Friday, 30 November 2012

Morality Behind the Wheel

I was behind the wheel again one recent morning and travelling along a section of dual carriageway near to home when two birds flew directly towards me at about grill height.

As far as I know, I hit the birds at around 100km/hr, killing them instantly.

This little event reminded me of a couple of things, the injured parrot that my father brought home one day (a sweet but otherwise irrelevant memory) and the first time I hit something substantial with a car – a kangaroo.

Hitting the kangaroo was pretty devastating, not only because the car was important to me at the time but also because rather than killing the kangaroo outright, the collision merely shattered both its legs.  After some initial flustering, I dispatched the kangaroo with a handy stick, bemoaned the damage that it did and got on my way.  Since then, I’ve hit another kangaroo, plenty of rabbits, a bluetongue or two, a deer, quite a few birds and about a bazillion insects.  And a taxi (I didn’t kill the taxi).

The thing that struck me as I pondered striking the birds was that of all the things I had struck, that first kangaroo affected me the most.  The remainder were little more than inconveniences.

But even when I think about my unfortunate interaction with that first kangaroo, my primary concern was ending its suffering – the fact that I had caused its death was a very minor concern.  The stupid thing did jump right out in front of me and I did try to avoid it to the best of my ability.  Note that I distinguish between being a party to causing its death, which I did when I hit it with my car, and killing it, which I did when I broke its neck with a stick.  I intentionally killed the kangaroo in order to prevent a lingering death that I had unintentionally caused.  It wasn’t the death that affected me so much as the suffering, which I brought to an end as quickly as I could.

Nowadays, whenever I see a dead kangaroo on the side of the road, if I think about it at all my thought will likely be something along the lines of “Oh, a dead kangaroo, better not drive over that”.

When I see a dead cat or dog beside the road, my reaction is quite different.

As an atheist, this may appear somewhat strange.  A dead animal is a dead animal, the species shouldn’t make much of a difference, but of course it does.

Part of the reason why I react differently is that I have pets myself, and so when I see a dead pet by the road, I can imagine one of my pets there and thus feel a vicarious sense of loss.  Plus, I have had pets die so the sight triggers memories of past losses.

This potential for emotion is amplified immensely if one considers what it would be like to hit a child.

To be honest, during my pondering, I indulged my rather black sense of humour with an attempt to come up with a joke along the lines of:

The first time I ran over a child, it was devastating … Active Suspension now comes standard with the 2013 Chevrolet Suburban – ‘No More Bumps in the Road’

or:

Friend – “It must be devastating to run over a child”

Me – “Yep, it’s pretty bad the first couple of times”

The point of this black humour, of course, is that for any normal person it would be absolutely devastating to hit a child and, at least when one tries to predict the experience, running over a child would be devastating every single time.  The nonchalance that one gains with respect to running over random wildlife, or even pets for the more hardhearted among us, doesn’t seem to apply to running over children.

Those who think humans are something totally separate from the animals might not be happy about it, but I think that this inability to normalise the death of children is related to our tendency to mourn the loss of someone else’s pet, even if we know neither the animal nor their owner.

The scale of the emotional power associated with the death of a child derives from the huge personal (and genetic) investment that K-selecting humans have in their children together with the fact that the vast majority of us ourselves have been children at some stage.  In the awful circumstances of seeing child killed, we can conceptually put ourselves in the position of the parents (modern day Western parents are, on average, likely to have slightly less than two children and a lot less likely to have witnessed infant mortality than in the past).  But not only that, we can also put ourselves in the position of the child.  The death of another, be it a child or any other human, is an uncomfortable reminder of our own mortality.  Death by some other means is also horrible, but a random death such as in a car accident is disturbingly democratic – it could happen to any one of us.

Despite the attempts of people like D’Souza to claim otherwise, uttering “Dead kid, well it’s not my kid, so why should I care?” is not the normal reaction of an atheist, irrespective of how devoted she is to the idea that we evolved rather than being created.  It’s the reaction of a psychopath.

