What is the probability that Jesus rose from the dead?
Here I'm going to construct a foolish partner to advance certain arguments. This is just a rhetorical device. I have to be careful to not commit a straw man here, nor do I wish to insult anyone. I don't intend to imply that anyone actually thinks like my partner. But while he's too foolish to actually stand in for any real person, he can therefore be useful, by standing in as the lower bound on what a reasonable person may think. Please, just understand him as the artificial rhetorical construction that he is.
Now, my foolish partner may say, "the probability that Jesus rose from the dead is zero. What's there to talk about?" But by doing so, he has committed the cardinal sin in Bayesian reasoning. Any real, non-theoretical probability CANNOT be absolutely zero or one. Think about what a zero probability value means: this represents a state of mind where absolutely NOTHING - no amount of even theoretically possible evidence - can alter their beliefs. There is no possible reasoning with such a person.
I am very certain that the sun will rise tomorrow. I may be 99.9999...% certain, but I cannot be 100% certain. That tiny difference between 99.9999...% and 100% represents possibilities like a super-advanced alien race stopping the rotation of the earth, or me being momentarily confused about what is meant by "the sun". And I am not 100% certain, because I can, at least in theory, be shown evidence that such an alien race exists, or that I had momentarily confused "the sun" with "the north star".
My partner may then say, "well, the probability may not be actually zero, but it's very close to it. Like, 0.000.....001%. Nobody has ever come back from the dead before." But actually, isn't that the very thing we're talking about? Whether Jesus had come back from the dead? Furthermore, it's presumptuous to think Jesus was just like everyone else, that he wasn't special in any way. Even if nobody else came back from the dead, we would need to do some additional thinking in the case of Jesus.
My partner would reply, "see, that's just special pleading. I don't see why Jesus should be special. Empirically, people do not come back from the dead. Therefore it's also highly unlikely that Jesus came back."
At this point, I'm going to simply give away the point about whether Jesus was special or not. I obviously believe that he was - but quite frankly, the argument for the resurrection is so strong that I can just handicap myself in several different ways like this without materially affecting the conclusion. I'll be doing this multiple times throughout this post.
Now, let's talk about how many people came back from the dead, "empirically". How many different people have you seen die and stay dead? Remember, we're talking about "empirical" evidence here, meaning that we only count people that you, yourself, have seen die in person. For many people, that number is probably zero. It might be one or two - maybe you've seen a grandparent pass away. Maybe more, if you're a health care worker or something like that.
My partner may say, "Even if I didn't see someone die in person, if there was a real resurrection, it would be all over the news. And there hasn't been any such reports, because people do not rise from the dead."
Well, at this point, my partner is begging the question on whether there has in fact been such reports, and is becoming slippery about what "empirically" means. But again, I will simply handicap myself and give away these points. "Empiricism" in the sense of "I only believe what I can see" is fundamentally flawed, anyway (It's self-defeating). So let's say news reports are enough, that a direct observation is not necessary. So, how many people have been covered in the news that you've seen? Thousands? Millions? If the argument is that Jesus was no different than these thousands or millions of other people, then I freely acknowledge that this does in fact establish an upper bound on the probability of the resurrection. However, this does NOT prove that the probability is zero, no more than a dozen coin flip of heads proves that the coin will always land heads. Instead, it merely says that the probability for the resurrection is likely to be below a certain level.
For example, say that you've examined a thousand swans, and they all turned out to be white. You want to use this fact to investigate the report of a black swan. Now, your thousand white swans don't prove that the probability of the reported swan being black is zero. Instead, combined with that report, it does say that the probability is likely to be below 1/1000. If you've examined a million swans, and all of them have been white, then your probability of observing a black swan would correspondingly drop to around 1/1 000 000 as the upper limit.
Now, the modern media is pretty comprehensive, so my partner may say, "The world news covers at least millions of other people. And none of them have come back from the dead. So the chance that Jesus came back from the dead is, at best, one in a million. That's basically zero. How could you believe in something that has only one in a million chance of being true? That's irrational."
Well, one in a million is a pretty small probability. But actually, I think we can just go ahead and say that out of the entire world population of 7 billion people, none of them are going to be raised from the dead. So, the probability for the resurrection has now dropped to 1 in 7 billion. I'm just giving away everything here. I've almost dropped the condition about an "empirical" probability. I'm making a blanket statement that absolutely nobody in the world, independent of anything that may be know about them, will rise from the dead. So, if we apply this general "observation" to the likelihood of Jesus's resurrection, that probability must be below 1 in 7 billion.
My partner may respond, "Um... So now you're making my argument for me. So yeah. The probability of the resurrection is less than 1 in 7 billion. Obviously you can't believe in something that unlikely to be true. This is why any naturalistic explanations must always be preferred to a supernatural one in these discussions of miracles, because the supernatural is always so unlikely."
Oh, but I'm not done yet. I'm going to give away even more of the argument. Why not just drop all pretense of an "empirical" probability? Why not say that everyone who has EVER lived - about 100 billion people in total - have all died, without a single one of them being raised from the dead? Forget saying anything about "empirical observations". Forget any semblance of reasoning from our direct experiences. I will simply grant that every single one of these 100 billion people have died and stayed dead. And against the weight of those 100 billion people, we'll estimate the probability of Jesus's resurrection. According to our previous line of thinking, this puts that probability at 1 out of 100 billion.
My foolish partner may say, upon the strength of this evidence that I have made up for him, "One in a hundred billion! Do you know how unlikely that is? That's 1 out of 100 000 000 000. That's a probability value of 0.000 000 000 01. That's basically zero. Just concede the argument - it's virtually impossible that Jesus rose from the dead. Absolutely any naturalistic explanation is going to be more likely than that." Oh, but I'm not done yet.
I'm going to give away another multiplier in the probability. I'm going to make it even smaller - not by an additional factor of ten, or even by a factor of a thousand. No, I'm going to give away far more. I'm going to SQUARE that tiny previous probability of 1 in 100 billion, and use that as the probability of Jesus's resurrection. One in a hundred billion, squared, is this:
probability = 1/10 000 000 000 000 000 000 000, or 0.000 000 000 000 000 000 000 1
There is no reason to do this. Squaring the probability makes no rational sense. I did it just to make the probability smaller, to handicap my argument further. I started with the "Nobody rises from the dead. It's never happened before. So Jesus also didn't rise from the dead" argument. I then stretched it to its strongest form, then started making stuff up to make it even stronger. I then ran out of stuff to make up, but I still wanted it to be even stronger - so I simply squared the already tiny probability value, with no possible rationale, to arrive at the absurdly minuscule probability value above. So now, as it stands, the probability of Jesus actually having risen from the dead is 1 out of 10 000 000 000 000 000 000 000 - essentially zero. That's game over, right? How could I, or anyone, believe in something so unlikely to be true? How could any hypothesis with a probability of 0.000 000 000 000 000 000 000 1 ever be taken seriously?
"Um... so yeah. What are you doing?", my partner may ask.
You'll see. Next week, you'll behold and understand the power of evidence.
You may next want to read:
Basic Bayesian reasoning: a better way to think (Part 1)
Miracles: their definition, properties, and purpose
Another post, from the table of contents
Bayesian evaluation for the likelihood of Christ's resurrection (Part 1)
Let's say that you're meeting someone new. You talk for a while, and the conversation turns to birthdays. You reveal that you were born in January, and your new friend says, "Oh, really? I was born in January too!" He seems earnest - he's not obviously joking, sarcastic, or ingratiating. From the little you know of him, he's not any more likely to be delusional or deceptive than anyone else.
Now, based only on his earnest word, would you be willing to believe that your new friend really was born in January? Note that I'm not looking for 100% certainty here. A willingness to entertain the idea, to give it at least a 50-50 shot of being true, is all that's required.
Also note that I'm not asking whether this event is likely to happen. Obviously, the probability that you and a random other person shares the same birth month is about 1/12, so it may be said to be "unlikely". Rather, I'm asking whether you would believe this person, given that this unlikely event has already occurred.
So, how would you respond? Would you say, "I find your claim to be highly dubious. There's only 1/12 chance that you were born in the same month as me"? Or would you simply reply, "Oh, hey, that's neat!"
I'm going to assume that you're willing to believe your new friend. I think you'll agree that it takes a special kind of jerk to say "I don't believe you. You must be lying or mistaken. It's just too unlikely for us to share the same birth month". In that case, what if it turns out that you share the exact same birthday? You mention that you were born January 23rd, and he claims the same. Would you still believe him?
Let's continue the same line of thought: what if you then tell him your mother's birthday, and what do you know - it turns that the date is also his mother's birthday! Your fathers, too, turn out to share a birthday. "Wow", he says, "so the three members of our family all share the same birthdays - amazing!" Would you be willing to believe him on this?
If so, at what point in comparing family birthdays would it become too unlikely for you to believe? That is, if you continued on to compare your grandparents and uncles and cousins, and they all continued to have the same birthdays, at what point would you say "I cannot believe this - this is too unlikely to be true", in spite of your friend's earnest insistence?
Decide on an answer, and remember it. Write it down somewhere. We'll come back to this answer in the coming weeks. Make a firm statement like, "I would be willing to believe up to 3 shared birthdays - myself, mother, and father - but if he claims 4 or more shared birthdays I would begin to be skeptical".
