"Now to him who is able to do immeasurably more than all we ask or imagine..." - Ephesians 3:20.

I have this tool that guides my thinking, which I privately call the principle of least awesomeness. It simply says that God is at least as awesome as anything we can imagine. Straightforward idea, right? Let's see some applications of it.

At a basic level, this principle says that God is at least as awesome as our sciences have discovered our universe to be - for he created it. This is why Christians should be happy whenever science discovers something new about the universe. The more elegant and beautiful the universe turns out to be, the stronger evidence we have for God's existence and glory.

The principle also helps me deal with propositions of a more speculative nature. For example, when I was much younger I once wondered, "what if there was a super-being that's so powerful, so beyond everything, that God himself was unaware of its existence?" Could something like that really exist, and if so, would that mean that my worship towards God was fundamentally misdirected? The key out of this dilemma was that I - a mortal, and a rather immature one at that - had imagined it. But it is impossible for my imagination to exceed God's own knowledge. So, God was already aware of my concept of this super-being, since I had been able to imagine it. Yet he preemptively declares himself to be supreme over all others, thereby eliminating my fictitious super-being as a possibility. God is at least as awesome as anything I can possibly imagine.

Now, I generally don't find such childish speculations to be all that helpful. Certainly there's no shortage of possibilities if we're willing to engage in wild hypotheses with no biblical or scientific basis. However, the principle of least awesomeness allows some good to come out of these speculations. In my super-being case above, it gave me a new appreciation of God's power and transcendence. Other "what-if" scenarios can help us appreciate other aspects of his being.

In this way, wild speculations can be somewhat productive - not because it's likely to be the truth, but because it helps us to exercise our mind and imagination, in stretching to a higher understanding for the lower bound on God's glory. God is at least as awesome as anything we can imagine.

As another example, I once read this idea about how each of us are all reincarnated as all other humans that have or will have existed, and once that process is complete we then become God, who then sends our human selves through all that process of reincarnation. Is the idea likely to be true? Of course not. It's just a wildly speculative idea with no backing. But it has a certain poetic appeal, no? In this system, anything nice or mean you do to someone else, you're really doing to one of your reincarnated selves. We're all one, with one another and with God. The principle of least awesomeness allows us to reject this system as any kind of accurate description of truth, yet allows us to redeem some of its poetic beauty. For the truth - the system that God has actually set up - is at least as beautiful as this system that we cooked up in our imagination. We are, in truth, truly one with each other and with God, in a profound unity that is at least as awesome as the one dreamt up of in any our imaginary systems.

Another idea I had concerns the multiverse theory - in particular, Everett's many-worlds interpretation of quantum mechanics. This hypothesis, in short, says that everything that's quantum-mechanically possible happens in a different universe of the multiverse. Now, I believe that multiverse theories are all very wrong (a point I will expound on in a future series of posts). But they are fertile grounds for wild speculation. Specifically, I thought that a many-worlds interpretation would be a neat way to resolve the issue of predestination. If all possibilities are actualized in a parallel universe, so would an individual's decision to either accept or reject Christ. Those who accept Christ would go on to their eternal salvation, having freely chosen to do so as manifested by them being on the "right" branches of the many-world multiverse. Those who reject Christ would be damned, having freely chosen their destiny, again as manifested by being on the "wrong" branches of the multiverse. Yet the whole picture, from beginning to end, would have been timelessly predestined, as the whole history of every possibility, choice, and outcome would have been written into the multiversal quantum-mechanical wavefunction at the foundation of the world. It would be preordained that every possibility comes into being, yet each person would justly receive the destiny that they had freely chosen - and there would truly be a grand number of rooms in our Father's house to accommodate all the possible ways that each person could have accepted Christ.

Now, is all that likely to be true? Of course not - I've already expressed my rejection of the multiverse theories, to say nothing of the wild, baseless leaps that this theory makes about predestination. Are there great difficulties with this hypothesis? Of course there is - not the least of which is the innumerable universes in which Christ never existed. I intend to be absolutely clear that such speculations are worthless as representations of what God really has wrought.

And yet, according to the principle of least awesomeness, such speculations can be redeemed in some ways. In case of the multiversal predestination theory, we can say that however predestination actually turns out to work, it will be at least as awesome as how it is presented to work in that theory. God is sovereign and all-knowing, to at least the degree he would be if he had created the many-worlds multiverse. Yet his sovereignty will allow all who freely choose him to come to him: again, the many-worlds multiverse represents the minimum degree to which this would be true. And however we choose to conceptualize heaven, it will be at least as awesome as Hilbert's Grand Hotel, that can accommodate all possible ways that every possible person can come to Christ. For it is a place that Christ himself has prepared for us.

So, feel free to speculate about God and his works. Not with the delusion that you're going to arrive at the truth, of course. You are quite safe from that. You needn't worry about exhausting the limitless riches of God with your dreams. So exercise your mind and stretch out your thoughts: you are safe in God's guarantee that he is at least as awesome as anything that we can imagine.

You may next want to read:

How is God related to all other fields of study?

Isn't the universe too big to have humans as its purpose?

Another post, from the table of contents

# NaClhv

Theology, philosophy, science, math, and random other things.

