord is
represented by a unique point on the plane. Let's call that region CHORD. Clearly
the area of CHORD = 2 * pi^2.
In all but discrete (finite) cases, we have to think about probability as a measure of
area. Here, we want to know the relative area of CHORD for which the length of a
chord (x',y') is greater than one. But to state that restriction, we have to revert back
to our original co-ordinate system and think of points on the circle as ordered pairs,
where the distance between two points (a,b) and (c,d) on the unit circle is the square
root of (a - c)^2 + (b - d)^2.
The bottom line is that we want to measure the ratio between the cumulative area of
egions in which chords (represented as single points on the x' y' plane) have the desired
property and the area of CHORD itself.
Of course, this just restates the problem. But I think it's enough to see whether
progress has been made. For example ...
> David Kawecki offers the simplest solution. His space of distribution is
> the possible range of values for a chord, all of which he suggests are
> equally likely. The largest chord is the diameter equal to 2 times the
> radius, and the smallest is one point with zero length, so half the
> time, a random chord ought to be larger than a radius.
You can't treat a problem like this with discrete (finite) methods. The likelihood
that an adult is 5' tall equals the likelihood that the same person is 1' tall. (In both
cases the probability is zero.) And similarly for all heights between 0 and 6'. But
that doesn't mean that the likelihood of an adult being less than 3' feet tall equals
the likelihood that an adult is between 3' and 6' tall ...
There is a real intuition behind this thought, but it's one that must be handled with
care. In fact, we relied on the same intuition when we made the implicit assumption
that the chances that a chord drawn at random would fall in a given region of CHORD
depend only on the area of that region ....
Or from Jim Piat ...
> So the probability of randomly doing so is proportional to the
> area of the smaller circle verses the area of the larger circle
> minus the area of the smaller circle. N'est pas?
I followed the argument right up to this point. But the force of the "so" is lost on
me. I understand the interest in measuring relative area, but these seem to be
the wrong regions to compare .... And if this is an indirect measure of the ratio
that interests us, we haven't been told why or how ...
Anyways ... this is as far as I could get with my fuzzy memories about
highschool math. Best,
---
John Devlin
Department of Philosophy
The University of Michigan
Ann Arbor, MI 48109 - 1003
------------------------------
Date: Thu, 3 Jul 1997 21:02:16 -0400 (EDT)
From: Everdell[…]aol.com
Subject: Re: RE: Challenge: Random chord
Andy Loewy writes: "there are a lot more angles in an arc than real numbers
on a line segment."
Are you sure, Andy. Neither set is "countably infinite." Both, I think,
have the cardinal Cantor called "C" - the cardinal of the continuum.
-Bill Everdell, Brooklyn
------------------------------
----------------------------------------------------------------------
Date: Sat, 5 Jul 97 02:58:49 UT
From: "Andrew Loewy" AndyLoewy[…]msn.com
Subject: RE: Challenge: Random chord
Tom,
i wrote my post intemperately.
I impulsively "saw" that the angles in an arc had to be more than there
were real numbers on a line because of the way the arms of the angles had to
fan out and how that had that to mean than there were tons more angles between
the arms of such an angle.
As i was driving to work, it hit me as to how wrong this view was...
take a line and have it "bite its tail" forming a circle. Every real number
point on that now circular line (perimeter) can be connected to the center of
the circle forming exactly that many angles ( ie.aleph1),
I guess, Tom, that people that dont 'eat this stuff for breakfast' can easily
fall into these sorts of impulsive traps..anyway, I'm sorry...hopefully it
allowed some of the readers to think through this stuff a little more and thus
be just that little more familiar with it. Still its a far cry from
expertise...
Andy.
------------------------------
Date: Sat, 5 Jul 97 03:04:37 UT
From: "Andrew Loewy" AndyLoewy[…]msn.com
Subject: RE: RE: Challenge: Random chord
My apologies, Bill, you are exactly right..the angles of an arc and the real
numbers on a line both equal aleph1...see my note to Tom.