Let's try another example. Let's say that you run into an acquaintance whom you haven't seen in a while. You exchange greetings and ask how he's been recently, and he excitedly tells you - "Guess what! I've actually won the jackpot in the lottery last month! I'm rich!" As before, he seems earnest - he's not obviously joking, sarcastic, delusional, or deceptive. Would you believe him, based only on his earnest word? Again, only a willingness to entertain the idea, just granting a 50 - 50 chance of it being true, is all we're looking for here. Would you give him at least even odds that he's telling the truth?
And if you would, how about if he claims to have won two consecutive jackpots? How about three? At which point would you say "That's just too much for me to believe"?
Next, let's switch over to other gambling games. Say that a friend claims to have had a very lucky night at the card tables. He says that he got a royal flush in a 5-card stud poker game. Would you believe him? What if he claims to have gotten two royal flushes last night? What if he claims three? At what point would you say, "I don't believe you. You seem earnest and all, but the chances of that happening are just too small"?
How about if he were playing Texas Hold'em, and claims to have had multiple pocket aces? Say that he claims to have had two, three, four, or five pocket aces last night. At what number does it become too unlikely to be true, despite your friend's earnest claim?
We can ask similar kinds of questions in many different ways. What if someone claims to be born as a part of twins, triplets, quadruplets, or quintuplets? What if someone claims to have recently been struck by lightning? Or that they were a victim of two or three such strikes?
Remember, in all these cases, that we're not looking for certainty. Just a willingness to grant even odds - a 50-50 likelihood for the statement is true - is enough to say that you'd believe your friend. Also, we're not asking whether these scenarios are likely; rather we're asking if you'd continue to believe this earnest person, despite the fact that he's claiming that an unlikely event happened.
Answer these questions. Give a specific number in each case: we want answers like "four royal flushes" and "two lightning strikes". Write them down somewhere - we'll come back to them later.
Next week, we'll turn to the question of Jesus's resurrection.
You may next want to read:
Basic Bayesian reasoning: a better way to think (Part 1)
The role of evidence in the Christian faith
Another post, from the table of contents
Now, based only on his earnest word, would you be willing to believe that your new friend really was born in January? Note that I'm not looking for 100% certainty here. A willingness to entertain the idea, to give it at least a 50-50 shot of being true, is all that's required.
Also note that I'm not asking whether this event is likely to happen. Obviously, the probability that you and a random other person shares the same birth month is about 1/12, so it may be said to be "unlikely". Rather, I'm asking whether you would believe this person, given that this unlikely event has already occurred.
So, how would you respond? Would you say, "I find your claim to be highly dubious. There's only 1/12 chance that you were born in the same month as me"? Or would you simply reply, "Oh, hey, that's neat!"
I'm going to assume that you're willing to believe your new friend. I think you'll agree that it takes a special kind of jerk to say "I don't believe you. You must be lying or mistaken. It's just too unlikely for us to share the same birth month". In that case, what if it turns out that you share the exact same birthday? You mention that you were born January 23rd, and he claims the same. Would you still believe him?
Let's continue the same line of thought: what if you then tell him your mother's birthday, and what do you know - it turns that the date is also his mother's birthday! Your fathers, too, turn out to share a birthday. "Wow", he says, "so the three members of our family all share the same birthdays - amazing!" Would you be willing to believe him on this?
If so, at what point in comparing family birthdays would it become too unlikely for you to believe? That is, if you continued on to compare your grandparents and uncles and cousins, and they all continued to have the same birthdays, at what point would you say "I cannot believe this - this is too unlikely to be true", in spite of your friend's earnest insistence?
Decide on an answer, and remember it. Write it down somewhere. We'll come back to this answer in the coming weeks. Make a firm statement like, "I would be willing to believe up to 3 shared birthdays - myself, mother, and father - but if he claims 4 or more shared birthdays I would begin to be skeptical".
Let's try another example. Let's say that you run into an acquaintance whom you haven't seen in a while. You exchange greetings and ask how he's been recently, and he excitedly tells you - "Guess what! I've actually won the jackpot in the lottery last month! I'm rich!" As before, he seems earnest - he's not obviously joking, sarcastic, delusional, or deceptive. Would you believe him, based only on his earnest word? Again, only a willingness to entertain the idea, just granting a 50 - 50 chance of it being true, is all we're looking for here. Would you give him at least even odds that he's telling the truth?
And if you would, how about if he claims to have won two consecutive jackpots? How about three? At which point would you say "That's just too much for me to believe"?
Next, let's switch over to other gambling games. Say that a friend claims to have had a very lucky night at the card tables. He says that he got a royal flush in a 5-card stud poker game. Would you believe him? What if he claims to have gotten two royal flushes last night? What if he claims three? At what point would you say, "I don't believe you. You seem earnest and all, but the chances of that happening are just too small"?
How about if he were playing Texas Hold'em, and claims to have had multiple pocket aces? Say that he claims to have had two, three, four, or five pocket aces last night. At what number does it become too unlikely to be true, despite your friend's earnest claim?
We can ask similar kinds of questions in many different ways. What if someone claims to be born as a part of twins, triplets, quadruplets, or quintuplets? What if someone claims to have recently been struck by lightning? Or that they were a victim of two or three such strikes?
Remember, in all these cases, that we're not looking for certainty. Just a willingness to grant even odds - a 50-50 likelihood for the statement is true - is enough to say that you'd believe your friend. Also, we're not asking whether these scenarios are likely; rather we're asking if you'd continue to believe this earnest person, despite the fact that he's claiming that an unlikely event happened.
Answer these questions. Give a specific number in each case: we want answers like "four royal flushes" and "two lightning strikes". Write them down somewhere - we'll come back to them later.
Next week, we'll turn to the question of Jesus's resurrection.
You may next want to read:
Basic Bayesian reasoning: a better way to think (Part 1)
The role of evidence in the Christian faith
Another post, from the table of contents
Finding pi in a square grid: or, why you can have square brownies for pi day
This is a picture I took on the side of a road:
The trees are planted in a regular pattern, in basically square grids. This creates the fascinating picture you see when you then look at the trees from the sides. You can see far off into the distance along some angles, but are blocked from seeing too far in other directions. The effect can be mesmerizing, especially when you drive past these fields on the highway.
While I was on one such drive, I began to wonder: what exactly is the pattern I'm looking at here? In particular: what is the probability that you can see one tree from another tree?
Let's nail down the specifics of the question: If you and a friend are in an infinite square grid of trees, and each of you choose to stand at a random tree, what is the probability that you two will be able to see each other? For the purposes of this question, we'll say that you can see each other if there are no trees between the two trees you've chosen. That is, one must be able to draw a straight, line-of-sight segment from one point on the square grid, to the other point on the grid, without any intervening points getting in the way.
I'm going to give away the answer, then explain how I got it. The probability is 6/π2, or about 61%.
That's right - you can get pi out of a problem that only has squares in it. There is no need to involve circles.
Let's begin the solution by saying that the tree you're standing at is the origin, represented by the coordinate (0,0). The x and y axis can then be arbitrarily assigned along the edges of the squares that make up the grid. All this can be done with no loss in generality.
The tree that your friend is standing on can then be assigned the coordinates (x,y). Now, can you see the point (x,y) from the origin? Well, is there a straight line connecting the two with no intervening points?
This is possible if x and y are coprime - that is, if they share no common factors other than 1. For example, 8 and 15 are coprime. So if your friend was standing at (8,15) or (15, 8), you would be able to see each other. But 2 and 2 are not coprime, because they're both divisible by 2. And if your friend were standing at (2,2), the point at (1,1) would get between you two, and you would not be able to see each other.
So now, the question has become "what is the probability that two randomly chosen integers are coprime"? Well, let's start by remembering the definition of "coprime" - that there can't be any natural number other than 1 that are factors of both integers.
Se we start by looking at 2 as a factor. We eliminate all the number pairs that are both divisible by 2, like (4,8) or (2,10). How many such pairs are there? Well, since the probability of a number being divisible by 2 is 1/2, the probability of both numbers being divisible by 2 is 1/22, or 1/4. After eliminating these, we have a remaining probability of 1 - 1/4 = 3/4. This is the probability of a number pair passing through the first filter, of them not both having 2 as a factor.
We next look at 3 as a factor, and eliminate all the number pairs that are both divisible by 3, like (30,81). As before, the probability of one number being divisible by 3 is 1/3, so the probability of both being divisible is 1/32 = 1/9, and after eliminating these, we have a probability of 1 - 1/9 = 8/9 left. This is the probability of a number pair passing through the second filter, of them not both having 3 as a factor.
Now, to combine both of the above to get a single probability value, we just multiply them: a number pair has a probability of 3/4 for passing through the first filter (having 2 as a factor), and 8/9 for passing through the second filter (having 3 as a factor). So the probability of passing through both would be 3/4 * 8/9.
But wait - can we do that? Sure, the probability of both numbers being multiples of 3 is 1/9, but doesn't that change when we condition it upon them also not being multiples of 2? We can't confuse conditional and unconditional probabilities, otherwise we'll make a mistake!
Actually, in this case we don't have to worry. In general, if p and q are prime numbers, the only way for a number to be a multiple of p and q is for it to be a multiple of p*q. In other words, if we look at the multiples of p in all the numbers going up to p*q, there are q such numbers, but only 1 number that's also a multiple of q: p*q itself. So there's q -1 other multiples of p that are NOT multiples of q.
Now, again looking at numbers up to p*q, there are p multiples of q. That means there are p*q - p numbers that are NOT multiples of q.
For the numbers larger than p*q, the pattern just repeats itself.