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I have witnessed the catastrophe of infant mortality from a closer vantage point than is comfortable.  I can tell you that there are few things more heart-breaking than watching grieving parents bear an infant’s coffin towards a grave.

I should also point out that in this article my focus was intended to be more on inadvertently causing death, not on causing suffering.  I do understand that when we strike an animal with our cars, it is entirely likely that the death we cause will not be instantaneous.  We can also imagine a moment of terror that comes with some level of realisation that death is looming in the form of a metallic box on wheels.

When discussing death, we sometimes reassure ourselves with the idea that death came quickly: “At least she didn’t suffer”.

It seems that, in a sense, we generally consider the causing of suffering to be a greater wrong than the causing of death.  Again, the idea that an atheist may be oblivious to the suffering of another animal, be it human or other, a pet or a wild animal, is nonsense.

Friday, 23 November 2012

When I spake as a child

When I was a child, I spake as a child, I understood as a child, I thought as a child: but when I became a man, I put away childish things.

So sayeth Paul in his first epistle to the Corinthians.

What does it mean to put away childish things, specifically childish understanding and childish thoughts?

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First, we might want to identify childish understanding and childish thoughts.  There’s some fabulous work done by Rebecca Saxe on the development of children’s understanding of the motivations of others.

Basically, up until a certain age, children at not able to clearly distinguish between what they themselves know and what others might know.  A young child’s cognitive boundaries are so vaguely defined that when a child witnesses an event, it is not immediately obvious to that child that other people might not have knowledge of that event – even if they were not present when the event occurred.  In other words, the fundamental assumption of a young child is “what I know, you know”.

Think about that for a moment.

At that stage of development, a child thinks that other people are effectively omniscient.

Fortunately, you don’t have to wait until adulthood to put that particular childish thing aside.

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I was deep in conversation with a six year old child a few years ago and the subject of death came up.  The exchange went a little like this:

neopolitan (neo) – What do you think happens when you die?

Little Batman Fan (LBF) – Batman comes and takes you to hospital.

neo – Ah.  You do know that Batman’s not real, don’t you?

LBF – Of course I know Batman’s not real!

neo – Good.

LBF – Bruce Wayne’s real.

neo – I see what you mean.  What happens when Batman takes you to hospital?

LBF – They make you better.

At this stage of his development, this particular child thought that death was fixable.  (I should point out that this child was never under any illusion that death could be avoided, having been told from the very first time the issue was raised that everybody dies.)

Fortunately, the child did not need to wait until adulthood to put that childish thing aside.

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I never claimed that my father could beat up other fathers.  He probably could, of course, but the topic never came up.  I don’t quite recall the moment when I realised that, despite being at least partly responsible for my creation (pending DNA tests), my father was somewhat short of perfect.  It may have been when we were all cowering behind a wall because he was doing a little experiment involving a largely empty petrol can and a bonfire.

The point is, there was a point in time when my father was pretty amazing – he provided clothing and food for his family, he built a house in front of our very eyes, he caught lizards and served them up for lunch (although he had failed to kill, skin and cook said lizards, which invariably escaped before being served), he could control this enormous machine to transport us around and if it broke down for any reason he could fix it, he set up and filled a swimming pool one Xmas Eve without any of us noticing until the next day, occasionally he could even exert some small measure of influence on my mother … he could do anything if he put his mind to it (within reason, of course). 

Then, later on, it came to my attention that he was not really omnipotent.  So I put aside my childish, unquestioning adoration of someone who was entirely human.

(Unlike some other adult children, I accept that this adoration was my doing, not something imposed on me.  The fact that my unrealistic expectations were not maintained by either parent is hardly something that I can hold them accountable for.)

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Children are exceedingly, and often excruciatingly, skilled at asking questions.  They ask a lot of questions.

What they aren’t quite as good at is answering questions or working out whether a question has an answer or not. 