------------------------------
Date: Sun, 6 Jul 1997 11:50:09 -0900 (PDT)
From: John Oller joller[…]unm.edu
Subject: Re: Challenge: Random chord
On Thu, 3 Jul 1997, Tom Anderson wrote:
> Date: Thu, 3 Jul 1997 08:39:10 -0500
> From: Tom Anderson tsander[…]ix.netcom.com
> To: Multiple recipients of list peirce-l[…]ttacs6.ttu.edu
> Subject: Re: Challenge: Random chord
>
> I'd like to offer a quote from Peirce about randomness, from REASONING AND THE
> LOGIC OF THINGS, ed. Ketner, Cambridge, 1992: Harvard University Press,
> starting on page 204:
>
> ". . . It is the operation of chance which produces the retardation of the
> upper layer of air we were just considering; but surely it is no ignorance of
> ours that has that effect. Chance, then, as an objective phenomenon, is a
> property of a _distribution_. That is to say, there is a large collection
> consisting say, of colored things and of white things. Chance is a particular
> manner of distributions of color among all the things. But in order that this
> phrase shall have any meaning, it must refer to some definite arrangement of
> all the things.
>
> "Let us begin by supposing that the multitude of colored things is _denumeral_,
> and that that of the white things is likewise _denumeral_. The denumeral
> multitude, as I explained in a former lecture, is that of all the whole
> numbers. Every denumeral multitude may be numbered. That is, the number 1
> may be affixed to one of its objects, 2 to another and so on in such a way
> that every object of the collection receives a number. When that is done I
> call the relation of an object receiving any number but 1 to the object
> receiving the next lower number a _generating relation_ of the collection. It
> is by no means indispensable to introduce any mention of numbers in defining a
> generating relation. I do so for the sake of using ideas which which you are
> familiar and thus save time and trouble. .
>
> "Now in our collection of denumeral colored things and denumeral white things,
> let F signify a particular generating relation, so that when the objects are
> numbered according to that relation the object numbered n + 1 is F of the
> object numbered n. Then, I say that a fortuitous distribution of color and
> whiteness in the collection consists in this that any object of the collection
> being colored or not is independent of it being an F of a colored thing, and is
> also independent of its being an F of an F of colored things, and is also
> independtend of its being at once an F of a colored thing and and F of an F of
> a white thing; and in short that an object's being colored of not is
> independent of its having or not having any character definable in terms of F,
> color and whiteness. That satisfactorily defines a _fortuitous distribution_
> when the colored things and white things are both denumeral.
>
> " . . .
>
> "If of the two subcollections, the colored things and the white things, one is
> denumeral and the other is more than denumeral, we may still speak, and
> sometimes do speak, of a fortuitous distribution. It is true that for a
> collection more than denumeral there can be no generating relation. But still,
> unless the total collection is a continuum of more than one dimension, with or
> without topical [Peirce's word for topological] singularities, all the objects
> in it may be placed in a sequence, at any rate by means of a relatively
> insignificant multitude of ruptures and junctions. It must be understood that
> the fortuitousness refers to the particular way in which the objects are placed
> in sequence. It must furthermore be understood that by a definite mode the
> whole sequence is broken up into a denumerla collection of subcollections and
> the fortuitousness is further relative to that mode of breaking up. [And]
> moreover this mode of dissection must be capable of a particular mode all at
> once inclusive of less and less without limit, and fortuitousness is still
> further relative to that mode of shrinking. If, then, no matter how small
> these subcollections are taken the character of the subcollection having any
> character definable in terms of the generating relation of the denumeral
> collection, of containing a blue thing, and of not containing a colored thing,
> then the distribution is fortuitous. For example, we may say that certain
> marked points are fortuitiously distributed upon an infinitely long line,
> meaning that if the line is cut up into a denumeral series of lengths, no
> matter how small, the lengths containing marked points will be fortuitously
> distributed along the whole series of lengths.
>
> "We might speak of a finite number of points being fortuitously distributed
> upon the circumference of a circle, meaning an approximate fortuitous
> distribution. When we say that a finite number of points are distributed _at
> random_ on the circumference, that is quite another matter. We then have in
> mind a fortuitous distribution, it is true, but it is a fortuitous distribution
> of the denumeral cases in which a man might in the course of all time. throw
> points down upon the circumference."
>
> What do people think? Do these ideas apply to our problem? I think they do.
>
> Tom Anderson
>
Yes, for what it is worth, I certainly think they do. Also, I note that
TNR-theory shows explicitly why Peirce's somewhat intuitive argument about
"pre-designation" must also be correct. Since no material particular (real or
imagined) can determine anything apart from an abstractive representation of
that particular (per the omega-perfection of TNRs; i.e., the third syntactic
perfection), Peirce's claim is explicitly justified and your extension to
the somewhat indeterminate problem you have posed is valid.
