Am I skeptical of the skeptics?

I may be skeptical about a lot of things, but shouldn't I also be skeptical of skepticism itself? Yeah, I am. Duh. We should be skeptical about everything.

But what does that mean?

Skepticism, at least the way I mean it, is not the same as doubting. What kind of position would that be if I doubted everything? Not only would I doubt psychic predictions, I would also have to doubt the impossibility of psychic powers. Not only would I doubt the link between the MMR vaccine and autism, I would also have to doubt the safety of vaccines. Not only would I doubt the 9/11 truth movement, I would also doubt that terrorists attacked the World Trade Center without any help from our government. What kind of self-defeating position is this?

But of course, I do doubt all these things. But doubt is only where skepticism begins. Following initial doubt, we carefully evaluate these claims in order to determine exactly how much doubt is warranted. It is not always the case that the public doubts too little. It is just as often the case that the public doubts too much.

And why might this be the case? One common reason is that people are not aware of or undervalue scientific evidence. A good skeptic realizes that a well-conducted scientific study can be worth more than a thousand heart-wrenching anecdotes. And an established scientific theory is supported by many such scientific studies. I doubt scientific theories too, but only as much as is warranted, which is not a whole lot.

So when I say I am skeptical of skepticism, this does not mean that I have serious doubts about it. It means that I have initial doubts, which I have evaluated and given careful consideration. And yes, some doubt is warranted towards skeptics. Skeptics will make mistakes and missteps. They can accidentally mine quotes. They can disagree about the sources of superstitions. They can make straw men of their opponents. They are human. But the basics of the skeptical method, those are sound.

Handshakes puzzle

There is a party with five couples. There is much handshaking at the beginning of the party. No one shakes hand with their own partner, and no one shakes hands twice with the same person.

Later, the host, Martin, asks everyone how many times they shook hands. All of the other nine people remembered exactly how many times, and answered him. Each of the nine answers was a different number!

How many handshakes did Martin himself have at the party?

solution posted

Mutual friends solutions

See the original puzzle

This one was first solved by Secret Squïrrel.

There are six people in the group. Choose one person, named person A. Person A has a relationship with each of the five other people. Each relationship is one of two types, friend or stranger. If there are five relationships of two kinds, then there must exist at least three relationships which are of the same kind. (This is called the pigeonhole principle.)

So now we can say that person A has the same kind of relationship with persons B, C, and D. Suppose that this relationship is friendship. If the relationship between B and C is a friendship, then A, B, and C are a group of three mutual friends. Similarly, if the relationship between B and D is a friendship, or if the one between C and D is a friendship, then you will be able to find a group of three mutual friends. However, if none of these pairs (B-C, B-D, C-D) are friendships, then they must all be strangers. And so, B, C, and D must constitute a group of three mutual strangers.

If A is stranger to B, C, and D, then a similar proof holds. No matter what, there must exist either a group of three mutual friends or three mutual strangers.

There are seventeen people in a group. Choose one person, named person A. Person A has a relationship with sixteen other people. Each of those relationships is one of three types: friends, strangers, or rivals. Therefore (by the pigeonhole principle), there must exist six relationships which are of the same type. I will refer to this relationship as "rivals", but the same argument holds even if they are friends or strangers.

So now we have person A, who is rivals with persons B, C, D, E, F, and G. If any two people out of B, C, D, E, F, and G are rivals with one another, then together with person A, they form a group of three mutual rivals. However, if none of the relationships between B, C, D, E, F, and G are rivalries, then they must all be friends or strangers with one another. But, as we proved earlier, if there are six people who are only strangers or friends with one another, then there must exist a group of three people who are either mutual friends or mutual strangers.

So I hope my proof wasn't too hard to understand. You may also see Secret Squïrrel's wording of the proof.

Buses and equilibria

If you've ever taken a public city bus, you know that that's where all of life's greatest lessons are learned. As you wait for the bus to arrive, especially when the bus comes late, you contemplate the mysteries of the universe. As you sit (or stand) in the moving bus, in a mostly silent crowd, you ponder what it truly means to be human.

Perhaps as a result of my physicist's intuition, I've noted that buses are frequently in an unstable equilibrium. Most buses will either be behind schedule, or ahead of schedule. The bus which is exactly on time exists in a precarious position, and there are opposing forces which threaten to slow it down or speed it up.

All it really takes to knock a bus off from this precarious position, this unstable equilibrium, is a red or green light. There are plenty of random factors involved in the speed of a bus, and any one of these could make the bus just a little late, or just a little early.