So, the probability of a number being a multiple of p, conditioned upon it not being a multiple of q, is given by (q - 1) / (pq - p) = 1/p. This is the same as the unconditioned probability.
Confused? Try it with 2 and 3. In numbers going up to 6, there are 2 numbers that are multiples of 3 (3 and 6). Among these, only one of them (6 itself) is also a multiple of 2. That leaves 2 - 1 = 1 number that's a multiple of 3 but not of 2.
But we also know that there are 3 numbers that are a multiple of 2 (2, 4, and 6). That means there are 6 - 3 = 3 numbers that are NOT a multiple of 2.
So, the probability of a number being a multiple of 3, conditioned upon it not being a multiple of 2, is 1/3. Exactly the same as the unconditioned probability.
Therefore the probability of having a particular prime number as a factor can be treated as independent probabilities, and since this is true for any two pair of prime numbers, the values derived from unconditioned probabilities can all be multiplied together.
So now, the only thing left to do to find the probability of two numbers being coprime is to simply go through all the possible numbers that can factor them both and eliminate those possibilities. We've already said that the probability of both numbers having 2 as a factor is 1/22, and the probability of both numbers having 3 as a factor is 1/32. The probability then of NOT having either 2 or 3 as a common factor is:
(1 - 1/22)(1 - 1/32)
We then extend this pattern for all numbers beyond 3 - except we can skip composite numbers, because if a number is not divisible by a prime number, it's guaranteed to not be divisible by any composite number with that prime number as a factor. So we don't have to check non-divisibility by 4, because we've already checked non-divisibility by 2.
Then the extended expression becomes:
probability = (1 - 1/22)(1 - 1/32)(1 - 1/52)(1 - 1/72)(1 - 1/112)(1 - 1/132)(1 - 1/172) ....
= product(1 - 1/p2), where p is an element in the set of all prime numbers.
This infinite product is a well-known expression that you can just look up, and it evaluates to 6/π2. That evaluation was first solved by Leonhard Euler, who was one of the greatest mathematician to have ever lived.
So, pi is far more profound than simply being the ratio of a circle's circumference to its radius. It comes up in all kinds of unexpected places, like in problems relating square grids or prime numbers.
I wish you a happy pi day! And feel free to eat square-shaped things today, like some square brownies, or a rectangular pizza you bring to a pi day potluck. You're mathematically justified!
You may next want to read:
The two envelopes problem and its solution
How to think about the future (Part 1)
Another post, from the table of contents
The trees are planted in a regular pattern, in basically square grids. This creates the fascinating picture you see when you then look at the trees from the sides. You can see far off into the distance along some angles, but are blocked from seeing too far in other directions. The effect can be mesmerizing, especially when you drive past these fields on the highway.
While I was on one such drive, I began to wonder: what exactly is the pattern I'm looking at here? In particular: what is the probability that you can see one tree from another tree?
Let's nail down the specifics of the question: If you and a friend are in an infinite square grid of trees, and each of you choose to stand at a random tree, what is the probability that you two will be able to see each other? For the purposes of this question, we'll say that you can see each other if there are no trees between the two trees you've chosen. That is, one must be able to draw a straight, line-of-sight segment from one point on the square grid, to the other point on the grid, without any intervening points getting in the way.
I'm going to give away the answer, then explain how I got it. The probability is 6/π2, or about 61%.
That's right - you can get pi out of a problem that only has squares in it. There is no need to involve circles.
Let's begin the solution by saying that the tree you're standing at is the origin, represented by the coordinate (0,0). The x and y axis can then be arbitrarily assigned along the edges of the squares that make up the grid. All this can be done with no loss in generality.
The tree that your friend is standing on can then be assigned the coordinates (x,y). Now, can you see the point (x,y) from the origin? Well, is there a straight line connecting the two with no intervening points?
This is possible if x and y are coprime - that is, if they share no common factors other than 1. For example, 8 and 15 are coprime. So if your friend was standing at (8,15) or (15, 8), you would be able to see each other. But 2 and 2 are not coprime, because they're both divisible by 2. And if your friend were standing at (2,2), the point at (1,1) would get between you two, and you would not be able to see each other.
So now, the question has become "what is the probability that two randomly chosen integers are coprime"? Well, let's start by remembering the definition of "coprime" - that there can't be any natural number other than 1 that are factors of both integers.
Se we start by looking at 2 as a factor. We eliminate all the number pairs that are both divisible by 2, like (4,8) or (2,10). How many such pairs are there? Well, since the probability of a number being divisible by 2 is 1/2, the probability of both numbers being divisible by 2 is 1/22, or 1/4. After eliminating these, we have a remaining probability of 1 - 1/4 = 3/4. This is the probability of a number pair passing through the first filter, of them not both having 2 as a factor.
We next look at 3 as a factor, and eliminate all the number pairs that are both divisible by 3, like (30,81). As before, the probability of one number being divisible by 3 is 1/3, so the probability of both being divisible is 1/32 = 1/9, and after eliminating these, we have a probability of 1 - 1/9 = 8/9 left. This is the probability of a number pair passing through the second filter, of them not both having 3 as a factor.
Now, to combine both of the above to get a single probability value, we just multiply them: a number pair has a probability of 3/4 for passing through the first filter (having 2 as a factor), and 8/9 for passing through the second filter (having 3 as a factor). So the probability of passing through both would be 3/4 * 8/9.
But wait - can we do that? Sure, the probability of both numbers being multiples of 3 is 1/9, but doesn't that change when we condition it upon them also not being multiples of 2? We can't confuse conditional and unconditional probabilities, otherwise we'll make a mistake!
Actually, in this case we don't have to worry. In general, if p and q are prime numbers, the only way for a number to be a multiple of p and q is for it to be a multiple of p*q. In other words, if we look at the multiples of p in all the numbers going up to p*q, there are q such numbers, but only 1 number that's also a multiple of q: p*q itself. So there's q -1 other multiples of p that are NOT multiples of q.
Now, again looking at numbers up to p*q, there are p multiples of q. That means there are p*q - p numbers that are NOT multiples of q.
For the numbers larger than p*q, the pattern just repeats itself.
So, the probability of a number being a multiple of p, conditioned upon it not being a multiple of q, is given by (q - 1) / (pq - p) = 1/p. This is the same as the unconditioned probability.
Confused? Try it with 2 and 3. In numbers going up to 6, there are 2 numbers that are multiples of 3 (3 and 6). Among these, only one of them (6 itself) is also a multiple of 2. That leaves 2 - 1 = 1 number that's a multiple of 3 but not of 2.
But we also know that there are 3 numbers that are a multiple of 2 (2, 4, and 6). That means there are 6 - 3 = 3 numbers that are NOT a multiple of 2.
So, the probability of a number being a multiple of 3, conditioned upon it not being a multiple of 2, is 1/3. Exactly the same as the unconditioned probability.
Therefore the probability of having a particular prime number as a factor can be treated as independent probabilities, and since this is true for any two pair of prime numbers, the values derived from unconditioned probabilities can all be multiplied together.
So now, the only thing left to do to find the probability of two numbers being coprime is to simply go through all the possible numbers that can factor them both and eliminate those possibilities. We've already said that the probability of both numbers having 2 as a factor is 1/22, and the probability of both numbers having 3 as a factor is 1/32. The probability then of NOT having either 2 or 3 as a common factor is:
(1 - 1/22)(1 - 1/32)
We then extend this pattern for all numbers beyond 3 - except we can skip composite numbers, because if a number is not divisible by a prime number, it's guaranteed to not be divisible by any composite number with that prime number as a factor. So we don't have to check non-divisibility by 4, because we've already checked non-divisibility by 2.
Then the extended expression becomes:
probability = (1 - 1/22)(1 - 1/32)(1 - 1/52)(1 - 1/72)(1 - 1/112)(1 - 1/132)(1 - 1/172) ....
= product(1 - 1/p2), where p is an element in the set of all prime numbers.
This infinite product is a well-known expression that you can just look up, and it evaluates to 6/π2. That evaluation was first solved by Leonhard Euler, who was one of the greatest mathematician to have ever lived.
So, pi is far more profound than simply being the ratio of a circle's circumference to its radius. It comes up in all kinds of unexpected places, like in problems relating square grids or prime numbers.
I wish you a happy pi day! And feel free to eat square-shaped things today, like some square brownies, or a rectangular pizza you bring to a pi day potluck. You're mathematically justified!
You may next want to read:
The two envelopes problem and its solution
How to think about the future (Part 1)
Another post, from the table of contents
The want of a mate
"I hate this", said Adam, to his empty dining room. It was late at night. He was alone. His only company at his table were several bottles of wine and his glass. He was just a drunk man, talking to himself out of the anguish of his latest break-up.
So it was with mild surprise that he heard someone reply, "hate enough to do something about it?" He looked up, and he would have sworn that the alcohol was making him hallucinate - except that drinking had never affected him that way before. What he saw across his table was a man - dressed in red tights with horns on his head and a pitchfork on his lap, twirling both his black mustache and his pointed tail. He looked like he'd just stepped out of a cartoon.
The alcohol might not have been causing a hallucination, but it did alter Adam's response. Rather than screaming or freaking out, he instead found himself replying, "what do you know, man? What do you know about how hateful women can be?"
"Oh, believe me, I do know. Let's say that... I'm not unfamiliar with the nature of women. Women and I go way back." said the other man, in a smarmy yet coarse and haughty voice. Adam went on with his drunken rant: "But Ashley, man, Ashley. How could she... She's not just any other women. She's stupid. And cruel. More than any other women. I hate her."