Cheers,
John Oller
********************************************************
John Oller Phone 505-277-7417 office
Department of Linguistics 505-856-6078 home
University of New Mexico Fax 505-277-6355
Albuquerque, NM 87131-1196 e-mail joller[…]unm.edu
********************************************************
P. S. My address will be changing shortly as I have accepted a new
appointment as Head of Communicative Disorders and Director of the Doris B.
Hawthorne Center for Communicative Disorders and Special Education at the
University of Southwestern Louisiana. Meantime, I will still continue to
receive e-mail at the above UNM address for the next several months. My
surface mail and other particulars, however, will change in the third
week of August, 1997 to
********************************************************
John Oller Phone 318-482-6721 office
Department of Communicative Disorders
University of Southwestern Louisiana Fax 318-482-6195
P.O. Box 3170
Lafayette, LA 70504-3170 e-mail joller[…]usl.edu
********************************************************
The USL address for e-mail should be valid in the last week of August,
1997.
------------------------------
Date: Mon, 07 Jul 1997 10:39:23 +0300
From: Antti Laato antti.laato[…]abo.fi
Subject: Re: conception of "modern..."
>Antti, how would you define "the Cartesian understanding of continuity"?
>
>Cheers,
>Cathy.
>
Dear Cathy!
It is an ordinary mathematical definition of continuity which you learn at
school. It is a discrete Cartesian mathematics which everyone learn but it
is only one possibility to define "continuity" and in the case of Peirce it
does not work very well.
Look further "Reasoning and the Logic of Things: The Cambridge Conferences
Lectures of 1898" (ed by K.L. Ketner) and its Introduction (by Ketner and
H. Putnam).
Best regards, Antti
Antti Laato, Dr (theology), Phil.Cand. (mathematics)
Senior Research Fellow, The Academy of Finland
The Department of the Biblical Studies
Abo Academy, Biskopsg. 16 Turku FIN-20500 Finland
e-mail: alaato[…]abo.fi
------------------------------
Date: Tue, 8 Jul 1997 05:34:36 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord
Hugo Alroe has pointed out, ""The tricky part in your challenge, Tom, is
in the meaning of 'a randomly drawn chord'.
What interests me here is the meaning of meaning. I take meaning to mean
the consequences of an action. Thus, meaning is not something intrinsic
to an entity but instead is something that refers to the relational
consequences between two or more entities. For example continuity is not
a property of lines, circles or areas but is the practical consequence of
a particular approach to dividing the line, circle or area. Moreover
continuity is not a property of "mind" as I think Peirce seems to be
saying but a property of a physical interaction between a subject and an
object. We call the subject's meaning it's intention and the object's
meaning its implication. In short, meaning is a property of a
relationship. Is this of any consequence?
Jim Piat
------------------------------
Date: Tue, 8 Jul 1997 11:25:35 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord (fwd)
Jeff Kasser wrote:
Jeff, this fascinates me. I'm very much an outsider, and know philosophy of
science only from sporadic forays over the years. I'd be very interested to
have some references to any philosophical discussion about Bertrand's paradox.
Uri Wilensky, who works with mathematics education and favors a method that
encourages and supports exploration, reports that typically, even very
well-trained and statistically sophisticated people have very poor intuitions
about distributions. He argues that this problem -- in spite of having no
clear 'correct' answer -- or because of that -- is an excellent vehicle for
developing such intuitions.
Personally, I've found it extremely helpful. My first intuition was to take a
line and run in down the circle in parallel movements, and to view the
distribution as a function of the distance along the perpendicular diameter, so
that half the square root of three would be the proportion of randomly selected
chords that were longer than a radius. I was amazed as I read on in Wilensky's
article to learn of alternative approaches -- and it almost immediately hit me
to ask: "What do all these approaches have in common?" Each offers a method
of generating chords that has implicit consequences for a view of the
distribution of chords -- so from an operational perspective, each gives a
defensible answer in the context of laying out a procedure for generating a
distribution. I found this amazing and fascinating! I like the problem
because it shows a nest of conceptual linkages, between randomness and
distribution. I also like the problem because of the way it plays with
emotions. Of course, people react in different ways, but I think everyone
shares at least a little bit of vulnerability in the situation when they
encounter a plausible account from someone else that differs from the way they
framed the problem. Exploring this set of emotions is an important way of
learning about how one learns, I believe.
