
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 'onepoint 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 honeybees, 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 nonradii 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 FIN20500 Finland email: 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 xx (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 onethird. 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 nonradii 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 lawabiding 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 counterclockwise 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/(1801) or .67039. signing off, gerald =====+======+======+======+======+======+======+======+======+======+===== Gerald Mc Collam (212) 6961476 home Center for Neural Science (212) 9983928 work 6 Washington Place, RM 809 New York, New York 10003 email: 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 perpendiculartothediameter 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 onetoone 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 honeybees, 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 honeybees, 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 vertextovertex 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 187778 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: 8037773733 University of South Carolina Fax: 8037779178  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 > > > > > BeyondBoundary_1_Wed_Jul_02_12:33:10_1997__29 ContentType: application/octetstream; name="ATTRIBS.BND" ContentTransferEncoding: Base64 ContentDisposition: attachment; filename="ATTRIBS.BND" QmV5b25kIFBhY2tlZCBBdHRyaWJ1dGVzAD4oABYAKAAAAAAAUmU6IENoYWxs ZW5nZTogIFJhbmRvbSBjaG9yZAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AHRzYW5kZXIAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA8Qk1TTVRQODY3ODcxMTE4MHRz YW5kZXIAQmV5b25kIFByb3ByaWV0YXJ5IERhdGEaAAAAAAQAAAAAAAAADwAo AAAAAAAAAAAAAAAAAAAAAAAAAAAAQ29udmVyc2F0aW9uIElkJzx2MDMxMDI4 MDFhZmRlMGIwMDIyMGFAWzE5OC41My4xNzUuMTddPgQAAAAAAAAAEAAJAAAA AAAAAAAAAAAAAAAAAAAAAAAATWVzc2FnZSBFbmNvZGluZwhJU08tODg1OQEA AAAAAAAAFQADAAAAAAAAAAAAAAAAAAAAAAAAAAAAVXNlIFByb3BvcnRpb25h bCBGb250AQABCQAAAAAAAAANACUAAAAAAAAAAAAAAAAAAAAAAAAAAABQcmV2 aW91cyBGcm9tJEpvZSBXaGVlbGVyIDxqd2hlZWxlckBpc2xhbmRuZXQuY29t Pg0AAAAAAAAACwBHAAAAAAAAAAAAAAAAAAAAAAAAAAAAUHJldmlvdXMgVG9F AJAPAQAJAP//NgAAAAI2ADVNdWx0aXBsZSByZWNpcGllbnRzIG9mIGxpc3Qg PHBlaXJjZS1sQHR0YWNzNi50dHUuZWR1Pg0AAAAAAAAACwBHAAAAAAAAAAAA AAAAAAAAAAAAAAAAT3JpZ2luYWwgdG9FAJAPAQAJAP//NgAAAAI2ADVNdWx0 aXBsZSByZWNpcGllbnRzIG9mIGxpc3QgPHBlaXJjZS1sQHR0YWNzNi50dHUu ZWR1PhEAAAAAAAAADQAGAAAAAAAAAAAAAAAAAAAAAAAAAAAAT3JpZ2luYWwg dGV4dAAAAAAAABEAAAAAAAAABAC7BgAAAAAAAAAAAAAAAAAAAAAAAAAAVGV4 dBAFSm9lLAoKVGhpcyBpcyB0aGUgVVJMOiAgaHR0cDovL3d3dy50dWZ0cy5l ZHUvJTdFdXdpbGVuc2svcGFwZXJzL3BhcmFkb3gvbHBwcC9tc3BwcC5odG1s CgpZb3UgY2FuIGZpbmQgbW9yZSBpbmZvcm1hdGlvbiBvbiB0aGUgcHJvamVj dCBVcmkgV2lsZW5za3kgcmVwb3J0cyBvbiBhdCB0aGlzIFVSTDoKCmh0dHA6 Ly9lbC53d3cubWVkaWEubWl0LmVkdS9ncm91cHMvZWwvZWxwcm9qZWN0cy5o dG1sCgpVcmkgV2lsZW5za3kgZGlzY3Vzc2VzIHRoZSB1c2Ugb2YgY29tcHV0 ZXIgcHJvZ3JhbSAiU3RhckxvZ28iIHRvIGV4cGxvcmUgdGhpcyBwcm9ibGVt LiAgSWYgeW91IGZvbGxvdyB0aGUgbGlua3MgZnJvbSB0aGUgImVscHJvamVj dHMuaHRtbCIgeW91IGNhbiBkb3dubG9hZCBhIE1hY2ludG9zaCB2ZXJzaW9u IG9mIHRoYXQgcHJvZ3JhbS4gIEEgUEMgdmVyc2lvbiBpcyBpbiB0aGUgd29y a3MuICBZb3UgY2FuIGFsc28gZmluZCBhIFBDIHZlcnNpb24gb2YgTE9HTy4g IFN0YXJMb2dvIGlzIGEgTE9HTyB0aGF0IGFsbG93cyBncmFwaGljIG1hbmlw dWxhdGlvbiBvZiBhIGxhcmdlIHBvcHVsYXRpb24gb2YgJ3R1cnRsZXMnIC0t IFdpbGVuc2t5IGFyZ3VlcyB0aGF0IHRoaXMgYWxsb3dzIHRoZSBsZWFybmVy IHRvIGRldmVsb3AgaW50dWl0aW9ucyBhYm91dCBkaXN0cmlidXRpb25zIGlu IHByb2JhYmlsaXR5IGFuZCBzdGF0aXN0aWNzIGluIGEgbmljZWx5IGZ1bmRh bWVudGFsIHdheS4gIAoKVG9tIEFuZGVyc29uCgo+ICBGcm9tOiBKb2UgV2hl ZWxlciA8andoZWVsZXJAaXNsYW5kbmV0LmNvbT4sIG9uIDcvMS85NyA0OjI3 IFBNOgo+ICA+SSBwb3N0ZWQgYSBjaGFsbGVuZ2UsIGFuZCBJJ2QgbGlrZSB0 byBzYXkganVzdCBhIGxpdHRsZSBtb3JlIGFib3V0IG15Cj4gID5tb3RpdmF0 aW9uIGluIGRvaW5nIHNvLiAgSSBnb3QgdGhpcyBwcm9ibGVtIGZyb20gYSB2 ZXJ5IGludGVyZXN0aW5nIFdlYiBwYWdlCj4gID5hYm91dCBtYXRoZW1hdGlj YWwgcGVkYWdvZ3kgdGhhdCB1cmdlcyBleHBsb3JhdGlvbiB1c2luZyBmdW5k YW1lbnRhbHMgYXMgYSB3YXkKPiAgPm9mIGxlYXJuaW5nIG1hdGguCj4gIAo+ ICBUb20sIHdoZXJlIGlzIHRoaXMgV2ViIHNpdGU/IFdoYXQgaXMgdGhlIFVS TD8KPiAgCj4gIEpvZSBXaGVlbGVyLCBWaWN0b3JpYSwgQi5DLgo+ICAKPiAg