One day, I asked the LBF (aged four) whether he’d ever seen a tree as large as the Giant Sequoia we were standing under.  His immediate answer, without any pause, was “Yes.”  He also accepted responsibility for invading Poland.  Many children seem to go through a stage of answering every question with the same answer.  For the LBF it was “yes”.  For another it was “don’t know”.  Perhaps for others it is “no”.  Again, answering all questions with the single answer is a thing that children usually put aside quite quickly.

The same applies for poorly thought-out questions.  What, I hear you ask, is a poorly thought-out question?  A poorly thought-out question is one which is inadvertently directing the answer or the type of answer, or one to which there simply is no answer, or one that fails to provide enough background information to formulate an answer.

Note that a poorly thought-out question is not necessarily a stupid question.  Children can be asking a question that seems totally reasonable from within their restricted world view.  For example:

Say it’s been raining and a child notices that the trees are wet and asks:

·         Who put all the water on the trees?

o   This is not necessarily a silly question.  The child may have previously observed a parent hosing down the car and observed that after the hosing down process, the car was wet.  While engaging in idle chit-chat with its parent, the child may have asked “Why are you putting water on the car?”  The parent may have reinterpreted the question as meaning “Why are you hosing down the car?” and answered with “Because we are going to visit some friends.”  Note that the parent has now answered a mechanism question (what function does hosing down the car with water serve) with a motivation answer (for what reason are you putting perfectly good drinking water on the car).

§  (A more reasonable exchange could go like this “Why are you putting water on the car?” “Because I’m cleaning the car” “Why?”  “Because I want the car to be clean.” “Why?” “Because I want our friends to be impressed with how clean our car is when we visit them.” “Why?” “Because adults are sad and pathetic.”)

o   If the parent answers with “the clouds” this leads to two related problems.  First, the clouds didn’t put water on the trees, so the answer is wrong.  Second, the answer follows the child’s lead by implying agency and this allows the child, quite reasonably, to ask “Why?”  This is a smart question being asked by an ignorant person of a thoughtless person.

o   A smarter, loving parent would treat this as a learning experience and ask “What do you mean?” to which the child will either rephrase the question (perhaps with “Why are the trees wet?” to which the parent can answer “Because it’s been raining.”) or re-ask the same question.  At this point the parent can make clear to the child that there was something wrong with the question – “Nobody put water on the trees, they’re wet because it’s been raining.”

The child could also ask:

·         Why do the trees want to be wet?

o   Again, this is not necessarily a silly question.  On warm days, humans quite often want to get wet, either going swimming or spraying water on themselves.

o   Parents can easily lead their children astray here by responding to the question as if it were valid:

§  “They want to get cool.”

§  “They drink water through their leaves.”

§  “They didn’t have any choice.  It was the clouds that did it.”

Or:

·         Which is better, a wet tree or a dry cactus?

o   This is starting to get a bit silly, but from the viewpoint of a child, it’s sometimes difficult to know what things can and cannot be validly compared.

These three questions represent the types of “problem” questions that children might ask:

·         questions that assume that there is an unknown agent behind natural phenomena

·         questions that assume agency in things that don’t have agency, and

·         questions based on comparison of things which cannot be validly compared

Now I don’t want to imply that asking questions is bad – far from it.  What I am saying is that children put aside asking poorly thought-out questions.  Well thought-out questions should continue to be asked.  The childish thing that should be put aside here is the acceptance of simple answers to complex questions.

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When I was a child, I believed that there was an omniscient being (or two).

When I was a child, I believed that there was an omnipotent being (or two).
 
When I was a child, I believed that death was fixable.

When I was a child, I gave the same simple answer to a lot of complex questions.

When I was a child, I believed that there were unknown agents or an unknown agent behind natural phenomena.

When I was a child, I believed that inanimate things had agency.

When I was a child, I compared things that could not validly be compared.

When I was a child, I accepted simple answers to complex questions.

Now that I am grown, I have put aside these childish things.

Shouldn’t we all?

Sunday, 18 November 2012

A Farewell to the Bertrand Paradox


Okay, this will be the last one on the Bertrand Paradox.