Finally, I found Peirce's comments on the topic of drawing random samples to be
written almost as if he had this problem in mind. He was able to lay out in
very clear terms how the infinite quantity from a which a given discrete
distribution must be defined has nothing within it that will assist the seeker
in finding an appropriate distribution -- so that if you want to define a way
of drawing random samples of chords, you must specify the method you will use
to generate the distribution. The answer then depends on how the method is
specified, and more than one method is possible -- the answer is undetermined
until the method is specificied.
Thanks for your comment, and thanks also if you can give some references on
philosophical reflections on Bertrand's paradox.
Tom Anderson
> I haven't had a chance to give Tom's challenge the attention it deserves
> (in part because I suspect I'd have little to contribute), but I have
> forwarded most of the recent postings to a colleague who shares
> Tom's interest in mathematical pedagogy, and he thought that the following
> post might prove helpful. I will forward any replies to John, of course,
> or he can be cc'd if you wish.
>
> Bertrand's Paradox has taken on a life in the philosophy of science
> literature, by the way, in ways that will be unsurprising to those of
> y'all who have wrestled with the problem. The Paradox is usually deployed
> to show how difficult it is to defend particular initial probability
> distributions in accordance with the principle of indifference. In
> general, there are many reasonable-sounding but incompatible ways of
> describing states to which we might assign equal initial probability.
> This probably isn't much news to many of you, but the problem with which
> you are wrestling is of pretty general philosophical and methodological
> interest.
------------------------------
Date: Tue, 8 Jul 1997 11:29:08 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord
John Oller wrote:
> Yes, for what it is worth, I certainly think they do. Also, I note that
> TNR-theory shows explicitly why Peirce's somewhat intuitive argument about
> "pre-designation" must also be correct. Since no material particular (real
or
> imagined) can determine anything apart from an abstractive representation of
> that particular (per the omega-perfection of TNRs; i.e., the third
syntactic
> perfection), Peirce's claim is explicitly justified and your extension to
> the somewhat indeterminate problem you have posed is valid.
I appreciate your comment, John -- it's just this kind of distinction in Peirce
that I think playing with this problem helps to make clear.
Tom Anderson
------------------------------
Date: Tue, 8 Jul 1997 11:31:52 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: RE: Challenge: Random chord
> From: "Andrew Loewy" AndyLoewy[…]msn.com, on 7/4/97 10:01 PM:
> Tom,
> i wrote my post intemperately.
> I impulsively "saw" that the angles in an arc had to be more than there
> were real numbers on a line because of the way the arms of the angles had to
> fan out and how that had that to mean than there were tons more angles between
> the arms of such an angle.
> As i was driving to work, it hit me as to how wrong this view was...
> take a line and have it "bite its tail" forming a circle. Every real umber
> point on that now circular line (perimeter) can be connected to the center f
> the circle forming exactly that many angles ( ie.aleph1),
> I guess, Tom, that people that dont 'eat this stuff for breakfast' can asily
> fall into these sorts of impulsive traps..anyway, I'm sorry...hopefully it
> allowed some of the readers to think through this stuff a little more and thus
> be just that little more familiar with it. Still its a far cry from
> expertise...
> Andy.
But, Andy,
What I love about this problem is that anyone approaching it who keeps his eyes
and ears open rapidly comes the same point the experts come to! They may clear
out some weeds that the rest of us stumble on more quickly, but the problem
exposes some fundamentals that more structured problems hide from the learner
and the expert alike.
Distributions!! How are they generated??
------------------------------
Date: Tue, 8 Jul 1997 13:27:29 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Cc: John Devlin john[…]ghostrider.philosophy.lsa.umich.edu
Subject: Re: Challenge: Random chord (fwd)
Jeff Kasser posted a way of approaching the random chord problem from John
Devlin. John's approach has the merit of laying out 'from first principles' a
way of viewing the problem.
John, in the course of developing his approach, wrote:
> You can't treat a problem like this with discrete (finite) methods.
> The likelihood that an adult is 5' tall equals the likelihood that the
> same person is 1' tall. (In both cases the probability is zero.) And
> similarly for all heights between 0 and 6'. But that doesn't mean
> that the likelihood of an adult being less than 3' feet tall equals the
> likelihood that an adult is between 3' and 6' tall ...
This is a very good point -- and it properly focuses our attention on the
question of distribution. That is, if you ask, "What are the odds that a
randomly chosen adult will be 6 feet tall?" you are implicitly asking, "What is
the distribution of height among adults?"
