A Practical Man Ponders Schrodinger

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A practical man walks into a room full of theoretical men who are deeply pitched in argument.  There is a box, with its door open and there is a dead cat. The practical man walks up and asks, 'What are you guys arguing about?  Who killed the cat?'

One of the theoretical men answered, 'No, we know how the cat died.  It was in a box, a piece of radioactive material emitted a particle that triggered a hammer that broke a bottle of cyanide gas which killed the cat'.

'Well, that is pretty straightforward, if a bit odd' answered the practical man, 'So what is the argument about?'

'Well', the theoretical man continued, 'We differ on what happened to the alive cat'.

'Wait', said the practical man, 'There were two cats in the box?  And one is missing?'

'No, there was only one cat.  But we have a mathematical equation that says before we opened the door, the cat was simultaneously dead and alive.' replied the theoretical man.

Very skeptically, the practical man asked, 'So, you observed the cat simultaneously dead and alive?'

'No', replied the intellectual man, 'the equation strictly forbids that.  If we try to look at it, it instantaneously is all dead or all alive'.

'So, you are just trusting the equation that it is true, even though you can never see it?' the practical man asked.

'Yes, that is the situation' the intellectual man assured him.

'OK, it seems to me that the cat was half alive and half dead and when the door was opened, as odd as it sounds, the half that was alive become dead and voila' one whole dead cat.'  the practical man opined.

'Well, that was what the original interpretation was.  A cat that was both alive and dead collapsed into one dead cat'.  the theoretical man said, 'For mathematical reasons they say it that way, but basically they agree with you.'

'Then, what is the argument about?' the practical man asked.

'Recently, more and more theoretical men are arguing that when the door was opened, there became a universe where we opened the door and the cat was alive' the theoretical man answered.

The practical man pondered that for awhile and then asked, 'Where is this other universe?'

The theoretical man answered, 'Nowhere that we can ever locate.  That is a requirement of the equation.  No part of this universe can ever interact with any part of the other universe.'

'OK.  Let's see if I got this.  You are arguing that the cat inside the box was both dead and alive though you can never see that.  And when you opened the box, a second universe formed for the alive cat, though you can never see that, either', asked the practical man.

The theoretical man verified, 'Yes, you have the basic principle down.  Though there is an argument that both universes existed and rather than a new universe being formed, the alive cat universe became closed off to us.'

The practical man, obviously getting disdainful, challenged, 'And, before the cat became both dead and alive, was there a dead cat universe and an alive cat universe?'

The theoretical man answered, 'We don't think so.'

'OK,' the practical man replied, 'it seems to me that there was just an alive cat universe until you put the cat in the box and it became simultaneously alive and dead.  The cat then needed a universe to be alive in and a universe to be dead in.'

'Well, that seems reasonable.  Though, I'm not sure anyone has ever thought about it that way' pondered the theoretical man.

'Why would anyone subscribe to this exploding number of universes theory?' asked the practical man

'Because they think it is easier', answered the theoretical man, 'However, they would say that the universe is a huge mass of these kinds of events and that, being the way we are, we are constantly limiting ourselves to one of them.'

'Wait, are you saying that in a totally inaccessible part of the universe, there is another me discussing with another you where the DEAD cat went?'  the practical man asked incredulously.

The theoretical man chuckled and answered, 'As I understand it, yes'.

The practical man, after pondering for a minute or two, answered, 'Look it, it seems to me that the choice is between a hypothetical cat disappearing and a real cat hopping over to the next universe over.  The hypothetical cat seems like the lesser of two evils'.

The practical man continued, 'Now, suppose I am going to flip a coin.  But, I flip it into a box so that I don't know if it came up heads or tails.  Are you saying that until I look inside the box, the coin is both heads and tails, simultaneously?'
The theoretical man replied, 'Well, that kind of depends upon whether your coin flip was a quantum event.  If so, then, yes, in theory, the coin was in superposition, both heads and tails simultaneously.  If not, well, then it was either heads or tails'.

The practical man stared at the theoretical man, dumbfounded, 'Are you listening to yourself?'

'OK', he continued, 'let's go back to the cat.  Suppose we put one more thing into the box - a remote camera.  You and your other theoretical friends are in this room doing your experiment and I am in the next room with a monitor connected to the camera.  Now, suppose I turn on the monitor, before you open the door.  Am I going to see a dead cat or a simultaneously alive and dead cat?  And, either way, how does the cat or the universe or whatever controls this know whether I've turned on the monitor or not?'

The theoretical man chucked and replied, 'Well, that is called the measurement problem.  We aren't quite sure what constitutes an observation that will cause the cat to stop being simultaneously dead and alive and choose one or the other.  Or, alternatively, branch into two universes'.

The practical man is clearly becoming frustrated, 'It seems to me that you theoretical guys went off the rails earlier on, when you decided that your equation describes the actual universe.  Let's go back to the coin flip.  I can write an equation, (1/2H + 1/2T)=1 that describes the probability of the flip, assuming that no cheating is going on.  It doesn't describe the universe after I flip the coin but before I look into the box.  It describes my expectations of what I am going to see.  Now, if I look into the box I am going to see either heads or tails and I'm not going to ask myself, if it is heads, what happened to the tails coin.  Or imagine that a second universe exists where I observed tails.'