If the bus is just a little late, then there will be more time for people to accumulate at each of the bus stops. Therefore, it becomes more likely that the bus will have to stop to let people board. And the more people who board, the longer the bus must stop. When there are more people on the bus, they will make more frequent stop requests, which take more time. Similarly, if the bus is just a little early, then there will be less people to board, and the bus will go faster.

The result is that buses tend to be very late or very early. In fact, it tends to be that if one bus is late, then the bus behind it is early, and the bus behind it is late again. If one bus is late, it starts to pick up passengers who expected to board the next bus. And so the next bus has less people to board, and ends up going faster. Sometimes, one bus becomes so much faster than the bus in front of it, that it will overcome it. Usually, at this point, the buses will stop, and they'll tell all the passengers in the slow bus to switch to the fast bus. The bus drivers must be aware of the positive feedback mechanisms which are at work here.

Unfortunately, though there are slow buses and fast buses, you are more likely to end up boarding the slow, crowded bus. The natural laws which govern buses can be unkind, though I prefer to think of them as simply indifferent.

A modal ontological argument

If you wish to understand this post, you should probably read my exposition on modal logic first. You may also wish to read my analysis on a much simpler ontological argument.

To briefly review the simpler ontological argument: The argument says that we can define God as a necessarily existing being. Therefore, by definition, God exists. However, this takes the power of definition too far. The most we can say is that if there exists an object which we can properly call God, then that object, by definition, exists. If God exists, then God exists.

The problem with the argument is that it's rather useless to define God as a being which exists. But hold on! I never defined God as a being which exists. I defined God as a being which necessarily exists. This definition is not quite as useless. We can say that if there exists an object which we can properly call God, then that object, by definition, necessarily exists. If God exists, then God necessarily exists.

By definition of God: g g

The letter "g" represents the specific statement "God exists". Note, however, we did not completely define God. In fact, we could replace God with all kinds of absurd objects, and the proof would still hold. This is actually quite a problem for the ontological argument (as well as many other proofs of God). The proof is valid for any statement p as long as pp is necessarily true. For instance, we could replace the object God with "the invisible pink unicorn which necessarily exists" or "a left shoe which necessarily exists".

Nonetheless, let's continue onward. I'm going to go through the proof, step by step.

• Theorem 1: ¬g ¬g (The contrapositive of the definition of g)
  
  
• Theorem 2: ¬g ¬g (Using the definition of to substitute into Theorem 1)
  
  
• Theorem 3: (¬g ¬g) (Application of Axiom N to Theorem 2)
  
  
• Theorem 4: ¬g ¬g (Application of Axiom K to Theorem 3)
  
  
• Theorem 5: ¬g¬g (Axiom 5 applied to ¬g)
  
  
• Theorem 6: ¬g ¬g (Combining Theorems 4 and 5)
  
  
• Theorem 7: ¬¬g ¬¬g (The contrapositive of theorem 6)
  
  
• Theorem 8: gg (Using the definition of to substitute into Theorem 7 twice)
  
  
• Theorem 9: g g (Combining Theorem 8 with Axiom T)

You may have noticed that I did not prove what I set out to prove. I only proved g g, which says that if g is possible, then it is true. That's not quite the same as saying that g is true, but it's still something. Now, all we have to do is prove that g is at least possibly true.

And here is where the problems begin. All the previous work was purely logical manipulation, and is necessarily true if you accept the axioms of modal logic. I thought the axioms were pretty reasonable, and rejecting them would be too high a price to pay. However, it seems we need another premise, g. To support this premise, lots of arguments have been offered by various people, but I don't think they're nearly as fun or rigorous as the modal logic section.

One common argument for g is that g is self-consistent. I can conceive of a God without having any contradictions. Based on my current knowledge, it is entirely possible that God exists. It's possible that God exists, therefore God exists.

The problem is that this same argument seems to apply to the statement ¬g. ¬g is a self-consistent statement. I can conceive of a world in which God does not exist without having any contradictions. Indeed, I can conceive of a world where nothing at all exists, where there could not possibly be any contradictions since there is nothing around to contradict. Based on my current knowledge, it is entirely possible that God does not exist. Therefore we can take the premise ¬g. By Theorem 2, we conclude ¬g: God does not exist.

Obviously, the premises g and ¬g cannot both be true. Either God exists in all possible worlds, or God exists in none of them. Since we used the same argument for both both g and ¬g, that argument must be fallacious. But where exactly did it go wrong?

The error, I think, is in the concept of . g does not quite mean "g is possibly true." In fact, it means "Among all possible worlds, there exists at least one in which g is true." The concept of "All possible worlds" was never exactly defined. In fact, the definition is arbitrary. I could have declared "all possible worlds" to be our world and our world alone, and you never would have been able to prove me wrong from the axioms.