### How to make a fractal: version 2.4

My fractal program has been updated. It can be found at:

How to make a fractal (http://www.naclhv.com/2014/06/how-to-make-fractal.html)

I've made a video tutorial for how to use the program, which can be found at:

https://youtu.be/8VMuW83eUXk

You may next want to read:

15 puzzle: a tile sliding game

Sherlock Bayes, logical detective: a murder mystery game

Another post, from the table of contents

How to make a fractal (http://www.naclhv.com/2014/06/how-to-make-fractal.html)

I've made a video tutorial for how to use the program, which can be found at:

https://youtu.be/8VMuW83eUXk

You may next want to read:

15 puzzle: a tile sliding game

Sherlock Bayes, logical detective: a murder mystery game

Another post, from the table of contents

### The intellect trap

I have another math brainteaser for this week. It's easier than last week's problem.

You're taking a long drive, and you want your average speed for the trip to be 50 miles per hour (mph). But you run into some traffic, and when you've traveled half the distance to the destination you notice that your average speed so far has only been 25 mph. How fast do you have to drive during the latter half to still meet the initial goal of averaging 50 mph for the whole trip?

That is a rough graph representing the probability that you will give the correct answer, as a function of the intelligence, education, and effort you bring to the problem. Note the dip in the middle, where you're LESS likely to get the question right than if you were stupid and ignorant. That dip is what I'm calling the intellect trap.

Now, are you ready for the answer?

The correct choice is "f) infinitely fast". To see why, imagine that the length of the total trip is 50 miles. Your goal is to averaging 50 miles per hour, so you're hoping to arrive at the destination in 1 hour. Then the half-way point would be at 25 miles, and it's here that you've realized that you've only been averaging 25 mph. Meaning, you've traveled 25 miles at 25 mph, using up the entire 1 hour that you had allocated for the whole trip. So you have no more time - zero hours - left to travel the remaining 25 miles, and therefore need to travel at an infinite speed.

I once gave this problem to a group of 9th grade geometry students. They knew what "average speed" meant. They knew that "distance = rate * time". They had done well in algebra. They were bright students. A five-year-old would have chosen the right answer 1 out of 6 times by chance. Yet, out of that class of about 30 students, only one got the right answer, and he had chosen his answer as a joke. Most of the class had mistakenly chosen 75 or 100 mph as the answer, which are the seemingly correct values upon a superficial examination. They had fallen for the intellect trap.

That, I think, is the most salient feature of this problem. There are situations where more intellect, education, or effort actually DECREASE your chances of getting the correct answer. And upon a moment's reflection, you'll see that this phenomenon is actually quite common. Consider one of your personal areas of expertise, a subject you know well enough to have taught or supervised others in. Isn't there something like a list of common beginner mistakes, or a "gotcha" moment in the flow of progress one makes?

Now, we would be fools if what we took away from this was "look at these stupid 9th graders, so dumb that even when they try harder it only makes them more likely get the wrong answer". No: what concerns me is "what intellect traps am I falling into? Which of my ideas are only half-baked without me realizing it? What traps out there are so large, that my whole field or segment of society has fallen into it? And how could we tell when we're in one?"

Note that I'm NOT just saying "be careful what you believe", "think about your opinions", or "do some research to back up your positions". All that's just common sense. I'm concerned here about when that common sense FAILS, when more care, more thought, and more research only leads more to the WRONG conclusion. How can we detect or prevent this?

One thing that comes to mind is to beware the feeling of contempt, which exacerbates the issue. Look at the structure of the multiple choices given in the initial average speed problem. The first three options are obviously, contemptibly wrong. If you're already behind schedule, how could you possibly catch up by doing nothing or slowing down? After dismissing these answers as obviously wrong, it becomes easier to also dismiss the last, correct answer this way when it doesn't conform to one's intuition.

A second precaution is to think past where you think you've reached the right answer. This is what I've tried to demonstrate in my solution to the two envelopes problem (which you may want to read first). First, there is the intuitive understanding that your opponent (the wild statistician) must be wrong, because what he proposes (that you continually switch the envelopes) is clearly nonsense. But beware the feeling of contempt: you cannot simply stop here. You must also propose the right way to think about the problem. "It is better to light a candle than to curse the darkness". Next, you should think further until you identify the exact nature of your opponent's error. And ideally, you should then be able to correct their error and redeem your opponent's thought process, so that it reaches the same conclusion as your initial way of thinking. In short, you'll know you've reached the end of this line when your thoughts are a superset of your opponent's thoughts. In doing so, you may be able to identify the intellect trap as the mistake that your opponent made.

This is, of course, not enough to completely guarantee that you're right. Nothing ever is. However, thinking about it from two different perspectives - such as yours and your opponents - and reaching the same conclusion greatly enhances your chances. Three or more perspectives are better still. I leave it as an exercise for the reader to find multiple other ways to solve the initial average speed problem, which I've only solved in a very limited way above.

I also realize that, looking at this post self-referentially, I've tackled this "intellect trap" problem in an incomplete way. I don't yet have a complete solution to which procedure will let you best detect and escape the trap. I only have several rules of thumb. If anyone has the complete solution (say, a complete Bayesian formulation of the problem and the answer), I'd be happy to hear from you.

But meanwhile, I feel that this is a common problem, and especially pernicious and difficult to detect at the society-wide level. With the benefit of hindsight, we can sometimes see when entire cultures fell victim to it, such as when communism gained favor among the intelligentsia of many countries near the beginning of the 20th century. But what such traps lurk in our world today? I worry about the polarization of politics, or the paucity of Christians in academia. These phenomena seem to be driven in part by this intellect trap. In these issues, I unfortunately see a great deal of contempt, few attempts at an improved solution, little attempt at engagement, and virtually no efforts to redeem the opponent's thoughts.