The theoretical man, shaking his head, replied, 'No, that doesn't work.  Because we have set up experiments that tested whether a particle was here or it was there and the results indicate that it was in both places simultaneously.'

The practical man looked shocked, 'How could that be?'

The theoretical man replied, 'Precisely.  That is what caused one of the founders of this theory to say that if it doesn't shock you, you don't understand it yet.  The problem is that this indeterminacy that exists at the particle level may have effects on larger scales.  Our simultaneously alive and dead cat is an example.'

After some pondering, the practical man responded, 'Well, I know that I am a practical man and not a theoretical man, but if what you say is true, it seems that reality must, in some sense, not be real.  Because, whether electrons or cats, the notion of things being simultaneously X and not X is not compatible with reality as I understand it.'

Nodding, the theoretical man replied, 'Exactly!  In fact fifty years ago, a Physicist proved that phenomena such as this simultaneously alive and dead cat force us into one of three conclusions, all of which are profoundly disturbing.  One way or another, they all call into question what we consider to be reality'.

With a huge sigh, the practical man said, 'Then, OK, I can live with that.  My world does not have simultaneously alive and dead cats.  You tell me that I can't ever observe such a cat.  So, my world behaves as if it is real.  Since I am a practical man, I can continue to behave as if the world is real while simultaneously accepting that, on some level it is not.  I may have trouble with simultaneously alive and dead cats, but I don't really have trouble with a simultaneously real and unreal universe as long as the unreal one doesn't affect my observed universe.  For all practical purposes, the unreal universe is theoretical.  Right?'

"Not entirely', replied the theoretical man, 'but that would require me to get into quantum tunneling or something similar.  While nobody will ever observe a simultaneously dead and alive cat, we do observe, for example, electrons being places that they should not be able to be and that can affect how computers behave.'

'Same equation?', asked the practical man?

'Pretty much.  A related one', the theoretical man agreed.

'Well, OK, then,' the practical man said with finality, 'I'll take my counsel from Scarlett O'Hara and I'll think about this tomorrow.  And, as we all know, practically speaking, tomorrow never comes.' 

Revisiting Monty Hall

I was introduced to the Monty Hall problem at a Mensa meeting by Cyd Bergdorf, the then head of TNS and Ron Hoeflin.  The problem goes like this.  

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?"

The problem is trivial and I said immediately, 'You switch, of course'.  However, then I added, 'it is irrelevant whether the host knows what is behind the doors'.  This is a statement that has not been agreed with by some of the smartest people in the world.  In a private message with Garth Zietsman and Rick Rosner, I found I could convince neither of them of my assertion.  In a way that is good, because it required me to work very hard at creating a proof for what I saw as obvious, but difficult to express convincingly.

The argument for its necessity goes like this.  If the host knows what is behind each door (and as Rick correctly pointed out wasn't inclined to thwart your attempt to win a car), by opening a door with a goat behind it, the host has inserted information.  Essentially he has collapsed the Bayesian prior and changed the probability to 50% that there is a car behind the unchosen and unopened door.  The Wikipedia article does a pretty good job of explaining this if you aren't unfamiliar with the problem.

If the problem said that you were going to play the game 100 times and asked what strategy would be best, then whether the host knew what was behind the doors would be relevant.  Either, in that sequence of 100 games, the host would or would not open doors with cars behind them based upon whether he knew what was behind the doors and was inclined to open only goat doors.  Again, Rick made the very correct point that it matters what was done in the cases when the host did open a car door, if we are in the scenario where that can happen.

But, of course, that is not the problem as stated.  One is presented with a unique event and, as such, the problem surfaces one of the peculiarities of probability.  I recognized this when I was about 8 and I started asking people, 'what is the probability that, if I flip a coin, it will land on heads?'  Everyone, of course, answered 50%.  I then replied, 'It is not.  It is either 0% or 100%'.  To my amazement, nobody 'got it'.  In essence, you really can't average one event.  The Monty Hall problem is of that type.  It is played once.

PROOF:
Consider two sets.  One is a set of n games where the host always opens a goat door.  In the other set of n games, the host opens a car door 1/3 of the time.  In the first set of n games, switching will win a car 1/2 of the time and staying with the door first chosen will win a car 1/3 of the time. One should switch.  In the second set, one will win a car 1/3 of the time if one switches or if one stays with the door first chosen.

Here is the key.  The result is not dependent upon the value of n.  If each set contains 100 games, it will be 1/3 and 1/2 for the goat door only set and 1/2 and 1/2 for the goat and car door set.  If n=200, it will be the same.  If it is 50 it is the same.  In the first set, one should switch and in the second, it doesn't matter.

Also, we will note that in the first set, it doesn't matter if the host is opening only goat doors by happenstance or by artifice.  One can argue, convincingly, that if the host opens only goat doors 100 times and never opens a car door, he almost certainly knows what is behind the doors.  However, whether he knows or just got insanely lucky doesn't matter.  The answer is based upon the elements in the set, not why the elements are as they are.

In the problem, as stated, n=1 and it belongs to the set where only goat doors are opened.  In that set, one should switch doors.  As we see, it doesn't matter whether a goat door was opened because of chance or because the host knew it was a goat door.

I invite questions or refutations.