I attend a church called Tribe, in Berkeley, California. You can find our website at tribesites.org.

I obviously think that my church is pretty great - I've been going there for about a decade now. I'd like to invite anyone who reads this post to come by for a Sunday service. Everyone is welcome, and you're especially welcome if you're a student at U.C. Berkeley. If you can't make it yourself but know a friend or a relative who's new to the Bay Area, feel free to suggest us to them.

We have lots of families, and a good number of young professionals and college students. The large number of families also mean a large number of kids - we have babies and toddlers and students at every level from elementary school to high school. All told, I think we have a little less than a hundred people in the congregation.

We're located at 2509 Hillegass Ave, Berkeley, CA. That's literally just a stone's throw away if you're a Berkeley student living in Unit 2 (a U.C. Berkeley Dorm). Our service starts at 11AM, and we sometimes serve lunch afterwards.

We are a church that believes in the centrality of Christ, of his person and his teachings. We furthermore believe that the Bible is unequivocally true and reliable - a revelation from God concerning Christ and everything else we need to know about what God wants from us.

There are ample opportunities to serve at our church, from helping out with the kids in Sunday School to working with an independent non-profit group that works closely with our church. We're always looking for volunteers and there's always work to be done.

Also, now would be a good time to point out that this blog has no official connection to the church. This is just my personal blog, and I'm just a guy who goes to my church. Although you can obviously infer things about a church from the kind of people who go there, you shouldn't assume that I speak for the church in any of the things I've said here on my blog.

If my description of the church sounds right for you, come on by!

"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 likewise 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 riches of God with your thoughts. Still, exercise and stretch out your mind: you are safe in God's guarantee that he is at least as awesome as anything that we can imagine.

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?

a) 0 mph
b) 25 mph
c) 50 mph
d) 75 mph
e) 100 mph
f) infinitely fast

Think 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:

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.

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 for the problem.

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, after you've picked an envelope but 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 $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 f(x) can be completely general, but 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:

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,