You may next want to read:

Basic Bayesian reasoning: a better way to think (Part 1)

Science as evidence for Christianity against atheism (introduction)

Another post, from the table of contents

You're taking a long drive, and you want your average speed for the trip to be 50 miles per hour (mph). But you run into some traffic, and when you've traveled half the distance to the destination you notice that your average speed so far has only been 25 mph. How fast do you have to drive during the latter half to still meet the initial goal of averaging 50 mph for the whole trip?

a) 0 mphThink about it, and choose your answer before you scroll down. Meanwhile, let me show you this graph to take up some screen space, so you won't see the answer right away:

b) 25 mph

c) 50 mph

d) 75 mph

e) 100 mph

f) infinitely fast

That is a rough graph representing the probability that you will give the correct answer, as a function of the intelligence, education, and effort you bring to the problem. Note the dip in the middle, where you're LESS likely to get the question right than if you were stupid and ignorant. That dip is what I'm calling the intellect trap.

Now, are you ready for the answer?

The correct choice is "f) infinitely fast". To see why, imagine that the length of the total trip is 50 miles. Your goal is to averaging 50 miles per hour, so you're hoping to arrive at the destination in 1 hour. Then the half-way point would be at 25 miles, and it's here that you've realized that you've only been averaging 25 mph. Meaning, you've traveled 25 miles at 25 mph, using up the entire 1 hour that you had allocated for the whole trip. So you have no more time - zero hours - left to travel the remaining 25 miles, and therefore need to travel at an infinite speed.

I once gave this problem to a group of 9th grade geometry students. They knew what "average speed" meant. They knew that "distance = rate * time". They had done well in algebra. They were bright students. A five-year-old would have chosen the right answer 1 out of 6 times by chance. Yet, out of that class of about 30 students, only one got the right answer, and he had chosen his answer as a joke. Most of the class had mistakenly chosen 75 or 100 mph as the answer, which are the seemingly correct values upon a superficial examination. They had fallen for the intellect trap.

That, I think, is the most salient feature of this problem. There are situations where more intellect, education, or effort actually DECREASE your chances of getting the correct answer. And upon a moment's reflection, you'll see that this phenomenon is actually quite common. Consider one of your personal areas of expertise, a subject you know well enough to have taught or supervised others in. Isn't there something like a list of common beginner mistakes, or a "gotcha" moment in the flow of progress one makes?

Now, we would be fools if what we took away from this was "look at these stupid 9th graders, so dumb that even when they try harder it only makes them more likely get the wrong answer". No: what concerns me is "what intellect traps am I falling into? Which of my ideas are only half-baked without me realizing it? What traps out there are so large, that my whole field or segment of society has fallen into it? And how could we tell when we're in one?"

Note that I'm NOT just saying "be careful what you believe", "think about your opinions", or "do some research to back up your positions". All that's just common sense. I'm concerned here about when that common sense FAILS, when more care, more thought, and more research only leads more to the WRONG conclusion. How can we detect or prevent this?

One thing that comes to mind is to beware the feeling of contempt, which exacerbates the issue. Look at the structure of the multiple choices given in the initial average speed problem. The first three options are obviously, contemptibly wrong. If you're already behind schedule, how could you possibly catch up by doing nothing or slowing down? After dismissing these answers as obviously wrong, it becomes easier to also dismiss the last, correct answer this way when it doesn't conform to one's intuition.

A second precaution is to think past where you think you've reached the right answer. This is what I've tried to demonstrate in my solution to the two envelopes problem (which you may want to read first). First, there is the intuitive understanding that your opponent (the wild statistician) must be wrong, because what he proposes (that you continually switch the envelopes) is clearly nonsense. But beware the feeling of contempt: you cannot simply stop here. You must also propose the right way to think about the problem. "It is better to light a candle than to curse the darkness". Next, you should think further until you identify the exact nature of your opponent's error. And ideally, you should then be able to correct their error and redeem your opponent's thought process, so that it reaches the same conclusion as your initial way of thinking. In short, you'll know you've reached the end of this line when your thoughts are a superset of your opponent's thoughts. In doing so, you may be able to identify the intellect trap as the mistake that your opponent made.

This is, of course, not enough to completely guarantee that you're right. Nothing ever is. However, thinking about it from two different perspectives - such as yours and your opponents - and reaching the same conclusion greatly enhances your chances. Three or more perspectives are better still. I leave it as an exercise for the reader to find multiple other ways to solve the initial average speed problem, which I've only solved in a very limited way above.

I also realize that, looking at this post self-referentially, I've tackled this "intellect trap" problem in an incomplete way. I don't yet have a complete solution to which procedure will let you best detect and escape the trap. I only have several rules of thumb. If anyone has the complete solution (say, a complete Bayesian formulation of the problem and the answer), I'd be happy to hear from you.

But meanwhile, I feel that this is a common problem, and especially pernicious and difficult to detect at the society-wide level. With the benefit of hindsight, we can sometimes see when entire cultures fell victim to it, such as when communism gained favor among the intelligentsia of many countries near the beginning of the 20th century. But what such traps lurk in our world today? I worry about the polarization of politics, or the paucity of Christians in academia. These phenomena seem to be driven in part by this intellect trap. In these issues, I unfortunately see a great deal of contempt, few attempts at an improved solution, little attempt at engagement, and virtually no efforts to redeem the opponent's thoughts.