John's method of laying out the possible chords via a coordinate system with 2
pi on one axis and 2 pi on the other axis, and reading a point as a code for a
chord from x to y, is a good way of displaying the problem. But viewing the
AREA of the 4 pi-squared space as a good representation of the distribution of
chords makes a very big assumption. When you do make that assumption, then you
come up via some simple geometry and arithmetic with the conclusion 2/3. But
it's very important to go slowly enough to see that you are making an
assumption about distribution. It's so easy to miss that, and thereby not give
appropriate credit to other solutions that differ only in projecting a
different distribution.
For example, pose the question this way, and you'll find that John's and Cathy
Legg's and Andy Loewy's answer doesn't work: "Draw a circle on the ground one
meter in radius, and take a package of needles exactly five centimeters long.
Stand about a pace and an half away from the circle, and throw a needle into
the circle. The only rule is that you have to get the needle inside the
circle. What are the odds that a chord described by extending a line in both
directions from the needle will be longer than one radius of the circle?"
I have an answer, and I'm working on a program to test my answer. John, Cathy,
and Andy -- can you see how you can reason -- following Jim Piat's logic --
that 2/3 isn't the right answer when you describe the distribution this way?
I'm not sure Jim's answer is correct for that method of selecting random
chords, either. I have a rough reason why it's not, but I need to work a
little more at it.
So what I'm suggesting is that inadvertently John made the same mistake as he's
pointing out using height of adults as an example.
Tom Anderson
> From: John Devlin john[…]ghostrider.philosophy.lsa.umich.edu
> Reply-To: jdevlin[…]umich.edu
> To: Jeff Kasser jleek[…]umich.edu
> Subject: Re: Challenge: Random chord (fwd)
>
> Hello Jeff Kasser. Thanks for your note dated Tue, 1 Jul 1997:
>
> Thought a little bit about the problem. I don't have an answer; not even a
guess. But I think the only way to tell if anyone is making progress is to
think about it from first principles. Otherwise you just have elaborate
hunches and no way to test them.
>
> Suppose you have a circle with radius 1. There are two ways to think about
points on the circle. If you graph it on a plane and center it at the point
(0,0), you can think of the points on the circle as ordered pairs (a,b) where
a^2 + b^2 = 1. Alternatively, you can pick a point, say the point (1,0), and
measure the length of segments going clockwise round the circumference, in
which case the points would range from 0 to 2 * pi. It turns out, I think,
that you need to keep both in mind.
>
> So we have
> y
> |
> | B
> _ |_
> / | \
> | | |
> ------------------------------- x
> C | | | A
> \ | /
> -|-
> | D
> |
>
> Where the points A, B, C, and D on the circle are
>
> A = (1,0), or 0 * pi (or 2* pi)
> B = (0,1), or 1/2 * pi
> C = (-1,0), or pi
> D = (0, -1), or 3/2 * pi
>
> depending on your choice of co-ordinates.
>
>
> A chord is just a pair of points on the circumference of the circle -- for
example, the chord (A,B), the chord (A,C), the chord (A,D), etc. And the
simplest way to represent ordered pairs is on a plane. But to do that we need
to think of chords as *points* on some plane -- ie as ordered pairs of real
numbers. This requires a coordinate system that represents *each* point on the
circle as a *single* real number, which is what our second system of
co-ordinates does. So draw a new graph and let both the x' and y' axis
represent points on the circumference from 0 to 2 * pi. Now we have a square
with area = 4 * pi^2.
>
> Here's how it would look, roughly, with some points labelled ...
>
> y'
>
> |
> |
> |------------------------------------
> 2* pi | (B,A) (C,A) (D,A) (A,A) |
> | |
> | |
> | (B,D) (C,D) (D,D) (A,D) |
> | |
> | |
> pi | (B,C) (C,C) (D,C) (A,C) |
> | |
> | |
> | (B,B) (C,B) (D,B) (A,B) |
> | |
> | |
> ------------------------------------------ x'
> (A,A) (B,A) (C,A) (D,A) (A,A)
> 0 pi 2 * pi
>
>
>
> Notice that we just get symmetry along the line x' = y' ... since the chord
(A,B) is just the chord (B,A). But within the triangular region bounded by y'
= 0, x' = 2 * pi and x' = y', every point on the plane represents a unique
chord and every chord is represented by a unique point on the plane. Let's
call that region CHORD. Clearly the area of CHORD = 2 * pi^2.