I also could have declared "all possible worlds" to be the set of worlds which are metaphysically possible (a rather complex philosophical concept). Under this definition, we would not be able to prove g or ¬g.

However, if I had declared "all possible worlds" to mean the set of all worlds which are self-consistent, then we would run into problems. Because under this definition, both g and ¬g appear to be true (unless the concept of God is inconsistent). And they contradict each other.

Similarly, if I had declared "all possible worlds" to mean the set of all worlds which are epistemologically possible (meaning, it may be true, for all we know), then we would have the same contradiction.

And what happens if we construct a pathological definition of "all possible worlds" such that g is true and ¬g is not true? Then I might question Axiom T, since it is no longer obvious that our world is included among this so-called set of "all possible worlds."

And it goes on and on. Well, wasn't it clever, at least at first? I think so. It seemed for a moment that we arrived at a paradox, a sort of 1=2 moment. Ontological arguments tend to be that way. Most people immediately recognize that it is a little too clever, that it proves a statement which is a little too strong to be true. Similarly, most philosophers think that ontological arguments fail, though they may disagree on exactly why they fail.

In my opinion, ontological arguments are merely interesting philosophical curiosities. It's rather silly when an apologist actually tries to use one as a serious argument.

Daily Bruin on religion

The campus' official newspaper, The Daily Bruin, had two articles this week about the recently conducted American Religious Identification Survey. Number one is Religion Continues to Thrive in US, and number two is Overly Secular Government is Against Founders’ Intent.

The first article is... okay, considering that it is a college newspaper, and the overall quality is low. The article's main point is that though the American Religious Identification Survey showed decline in religious identification, most of the decline occurred between 1990 and 2001, not between 2001 and 2008. These results suggest that contemporary atheist authors like Richard Dawkins are not the main cause of the trend, since the main change occurred before these books were even written. However, aside from the main point, there are several flaws in the article. In particular, Roy Natian told me himself that he felt misrepresented and quoted out of context.

However, I'm willing to ignore all that so I can do a thorough fisking of the second article.

Religion is under siege in America, when we are in the greatest need of hope that can come only from belief in a higher power.

The unfortunate implication of this statement is that atheists do not have hope, or that it does not matter whether they have hope.

Just in time for Passover and Easter, probably the most significant religious holidays for Jews and Christians, respectively, the cover of Newsweek magazine bears the dolorous title, “The Decline and Fall of Christian America.”

He makes it sound as if Newsweek is intentionally delivering bad news just in time for a day of celebration. There's a subtle persecution complex going here.

The lead article focuses on the new American Religious Identification Survey, which has shown that the number of unreligious Americans has been steadily increasing since 1990.

This directly contradicts the other Daily Bruin article's point that the increase was not steady. (To be fair, he might just be repeating a claim by Newsweek.)

Though the author of the article – Jon Meacham, who is also the editor of the magazine – first attempts to hide behind his own religion – as many closet seculars do – to convince his Christian audience that he is troubled by the statistics in the survey, it is not difficult to see that he is actually gleeful about it.

"Closet seculars"? It seems the Daily Bruin writer just cannot comprehend the existence of a Christian who believes in separation of church and state. Funny, I thought it was supposed to be the atheists who oversimplify religion.

Like many liberal Americans, [Jon Meacham] repeatedly confuses the idea of separation of church and state with the idea that religion has no place at all in our political system. While insisting that the presence of any religious ideas in government would necessarily “compel or coerce religious belief or observance,” he blinds himself entirely to the pivotal role that faith has in our political system.

I wouldn't say that separation of church and state implies that religion has no place at all in politics. I mean, it would be wrong to discriminate against politicians solely on the grounds of their religion. But a pivotal role? It almost sounds as if he's arguing that because religion is so pivotal in politics, we are justified in compelling religious belief. (At the end of the article, he says he does not support compelling religious belief.)

Contrary to popular belief, there is no reference to “separation of church and state” in the Constitution – it is the invention of people (like the Newsweek author) who want desperately to accommodate the narrow interests of the relatively slight number of non-believers in the country who represent anti-traditionalism – or “progressivism.” According to the Constitution, the only prohibition on government is that it cannot promote the formal “establishment” of a particular religion.

Of course, the Constitution never uses the exact phrase “separation of church and state”. But the main idea is right there in the First Amendment: "Congress shall make no law respecting an establishment of religion, or prohibiting the free exercise thereof." The separation of church and state is the interpretation of the First Amendment. Furthermore, contrary to popular belief, there are plenty of believers who also believe in separation of church and state – their existence is denied by people who want desperately to accommodate the narrow interests of the relatively slight number of conservatives in the country who represent anti-progressivism – or “traditionalism.” (Doesn't it sound horrible when I frame it this way?)