You may next want to read:

Basic Bayesian reasoning: a better way to think (Part 1)

Science as evidence for Christianity against atheism (introduction)

Another post, from the table of contents

### The two envelopes problem and its solution

A job I was looking at had a requirement that read: "Inability to stop thinking about the two envelopes problem unless you’ve truly come to peace with an explanation you can communicate to us". So I thought I'd post my explanation to the problem.

The setup to the problem goes like this:

But now, a wild statistician appears, and makes the following argument:

That can't possibly be the right choice. Now, here is the real two envelopes problem: something must be wrong with the wild statistician's argument - but what exactly is the nature of his error?

The solution to the problem goes as follows:

If we start by assuming there's $20 in your envelope, it is NOT equally likely that the other envelope contains $40 or $10. This is where the wild statistician goes wrong. In general, given a value x in your current envelope, it is NOT equally likely for the other envelope to contain 2x or x/2.

Before we get more mathematical, let's examine the problem intuitively, by grounding it in a solid example. Say that you're on a television game show, and you're playing this two envelopes game. You know that American TV game shows typically give prizes from hundreds to tens of thousands of dollars. Now, if the host of the show lets you know that your envelope contains $50, should you switch? I certainly would. I know that, given the typical payout of TV shows, the two envelopes were more likely set up to contain $100 and $50 rather than $50 and $25. The two probabilities are NOT EQUAL.

Oh the other hand, imagine that you're a high school statistics student, and your teacher is playing this two envelope game with you for a class lesson. Your envelope contains the same $50 as in the previous example. Should you make the switch? No way. You seriously think your teacher put in $100 in the other envelope to give to a high school student, for a single lesson? If your teacher has 5 statistics classes, he stands to lose up to $500 on that one lesson - likely far exceeding his pay for the day. It is much more likely that your teacher chose $50 and $25 for the values rather than $100 and $50. Again, the two probabilities are NOT EQUAL.

Now, if the two probabilities were equal, then the wild statistician would be right, and you should switch. And you should continue to do so as long as the probabilities remained equal. But the problem described by that situation is not the two envelope problem. It's actually a 50-50 bet where if you win, you double your money, but if you lose, you only lose half your money (compared that to most casino games, where you lose your entire bet). If you find a game like that, you should continue playing it for a very long time.

But for the two envelope problem, the chances of either doubling or halving your money are generally not equal. This will be true for ANY reasonable probability distribution of possible values of money in the envelops. "Reasonable" here means that the probability distribution must sum to one, and that it must have a finite expectation value. Consider any of the following probability distributions (or any other reasonable distribution you wish to think up) for the money in the envelopes:

The orange line the probability distribution for the smaller amount money in one of the envelopes. The green line is the probability distribution for double that value, in the other envelope - it's been stretched horizontally by 2 to represent the doubling, and compressed vertically by 0.5 to keep the probability normalized. You see that the two probabilities are equal (where the lines cross) only for very rare, special amounts of money. In general, if you see a small amount of money in your envelope, you're more likely to have the "smaller" of the two envelopes, and if you see lots of money, you're more likely to have the "greater" of the two. You should be able to understand this intuitively, in conjunction with the game show / statistics teacher examples given above.

Whether you should switch or not depends on the expectation value of the money in the envelopes. If the amount in the "smaller" envelope is A, then the amount in the "greater" envelope would be 2A, and the expectation value for choosing them with 50-50 chance would simply be 3A/2. Since the envelopes are indistinguishable, this is in fact the expectation value of choosing either one, so it doesn't matter which one you choose. This is nothing more than the original, simple argument presented at the very beginning.

However, what if the wild statistician insists on putting the problem in terms of expected gain conditioned on the different possible values of the money in your current envelope? This is how his original flawed argument was framed. It's an overly complicated way of thinking about the problem, but shouldn't we also be able to come to the correct solution this way?

We can. (Beware, calculus ahead) Let:

x = amount of money in your current envelope,

f(x) = probability distribution of the money in the "lesser" envelope, and

g(x) = probability distribution of the money in the "greater" envelope.

Then g(x) = 0.5 f(0.5x) due to the stretch/compression transformations. Also, the overall distribution for the amount in your current envelope, given that you chose one of the two envelopes with equal chance, is:

p(x) = 0.5( f(x) + g(x) ).

Then, the expectation value for switching is given by the following integral:

Expectation value for switching = ∫ E(x) p(x) dx

Where E(x) is the expectation value of switching when the money in your current envelope is x. This is given by:

E(x) = x * p("smaller" envelope|x) - 0.5x * p("greater" envelope|x)

That is to say, upon switching, you'll gain x if you currently have the "smaller" envelope, but lose 0.5x if you currently have the "greater" envelope. Furthermore, the p("smaller" envelope|x) and p("greater" envelope|x) values can easily be calculated by the definition of conditional probability as follows:

p("smaller" envelope|x) = f(x)/p(x),

p("greater" envelope|x) = g(x)/p(x)

putting this all together, we get:

Expectation value for switching = ∫ E(x) p(x) dx =

∫ (x * f(x)/p(x) - 0.5x * g(x)/p(x)) p(x) dx = ∫ x * f(x) - 0.5x * g(x) dx = ∫ x f(x) - ∫ 0.5x g(x) dx

However,

∫ 0.5x g(x) dx = ∫ 0.5x 0.5 f(0.5x) dx = ∫ 0.5x f(0.5x) 0.5dx = ∫ u f(u) du = ∫ x f(x)

Where we used a u-substitution and took advantage of the fact that the integral goes from 0 to infinity in the last two steps. Therefore:

Expectation value for switching = ∫ E(x) p(x) dx = ∫ x f(x) - ∫ x f(x) = 0

So there is no expected gain or loss from switching, which is the same conclusion we reached at the very beginning.