"Oh, you mean me?" There was a 'poof' across the table, and the man in red tights had transformed himself into a pretty young women. "Ashley!" shouted Adam, for it was her - with that arc in her nose and her hair framing her cheekbones. "Ashley! I love you! How could you - just dump me like that? And why on Valentine's day? What did I do? Why can't we stay together?"
'Ashley' opened her pink lips and spoke, but the voice that came out was the same smarmy, coarse, and haughty voice of the man in red. And it said, "Because, hon, you're just not good enough. You're kinda cute from some angles, but not really good looking, you know? And I regularly get hit on by other guys who have more money than you. They're always more interesting, too - you got boring pretty quickly. Basically, I just thought I could do much better."
Adam was momentarily stunned into silence. He didn't know whether he would next explode in anger or start crying his eyes out. But before anything else could happen, 'Ashley' transformed back into the man in red, and he said, "Oh come on, I did you a favor. Would you rather that I tell you again all the lies you've already heard? 'Oh, it's not you, it's me', 'Oh, I just don't think we'll work out', 'Oh, let's stay friends'? Bah. You needed to hear the truth, and that's what I just told you. I did you a favor!"
Adam's eyes narrowed, and his mind seemed to focus a bit through his drunken haze. He said:
"What do you want? You're not in the business of doing favors."
"Oh? What have you heard about me?"
"Nothing good. That you tempt people to destroy their souls."
"Oh please. You can't believe everything you hear. Is it really so hard to believe that I've come to help you?" As the man said this, he grow white wings and a halo appeared over his head, and his clothing became white. But upon closer inspection, the halo was clearly held up by some metal wires extending up from his back, and the wings were poorly fitted props that was cramped awkwardly between the chair and the man's back. The man, now in white, looked like he was at a masquerade party in a cheap angel's costume.
"Look, I really did come here to do you a favor - to offer you a gift!" continued the man, in his smarmy, coarse, and haughty voice, twirling his mustache the whole time. "How would you like to have any woman you want?"
"What's the catch?"
"Come on now. You're hurting my feelings! There is no catch."
"You're lying. I bet it's something like all the women will go completely crazy fighting over me, so that my life is ruined because of the madness and the drama. Right?"
"No, not at all. Not all women will go after you - just the ones that you show affection to. And they'll only be in love with you "normally" - there's no diabolical super-powered obsession involved."
"Oh? Then is this some scheme to trap me or corrupt me through sex? Where I can have any women that I want so that I get completely addicted to perpetual sex with supermodels and somehow ruin myself?"
"Not at all. I mean, you can choose to use your gift in that way, and I won't stop you. To be honest, perpetual sex with supermodels is pretty nice. But the gift will also let you settle down with a nice family, if that's what you want. The choice is yours."
"'Nice family'? You probably mean that my kids will be your spawn and they'll grow up to be mass murderers who begin their killing spree by stabbing me. That's your idea of a 'nice family', isn't it?"
"Please. Any wife or kids you choose to have will be perfectly normal. I mean, they might be crazy, but that's not any more likely than for anyone else. There's nothing in my gift that will specifically turn them that way. You want your nice, suburban house and your white picket fence and a devoted, loving wife to grow old with in material comfort? My gift allows for that. It will even let you have up to two kids and a dog and a well-manicured lawn if you want. Nothing in my gift prevents any of that."
"What if my ex-girlfriends go crazy because they're still in love with me? Are you going to make them go insane with jealously and become ax-murdering lunatics who come after me?"
"Again, no. The exes you acquire through this gift are no more likely to go crazy than anyone else's ex-girlfriends. Look, do you know who I am? Do you honest think that if I wanted to screw you, I'd have to trick you in some way? I'm offering you this gift, because I really want you to have it. Because I really want to see you embrace it wholeheartedly, for your own enjoyment and benefit. Is that really so hard to believe?"
"Yes. Why are you being so - generous? Why offer me this gift at all?"
"I already explained why. I earnestly want to see you accept the gift and use it. And why shouldn't I be generous? I am powerful: I can give what gifts I please, if only to just demonstrate that I can. In fact, I think I'll do that right now: in addition to giving this gift to you, I'll pass on your exact same gift to any children you might have. They'll be marked by the gift from birth, and when they reach the age that you are now, I'll activate for them the ability to have any sexual partner they want. How's that for generous?"
"Seriously? There's no catch? The women I show affection towards won't suddenly become monstrously ugly? This "mark" on my future kids isn't some hideous birthmark, and it won't make them turn evil? I won't become impotent or change sexual orientation as soon as I get in bed with these women? My natural life span won't be shortened as a result of this gift?"
"No, no, no, and no. I've told you exactly what my gift will do. There are no such unexpected, additional side effects you need to fear."
"You're still hiding something. There's some fine print you're not telling me."
"No, I've told you everything, in perfectly clear language. We've gone over all of the abilities, powers, restrictions, limitations, and conditions on your gift. The only thing you have to do, is to say that you accept. Do you?"
The question hung in the air. Adam wasn't sure whether he had become more sober or more confused since this conversation began. He thought about his answer for a bit - but it was late, and he was tired, drunk, and emotionally unstable. He thought he might not remember anything the next morning.
As it turned out, he ended up remembering quite a bit - he remembered saying "I accept". He remembered much of the details of the deal. But he could not remember what it had been that was nagging at the back of his mind about those details. And so, he never came to understand the nature or the magnitude of his mistake.
You may next want to read:
How to think about the future
Make the most of your time and your life. Number your days.
Another post, from the table of contents
So it was with mild surprise that he heard someone reply, "hate enough to do something about it?" He looked up, and he would have sworn that the alcohol was making him hallucinate - except that drinking had never affected him that way before. What he saw across his table was a man - dressed in red tights with horns on his head and a pitchfork on his lap, twirling both his black mustache and his pointed tail. He looked like he'd just stepped out of a cartoon.
The alcohol might not have been causing a hallucination, but it did alter Adam's response. Rather than screaming or freaking out, he instead found himself replying, "what do you know, man? What do you know about how hateful women can be?"
"Oh, believe me, I do know. Let's say that... I'm not unfamiliar with the nature of women. Women and I go way back." said the other man, in a smarmy yet coarse and haughty voice. Adam went on with his drunken rant: "But Ashley, man, Ashley. How could she... She's not just any other women. She's stupid. And cruel. More than any other women. I hate her."
"Oh, you mean me?" There was a 'poof' across the table, and the man in red tights had transformed himself into a pretty young women. "Ashley!" shouted Adam, for it was her - with that arc in her nose and her hair framing her cheekbones. "Ashley! I love you! How could you - just dump me like that? And why on Valentine's day? What did I do? Why can't we stay together?"
'Ashley' opened her pink lips and spoke, but the voice that came out was the same smarmy, coarse, and haughty voice of the man in red. And it said, "Because, hon, you're just not good enough. You're kinda cute from some angles, but not really good looking, you know? And I regularly get hit on by other guys who have more money than you. They're always more interesting, too - you got boring pretty quickly. Basically, I just thought I could do much better."
Adam was momentarily stunned into silence. He didn't know whether he would next explode in anger or start crying his eyes out. But before anything else could happen, 'Ashley' transformed back into the man in red, and he said, "Oh come on, I did you a favor. Would you rather that I tell you again all the lies you've already heard? 'Oh, it's not you, it's me', 'Oh, I just don't think we'll work out', 'Oh, let's stay friends'? Bah. You needed to hear the truth, and that's what I just told you. I did you a favor!"
Adam's eyes narrowed, and his mind seemed to focus a bit through his drunken haze. He said:
"What do you want? You're not in the business of doing favors."
"Oh? What have you heard about me?"
"Nothing good. That you tempt people to destroy their souls."
"Oh please. You can't believe everything you hear. Is it really so hard to believe that I've come to help you?" As the man said this, he grow white wings and a halo appeared over his head, and his clothing became white. But upon closer inspection, the halo was clearly held up by some metal wires extending up from his back, and the wings were poorly fitted props that was cramped awkwardly between the chair and the man's back. The man, now in white, looked like he was at a masquerade party in a cheap angel's costume.
"Look, I really did come here to do you a favor - to offer you a gift!" continued the man, in his smarmy, coarse, and haughty voice, twirling his mustache the whole time. "How would you like to have any woman you want?"
"What's the catch?"
"Come on now. You're hurting my feelings! There is no catch."
"You're lying. I bet it's something like all the women will go completely crazy fighting over me, so that my life is ruined because of the madness and the drama. Right?"
"No, not at all. Not all women will go after you - just the ones that you show affection to. And they'll only be in love with you "normally" - there's no diabolical super-powered obsession involved."
"Oh? Then is this some scheme to trap me or corrupt me through sex? Where I can have any women that I want so that I get completely addicted to perpetual sex with supermodels and somehow ruin myself?"
"Not at all. I mean, you can choose to use your gift in that way, and I won't stop you. To be honest, perpetual sex with supermodels is pretty nice. But the gift will also let you settle down with a nice family, if that's what you want. The choice is yours."
"'Nice family'? You probably mean that my kids will be your spawn and they'll grow up to be mass murderers who begin their killing spree by stabbing me. That's your idea of a 'nice family', isn't it?"