IkdyZWF0IHNwaXJpdHMgaGF2ZSBhbHdheXMgZW5jb3VudGVyZWQgdmlvbGVu dCBvcHBvc2l0aW9uIGZyb20gbWVkaW9jcmUKPiAgbWluZHMuIgo+ICAgICAg ICAtLUFsYmVydCBFaW5zdGVpbgo+ICAKPiAgCj4gIAo+ICAKPiAgpQEDABAF ngEFAAIAAAAeAAEAAQABABgAAAAAAAAAAgAZAEAAAAAAAAAAAQBZAFIAAAAA AAAAAgCrADgAAAAAAAAAAQDjAC4EAAAAAAAAOP8AAAAAAAAAAAAAAAAAAAAA TVMgU2FucyBTZXJpZgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAAOP8AAAAAAAC8 AgABAAAAAAAATVMgU2FucyBTZXJpZgAAAAAAAAAAAAAAAAAAAAAAAAAAAAAA AQABAAUAAQAGAAYAAQAHAFgAAQBZAFkAAQBaAKsAAQCsAKwAAQCtAOIAAQDj AOMAAQDkANsCAQDcAtwCAQDdAukCAQDqAuoCAQDrAiwDAQAtA3YDAQB3A8cD AQDIAxoEAQAbBDAEAQAxBDQEAQA1BGUEAQBmBGkEAQBqBIgEAQCJBIwEAQCN BNcEAQDYBOIEAQDjBP0EAQD+BAEFAQACBQUFAQAGBQkFAQAKBQ0FAQAOBREF AAAAAAAAAAAAAAAAZAABpAEBSAMB7AQBkAYBNAgB2AkBfAsBIA0BxA4BaBAB DBIBsBMBVBUB+BYAAAAAAAAAAAAAABAABgAAAAAAAAAAAAAAAAAAAAAAAAAA AEF0dGFjaG1lbnQgQ291bnQEAAAAAAA= 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 reconsider? 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 perpendiculartothediameter 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 onetoone > 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 perpendiculartothediameter 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 onetoone > 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 ngon, 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 onethird. 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 reconsider? 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 reconsider? 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 reasonablesounding 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 ReplyTo: 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 coordinates. 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 coordinates 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 coordinate 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 peircel[…]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 TNRtheory shows explicitly why Peirce's somewhat intuitive argument about "predesignation" must also be correct. Since no material particular (real or imagined) can determine anything apart from an abstractive representation of that particular (per the omegaperfection 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 5052777417 office Department of Linguistics 5058566078 home University of New Mexico Fax 5052776355 Albuquerque, NM 871311196 email 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 email 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 3184826721 office Department of Communicative Disorders University of Southwestern Louisiana Fax 3184826195 P.O. Box 3170 Lafayette, LA 705043170 email joller[…]usl.edu ******************************************************** The USL address for email 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 FIN20500 Finland email: 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 welltrained 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 reasonablesounding 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 > TNRtheory shows explicitly why Peirce's somewhat intuitive argument about > "predesignation" must also be correct. Since no material particular (real or > imagined) can determine anything apart from an abstractive representation of > that particular (per the omegaperfection 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 pisquared 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 > ReplyTo: 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 coordinates. > > > 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 coordinates 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 coordinate 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 pickup 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 counterclockwise 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 counterclockwise 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 pickup > 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 redistributions 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 NeymanPearson 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  NeymanPearson 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 illformed 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. 161174, and another "Did Pearson reject the NeymanPearson philosophy of statistics?" in SYNTHESE, 1992, v.90, pp233262. 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