Despite the contention of many liberals that we have always been a secular country – which ignores the omnipresence of religious symbols in our government, such as state seals, currency and the Pledge of Allegiance, to name a few – there is no escaping that America was founded by deeply religious men who envisioned that their society would be based on their Judeo-Christian beliefs.

My understanding is that some of the founding fathers were deists, a few were anti-clerical, and some were quite religious. It's sort of irrelevant, since they obviously didn't intend for the government to establish their religion. These religious symbols, including the mention of God on our currency and in the Pledge of Allegiance, were mostly added in during the McCarthy era.

The founders clearly thought that liberty was a blessing from God. After all, the Declaration of Independence tells us that our rights are “endowed by (our) Creator.” Nevertheless, people continue to assert that the U.S. is not a Judeo-Christian nation but rather one that just happens to be home to many Christians and Jews.

The wording of the Declaration of Independence is entirely consistent with Deism, since it only refers to a "Creator". Anyways, the document has no real legal standing.

But in a free and open society, how is order maintained if values are not based on a particular faith or book of faith, which protects us from the moral whims of people and culture? National issues cannot be arbitrated if we accept moral relativism. Atheists will proclaim that they get their values from their hearts, but this view is a subjective kind of morality.

On what basis could you claim that values based on a sacred text are any better than the accumulated "moral whims of people and culture"? Why do we need protecting from these so-called whims? Just because you think you've found a black-and-white method to determine your values doesn't mean it's correct.

Every great question, whether it be abortion, school prayer or even taxation, comes down to the views of the people making the decision. How would they be able to decide these questions if there were no commonly agreed upon standard of right and wrong, as in the Bible?

If only we could go back to the good old days when everybody agreed on everything.

And there is another problem: If people happen to have religious convictions, should they have to choose between politics and their faith – that is, would they either have to ignore their beliefs or stay out of politics?

Two answers: 1) No. It's wrong to exclude people of any religious category from holding office.
2) Yes, if their religious beliefs affect their politics in such a way that they are unelectable. For instance, if you have the religious conviction that abortion should be illegal, and if voters disagree with you, then they are justified in keeping you from being elected.

In the end, we have to remember that the disestablishment of religion is just as dangerous as the establishment thereof. The dangers that secularism poses to a free society are historical and manifold – just think, it was the decline of religion in Western Europe that inspired fascism and led to the extermination of millions of Jews.

Correlating such broad trends such as secularization and fascism is a pretty sketchy argument. Did you ever hear the one about pirates and global warming?

Nobody is arguing for a state-sponsored religion or for one religion to be “compelled” on anyone. Instead, we are only arguing for the preservation of some religious sway in our government that will inform our politics. That should not infuriate nonbelievers, and it would be perfectly reconcilable with the founders’ intentions.

No, it does not infuriate me. Secularism, in the context of politics, does not mean political hostility to religion, it means political indifference to religion. Politicians are entirely allowed to make decisions which are informed by their religion. But this does not make them better politicians. It diminishes their capability to represent the diverse views in the US.

Some modal logic

In 2007, I wrote a post that explained a very simple ontological argument for the existence of God, and why it is fatally flawed. However, ontological arguments are a whole category of arguments for the existence of God. From that point, they only get more complicated and sophisticated. I intend to present an ontological argument which is one step more sophisticated, using modal logic. Modal logic is a system of logic which has incorporated the concepts of "necessity" and "possibility". These concepts are represented symbolically by squares and diamonds and... well, it's a whole lot of fun!

There are actually many different kinds of modal logics, depending on what axioms you choose, and what kind of extra concepts are incorporated. However, I will choose a fairly basic set of axioms, and explain them as I go.

What we are trying to do here is model the philosophical concepts of necessity and possibility. If we say a proposition is "necessary", that means that it is true in all possible worlds. The world which we happen to live in is one of those possible worlds. Therefore, if a proposition is necessarily true, then it is true. We will take this as an axiom, which I will state symbolically.

Axiom T: p p

The letter "p" represents a proposition, a statement which is either true or false. "p" could be the proposition "God exists", or it could be the proposition "The moon is made of cheese." When we put a "" in front of the proposition, we are modifying the proposition with the concept of necessity. For example, if "p" is the statement "Snow is white", then "p" is the statement "Snow is necessarily white", meaning that snow is white in all possible worlds. Stated in English, Axiom T says that if a proposition is necessarily true, then it is true.

However, it is not the case that if a proposition is true that it is necessarily true. For instance, the statement "I have black hair" happens to be true in this world, however, I may have dyed my hair blue in other possible worlds. Therefore, we cannot say without qualification that pp. However, it is a general truth in logic that the statement "pq" is equivalent to "not-qnot-p". (From this point forward, I will replace the word "not" with the symbol "¬".) "¬q¬p" is called the contrapositive of statement "pq". If any statement is true, then its contrapositive must be true as well.