"Please. Any wife or kids you choose to have will be perfectly normal. I mean, they might be crazy, but that's not any more likely than for anyone else. There's nothing in my gift that will specifically turn them that way. You want your nice, suburban house and your white picket fence and a devoted, loving wife to grow old with in material comfort? My gift allows for that. It will even let you have up to two kids and a dog and a well-manicured lawn if you want. Nothing in my gift prevents any of that."
"What if my ex-girlfriends go crazy because they're still in love with me? Are you going to make them go insane with jealously and become ax-murdering lunatics who come after me?"
"Again, no. The exes you acquire through this gift are no more likely to go crazy than anyone else's ex-girlfriends. Look, do you know who I am? Do you honest think that if I wanted to screw you, I'd have to trick you in some way? I'm offering you this gift, because I really want you to have it. Because I really want to see you embrace it wholeheartedly, for your own enjoyment and benefit. Is that really so hard to believe?"
"Yes. Why are you being so - generous? Why offer me this gift at all?"
"I already explained why. I earnestly want to see you accept the gift and use it. And why shouldn't I be generous? I am powerful: I can give what gifts I please, if only to just demonstrate that I can. In fact, I think I'll do that right now: in addition to giving this gift to you, I'll pass on your exact same gift to any children you might have. They'll be marked by the gift from birth, and when they reach the age that you are now, I'll activate for them the ability to have any sexual partner they want. How's that for generous?"
"Seriously? There's no catch? The women I show affection towards won't suddenly become monstrously ugly? This "mark" on my future kids isn't some hideous birthmark, and it won't make them turn evil? I won't become impotent or change sexual orientation as soon as I get in bed with these women? My natural life span won't be shortened as a result of this gift?"
"No, no, no, and no. I've told you exactly what my gift will do. There are no such unexpected, additional side effects you need to fear."
"You're still hiding something. There's some fine print you're not telling me."
"No, I've told you everything, in perfectly clear language. We've gone over all of the abilities, powers, restrictions, limitations, and conditions on your gift. The only thing you have to do, is to say that you accept. Do you?"
The question hung in the air. Adam wasn't sure whether he had become more sober or more confused since this conversation began. He thought about his answer for a bit - but it was late, and he was tired, drunk, and emotionally unstable. He thought he might not remember anything the next morning.
As it turned out, he ended up remembering quite a bit - he remembered saying "I accept". He remembered much of the details of the deal. But he could not remember what it had been that was nagging at the back of his mind about those details. And so, he never came to understand the nature or the magnitude of his mistake.
Tina did not talk to her father often. In fact, she'd probably describe their relationship as "strained", and she was pretty sure that her father would not disagree. But she had to talk to him at length about this "gift".
"So it's all true? I can get any man I want now?" Asked Tina. She had recently been flirting with a guy, but not really expecting it to lead anywhere. So she was surprised when he suddenly seemed very earnest about his love for her, and absolutely shocked when her father told her that it was due to her new superpower.
"Well, it's been true enough for me. Since the stranger's visit, women just kept falling for me. I mean... I've always... you know. And now that you've also gotten this gift, it sounds like the part about it passing on to my kids is also true. So I guess your little sister will probably get it too in a few years?"
Tina hesitated a moment before her next question. "Is this... is it why you left mom?"
"Tina, you know that things were rocky between me and your mother long before the divorce."
"Yes, but is this why?"
Her father was silent for a long time. They were both remembering the past. Tina had heard whispered rumors of her father's infidelity, but they never seemed to amount to anything. None of the other women seemed to want to speak up about what had happened - as if they still had a stake in pleasing her father. And Tina's mother had always seemed to want to defend her father a little too much - she was completely smitten with him, and couldn't oppose him even in the face of infidelity. Tina thought it was pathetic. She had long ago told herself that she wasn't going to be like that. She wasn't going to waste her time or energy on defending or remembering anyone who left her.
"Look, Tina, this... gift, or curse, or whatever you called it, has many effects. But really, it's not so bad. You shouldn't think about how it affected things in the past between your mother and I. Hey, you really can get any man you want now! How could that be a bad thing? Remember how the last time you called me was when Billy broke off the engagement with you? We never have to go through anything like that again. Think of the future!"
Oh, yes. Tina was never going to go through anything like that again. She wasn't going to be pathetic like her mother. She wasn't going to ever tolerate any man like her father. She wasn't going to worry about her biological clock, or needing to feel wanted, or not getting married - ever again. She was going to find her perfect man and have a perfect family with him, with a perfect boy and a perfect girl. She was now going to be in complete control of her relationships. She was going to put her gift to good use.
Donny had been born under a lucky star. His father was rich and powerful, if a bit henpecked. He had his doting mother's full affection and attention. And he had always been popular with the ladies. In fact, good fortune in love seemed to run in his mother's side of the family. He had some cousins with some unbelievable stories to tell, who where far better with women than even "the Don" himself. That "good fortune in love" seemed outright ridiculous at times.
To be perfectly blunt, he actually sometimes wondered how a woman like his mother managed to hook a man like his father. It didn't make any sense, given his understanding of sexual relationships. But now, after being told about the gift, he understood why - his entire mother's side of the family really did have some unreal aid. And at last, that gift was finally active in him.
Well, it's not like he needed any extra help with women - but it certainly didn't hurt. He could compete with his cousins on an even footing now. He was going to live out all his wildest fantasies. He was never going to settle down or be satisfied. He was going to take all the women who caught his slightest fancy.
He didn't want any children - they were unfortunate side effects to his main goal in life as far as he was concerned, and he took some care to avoid them. But he did end up having two kids, through sheer low-probability accidents over many trials. Not that he would ever pay attention to them, of course - he died alone, never having wanted or understood fatherhood.
People had called Kat "confused" before. She supposed it was a fair description. After all, there was that "experimental" phase when she tried on various sexual identities. She had been married for a short time - then divorced. She had a kid, not from that marriage, but now she was wondering what the hell she was thinking at the time. She certainly felt confused.
So it came to be that when the stranger's gift finally kicked in for her, she achieved clarity at last. She could get anyone she wanted! She could have whatever family she wanted! It was all up to her!
She eventually settled on a somewhat "wild" and "unstable" guy in the end. He felt perfect for her - quite frisky and adventurous, willing to try anything. She didn't feel as confused anymore - and she certainly felt happier than before. She lived out her days doing everything she wanted, and if that ever turned out to be a bit too crazy, she always had a kid she could go back to, someone to whom she could play 'mother'. And she lived out her days that way - yo-yoing between "crazy" and "stable", between "sexually liberated" and "devoted mother".
People generally thought of Janus as a level-headed guy - rational, thoughtful, not given to erratic emotional impulses. Sure, some members of his extended family were a bit off the wall, but Janus himself was well known to be calm, if a bit cold. So it came as a complete surprise to everyone when he went on his killing spree not long after being told of the gift.
In Janus's family, informing their grown children of the gift had become a tradition to look forward to, almost like a rite of passage. His parents had told him clearly of their gift from the stranger, and how it will allow him and his descendants to have any mate they wanted. They told him all the details of how the gift operated and what he could expect. So there should have been no surprises for Janus when his gift activated. And yet, not long afterwards, Janus went on talking about the "work of the devil", the "end of the world", and the "original curse that afflicts us". He then murdered 16 people, starting with his younger brother, at a family reunion. Soon thereafter, he was killed by the police when they intercepted him in transit to his distant cousin's home with an arsenal of weapons.
The tragedy rocked the remaining members of Janus's extended family - they had lost so many of their kin. But still, they would carry on. There were plenty of survivors. And Adam's descendants, together with the stranger's gift, would endure for as long as the human race continued to exist.
"So it's all true? I can get any man I want now?" Asked Tina. She had recently been flirting with a guy, but not really expecting it to lead anywhere. So she was surprised when he suddenly seemed very earnest about his love for her, and absolutely shocked when her father told her that it was due to her new superpower.
"Well, it's been true enough for me. Since the stranger's visit, women just kept falling for me. I mean... I've always... you know. And now that you've also gotten this gift, it sounds like the part about it passing on to my kids is also true. So I guess your little sister will probably get it too in a few years?"
Tina hesitated a moment before her next question. "Is this... is it why you left mom?"
"Tina, you know that things were rocky between me and your mother long before the divorce."
"Yes, but is this why?"
Her father was silent for a long time. They were both remembering the past. Tina had heard whispered rumors of her father's infidelity, but they never seemed to amount to anything. None of the other women seemed to want to speak up about what had happened - as if they still had a stake in pleasing her father. And Tina's mother had always seemed to want to defend her father a little too much - she was completely smitten with him, and couldn't oppose him even in the face of infidelity. Tina thought it was pathetic. She had long ago told herself that she wasn't going to be like that. She wasn't going to waste her time or energy on defending or remembering anyone who left her.
"Look, Tina, this... gift, or curse, or whatever you called it, has many effects. But really, it's not so bad. You shouldn't think about how it affected things in the past between your mother and I. Hey, you really can get any man you want now! How could that be a bad thing? Remember how the last time you called me was when Billy broke off the engagement with you? We never have to go through anything like that again. Think of the future!"
Oh, yes. Tina was never going to go through anything like that again. She wasn't going to be pathetic like her mother. She wasn't going to ever tolerate any man like her father. She wasn't going to worry about her biological clock, or needing to feel wanted, or not getting married - ever again. She was going to find her perfect man and have a perfect family with him, with a perfect boy and a perfect girl. She was now going to be in complete control of her relationships. She was going to put her gift to good use.