I really only intended it to be a bit of light-hearted stab at William Lane Craig and his misuse of infinity.  I didn’t intend getting drawn into the nitty-gritty.  Anyways …

My parting shot is an attempt to state my position.  I apparently fall into the smaller of two schools of thought on the paradox, that the probability in question is determined at p=0.5 rather than being fundamentally undetermined.  If you don’t know what I am talking about now, and are vaguely interested, you have to go back to where I started three posts ago.

My argument for p=0.5, developed after the work reflected in recent post, is not the standard argument – as far as I can tell.  So, I will detail what that argument is here, as briefly as I can, then I’ll leave it at that.

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Fundamentally the argument is that if you restrict the options for selecting a random chord, you are less likely to get a properly random distribution of chords and it is this that we see manifested as the “paradox”.  The method for getting the best, most random distribution (I believe) is to follow this procedure for determining a chord for a given circle:

My Method

1.           Select a random point.

2.           Select a second random point.

3.           Does the line defined by those two points intersect a circle?

NO – return to step 1.

YES – continue.

4.           You now have two points on the circle, between which is a chord (or a vanishingly small chance of a tangent).

The three classic methods of determining chords are as follows (using the order in which they appear in the Wikipedia entry):

Classic Method 1:

1.           Select a random point on the circumference on the circle.

2.           Select a second random point on the circumference on the circle.

3.           Does the line defined by those two points intersect a circle?

NO – return to step 1.

YES – continue.

4.           You now have two points on the circle, between which is a chord (or a vanishingly small chance of a tangent).

Classic Method 2:

1.               Select a random point on the circumference on the circle.

1a.         Draw a line segment between the random point and the locus of the circle.

2.               Select a second random point on the radius just drawn.

2a.         Draw a line at the second random point, perpendicular to the radius.

3.           Does the second line drawn intersect a circle?

NO – return to step 1.

YES – continue.

4.           You now have two points on the circle, between which is a chord (or a vanishingly small chance of a tangent).

Classic Method 3:

1.           Select a random point inside the circle.

1a.        Draw the line defined by that point and the locus of the circle.

1b.        Draw the line perpendicular to the line just drawn at the random point.

2.           Select a second random point one of the points intersecting with the circle as the second point.

3.           Does the line defined by those two points intersect a circle?

NO – return to step 1.

YES – continue.

4.           You now have two points on the circle, between which is a chord (or a vanishingly small chance of a tangent).

Now, if you are an average person, this will mean little or nothing to you, but if you are someone who grapples with philosophy of mathematics, or otherwise has an interest in the Bertrand Paradox, you might notice that in all four methods (mine and the three classic methods), you end up with two points defining a chord in Step 4.  But in the three classic methods, you have restrictions on your selection of points and, in two methods, there are extra steps.

I am fully cognisant of the fact that you won’t get a chord every single time with my method, in fact almost all of the lines you draw won’t end up defining a chord – if your selection is sufficiently random.  (In my simulation, with a limit on points as follows: (-100.r,-100.r)<=(x,y)<=(100.r,100.r), 99.5% of all pairs failed to define a chord.)  But, since you are only interested in the lines that do define a chord, you can throw away the failed pairs of points without causing any harm to your random distribution.

The three classic methods, on the other hand, produce a chord every single time, which is convenient.  The cost of that convenience however, in the case of methods 1 and 3, is a skewing of the distribution.  Method 2 has the benefit of being convenient and also producing the result p=0.5, the result you arrive at with the inconvenient method.

I should add, as a clarification, that when Method 2 gives the correct result this appears to be a consequence of what I think of as “maximal uniformity”.  In other words, the result is not skewed because the method lends itself to a smooth distribution of chords.  Method 2 can be generalised to reflect an aspect my method:





Method 2 (Generalised):

1.               Select a random point.

1a.         Draw a line between the random point and the locus of the circle.

2.               Select a second random point.

2a.         Draw a second line which is perpendicular to the first and also passes through the second random point.

3.           Does the second line intersect the circle?

NO – return to step 1.

YES – continue.