>
> In all but discrete (finite) cases, we have to think about probability as a
measure of area. Here, we want to know the relative area of CHORD for which
the length of a chord (x',y') is greater than one. But to state that
restriction, we have to revert back to our original co-ordinate system and
think of points on the circle as ordered pairs, where the distance between two
points (a,b) and (c,d) on the unit circle is the square root of (a - c)^2 + (b
- d)^2.
>
> The bottom line is that we want to measure the ratio between the cumulative
area of regions in which chords (represented as single points on the x' y'
plane) have the desired property and the area of CHORD itself.
>
> Of course, this just restates the problem. But I think it's enough to see
whether progress has been made. For example ...
>
> > David Kawecki offers the simplest solution. His space of distribution is
> > the possible range of values for a chord, all of which he suggests are
> > equally likely. The largest chord is the diameter equal to 2 times the
> > radius, and the smallest is one point with zero length, so half the
> > time, a random chord ought to be larger than a radius.
>
> You can't treat a problem like this with discrete (finite) methods. The
likelihood that an adult is 5' tall equals the likelihood that the same person
is 1' tall. (In both cases the probability is zero.) And similarly for all
heights between 0 and 6'. But that doesn't mean that the likelihood of an
adult being less than 3' feet tall equals the likelihood that an adult is
between 3' and 6' tall ...
>
> There is a real intuition behind this thought, but it's one that must be
handled with care. In fact, we relied on the same intuition when we made the
implicit assumption that the chances that a chord drawn at random would fall in
a given region of CHORD depend only on the area of that region ....
>
> Or from Jim Piat ...
>
> > So the probability of randomly doing so is proportional to the
> > area of the smaller circle verses the area of the larger circle
> > minus the area of the smaller circle. N'est pas?
>
> I followed the argument right up to this point. But the force of the "so"
is lost on me. I understand the interest in measuring relative area, but these
seem to be the wrong regions to compare .... And if this is an indirect
measure of the ratio that interests us, we haven't been told why or how ...
>
> Anyways ... this is as far as I could get with my fuzzy memories about
highschool math. Best,
>
> ---
> John Devlin
> Department of Philosophy
> The University of Michigan
> Ann Arbor, MI 48109 - 1003
>
>
>
>
>
------------------------------
------------------------------
Date: Tue, 8 Jul 1997 19:08:55 -0400 (EDT)
From: Jeff Kasser jleek[…]umich.edu
Subject: Re: Challenge: Random chord (fwd)
Tom and anyone else interested in applications of Bertrand's Paradox in
the philosophy of science:
A good place to start, I think, would be Colin Howson and Peter Urbach's
*Scientific Reasoning: The Bayesian Approach*. This is published by Open
Court and is in its second edition. I only have the first, and the
discussion of Bertrand's Paradox and other paradoxes afflicting the
principle of indifference is at pages 45ff of that edition. Howson and
Urbach cite to Neyman's *Lectures and Conferences on Mathematical
Statistics and Probability* (1952, published by the U.S. Dept. of
Agriculture) as a locus classicus. I first encountered the paradox in
Henry Kyburg's *Probability and Inductive Logic* when a faculty member
here let me borrow his copy; I'm pretty sure it's out of print.
I don't want to do any false advertising for the interest of these
discussions, though. If memory serves, Peirce's discussion of Laplace
criticizes the principle of indifference in an informal way that
anticipates the more formal discussions in the works cited above. And I
think that the interest of Bertrand's Paradox was largely a matter of
generalizing such results to the continuous domain. So I appreciate your
enthusiasm Tom, and have really enjoyed the discussion you have catalyzed,
but I hope I haven't suggested more than I can deliver.
Best,
Jeff
Jeff Kasser
Dept. of Philosophy
University of Michigan
jleek[…]umich.edu
------------------------------
Date: Tue, 8 Jul 1997 22:01:12 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord (fwd)
Tom,
Is it your point that the meaning of "randomly sampling an infinite
population" is ambiguous. And, if so, what exactly do you take to be the
nature of the ambiguity. Do you think that mathematically there is more
than one distribution of all possible instances of chords in a circle?
Or, are you saying that there is more than one way to randomly sample an
infinite distribution? Personally I think there is a most best way to
specify the underlying distribution (run every possible chord through
every possible point on or within the circle). But , given this
distribution, I can't figure out how to randomly sample this population
or how to calculate the percent greater than the radius. Or, am I still
missing the point. Thanks. BTW, I'm off to the store to buy some pick-up
sticks.