Donny had been born under a lucky star. His father was rich and powerful, if a bit henpecked. He had his doting mother's full affection and attention. And he had always been popular with the ladies. In fact, good fortune in love seemed to run in his mother's side of the family. He had some cousins with some unbelievable stories to tell, who where far better with women than even "the Don" himself. That "good fortune in love" seemed outright ridiculous at times.
To be perfectly blunt, he actually sometimes wondered how a woman like his mother managed to hook a man like his father. It didn't make any sense, given his understanding of sexual relationships. But now, after being told about the gift, he understood why - his entire mother's side of the family really did have some unreal aid. And at last, that gift was finally active in him.
Well, it's not like he needed any extra help with women - but it certainly didn't hurt. He could compete with his cousins on an even footing now. He was going to live out all his wildest fantasies. He was never going to settle down or be satisfied. He was going to take all the women who caught his slightest fancy.
He didn't want any children - they were unfortunate side effects to his main goal in life as far as he was concerned, and he took some care to avoid them. But he did end up having two kids, through sheer low-probability accidents over many trials. Not that he would ever pay attention to them, of course - he died alone, never having wanted or understood fatherhood.
People had called Kat "confused" before. She supposed it was a fair description. After all, there was that "experimental" phase when she tried on various sexual identities. She had been married for a short time - then divorced. She had a kid, not from that marriage, but now she was wondering what the hell she was thinking at the time. She certainly felt confused.
So it came to be that when the stranger's gift finally kicked in for her, she achieved clarity at last. She could get anyone she wanted! She could have whatever family she wanted! It was all up to her!
She eventually settled on a somewhat "wild" and "unstable" guy in the end. He felt perfect for her - quite frisky and adventurous, willing to try anything. She didn't feel as confused anymore - and she certainly felt happier than before. She lived out her days doing everything she wanted, and if that ever turned out to be a bit too crazy, she always had a kid she could go back to, someone to whom she could play 'mother'. And she lived out her days that way - yo-yoing between "crazy" and "stable", between "sexually liberated" and "devoted mother".
People generally thought of Janus as a level-headed guy - rational, thoughtful, not given to erratic emotional impulses. Sure, some members of his extended family were a bit off the wall, but Janus himself was well known to be calm, if a bit cold. So it came as a complete surprise to everyone when he went on his killing spree not long after being told of the gift.
In Janus's family, informing their grown children of the gift had become a tradition to look forward to, almost like a rite of passage. His parents had told him clearly of their gift from the stranger, and how it will allow him and his descendants to have any mate they wanted. They told him all the details of how the gift operated and what he could expect. So there should have been no surprises for Janus when his gift activated. And yet, not long afterwards, Janus went on talking about the "work of the devil", the "end of the world", and the "original curse that afflicts us". He then murdered 16 people, starting with his younger brother, at a family reunion. Soon thereafter, he was killed by the police when they intercepted him in transit to his distant cousin's home with an arsenal of weapons.
The tragedy rocked the remaining members of Janus's extended family - they had lost so many of their kin. But still, they would carry on. There were plenty of survivors. And Adam's descendants, together with the stranger's gift, would endure for as long as the human race continued to exist.
Sarah was old now. But looking back, she couldn't help but feel satisfied with her life. She was glad of her gift.
She remembered how she had fantasized about the man of her dreams. How, in her youth before the gift kicked in, dating had been so incredibly difficult. How hard it had been to not settle for Joe, when she knew that in a few more years she'd be able to get any guy she wanted.
But after the gift, the only difficulty lay in finding the right man, with the knowledge that this would be the one - the one that had no chance of escaping her. She agonized for a long time between various CEO's and high-ranking politicians. She wanted a high-caliber man, but didn't want her family life to be too public. And in the end, when she finally selected her catch, she was sure that she had made the right choice.
And time bore out that choice. She was happily married. She was now in the twilight years of her life, and she had the perfect extended family - two capable children, who themselves were happily married to give her four promising grandchildren. She could even expect great-grandchildren soon. The gift had indeed been good to her and her family, and would continue to be good to her descendants.
Khan enjoyed watching his family grow. He had two children, and four grandchildren, and though he might not be around to see it, he'd surely have eight great-grandchildren and sixteen great-great-grandchildren. His entire extended family seemed to be doing equally well. And really, wasn't this the best that life has to offer? To see your children prosper and multiply? His descendants would be unstoppable. They had the gift. They would continue to prosper and multiply, until they filled the whole earth. It was an inevitability of exponential growth: 2, 4, 8, 16, 32, 64, 128 ... on and on, until eventually everyone on earth would be descended from him. It sent shivers down his spine to think of it.
Of course, there would be the rare few among his descendants that, for whatever reason, won't have their full allotment of two children. But that could hardly stop the overall trend. He did also wish that the gift would allow for more than two children, because his family would then grow faster. But he wasn't complaining. In due time, in either case, his descendants would inherit the whole earth - maybe even the whole universe.
Elizabeth was worried about her country. As the ruler, there were hundreds of thing that demanded her attention, but the latest hot-button issue was immigration. And it was giving Elizabeth nightly headaches.
Elizabeth's country was the envy of the world in a lot of ways. Yes, it lagged behind some other countries in scientific and economic development - but not by very much. However, in every measure of happiness and stability, it was miles ahead of the rest of humanity.
It was all thanks to the gift. A large majority of people in the country now had it, including Elizabeth
herself - although she was not aware of it. And once enough people in the population had it, the gifted could now often marry one another. And marriages from such pairs were ideal marriages. Both partner's gift would affect the other, so that they both loved each other. Such marriages were incredibly stable. They had virtually zero divorce rates, and made for a picture-perfect family - with happy, content parents, and their two children being raised in that loving environment. For many of them who were not aware of the gift, they simply thought they were a very lucky couple.
The country was therefore known as the country of families. And yes, that did perhaps limit the kind of national achievements that required 100 hour workweeks from some people. But the stability of the citizenry as a whole compensated for that in a myriad of other ways.
But because of that stable native population, with two parents nearly always having two kids, the proportion of the population growth that came from immigration was increasing rapidly. And many people wanted entry: the country was stable, happy, and prosperous, and therefore a popular destination for immigrants. Furthermore, many citizens would travel outside the country, find a marriage partner, and hook them with the gift and bring them back. And people were becoming xenophobic from all this immigration.
Elizabeth sighed. She supposed, on the whole, that things were not so bad. I mean, if the country was doing so well that people were worried about too many immigrants wanting to come in, that was a good problem to have, right?
Thomas wrote out the terms of the gift:
1. From the moment of the gift's activation, any sexual partner you choose will fall in love with you when you display affection towards them.
2. The gift will allow you to have up to two children.
3. The gift will not do anything else. It has no other supernatural effect. It will not drive people crazy with love or lust, it will not make anyone homicidal maniacs, it will not unnaturally "corrupt" you, or shorten your lifespan, or do or prevent anything of that sort.
4. Any children you have will also be born with the gift.
Those were the terms of the gift, as they were first given Adam, all those many years ago. And they were easily verified through the stories of many of Thomas's ancestors, as well as the undeniable demographic trends throughout history.
Thomas looked at the list of terms, and the graphs showing the demographic trends, and the map showing the dispersion of Adam's descendants throughout the whole of humanity. The terrible conclusion was inevitable. It had been, ever since that first day, the day of Adam's fateful decision. Everything was perfectly clear. Thomas did not want to speak his thoughts out loud. But in the end, the weight of despair upon him finally forced the words from him, in a sinking sigh:
"Humanity is doomed."
She remembered how she had fantasized about the man of her dreams. How, in her youth before the gift kicked in, dating had been so incredibly difficult. How hard it had been to not settle for Joe, when she knew that in a few more years she'd be able to get any guy she wanted.
But after the gift, the only difficulty lay in finding the right man, with the knowledge that this would be the one - the one that had no chance of escaping her. She agonized for a long time between various CEO's and high-ranking politicians. She wanted a high-caliber man, but didn't want her family life to be too public. And in the end, when she finally selected her catch, she was sure that she had made the right choice.
And time bore out that choice. She was happily married. She was now in the twilight years of her life, and she had the perfect extended family - two capable children, who themselves were happily married to give her four promising grandchildren. She could even expect great-grandchildren soon. The gift had indeed been good to her and her family, and would continue to be good to her descendants.
Khan enjoyed watching his family grow. He had two children, and four grandchildren, and though he might not be around to see it, he'd surely have eight great-grandchildren and sixteen great-great-grandchildren. His entire extended family seemed to be doing equally well. And really, wasn't this the best that life has to offer? To see your children prosper and multiply? His descendants would be unstoppable. They had the gift. They would continue to prosper and multiply, until they filled the whole earth. It was an inevitability of exponential growth: 2, 4, 8, 16, 32, 64, 128 ... on and on, until eventually everyone on earth would be descended from him. It sent shivers down his spine to think of it.
Of course, there would be the rare few among his descendants that, for whatever reason, won't have their full allotment of two children. But that could hardly stop the overall trend. He did also wish that the gift would allow for more than two children, because his family would then grow faster. But he wasn't complaining. In due time, in either case, his descendants would inherit the whole earth - maybe even the whole universe.
Elizabeth was worried about her country. As the ruler, there were hundreds of thing that demanded her attention, but the latest hot-button issue was immigration. And it was giving Elizabeth nightly headaches.
Elizabeth's country was the envy of the world in a lot of ways. Yes, it lagged behind some other countries in scientific and economic development - but not by very much. However, in every measure of happiness and stability, it was miles ahead of the rest of humanity.