4.           You now have two points on the circle, between which is a chord (or a vanishingly small chance of a tangent).

Again, this method will result in the vast majority of pairs of random points failing to produce a chord.  Chords will only be produced if the second line intersects the first within the circle.  The classic Method 2, therefore, by restricting the selection in step 2 merely eliminates the vast majority of pairs of random points that would not result in a chord without skewing the outcome.

The classic Method 3 can have “maximal uniformity” imposed on it by using polar co-ordinates rather than cartesian, as explained in an earlier article, resulting in p=0.5.


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I am aware that the Bertrand Paradox has value in that it demonstrates that poor selection of a methodology can result in incorrect modelling of a random distribution, and there may be other pedagogical or philosophical uses to which it can be put.  But I’m not convinced that that value means that we should not accept that there is in fact a uniquely correct, if somewhat inconvenient method of approaching a truly random distribution of chords.

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It struck me a few days after I posted this that I could describe my argument differently, as a sort of decision tree.  Just to ram the point home yet another way, here is the decision tree for my method (green), method 1 (light orange), method 2 (blue) and method 3 (dark brown):


As the count at Z approaches infinity, the values of E/D and F/D both approach zero.  (If you don’t believe me, get yourself a random number generator and try it.)
The restriction of hypothetical samples of chords per methods 1 and 3 is actually quite severe.  Imagining that such a restriction won’t have any effect at all is hubris of quite staggering proportions.

If this doesn’t persuade detractors, then I guess nothing will.

Thursday, 15 November 2012

Response to Melchior (in respect to the recent puzzle question)



Interaction between Melchior and me at reddit/math has got too long to continue in its original form.  Here’s what’s gone before in response to my recent article about the Bertrand Paradox (note that by some cruel twist of fate, the name neopolitan was not available to me):


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Melchior:


You pose an ill-defined problem to which there is no right or wrong answer, then you nominate one answer as "probably right" for dubious reasons. You stick with the vague "select at random from a set" concept and try to work with subset relationships, but this approach can't express the hidden assumptions that underlie the paradox.


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wotpolitan:


Thanks Melchoir.


I posed the question as "what is the probability, p, of a random chord in a circle being longer than the side of the largest equilateral triangle that can be drawn in that same circle?" Wikipedia has it expressed as "Consider an equilateral triangle inscribed in a circle. Suppose a chord of the circle is chosen at random. What is the probability that the chord is longer than a side of the triangle?"


I don't really think that either question is ill-defined.


It only appears ill-defined when you select chords from a biased set (thus making them not random).


My set was only biased so as to make sure the two random points result in a chord, I don't think that that is invalid, is it?


But, yes, I agree, I do have to make that a little more clear in the original article. Thanks again.


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Melchior:


| “It only appears ill-defined when you select chords from a biased set”


There's no such thing as a "biased set". Every probability distribution is "biased" relative to some other probability distribution.


| “(thus making them not random)”


The phrase "not random" is meaningless here. I think you mean that the distribution is not uniform, but that's also meaningless unless coordinates are specified, and it does depend on the coordinates.


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wotpolitan:


Ok, maybe this is why I should have paid a little more attention to statistics many years ago.


Re a “biased set” – if you are considering the probability of selecting a human over 180cm tall and you restrict your measurements to all American males, then you are not going to get the correct result for all humans. This is what I mean by a “biased set”. You might want to call this a “biased sample”, but note that I said ALL American males. If you, for some reason have got it in your head that the set of all humans is identical to the set of all American males, then you have a biased set.


This is similar to the case with the p=1/3 calculation of the probability of randomly selecting a chord of length greater than r.√3 in a circle of radius r. If you restrict your selection of points which define a line which then results in a chord to any two on the circumference of the circle, you get a “biased set”. The biased set results in p=1/3 while an unbiased set (which I think I’ve defined) results in p=1/2. I’m sorry if my terminology is not standard but if you can try to see past that it would be much appreciated.