You may next want to read:

The intellect trap

Basic Bayesian reasoning: a better way to think (Part 1)

A common mistake in Bayesian reasoning

Another post, from the table of contents

The setup to the problem goes like this:

You have two indistinguishable envelopes in front of you. They both contain money, but one envelope has twice as much money as the other.

You get to choose one of the envelopes to keep. Since the envelopes are indistinguishable, you have 1/2 chance of having chosen the one with more money.

But now, before your choice becomes finalized, you are given the opportunity to switch to the other envelope. Should you make the switch?Now, one sensible and easy reply is to say that you shouldn't bother. The envelopes are indistinguishable and you have no idea which one contains more money. Your chances of getting the bigger payout remains 50-50 regardless of your choice.

But now, a wild statistician appears, and makes the following argument:

"Let's say, for the sake of argument, that the envelope you have now contains $20. Then the other envelope might contain $40, or $10. Since these two possibilities are equally likely, your expectation value after switching would be half of their sum (0.5*$40 + 0.5*$10), or $25. That's 25% more than the $20 you have now.

But if we think about this more, the initial choice of $20 actually doesn't matter. You can make the same argument for any possible value of money in your envelope. You'll always gain 25% more on average by switching. So, even without knowing the amount of money in your envelope now, you should switch."Impressed by the wild statistician's use of numbers and such, and figuring that even if he's wrong you would at worst break even, you decide to make the switch. But then, as you're about to finalize your decision and take the new envelope home, the statistician repeats exactly the same argument, word for word. "Let's say, for the sake of argument..." He's now urging you to switch BACK to your original envelope. After all, the two envelopes are indistinguishable. If there is a rational reason to switch the first time, the same reason must equally apply for switching the second time. But at this point, it becomes obvious that if you continued to listened to the wild statistician, you would do nothing but switch the two envelopes for all eternity.

That can't possibly be the right choice. Now, here is the real two envelopes problem: something must be wrong with the wild statistician's argument - but what exactly is the nature of his error?

The solution to the problem goes as follows:

If we start by assuming there's $20 in your envelope, it is NOT equally likely that the other envelope contains $40 or $10. This is where the wild statistician goes wrong. In general, given a value x in your current envelope, it is NOT equally likely for the other envelope to contain 2x or x/2.

Before we get more mathematical, let's examine the problem intuitively, by grounding it in a solid example. Say that you're on a television game show, and you're playing this two envelopes game. You know that American TV game shows typically give prizes from hundreds to tens of thousands of dollars. Now, if the host of the show lets you know that your envelope contains $50, should you switch? I certainly would. I know that, given the typical payout of TV shows, the two envelopes were more likely set up to contain $100 and $50 rather than $50 and $25. The two probabilities are NOT EQUAL.

Oh the other hand, imagine that you're a high school statistics student, and your teacher is playing this two envelope game with you for a class lesson. Your envelope contains the same $50 as in the previous example. Should you make the switch? No way. You seriously think your teacher put in $100 in the other envelope to give to a high school student, for a single lesson? If your teacher has 5 statistics classes, he stands to lose up to $500 on that one lesson - likely far exceeding his pay for the day. It is much more likely that your teacher chose $50 and $25 for the values rather than $100 and $50. Again, the two probabilities are NOT EQUAL.

Now, if the two probabilities were equal, then the wild statistician would be right, and you should switch. And you should continue to do so as long as the probabilities remained equal. But the problem described by that situation is not the two envelope problem. It's actually a 50-50 bet where if you win, you double your money, but if you lose, you only lose half your money (compared that to most casino games, where you lose your entire bet). If you find a game like that, you should continue playing it for a very long time.

But for the two envelope problem, the chances of either doubling or halving your money are generally not equal. This will be true for ANY reasonable probability distribution of possible values of money in the envelops. "Reasonable" here means that the probability distribution must sum to one, and that it must have a finite expectation value. Consider any of the following probability distributions (or any other reasonable distribution you wish to think up) for the money in the envelopes:

Whether you should switch or not depends on the expectation value of the money in the envelopes. If the amount in the "smaller" envelope is A, then the amount in the "greater" envelope would be 2A, and the expectation value for choosing them with 50-50 chance would simply be 3A/2. Since the envelopes are indistinguishable, this is in fact the expectation value of choosing either one, so it doesn't matter which one you choose. This is nothing more than the original, simple argument presented at the very beginning.

However, what if the wild statistician insists on putting the problem in terms of expected gain conditioned on the different possible values of the money in your current envelope? This is how his original flawed argument was framed. It's an overly complicated way of thinking about the problem, but shouldn't we also be able to come to the correct solution this way?

We can. (Beware, calculus ahead) Let:

x = amount of money in your current envelope,

f(x) = probability distribution of the money in the "lesser" envelope, and

g(x) = probability distribution of the money in the "greater" envelope.

Then g(x) = 0.5 f(0.5x) due to the stretch/compression transformations. Also, the overall distribution for the amount in your current envelope, given that you chose one of the two envelopes with equal chance, is:

p(x) = 0.5( f(x) + g(x) ).