Jim Piat
------------------------------
Date: Wed, 9 Jul 1997 11:54:42 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord (fwd)
Jim Piat wrote:
> Is it your point that the meaning of "randomly sampling an infinite
> population" is ambiguous.
Vague -- not meaningless, but only calculable when some kind of operation is
specified. I got a couple of physical operations from a textbook last night.
Take a disk, say a foot in diameter, and take a big box, say twenty feet long,
four feet wide, and mark parallel lines one foot apart -- so you have nineteen
of them. Then throw the disk into the box. Where it falls, a line will define
a unique chord.
Another method: Fasten a disk to a surface at a point on the circumference.
Draw a tangent at that point on the surface, and spin the disk around the
point. Where it stops, a chord will be defined by the fixed point and the
line.
> And, if so, what exactly do you take to be the
> nature of the ambiguity.
Just what is the distribution in question? How is it defined, how is it
delimited? I know that the range of all possible chords is zero to 2, if the
radius is one unit. So I can create a distribution of chords by this method:
Define one point on the circumference as 0, and any other point as the
distance on the circumference measured clockwise from point 0, divided by pi,
so that the point directly opposite 0 would be 1, and the point halfway down
would be .5, and just before 0 going clockwise would be 1.999 . . .
Now define random chord in this manner: start at the origin, and using a
random number generator, generate a ray to the circumference using a random
angle from 0 to 360 degrees. Then, from that point draw a chord, in a
counter-clockwise direction exactly the length in units corresponding to that
point. Since we defined the point just opposite point 0 as point 1, if the
ray lands there, draw a chord in a counter-clockwise direction of one radius in
length. Thereby, we assign a unique and different chord to each point on the
circumference, and all possible chords will be hit. The odds of generating a
chord more than one radius long will be exactly 50%.
> Do you think that mathematically there is more
> than one distribution of all possible instances of chords in a circle?
Indeed I do!! You method assigned 3/4 of them to lengths longer than a radius.
The method I just described assigned 1/2 of them to that size. The first
method -- throwing the disk across some parallel lines -- describes half the
square root of two. The spinning method describes a distribution that give 2/3
of the chords longer than a radius. I'm still not sure -- my trig is rusty as
the nails on the Titanic -- about the distribution of chords given by throwing
a needle randomly onto a circle -- but I'm sure it's MORE than 3/4.
> Or, are you saying that there is more than one way to randomly sample an
> infinite distribution? Personally I think there is a most best way to
> specify the underlying distribution (run every possible chord through
> every possible point on or within the circle). But , given this
> distribution, I can't figure out how to randomly sample this population
> or how to calculate the percent greater than the radius. Or, am I still
> missing the point. Thanks. BTW, I'm off to the store to buy some pick-up
> sticks.
What do you think? I think the kicker is the fact that there are an
uncountably infinite number of chords -- and the very direct analysis of the
problem can be found, I believe, in the answer Peirce gives in the quote I
quoted from RLT, although I admit the point is put a little abstractly. I
don't know where it's put better, and I'd rather bet that no other commentator
on the problem connects the problem to the analysis of continuity.
One way to look at the problem is negative: Hey, damn it, this problem doesn't
specify enough what you are to do!! I think that Peirce would see that as both
an advantage and a disadvantage -- it's something that shows the power of
continuity. A continuous distribution is subject to varieties of further
re-distributions when you specify concrete methods of generating them. You CAN
say some worthwile things about various possible distributions without
specifying which one, and then go on to say specific things about particular
distributions once they are specified.
Incidentally, the probability textbook I found at home (my dad's legacy)
reported a German writer giving 6 different coherent plausible answers to the
problem!!
Jim -- I'm curious about your take on the emotions generated by working on this
problem? I think they are very important and interesting.
Tom Anderson
------------------------------
Date: Wed, 9 Jul 1997 20:17:08 -0400
From: piat[…]juno.com (Jim L Piat)
Subject: Re: Challenge: Random chord (fwd)
Tom Anderson asks about the emotions generated by working on the chord
problem. Speaking just for myself, I found the problem challenging,
stimulating, fun, puzzling (still do), surprising, frustrating, etc. All
the joys of trial and error. Likewise for the public exchange of ideas,
but here I should add that at times I grandiosely imagined that I had
come up with some unique insight only to later fear that I'd made a
terrible ass of myself. Grandiosity being the common element in both
cases. Overall it was fun and I appreciate the opportunity to have
participated.