It was all thanks to the gift. A large majority of people in the country now had it, including Elizabeth
herself - although she was not aware of it. And once enough people in the population had it, the gifted could now often marry one another. And marriages from such pairs were ideal marriages. Both partner's gift would affect the other, so that they both loved each other. Such marriages were incredibly stable. They had virtually zero divorce rates, and made for a picture-perfect family - with happy, content parents, and their two children being raised in that loving environment. For many of them who were not aware of the gift, they simply thought they were a very lucky couple.
The country was therefore known as the country of families. And yes, that did perhaps limit the kind of national achievements that required 100 hour workweeks from some people. But the stability of the citizenry as a whole compensated for that in a myriad of other ways.
But because of that stable native population, with two parents nearly always having two kids, the proportion of the population growth that came from immigration was increasing rapidly. And many people wanted entry: the country was stable, happy, and prosperous, and therefore a popular destination for immigrants. Furthermore, many citizens would travel outside the country, find a marriage partner, and hook them with the gift and bring them back. And people were becoming xenophobic from all this immigration.
Elizabeth sighed. She supposed, on the whole, that things were not so bad. I mean, if the country was doing so well that people were worried about too many immigrants wanting to come in, that was a good problem to have, right?
Thomas wrote out the terms of the gift:
1. From the moment of the gift's activation, any sexual partner you choose will fall in love with you when you display affection towards them.
2. The gift will allow you to have up to two children.
3. The gift will not do anything else. It has no other supernatural effect. It will not drive people crazy with love or lust, it will not make anyone homicidal maniacs, it will not unnaturally "corrupt" you, or shorten your lifespan, or do or prevent anything of that sort.
4. Any children you have will also be born with the gift.
Those were the terms of the gift, as they were first given Adam, all those many years ago. And they were easily verified through the stories of many of Thomas's ancestors, as well as the undeniable demographic trends throughout history.
Thomas looked at the list of terms, and the graphs showing the demographic trends, and the map showing the dispersion of Adam's descendants throughout the whole of humanity. The terrible conclusion was inevitable. It had been, ever since that first day, the day of Adam's fateful decision. Everything was perfectly clear. Thomas did not want to speak his thoughts out loud. But in the end, the weight of despair upon him finally forced the words from him, in a sinking sigh:
"Humanity is doomed."
Rose felt torn - she understood the arguments from those talking heads, but she also could not deny the feelings in her heart.
The vast majority of the human population had the gift now. They were, for the most part, happy with their lot in life. There was the issue, however, that two humans with the gift could only produce two children. Coupled with the fact that some pairs were bound to produce less than two children for whatever reason, there was a real problem with population decline.
Some extremist had in fact gone quite far in tackling this problem. They had taken to calling the gift "the curse", and advocated for total separation of the afflicted population from "the pure", the ever-diminishing fraction of humanity that still lacked the "curse". They wanted strict regulations on who can have sex with whom, with the purpose of keeping the still "pure" people away from the "accursed" ones. The really extreme "pure" ones even occasionally employed deadly force in order to keep their sons or daughters from "falling into the hands of the accursed". All of this could be very difficult, given that many people did not even know whether they had the gift - some were not yet old enough to have it be active, and some people assumed that they were just lucky in their perfectly happy marriage with two kids.
Of course, there was a counter-movement to all this as well. With the slogan "love conquers", they argued for allowing everyone the freedom to love whomever they wanted. They argued for governments staying out of people's bedrooms. They argued for not killing young people who were acting in love. And really, who could argue against that?
Rose herself was the product of two parents from the opposite sides of this debate. Her mother had the gift. Her father did not, and was initially opposed to the idea of intermingling with someone like his eventual wife. But he could not resist that very gift which he was trying to contain.
Rose herself somewhat bought into the position that her father had once held. She could see why population decline would be a problem. But that problem seemed far away and abstract, while her crush Carl - with his broad shoulders and piercing blue eyes - was in the here and now, and as real as the flesh and blood that beat and pumped in her chest. What good were all the arguments of the talking heads in the face of love? What did it matter that he and his lineage did not yet have the gift? After all, who could be against love? Would it not win in the end?
And so the remorseless course of history would continue, working itself through one more person.
By the time Hugh was born, nobody called the stranger's gift "the gift" anymore - since everyone had it. The last person without it had died some time ago. It was now simply a feature of the human condition. People generally found themselves locked into monogamous pairings which were virtually unbreakable, but there were also some polyamorous groupings where a group of people were all under the effects of one another's gift.
Hugh was in the latter group - the sex was good and he was happy. He didn't want any children, nor did anyone else in his group. That would disrupt their sex and happiness. He knew that there were people in high places that worried about the fate of humanity, but that was hardly his concern - the sex was good and he was happy. What more could he want?
As he lay in bed in a tangle with other bodies, he briefly looked out the window and stared into the night sky. He had wanted to be an astronaut when he was a boy - but nobody nowadays talked much about interplanetary travels or anything like that. Humanity was in decline. It had lost hope.
But what did that matter to Hugh? The sex was good and he was happy. So he and his group would continue on with their lives, choosing to enjoy their life unencumbered by things like the "humanity's future" or "having children".
Something like this would happen occasionally, where someone would not have their full allotment of two children. With the general malaise afflicting humanity, many people simply chose the childless lifestyle. Others had medical problems with conceiving. Some died in accidents before having children.
And every time this happened - every time a pair did not produce their full allotment of two children - the total potential human population would be permanently, irreversibly decreased. Two people, who would eventually die, were replaced by less than two people. Thus the human population would bleed away, one or two people at a time - inevitably, irresistibly, inexorably towards zero. Towards extinction.
Like many others, Hugh knew all this. But he didn't care. The sex was good and he was happy. He turned over in his bed, groping for one body part or another.
The vast majority of the human population had the gift now. They were, for the most part, happy with their lot in life. There was the issue, however, that two humans with the gift could only produce two children. Coupled with the fact that some pairs were bound to produce less than two children for whatever reason, there was a real problem with population decline.
Some extremist had in fact gone quite far in tackling this problem. They had taken to calling the gift "the curse", and advocated for total separation of the afflicted population from "the pure", the ever-diminishing fraction of humanity that still lacked the "curse". They wanted strict regulations on who can have sex with whom, with the purpose of keeping the still "pure" people away from the "accursed" ones. The really extreme "pure" ones even occasionally employed deadly force in order to keep their sons or daughters from "falling into the hands of the accursed". All of this could be very difficult, given that many people did not even know whether they had the gift - some were not yet old enough to have it be active, and some people assumed that they were just lucky in their perfectly happy marriage with two kids.
Of course, there was a counter-movement to all this as well. With the slogan "love conquers", they argued for allowing everyone the freedom to love whomever they wanted. They argued for governments staying out of people's bedrooms. They argued for not killing young people who were acting in love. And really, who could argue against that?
Rose herself was the product of two parents from the opposite sides of this debate. Her mother had the gift. Her father did not, and was initially opposed to the idea of intermingling with someone like his eventual wife. But he could not resist that very gift which he was trying to contain.
Rose herself somewhat bought into the position that her father had once held. She could see why population decline would be a problem. But that problem seemed far away and abstract, while her crush Carl - with his broad shoulders and piercing blue eyes - was in the here and now, and as real as the flesh and blood that beat and pumped in her chest. What good were all the arguments of the talking heads in the face of love? What did it matter that he and his lineage did not yet have the gift? After all, who could be against love? Would it not win in the end?
And so the remorseless course of history would continue, working itself through one more person.
By the time Hugh was born, nobody called the stranger's gift "the gift" anymore - since everyone had it. The last person without it had died some time ago. It was now simply a feature of the human condition. People generally found themselves locked into monogamous pairings which were virtually unbreakable, but there were also some polyamorous groupings where a group of people were all under the effects of one another's gift.
Hugh was in the latter group - the sex was good and he was happy. He didn't want any children, nor did anyone else in his group. That would disrupt their sex and happiness. He knew that there were people in high places that worried about the fate of humanity, but that was hardly his concern - the sex was good and he was happy. What more could he want?
As he lay in bed in a tangle with other bodies, he briefly looked out the window and stared into the night sky. He had wanted to be an astronaut when he was a boy - but nobody nowadays talked much about interplanetary travels or anything like that. Humanity was in decline. It had lost hope.
But what did that matter to Hugh? The sex was good and he was happy. So he and his group would continue on with their lives, choosing to enjoy their life unencumbered by things like the "humanity's future" or "having children".
Something like this would happen occasionally, where someone would not have their full allotment of two children. With the general malaise afflicting humanity, many people simply chose the childless lifestyle. Others had medical problems with conceiving. Some died in accidents before having children.
And every time this happened - every time a pair did not produce their full allotment of two children - the total potential human population would be permanently, irreversibly decreased. Two people, who would eventually die, were replaced by less than two people. Thus the human population would bleed away, one or two people at a time - inevitably, irresistibly, inexorably towards zero. Towards extinction.
Like many others, Hugh knew all this. But he didn't care. The sex was good and he was happy. He turned over in his bed, groping for one body part or another.
Humanity was inescapably doomed. There was simply no way to increase the population. And every time a couple failed to produce two children, humans would lose another sliver of hope, take one more step towards extinction.