The contrapositive of Axiom T is ¬p ¬p. Remember, p can be absolutely any proposition, so we're going to replace it with the proposition "¬q". Therefore, by Axiom T, q ¬¬q. The sequence of symbols "¬¬" may seem like meaningless jargon, but it in fact represents the concept of possibility. Saying ¬¬q is the same as saying that there exists a possible world where q is true. It may or may not be the world which we live in, but it's possible. Possibility is such an important concept that we're going to give it its own symbol, "".

Definition: p means ¬¬p

As we proved in the last paragraph, p p. This makes sense, since if a proposition is true in our world, then it must be true in at least one possible world, namely, our own.

We would like our concepts of necessity and possibility to be properties which pertain to the entire set of possible worlds, not just our own. For instance, if it's possible that dogs exist, then that just means that there exists at least one possible world where dogs exist. Therefore, no matter which of the possible worlds we lived in, we could still correctly state that it's possible that dogs exist. While the statement "dogs exist" may be true in some worlds but not others, the statement "dogs possibly exist" is either true or false in all possible worlds. That is the reasoning behind the following two axioms, known as Axiom 4 and Axiom 5.

Axiom 4: p p
Axiom 5: p p

Another thing we would like is for our axioms to be true in all possible worlds. It would be quite absurd if, for instance, Axiom T was true in our world, but not in others. And so we take this as a new axiom.

Axiom N: All axioms, as well as theorems proven from them, are necessarily true

Applying Axiom N to Axiom T, we could say (p p). But what useful things could we say from a statement of the form (p q)? The answer to that is our last axiom.

Axiom K: (p q) (p q)

This means that if p q is true in all possible worlds, and if p is true in all possible worlds, then q is true in all possible worlds.

A summary:

• Definition: p denotes the statement that p is "necessary", or true in all possible worlds.
  
  
• Definition: p, or ¬¬p denotes the statement that p is "possible", or true in at least one world.
  
  
• Axiom T: p p "If a statement is true in all possible worlds, then it is true in our world."
  
  
• Axiom 4: p p "Necessity is a property of all possible worlds."
  
  
• Axiom 5: p p "Possibility is a property of all possible worlds."
  
  
• Axiom K: (p q) (p q) "If p implies q in all worlds, and p is true in all worlds, then q is true in all worlds"
  
  
• Axiom N: All axioms, as well as theorems proven from them, are necessarily true

These axioms all sound reasonable, do they not? Using all these axioms (excluding Axiom 4), we will prove that God exists... in my next post.

A parable on induction: The Village Census

Bertrand Russell's Census Officer

A reader mentioned a parable by Bertrand Russell about "induction by simple enumeration". Through my infinite resources, it just so happens that I have access to the exact quote:

Induction by simple enumeration may be illustrated by a parable. There was once upon a time a census officer who had to record the names of all householders in a certain Welsh village. The first that he questioned was called William Williams; so were the second, third, fourth, . . . At last he said to himself: "This is tedious; evidently they are all called William Williams. I shall put them down so and take a holiday." But he was wrong; there was just one whose name was John Jones. This shows that we may go astray if we trust too implicitly in induction by simple enumeration.

- Bertrand Russell, The History of Western Philosophy, page 543

In my unrelenting quest to make everything more tediously mathematical, I wish to analyze this parable using Bayes' Theorem. But first, let us simplify it just a bit. Let us assume that there are only five villagers (all men), and the census officer has questioned four of them.

An aside on notation

I was originally planning to explain my point in plain English, but I found it highly cumbersome. Therefore, I will be using a little notation from probability theory. Firstly, it is useful to denote several claims using letters:

• W: all five villagers are named William Williams.
• J: only four of the villagers are named William Williams, and the last is not.
  
• E: The four villagers that the census officer questioned are named William Williams.

We will use the letter P to denote the probability of a claim. For instance, P(W) is the probability that all five villagers are named William Williams. P(W) is just a number between 0 and 1 (where 1 means that W absolutely must be true). P(W) is called the "prior" probability of W. It's called the "prior" probability because it is prior to any evidence. P(W) is the probability that W is true before we learn about the census officer's new observations.

When the census officer questions the villagers, we find that all four people he questioned were named William Williams. In other words, we find that claim E is true. If we want to know the probability of W before the census officer questions the villagers, then we need P(W). If we want to know the probability of W after the census officer questions the villagers, then we would denote this probability as P(W|E). P(W|E) is called "the probability of W given E".