Re “not random”, hopefully this is clarified by the above but to make sure, by this I mean that a random selection from the set of all possible chords is not going to give you the same result as the random selection of a pair of points on a circle, even if those points can then be used to define chords. A “random” selection from a subset of all chords is not necessarily equivalent to a random selection from the set of all chords. (It’s like trying to select people at random from a crowd and subconsciously only selecting females that you think are particularly good-looking, it might be a random selection of good-looking females, but it’s not random in terms of the task at hand.) Again, I might not have used the correct terminology for which I apologise.


By the way, are you a p=1/3 person? Or simply undecided?


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Melchior:


The set of humans is simply different from the set of American males. Neither one is biased.


The distinction is a red herring anyway, because we're talking about one and only one set: the set of chords in the circle. You keep saying "subset", but the set of chords defined by choosing two endpoints is precisely the same set as the set of all chords. There is no chord that exists in one set but not the other.


Subsets can't rescue you from this paradox. It's not just a matter of terminology; there's something very different going on here. This is a really important point, and you have to accept it in order to understand anything else about the problem.


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Dear Melchior.


This last response is quite interesting.


Wikipedia defines a paradox broadly enough to include puzzling results that can be rectified by demonstrating that one or more of the premises themselves are not really true, are a play on words, are faulty and/or cannot all be true together, going far enough to state that paradox is often used interchangeably with contradiction.  Even dictionaries seem to be rather vague on the topic, for example the first entry at merriam-webster.com isn’t even remotely close to what I personally would define as a paradox.


Let me clarify.  In this context I mean by paradox, as I am sure the people who originally named it the Bertrand Paradox meant, a problem which appears to have (at least) three alternative solutions which cannot all be correct.  It cannot be true that p=1/2 and p=1/3 and p=1/4 in this scenario, when p is the probability that a chord selected at random is greater than r.3.


I know that Bertrand originally presented the problem as an example of how different methods of producing a random variable can affect the result, but I think you might be interpreting this incorrectly.  A truly random selection of a chord is not necessarily equivalent to following a process which is (in a sense) random and which results in a chord.  You might think that, by following each of the three methods listed in the Wikipedia article, you are randomly selecting a chord, but my argument is that you are not.  You are close to it with method 2 (the radius method), but you are not with methods 1 or 3 – unless you modify them in the way that I have described in my earlier article.


I am curious as to your intent, Melchior.  It may be a tiny bit unfair to use this medium to analyse your responses, but life is always a tiny bit unfair, so I’ll carry on regardless.


You initially said the problem is ill-defined.  It’s not.  You also made mention of the “hidden assumptions that underlie the paradox”, but you’ve not made clear what these hidden assumptions are.  Accepted that I haven’t previously asked you to, but I am now:  What are the hidden assumptions underlying the Bertrand Paradox?


Then you totally misunderstood the point about the “biased set”.  The set of American males is a set and the set of humans is a set, it’s generally accepted that the former is a subset of the latter.  The problem arises when you take only American males and assume that the results of any measurement will be representative of humanity as a whole.  The distribution of chords as generated via two random points on the circumference of a circle is not the same as the distribution of chords as generated by two random points selected without the constraint.  The former is a subset of the latter.  Do you not agree?


I did ask you if you were a p=1/3 person or just undecided, which you didn’t respond to.  Could you please be open about that?


Finally, you say that subsets “can’t rescue (me) from the paradox” and “there’s something very different going on here”.  Then you talk about understanding “anything else about the problem”.


When you talk about rescue from a paradox, it seems to me that you are talking about inherent self-contradiction as in the following couplet:


                          The sentence below is a lie.


                          The sentence above is true.


That’s an inescapable paradox caused by indirect self-reference, you have to just walk away from it rather than try to work out whether one or both or neither of them are correct.


Are you implying that the Bertrand Paradox is a true paradox in the stricter sense?  Is the “something very different” that you are hinting at merely a reference to the problems associated with simulating randomness (which Bertrand was in essence talking about)?


And, as a last question, what is the “anything else about the problem” of which you speak?


Sincerely in anticipation of a fruitful continuation of this discussion,


neopolitan