Jim Piat
------------------------------
Date: Thu, 10 Jul 1997 12:32:06 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord (fwd)
Jeff Kasser wrote:
>
> A good place to start, I think, would be Colin Howson and Peter Urbach's
> *Scientific Reasoning: The Bayesian Approach*. This is published by Open
> Court and is in its second edition. I only have the first, and the
> discussion of Bertrand's Paradox and other paradoxes afflicting the
> principle of indifference is at pages 45ff of that edition. Howson and
> Urbach cite to Neyman's *Lectures and Conferences on Mathematical
> Statistics and Probability* (1952, published by the U.S. Dept. of
> Agriculture) as a locus classicus. I first encountered the paradox in
> Henry Kyburg's *Probability and Inductive Logic* when a faculty member
> here let me borrow his copy; I'm pretty sure it's out of print.
I appreciate and will track down the references. I've just been reading some
very interesting articles by Deborah Mayo, a philosopher of science at Virginia
Polytechnic, about Neyman-Pearson and Peirce. She examines Peirce's notion of
predesignation, about which he's remarkably clear in the 1883 'Theory of
probable inference', and equally clear but much more compressed in REASONING
AND THE LOGIC OF THINGS. She traces out how that notion is related to -- and
in many ways clarifies -- Neyman-Pearson ideas about confidence intervals and
related notions of statistical testing. Much of Pearson's concern was in using
the way experiments are designed to improve the value of the resulting
information, so that you can more rapidly make progress if you plan an
experiment well, which planning involves spelling out a prediction, determining
methods of data collection and statistical tests prior to carrying out the
investigation. While Bertrand's paradox isn't empirical in the same way that
the problems Peirce and Pearson discussed, I think the issues are the same, and
that predesignation is a way of settling the paradox. I also think that
Peirce's treatment of vagueness and continuity are very relevant to the
analysis of this problem, and allow one to preserve the way the problem is
originally put rather than simply rejecting it as ill-formed or badly put.
Rather, it's not put specifically enough to allow you calculate an answer, but
you still can say some things about the problem.
Mayo has one piece, "The test of experiment: C. S. Peirce and E. S. Pearson"
in E. Moore, ed., CHARLES S. PEIRCE AND THE PHILOSOPHY OF SCIENCE, 1993,
Tuscaloosa: U of Alabama Press, p. 161-174, and another "Did Pearson reject
the Neyman-Pearson philosophy of statistics?" in SYNTHESE, 1992, v.90,
pp233-262.
The paper I mentioned by Uri Wilensky has a number of references to Kolmogorov,
Savage and others about the Bertrand paradox, but I haven't looked them up.
Thanks very much,
Tom Anderson
----------------------------------------------------------------------
Date: Fri, 11 Jul 1997 16:54:10 -0700
From: Tom Anderson tsander[…]ix.netcom.com
Subject: Re: Challenge: Random chord
Jim Piat wrote:
> Hugo Alroe has pointed out, ""The tricky part in your challenge, Tom, is
> in the meaning of 'a randomly drawn chord'.
Yes! I'd like to report on the results of some computer experiments I ran,
using four different ways of specifying a randomly drawn chord.
>From seven runs of 4,000 chords.
Chord_A: Generate a segment of a radius, of random length. Generate a chord
perpendicular to the end of the segment.
Chord_B: Generate a segment of a radius, of random length, but adjust for
area. Generate a chord perpendicular to the end of the segment.
Chord_C: Generate a chord by selecting a random angle from a radius at the
circumference.
Chord_D: Generate a segment of a radius, adjust for area. Generate a chord by
selecting a random angle from the end of the segment.
. Average St. Deviation (of seven averages)
Chord_A 0.866431 0.005272
Chord_B 0.748187 0.006716
Chord_C 0.667631 0.00653
Chord_D 0.943379 0.003459
Just a comment on 'adjusting for area'. If you select a segment of a radius,
the points at the end of these will be disproportionately skewed to the center.
To compensate for that in experiments B and D, I first selected a random number
from 0 to 1 and multiplied it by pi, then generated the radius segment from
this by taking dividing that number by pi and taking the square root. I can
give more details on the math later.
The numbers reported as averages are the average of seven averages of 4000
random chords generated by these methods. The number represents the proportion
that are longer than one radius in length.
What do people think?
Tom Anderson
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