Mara was an elder in her tribe, which was one of the larger tribes remaining in the world. It was her responsibility to pass on the history of that doom onto the next generation. Although it seemed pointless to her, this was the way that it was done for many, many years. So she gathered up the children and told them this story:
"...so it came to be that everyone was accursed. But back then, people thought we could somehow escape our doom. There were a great many more of us humans, and we were quite skilled in our sciences - and that fueled our false hope."
"There were many attempts to get around the curse. Before everyone was afflicted, some had tried to escape by moving away from the Earth altogether, by flying far into the sky and colonizing other planets. But the curse went with them, in unwitting people who did not know that they were its carriers. Eventually, everyone was accursed, and it did not matter where they were."
"There were advances in midwifery technologies, but none of them could get around the curse. Nothing changed the fact that we could only have "up to two kids". We also tried to clone people - that's when they used science to make a copy of someone else - so as to circumvent that limit. But the curse counted the clone as a twin of the original person - that is, as a child of that original person's parents. So cloning would succeed only once if performed on someone who was an only child, but always fail otherwise. So it was useless as a way of increase the population."
"Others tried to have non-biological "children" through AI programs - that's when they used to use science to make a fake machine-person. But the curse extend there as well. These ancient peoples wanted a "true AI", a machine worthy to be called a "child" of those scientist. But they didn't know that the curse would count that one machine as one offspring for everyone involved in its creation. And these were monumental projects: they involved thousands of people. And absolutely none of them could have already had two children. Otherwise the project would fail because the resulting machine-person would then be a third child. Not realizing this caused many failures, for a very long time, to the point where many thought the task impossible."
"They eventually succeeded by only using childless scientists on the project - but it was not worth it. Can you imagine, thousands of people losing one of their two allotment for children, all so that they can produce just one fake machine-person? Yes, the machine-person was slightly smarter and more durable than an average person, and it had the potential to be improved upon. But it could not replace all the potential children it had consumed, the promising children of these very smart scientists. And they could not make more of these machine-persons without consuming even more potential children. So these projects were abandoned; they could not save us from our doom."
"We also tried to increase our lifespans, so that the increase in life from longer lifespans might outpace the decrease in life from failure to reproduce. But this was a forlorn hope. For this to work, we needed to extend our lifespans indefinitely forever, which is impossible."
"We tried many other things - but nothing worked. We are still cursed. As the repeated failures built up, humanity lost hope. The population dropped precipitously, and our past greatness and abilities have disappeared along with the people."
"And that, children, is how we - humanity - find ourselves here after these long years, diminished in number and power. We cannot hope to escape the curse, but we can defy it. That is why we are going to live on and resist our fate as long as we can. That is why you must know of our past history, and why you must all have your full allotment of two children. That is the only thing remaining for our species before extinction."
At this point, a messenger ran up to Mara and interrupted her story. "Elder Mara", he said, "I bring an important message from the hills clan."
"What is it?"
"Four of their people have been killed in a hunting accident, including their chief's wife. They are now too small to sustain their own population. Their chief now only has his near of kin as potential breeding mates; all their other women are locked to other men by the curse. But he still has one more potential child to produce, and he does not want to waste that potential. Therefore he formally requests for a full merger of his clan into our tribe."
"Yes, such breeding inefficiencies are the last thing humanity can afford. Notify the other elders and call for a council tonight. We must discuss this merger."
Mara then dismissed the children and the messenger, and prepared for the council. They must accept this merger: that was obvious. It was their best chance at leaving their full allotment of children.
But even as she went over the details, the thought came unbidden back to her mind: 'why bother? It's all pointless. It'll all end the same anyway'. But she shook it off. They were going to resist extinction for as long as possible. That was the only thing of any meaning left to be done. That was the way it had been done for many, many years now. It had to be done this way, until the bitter end.
There remains little to tell.
Mara was an elder in her tribe, which was one of the larger tribes remaining in the world. It was her responsibility to pass on the history of that doom onto the next generation. Although it seemed pointless to her, this was the way that it was done for many, many years. So she gathered up the children and told them this story:
"...so it came to be that everyone was accursed. But back then, people thought we could somehow escape our doom. There were a great many more of us humans, and we were quite skilled in our sciences - and that fueled our false hope."
"There were many attempts to get around the curse. Before everyone was afflicted, some had tried to escape by moving away from the Earth altogether, by flying far into the sky and colonizing other planets. But the curse went with them, in unwitting people who did not know that they were its carriers. Eventually, everyone was accursed, and it did not matter where they were."
"There were advances in midwifery technologies, but none of them could get around the curse. Nothing changed the fact that we could only have "up to two kids". We also tried to clone people - that's when they used science to make a copy of someone else - so as to circumvent that limit. But the curse counted the clone as a twin of the original person - that is, as a child of that original person's parents. So cloning would succeed only once if performed on someone who was an only child, but always fail otherwise. So it was useless as a way of increase the population."
"Others tried to have non-biological "children" through AI programs - that's when they used to use science to make a fake machine-person. But the curse extend there as well. These ancient peoples wanted a "true AI", a machine worthy to be called a "child" of those scientist. But they didn't know that the curse would count that one machine as one offspring for everyone involved in its creation. And these were monumental projects: they involved thousands of people. And absolutely none of them could have already had two children. Otherwise the project would fail because the resulting machine-person would then be a third child. Not realizing this caused many failures, for a very long time, to the point where many thought the task impossible."
"They eventually succeeded by only using childless scientists on the project - but it was not worth it. Can you imagine, thousands of people losing one of their two allotment for children, all so that they can produce just one fake machine-person? Yes, the machine-person was slightly smarter and more durable than an average person, and it had the potential to be improved upon. But it could not replace all the potential children it had consumed, the promising children of these very smart scientists. And they could not make more of these machine-persons without consuming even more potential children. So these projects were abandoned; they could not save us from our doom."
"We also tried to increase our lifespans, so that the increase in life from longer lifespans might outpace the decrease in life from failure to reproduce. But this was a forlorn hope. For this to work, we needed to extend our lifespans indefinitely forever, which is impossible."
"We tried many other things - but nothing worked. We are still cursed. As the repeated failures built up, humanity lost hope. The population dropped precipitously, and our past greatness and abilities have disappeared along with the people."
"And that, children, is how we - humanity - find ourselves here after these long years, diminished in number and power. We cannot hope to escape the curse, but we can defy it. That is why we are going to live on and resist our fate as long as we can. That is why you must know of our past history, and why you must all have your full allotment of two children. That is the only thing remaining for our species before extinction."
At this point, a messenger ran up to Mara and interrupted her story. "Elder Mara", he said, "I bring an important message from the hills clan."
"What is it?"
"Four of their people have been killed in a hunting accident, including their chief's wife. They are now too small to sustain their own population. Their chief now only has his near of kin as potential breeding mates; all their other women are locked to other men by the curse. But he still has one more potential child to produce, and he does not want to waste that potential. Therefore he formally requests for a full merger of his clan into our tribe."
"Yes, such breeding inefficiencies are the last thing humanity can afford. Notify the other elders and call for a council tonight. We must discuss this merger."
Mara then dismissed the children and the messenger, and prepared for the council. They must accept this merger: that was obvious. It was their best chance at leaving their full allotment of children.
But even as she went over the details, the thought came unbidden back to her mind: 'why bother? It's all pointless. It'll all end the same anyway'. But she shook it off. They were going to resist extinction for as long as possible. That was the only thing of any meaning left to be done. That was the way it had been done for many, many years now. It had to be done this way, until the bitter end.
There remains little to tell.
There was, at last, a man - or a boy, rather - with no name. Names, it turned out, are not really necessary when the only other person that he'd ever seen in his life was his mother. A simple "Hey, you" or just a grunt sufficed in place of a name.
There was nobody else. He supposed that he must have come from a father, and his mother acknowledged as much. But the boy never met anyone else, in all their long travels throughout the abandoned world. He supposed that they were the only humans left.
His mother had used her curse on him when he was eleven - as soon as he began to show signs of puberty. He was now irresistibly in love with her.
He copulated with his mother nightly. But it made him sad to see that she did not particularly enjoy these sessions. He loved her with a burning passion, but the feeling was not reciprocated. His mother's heart was not really into it, as if it really belonged to another. He hoped that when he grew up and his curse became activated, she could finally enjoy him.
After these copulation sessions, his mother would always tell him the same things: that he must get her pregnant. That the child must be a girl. That they had only one shot at having a girl and not a boy. That he must then use his curse on that girl - his daughter - even if he was still in love with his mother. That he must then get his daughter pregnant, and this time it must be a boy. He must then instruct his daughter to mate with her son. This was the way it was done. It was done this way for a while now. This the only way for a mother and a son, who can each only have two children, to propagate themselves. This chain of incest was the only way to stave off extinction.
Rarely, his mother told him of other things as well: that there used to be a lot more people in the past. That humans did not always have the curse - that once upon a time, humans could have more than two children each, just like dogs or rats can. That it all started from just one man who first accepted the curse. On such occasions they sometimes talked about what life must be like for a dog, and philosophized about the pros and cons of being human. But nothing changed the intense envy they felt whenever they came across dead dogs, especially when it was being eaten by the fellow dogs in their pack.
For, of course, their plan for the chain of incest was ultimately futile. Soon, someone in this chain would be born as the wrong sex, or die in an accident, or become barren due to the genetic defects caused by incest. And when that happened, that would be the bitter end. Their lineage - humanity's lineage - would then finally perish, extinct for the want of a mate.
You may next want to read:
How to think about the future
Make the most of your time and your life. Number your days.
Another post, from the table of contents