Then, the expectation value for switching is given by the following integral:

Expectation value for switching = ∫ E(x) p(x) dx

Where E(x) is the expectation value of switching when the money in your current envelope is x. This is given by:

E(x) = x * p("smaller" envelope|x) - 0.5x * p("greater" envelope|x)

That is to say, upon switching, you'll gain x if you currently have the "smaller" envelope, but lose 0.5x if you currently have the "greater" envelope. Furthermore, the p("smaller" envelope|x) and p("greater" envelope|x) values can easily be calculated by the definition of conditional probability as follows:

p("smaller" envelope|x) = f(x)/p(x),

p("greater" envelope|x) = g(x)/p(x)

putting this all together, we get:

Expectation value for switching = ∫ E(x) p(x) dx =

∫ (x * f(x)/p(x) - 0.5x * g(x)/p(x)) p(x) dx = ∫ x * f(x) - 0.5x * g(x) dx = ∫ x f(x) - ∫ 0.5x g(x) dx

However,

∫ 0.5x g(x) dx = ∫ 0.5x 0.5 f(0.5x) dx = ∫ 0.5x f(0.5x) 0.5dx = ∫ u f(u) du = ∫ x f(x)

Where we used a u-substitution and took advantage of the fact that the integral goes from 0 to infinity in the last two steps. Therefore:

Expectation value for switching = ∫ E(x) p(x) dx = ∫ x f(x) - ∫ x f(x) = 0

So there is no expected gain or loss from switching, which is the same conclusion we reached at the very beginning.

You may next want to read:

The intellect trap

Basic Bayesian reasoning: a better way to think (Part 1)

A common mistake in Bayesian reasoning

Another post, from the table of contents

### The lifetime of evil (part 2)

In the last post, I introduced the idea that an act - an evil act in particular - has a characteristic time-scale over which its consequences become clear. This time-scale can be determined from the mechanics of the act in question. I was initially inclined to call this the "half-life of evil", but that phrasing expresses an unwarranted precision. I've therefore settled on calling it the lifetime of evil. Let's look at some of the possible applications for this idea.

The Bible says that the sins of one generation will affect their children down to the third and forth generations - so, something like 60-120 years. I find that this time interval is the approximate lifetime for a national or cultural evil that infects a whole society (which is what the Bible is addressing in the passage mentioned above). Put another way: this is characteristically how long it takes for a fundamentally wrong form of government to fail.

The late Soviet Union is a good instance of this rule. The "evil empire" lasted from 1922 to 1991 - only 69 years. As the fall of communism is within living memory, you may remember how surprising and sudden this event was. For many of us, it seemed that the bifurcation of the world into communist or capitalist countries was a permanent feature of history. After all, how could a country just... stop? But a flawed system cannot last forever, and the Soviet Union collapsed within a human lifetime.

Maoist China underwent much the same story. Communists in China came to power in 1949, and while they are still in power, their policy has changed dramatically since their first days under Mao in embracing capitalistic reforms. While they still have a ways to go, the rule in question seems to bear out: a flawed national system typically only lasts several generations. It may collapse or change, but it cannot endure.

Along the same lines of thinking, how long do you think North Korea will last? I don't think anyone needs convincing that the government there is evil. I've heard it described as "the worst country in the world" and "more 1984 than 1984". It's been that way since at least the end of the Korean War - 62 years ago as of 2015. Such evil cannot endure. I therefore have hope, based on the idea of the lifetime of evil, that I will live to see North Korea fundamentally change.

It's not hard to see how this could happen. North Korea is so isolated and backwards that eventually, it must implode under the weight of the technological differential between it and the outside world. I mean, if it gets to the point where we have cheap fusion-powered laser-mounted microdrones, then a teenager, as a prank, might start tattooing American flags on the face of whichever spawn of Kim Il Sung sits atop their throne. Or South Korea could just bombard all their starving citizens with food packages that include a satellite-internet connected smartphone. That regime cannot withstand this kind of pressure.

So, the idea of the lifetime of evil has made a prediction, one that I think is likely to come to pass. Though it may still be decades away, policy makers in South Korea or the United States should be thinking about how to control and handle this coming change in North Korea.

Another way to validate the concept of lifetime of evil is to look at the lifetime of atheistic countries. All three of the examples cited above (Soviet Union, Maoist China, and North Korea) were officially atheistic, and right now it looks like North Korea may become the most successful atheistic country of the three, based on longevity. In general, atheistic states do not last. Throughout the entire history of the world, I have not heard of a single state that put its emphasis on atheism lasting more than a hundred years. From the anti-Christian movements in the French Revolution to the Khmer Rouge, none have stood the test of time. If atheism is true and good for human prosperity, where is the enlightened atheistic civilization that outproduces and outlives the other civilizations of the world? Where is the powerful atheistic technocracy that will conquer the world with its mighty army of nanobot terminators? It's not like atheism is a new or a complicated idea: there were atheists in the ancient world, so it's had plenty of time to rise and prosper if it's really the true path to prosperity. But the only places where atheists seem to be doing well nowadays is in the historically Christian nations of the West.

Are there other ways to validate the idea that evil has a finite lifetime, typically lasting several generation for a society-wide wickedness? I think so. One can hardly talk about a society-wide historical evil without talking about race-based chattel slavery. I do not intend to cover the whole sordid history of slavery in the Americas in this one post, but here are some important dates, from Wikipedia:

Why 60-120 years? Why three to four generations? That, as I said before, is mechanistically determined. And for a cultural, societal evil, the mechanism here is none other than the transmission of morals to the next generation. We know that parents play a huge role in how their children will turn out. We know that our experiences in our youth shape the rest of our lives. Is it any surprise that this is the key mechanism in determining the lifetime from an evil act to its full consequences, or that it would take about three generations for a new morality to be completely inculcated?