Back to the village

What we see in this parable is that a census officer has collected a piece of evidence: claim E is true. What should he conclude from this piece of evidence? In the parable, he concludes that all villagers are named William Williams, but is this conclusion reasonable? In other words, is P(W|E) high? We might compare the officer's conclusion with other conclusions he might have had. For instance, would it have been better to conclude that only four of the villagers were named William Williams? In other words, is P(J|E) high? Which is higher, P(W|E) or P(J|E)?

To compare these two probabilities, it might be best to take the ratio between the two. Let's find P(W|E)/P(J|E). If the value of P(W|E)/P(J|E) is greater than 1, that means that, given the census officer's observations, he was justified in concluding that all villagers are named William Williams. If the value is less than 1, then he was not justified in his conclusion. According to Bayes' Theorem, P(W|E)/P(J|E) = P(W)/P(J) * P(E|W)/P(E|J).

Basically, this means that the officer's conclusion depends on two factors. The first factor, P(W)/P(J), is the ratio of prior probabilities of W and J. This means that if we know that W is much more likely than J before learning about the results of the census, then will still be much more likely after learning about the census. After all, the census is just one piece of evidence, which may not be enough to overcome our previous prejudices. What could be the source of these prejudices? Perhaps we know from the last census that J was true last year. Or perhaps they all adhere to some religion that requires them to all share a surname. But in absence of anything like that, we may assume that P(W)/P(J) is not too big or too small. It's about equal to 1.

The second factor, P(E|W)/P(E|J), is easy to calculate. P(E|W) is nothing other than the probability of E given W. If W is true, then E is necessarily true, so P(E|W) is simply 1. P(E|J) is the probability of E given J. If J is true, then it is only by lucky coincidence that the census officer only found villagers named William Williams. Assuming that the census officer questioned four random villagers, then P(E|J) is equal to 1/5. Therefore, the second factor, P(E|W)/P(E|J), is equal to 5.

Going back to Bayes' theorem, we know that P(W|E)/P(J|E) = P(W)/P(J) * P(E|W)/P(E|J). Based on my analysis, this equal to 5. Since it's greater than 1, that means that the officer is justified in concluding that all the villagers are named William Williams.

But hold on! Recall that I said that P(W)/P(J) is "about" 1. I'm a physicist, so when I say something is "about" 1, I really mean that it could be anywhere between 0.01 and 100, maybe even more or less. There is a lot of uncertainty which I completely ignored. With this in mind, the value of P(W|E)/P(J|E) can be anywhere between 0.05 and 500. The census officer's conclusion doesn't look so solid now!

Another comparison

Previously, we compared the two claims W and J. But what happens if we introduce a third claim C?

• C: The four villagers which the census officer questioned are named William Williams. The last one has a different name.

Claim C is distinct from J because it's more specific. Claim J only states that one of the villagers is not named William Williams. Claim C states that a specific villager, the one which the officer skipped, is not named William Williams.

Let us compare claim C to claim W. As before, we will calculate P(W|E)/P(C|E),* which is equal to P(W)/P(C) * P(E|W)/P(E|C).

*The advanced reader would note that this is also exactly equal to P(W|E)/P(J|E), our previous result.

Our first factor in the equation is P(W)/P(C). We're comparing the prior probabilities of two claims. W claims that the last villager is named William Williams. C claims that the last villager is not named William Williams. All things considered, C seems more likely than W, since there are a lot of names out there, and only one of them fits W. Nevertheless, in absence of anything which would cause major prejudice, I will say that P(W)/P(C) is "about" 1, same as I did before.

The second factor, P(E|W)/P(E|C), is easy to calculate. Both W and C necessarily imply E, therefore P(E|W) = P(E|C) = 1. The ratio is just 1.

Therefore, when we compare the claims C and W, we find that P(W|E)/P(C|E) is "about" 1, same as P(W)/P(C). The ratio of probabilities is the same before and after the results of the census. The evidence tells us nothing about the comparison of claim C to W. The census officer's conclusion is looking even weaker than before!

Of course, the officer's argument was weak from the start. But before, the weakness was hidden in the word "about".

There moral of the story is that you have to be very careful with Bayesian analysis. Sometimes, the conclusion seems to depend on what angle you look at it. There is always a degree of uncertainty in the prior probabilities that you can never eliminate. The best way to do it is with more evidence, evidence, evidence. For instance, if the officer had simply questioned the last villager, then his evidence would likely overwhelm any previous uncertainty.

Searching in closets

I've joked with some of my friends that I get withdrawal symptoms from physics. After I’ve finished all my physics homework, that’s when the doubts begin. What am I doing in my life? Is there something I’m missing out on? Why can’t more of life’s problems be solved by elementary considerations of differential equations?

And so my thoughts turned to something that I usually completely ignore: romance. I’m afraid that I’m going to sound like some utterly confused teenager when I talk about this. I just don’t get any of it really. But then, what is the point of accumulating readers if I can't occasionally inflict my life upon them?

To be direct, I was entertaining the possibility that I’m gay. The answer, I believe, is no, I’m not. How do you even know if you’re gay? People just seem to know. They feel it inside. They feel an attraction to people of the same sex. I don’t really feel any of that. So I guess it was silly of me to ever worry about it in the first place.

But that was not immediately obvious to me. See, I deeply fear the possibility that I might be anything other than straight. I am very supportive of the LGBT movement, and I would never hate anyone for being queer. But if it were me, I might hate myself for it. It's not what I would want to be. I'm special and unique in quite a number of ways already, and I don't need more than that. I have a lot of respect for homosexuals who come to be comfortable with themselves, because I'm not sure I could accomplish that myself.

I strongly believe in critical thinking as a guiding principle in all of life's decisions. Therefore, when fear comes into play, rather than ignoring it, I keep a very close eye on it. You never know when a fear might lead to denial. If I fear a conclusion, I might dismiss it far too quickly, and then conjure up a rationalization after the fact. So when fear comes into play, I become more cautious, and do not immediately trust my own judgments. On the other hand, fear can also have the opposite effect. I could end up focusing disproportionately on a remote possibility, just because I fear it. I suppose that's exactly what happened this particular time.

But that’s not the end of it. There was a reason why I considered that I might be gay in the first place. I may not ever have felt attracted to males, but I’ve hardly felt any attraction to females either. So by the same logic which says I’m not gay, I could also argue that I’m not straight.

Story time! Remember back in middle school when everyone started getting crushes? I remember it happening to the other kids. I had a large group of good friends, about half girls, half boys. At first I was oblivious, but soon I realized there was all sorts of activity going on between various of my friends. Mostly, I just heard snippets of gossip while listening in on friends’ conversations. It was all about who made a cute couple, who got unexpectedly dumped, and so forth. I couldn’t really keep track of the drama myself, because I was rarely part of that particular conversation. I mostly concerned myself with other things.

Incidentally, I did have my own girlfriend one time. We didn't do a whole lot. I was too young. I couldn't figure out how I was supposed to react to anything. Nevertheless, I make a point of never regretting actions that are more than a few years old. Lots of young teens are plenty awkward, and nothing needs to be made of it.

But nothing has happened since, not that I’m aware of. I’ve always believed that romance is something that happens to you, and then you start worrying about it. Nothing has ever happened, so I’ve never worried about it. I've never had a crush, never thought anyone was "hot", or felt the urge to get to know any particular person intimately. It was not a matter of being too nervous to act upon my desires, it was a matter of not having them to begin with. In this respect, I have not changed much since middle school.

I suppose there are many factors I could blame. I am an introvert after all. I probably spend more time doing physics homework than partying (because I enjoy the former more than the latter). But being introverted does not mean being socially unsuccessful. Introverts are simply less concerned about seeking social activities. I may be introverted, but I am not shy, and I have plenty of friends. So I don’t think it’s for want of meeting new people.

With all that in mind, I’ve been considering a new possibility: am I asexual? The answer, I believe, is no, I’m not. But again, fear is in play, and I do not immediately trust my own judgment.

Asexuality is defined on AVEN, the Asexual Visibility and Education Network. The definition is someone who does not experience sexual attraction. However, many asexuals experience romantic attraction of a nonsexual nature, or experience sexual arousal which is not connected to other people. Asexuality first showed up in Kinsey’s famous studies of sexuality. In Kinsey’s study, he placed everyone on a scale from 0 to 6, where 0 is heterosexual, and 6 is homosexual. But he also placed about 1% of the population into a category “X” because they had "no socio-sexual contacts or reactions" (from Wikipedia). Since then, they’ve been ignored by all but a few studies, and by the public. I know I had never heard of it.

AVEN contends that asexuality is an intrinsic quality of a person. Just like homosexuality, it is not a choice, and it is unlikely (though possible) for it to change. But without much scientific study done on the subject, I am skeptical of these conclusions. I suppose it is plausible, and I believe it. On the other hand, in the case of a specific individual, how does one distinguish between asexuality and clueless romantic incompetence?

I honestly do not believe that I am asexual. The trouble is, I can't seem to express exactly why I feel this way. It sure makes me suspicious when I can't make my reasons explicit, not even to myself.