This process can perhaps most clearly be seen through an example about sexual morality. Sexuality, more so than our other activities, directly affects the next generation. For example, let's say that one generation somehow makes the mistake of thinking that vaginal sex is very wrong. It's considered "dirty" or "unnatural" compared to other forms of sex. Let's furthermore assume that this group of people persist in their mistake until something dramatic forces them to change or collapse. The first generation to make this mistake is presumably already sexually mature, and some of them already have children - the second generation. They then pass this twisted idea about sex on to the second generation. But do you expect to see any negative effects of the mistake at this point? Not really - the second generation already exists, but they're still just growing up, and won't become mature and sexually active for some time. The first real signs of trouble will come when this second generation starts trying to form serious relationships and having children of their own. The psychological hang-ups and the mechanical difficulties associated with their misconceptions will now directly affect their ability to produce kids - the third generation. And for any third generation children they manage to produce through their guilt and shame, this confusion about sexuality will be all they have known. They will grow up thinking that this mistaken viewpoint is normal, and that their second generation parent's difficulties were also normal. From here, it's a toss-up whether society collapses from population implosion, or whether they'll last long enough to have a fourth generation. Fifth and further generations are increasingly unlikely.

What are we to make of all this? In particular, what can we take away from the idea that the lifetime of evil is typically three or four generations? First, we can take comfort in the idea that evil will not endure forever. We can also remember this time frame, and measure out our responses accordingly. For example, if you believe that slavery was right after all, then you have virtually no hope - It's been abolished for 150 years with no sign that we'll ever go back to that model. You should give up. On the other hand, if you believe that gay marriage is wrong - well, we'll see. How will the debate look in 30, 60, or 120 years? Because that's the time period over which these things characteristically change. And lastly, if you do see the world making a terrible mistake, it tells you what you can expect in the future. If you're young and lucky, you may outlive the mistake, like a young Russian in 1922 possibly living 69 more years and outliving the Soviet Union. Even if you don't see something as dramatic as the collapse of communism, you may see enough to know that it will happen. But more practically, you should train up the next generation in the way they should go, in the discipline and instruction of the LORD. There is a very good chance that your children will live to see the mistake for what it is, and the world will need good men and women when it tries to recover.

You may next want to read:

The lifetime of evil (part 1) (Previous post of this series)

History, moral progress, and moral perfection (part 1)

Human laws, natural laws, and the Fourth of July

Another post, from the table of contents

The Bible says that the sins of one generation will affect their children down to the third and forth generations - so, something like 60-120 years. I find that this time interval is the approximate lifetime for a national or cultural evil that infects a whole society (which is what the Bible is addressing in the passage mentioned above). Put another way: this is characteristically how long it takes for a fundamentally wrong form of government to fail.

The late Soviet Union is a good instance of this rule. The "evil empire" lasted from 1922 to 1991 - only 69 years. As the fall of communism is within living memory, you may remember how surprising and sudden this event was. For many of us, it seemed that the bifurcation of the world into communist or capitalist countries was a permanent feature of history. After all, how could a country just... stop? But a flawed system cannot last forever, and the Soviet Union collapsed within a human lifetime.

Maoist China underwent much the same story. Communists in China came to power in 1949, and while they are still in power, their policy has changed dramatically since their first days under Mao in embracing capitalistic reforms. While they still have a ways to go, the rule in question seems to bear out: a flawed national system typically only lasts several generations. It may collapse or change, but it cannot endure.

Along the same lines of thinking, how long do you think North Korea will last? I don't think anyone needs convincing that the government there is evil. I've heard it described as "the worst country in the world" and "more 1984 than 1984". It's been that way since at least the end of the Korean War - 62 years ago as of 2015. Such evil cannot endure. I therefore have hope, based on the idea of the lifetime of evil, that I will live to see North Korea fundamentally change.

It's not hard to see how this could happen. North Korea is so isolated and backwards that eventually, it must implode under the weight of the technological differential between it and the outside world. I mean, if it gets to the point where we have cheap fusion-powered laser-mounted microdrones, then a teenager, as a prank, might start tattooing American flags on the face of whichever spawn of Kim Il Sung sits atop their throne. Or South Korea could just bombard all their starving citizens with food packages that include a satellite-internet connected smartphone. That regime cannot withstand this kind of pressure.

So, the idea of the lifetime of evil has made a prediction, one that I think is likely to come to pass. Though it may still be decades away, policy makers in South Korea or the United States should be thinking about how to control and handle this coming change in North Korea.

Another way to validate the concept of lifetime of evil is to look at the lifetime of atheistic countries. All three of the examples cited above (Soviet Union, Maoist China, and North Korea) were officially atheistic, and right now it looks like North Korea may become the most successful atheistic country of the three, based on longevity. In general, atheistic states do not last. Throughout the entire history of the world, I have not heard of a single state that put its emphasis on atheism lasting more than a hundred years. From the anti-Christian movements in the French Revolution to the Khmer Rouge, none have stood the test of time. If atheism is true and good for human prosperity, where is the enlightened atheistic civilization that outproduces and outlives the other civilizations of the world? Where is the powerful atheistic technocracy that will conquer the world with its mighty army of nanobot terminators? It's not like atheism is a new or a complicated idea: there were atheists in the ancient world, so it's had plenty of time to rise and prosper if it's really the true path to prosperity. But the only places where atheists seem to be doing well nowadays is in the historically Christian nations of the West.

Are there other ways to validate the idea that evil has a finite lifetime, typically lasting several generation for a society-wide wickedness? I think so. One can hardly talk about a society-wide historical evil without talking about race-based chattel slavery. I do not intend to cover the whole sordid history of slavery in the Americas in this one post, but here are some important dates, from Wikipedia:

1492: Columbus lands in the Americas.So, what are we to make of these dates? If we simply take the difference between the introduction of slavery into the Americas to the date of total emancipation, That comes to more than 300 years, and it seems to blow a hole in my theory of the lifetime of evil being 60-120 years. But remember, that time frame is for a comprehensive, society-wide evil, like the governments of Soviet Union or North Korea. Slavery leaked into America relatively slowly, and it seems to have not become a full-blown societal evil until the late 1600. Furthermore, early protest against it started as soon as became a major problem, and it was completely abolished in the Northern United States decades earlier than in the South. It was, on the whole, a complex phenomenon, with difficult dates to pin down for a whole geographical area. So, if we take the somewhat arbitrary date of 1690 as the threshold of "full-blown societal evil", we get about 110 years of slavery in the North, and 175 years for the whole of the U.S. - within the allowed time frame for a lifetime of evil, given the heinous and complicated nature of the slavery problem. Remember the analogy with a half-life: the 60-120 years is not an absolute time interval. It is rather a characteristic time frame.

1526: The first enslaved Africans came to what is now the United States, to a Spanish settlement.

1619: The first Africans are brought to English North America, to Virginia.

Throughout the 1500's and 1600's, slavery took on different forms - It was not all race-based chattel slavery, and many slaves were treated as indentured servants. It's also important to keep in mind the relatively smaller population of Europeans and Africans in these earlier days. Wikipedia says "evidence suggests that racial attitudes were much more flexible in 17th century Virginia than they would subsequently become".

1690: A census records 950 Africans in Virginia. Again, it's not yet a full-blown society wide problem, although by the late 1600's it was rapidly becoming so, with the loss of previously held rights and a dramatic increase in the importation of slaves.

1705: The basic legal framework for slavery is established in Virginia.

1710: The African population of Virginia increases to 23,100.

1776: American Independence. Around this time, many, but not all, colonies had banned or restricted slavery to differing degrees.

1787: Slavery is encoded into the U.S. Constitution.

1808: International importation of slaves into the U.S. is banned. All of the northern states pass anti-slavery laws by this time.

1865: End of the Civil War. Slavery is abolished throughout the U.S.

Why 60-120 years? Why three to four generations? That, as I said before, is mechanistically determined. And for a cultural, societal evil, the mechanism here is none other than the transmission of morals to the next generation. We know that parents play a huge role in how their children will turn out. We know that our experiences in our youth shape the rest of our lives. Is it any surprise that this is the key mechanism in determining the lifetime from an evil act to its full consequences, or that it would take about three generations for a new morality to be completely inculcated?

This process can perhaps most clearly be seen through an example about sexual morality. Sexuality, more so than our other activities, directly affects the next generation. For example, let's say that one generation somehow makes the mistake of thinking that vaginal sex is very wrong. It's considered "dirty" or "unnatural" compared to other forms of sex. Let's furthermore assume that this group of people persist in their mistake until something dramatic forces them to change or collapse. The first generation to make this mistake is presumably already sexually mature, and some of them already have children - the second generation. They then pass this twisted idea about sex on to the second generation. But do you expect to see any negative effects of the mistake at this point? Not really - the second generation already exists, but they're still just growing up, and won't become mature and sexually active for some time. The first real signs of trouble will come when this second generation starts trying to form serious relationships and having children of their own. The psychological hang-ups and the mechanical difficulties associated with their misconceptions will now directly affect their ability to produce kids - the third generation. And for any third generation children they manage to produce through their guilt and shame, this confusion about sexuality will be all they have known. They will grow up thinking that this mistaken viewpoint is normal, and that their second generation parent's difficulties were also normal. From here, it's a toss-up whether society collapses from population implosion, or whether they'll last long enough to have a fourth generation. Fifth and further generations are increasingly unlikely.

What are we to make of all this? In particular, what can we take away from the idea that the lifetime of evil is typically three or four generations? First, we can take comfort in the idea that evil will not endure forever. We can also remember this time frame, and measure out our responses accordingly. For example, if you believe that slavery was right after all, then you have virtually no hope - It's been abolished for 150 years with no sign that we'll ever go back to that model. You should give up. On the other hand, if you believe that gay marriage is wrong - well, we'll see. How will the debate look in 30, 60, or 120 years? Because that's the time period over which these things characteristically change. And lastly, if you do see the world making a terrible mistake, it tells you what you can expect in the future. If you're young and lucky, you may outlive the mistake, like a young Russian in 1922 possibly living 69 more years and outliving the Soviet Union. Even if you don't see something as dramatic as the collapse of communism, you may see enough to know that it will happen. But more practically, you should train up the next generation in the way they should go, in the discipline and instruction of the LORD. There is a very good chance that your children will live to see the mistake for what it is, and the world will need good men and women when it tries to recover.

You may next want to read:

The lifetime of evil (part 1) (Previous post of this series)

History, moral progress, and moral perfection (part 1)

Human laws, natural laws, and the Fourth of July

Another post, from the table of contents