And if that weren't enough, there's also Grey-A, which is the blurry boundary between sexuality and asexuality. Ugh, now I'm thinking I'm demisexual, meaning I never get attracted to people based on looks or other instantly available information. Life kinda sucks for demisexuals because they usually only become attracted to people who are already friends. No matter how friendly I am, that's a relatively small set of people. And then it's like you're crossing some sort of awkward friends boundary.

Sigh...

At least writing all that made me feel a little better (even if my parents may be reading this :-/ ).

Forks and Torques

Today is a good day I think to rationalize trivial personal quirks.

For example, I have the quirk that I hold forks and spoons in an improper way.

If you're holding a spoon or a fork alone, you're supposed to hold it with your thumb and index finger as it rests on the side of your middle finger. This really needs a picture, but unfortunately, the only one I could find is copyrighted, and I don't feel comfortable taking it. Hopefully, the lack of pictures on the net means that most people simply don't care about etiquette

Out of habit, I hold forks differently, and I could just leave it at that. But today let's take option B: indisputable proof that my way is superior.

The problem with the "proper" way to hold a fork is that it's inefficient. How are you supposed to balance the torque on the fork? If you got food on one end of the fork, it applies an off-center force. Thus, it applies a torque. Without an equal and opposite torque, the fork will simply tip over, resulting in an etiquette catastrophe. Imagine all the gasps which will be pointed in your general direction.

The best way to apply an opposite torque is to apply a downward force on the handle of the fork. The amount of torque is proportional to the distance from the center of mass times the strength of the force. Thus, if you hold it the "proper" way, only holding the fork near the center, you must apply a much stronger downward force. This makes the fork seem much heavier than it really is.

Put in plainer words, what's that long handle for if you're not going to use it for extra leverage?

I suppose if this really becomes a problem, you're probably putting way more food on your fork than is polite...

How do I hold it, you ask? Erm... I do it in a way that gives me more leverage. Do not question this!

Next time, I will discuss why it is always better to break your egg on the larger end.

No need to wait for the endtimes

"If people think the world is going to end on December 21, 2012, all we gotta do is make them commit. And then when December 22nd, 2012 rolls around, they'll realize how silly it all was."

I'm pessimistic about the effectiveness of this argument.

How do we know that there will be a 2013, anyways? You can't talk about future evidence. Either you've got present evidence, or you've got none at all. So what is the present evidence, and why aren't we talking about that instead?

I just don't buy the idea that belief in the endtimes will suffer a major blow after 2012. People will simply forget about it, and move on to the next date. What makes me think this? Well, remember Y2K? Neither do I. No one will remember for very long.

By patiently waiting for 2012, we are being complicit in this forgetfulness. We are acting as if December 12, 2012 is the first and only prediction of endtimes. In reality, the idea that the world as we know it will very soon come to an end is one of the most historically persistent delusions in existence. The point is, we do not have to wait for the prediction date to pass, because it has already passed. It has already passed thousands of times throughout the ages.

Here is a pretty good list of predictions of the end of the world as we know it.

So there is our present evidence. Our present evidence is that previous predictions have all been wrong. So if you believe in the endtimes, it is your turn to present the evidence. Show us why your scenario is far more likely than all other endtimes scenarios which have been predicted in the past.

Mutual friends and strangers

Another puzzle...

There are six people in a group. If you consider any two people in the group, they are either friends to each other, or strangers to each other. Prove that there must exist three people in the group who are mutual friends or mutual strangers.

And now for the challenge puzzle.

There are seventeen people in a much unfriendlier group. If you consider any two people in the group, they are either friends, strangers, or rivals to each other. Prove that there must exist three people in the group who are mutual friends, mutual strangers, or mutual rivals.

(See the solution)

Solution to measuring problems

These are the solutions to "Two more measuring problems".

Susan of Intrinsically Knotted solved the first one:

Start both ropes burning, the first rope just on one end, and the second rope burning from both ends. The second rope will burn completely in exactly 30 minutes (since it is effectively burning twice as fast), at the end of which you will have exactly 30 minutes of rope left on the first rope. Now you can burn the first rope from both ends and it will take exactly 15 minutes.

Eduard's friend Constantino solved the second one in eight steps. This is impressive, since I was only previously aware of a nine-step solution.

Notation is X/Y where X is number of cups of water, and Y is the number of cups of lemonade. The containers are listed in order from smallest to largest.

Start: 0/0, 0/10, 0/0

1. 0/6, 0/4, 0/0
2. 0/0, 0/4, 0/6
3. 6/0, 0/4, 0/6
4. 0/0, 0/4, 6/6
5. 0/4, 0/0, 6/6
6. 1/5, 0/0, 5/5
7. 0/0, 1/5, 5/5
8. 0/0, 5/5, 5/5

And there are lots of other solutions you can think up, I'm sure.