The n-Category Café

Ah, I can tell you exactly where that logic goes wrong.

If you look carefully, you’ll notice that the issue is the multiple meanings of the word ‘Ultimate’.

I’ll illustrate by example: When I start on a new topic of inquiry, I tend to make a list of all open questions I have, and roughly order them by “expected information gained from answering” that question. So whichever question goes to the top is the “ultimate” question for me for that topic, in the sense that I’ll gain the most information from answering it.

(I suppose… if you did that for an entire field of study across everyone working in that field then whatever came up at the top would be THE ultimate question for that field. Maybe that question would be “what is the complete theory for this field of study” or something).

So in other words there are two different questions that we can call “the ultimate question” -

Q1 - the question whose answer is “make a list of all the questions and select the one at the top”

Q2 - that top question whose answer is “the complete theory of this field”.

So by calling both of these questions “the ultimate question”, David makes it seem as if “the answer to UQ is UQ”, however in reality he is saying: “The answer to Q1 is Q2”

And there’s nothing strange about that.

Anyway, my point is: if people wanted to start collecting open questions and voting on them, then I’d actually help with writing a web app for that. “What are open questions for field X” is usually my first google search when learning a new thing, and given the groundbreaking work you guys are doing here, imagine how useful such a resource would be!

There are several problems with the logic. First, when he says the Ultimate Question is “What is the answer to the Ultimate Question,” that is true in one way, namely that it has the same answer, and false in another, namely in that the question is being described in a different way.

Second, he then changes it to “What is the Ultimate Question, and what is its answer,” and this is surely NOT the Ultimate Question, but a different question, since the answer will no longer be the same as the answer to the Ultimate Question or to “What is the answer to the Ultimate Question,” since the answer to the former will include saying what the Ultimate Question is, while the answer to the Ultimate Question will not include that.

Third, in order to answer the question he raised, he needs to provide the answer to the Ultimate Question, and when he says “the answer is what has just been given,” this is wrong, because it does not include this part of his answer.

I agree that it counts as funnier to the degree that the logic is more valid, but unfortunately it is not very valid.

How anticlimactic. ;)

Jokes aside, a contender would be, “why does there exist anything at all?” This goes for the universe and multiverse, and any other possible levels of physical manifestations. For our universe, the completion of quantum gravity can at least provide a tantalizing hint at the answer. The journey continues.

Over on G+, the logician Joel David Hamkins pointed out an earlier version of this puzzle:

• Ned Markosian, The paradox of the question, Analysis 57.2 (April 1997), 97–99.

I’ll quote the whole thing:

Once upon a time, during a large and international conference of the world’s leading philosophers, an angel miraculously appeared and said, ‘I come to you as a messenger from God. You will be permitted to ask any one question you want – but only one! – and I will answer that question truthfully. What would you like to ask?’ The philosophers were understandably excited, and immediately began a discussion of what would be the best question to ask. But it quickly became obvious that they needed more time to discuss the matter, so they asked the angel if he could get back to them. The angel was obliging, and said that he would return at the same time the next day. ‘But be prepared then,’ he warned them, ‘for you will only get this one chance.’

All of the philosophers gathered at the convention worked at a frenzied pace for the next twenty-four hours, proposing and weighing the merits of various questions. Other philosophers from around the world became involved as well, faxing and emailing their suggestions. Some were in favour of asking the kind of practical question that lots of people might like to know the answer to, such as this one:

(Q1) Is it better to check your oil when the car is hot or when it is cold?

But others said they should not squander this rare opportunity, which gave them a chance to learn something about a truly important and intrinsically interesting topic, and after some discussion it was generally agreed that this was right.

The philosophers were puzzled, however, about which truly important and intrinsically interesting topic they should address in their question. The problem was that they really needed to know in advance what would be the best question to ask, in order to make the most of their marvelous opportunity. One proposal was to try to sneak in two questions, by asking something like this:

(Q2) What would be the best question for us to ask, and what is the answer to that question?

But this proposal was quickly voted down when it was pointed out that the angel had explicitly said that they would get just one question.

Another proposal was simply to ask the first of the questions in Q2, in the hopes that some day they would have another opportunity similar to this one, when they could then ask the question they knew to be the best. This proposal was ruled out, however, on the grounds that if they adopted it then they would probably never get a chance to ask the best question once they knew what it was. For a while there was a growing consensus that they should ask this question:

(Q3) What is the answer to the question that would be the best question for us to ask?

That way, it was argued, they would at least have the all-important information contained in the relevant answer. But eventually concerns were raised about the possibility of receiving, in response to Q3, an answer such as ‘seven’, or ‘yes’, which would mean nothing to them unless they knew which question was being answered.

Finally, just as the philosophers were running out of time, a bright young logician made a proposal that was quickly and overwhelmingly approved. Here was her question:

(Q4) What is the ordered pair whose first member is the question that would be the best one for us to ask you, and whose second member is the answer to that question?

Nearly everyone (remember, these are philosophers we’re talking about) agreed that this was the ideal way to solve their little puzzle. By asking Q4 the philosophers could ensure that they would learn both what the best question was, and also what the answer to that question was. There was a great deal of celebrating and back-clapping, and as the minutes ticked down to the time when the angel had promised to return, the mood among philosophers throughout the world was one of nearly feverish anticipation. Everyone was excited about the prospect of learning some wonderful and important truth. They were also more than a little pleased with themselves for hitting upon such a clever way to solve the problem of how to find out what the best question was, and also get the answer to that question, when they had only one question to work with.

Then the angel returned. The philosophers solemnly asked their question – Q4 – and the angel listened carefully. Then he gave this reply:

(A4) It is the ordered pair whose first member is the question you just asked me, and whose second member is this answer I am giving you.

As soon as he had given his answer, the angel disappeared, leaving the philosophers to pull out their hair in frustration.

The above story leaves us with another little puzzle to solve. At the time the philosophers asked Q4, it seemed like that question was the ideal one for their peculiar situation. But as it turned out, Q4 was obviously not at all the right thing to ask. (They would have been better off asking whether one should check one’s oil when the car is hot or when it is cold.) The puzzle, then, is this: What went wrong?

Hamkins also recommended this response to Markosian:

• Theodore Sider, On the paradox of the question, Analysis 57 (1997), 97–101.

I like it a lot. It confirms my impression that the original ‘paradox’ is not very compelling. But it goes further and makes the whole issue more interesting. I’ll quote a very large initial segment, leading up to a serious proposal for the Ultimate Question.

Ned Markosian (1997) tells a story in which philosophers have an opportunity to ask an angel a single question. In order to circumvent their ignorance of what question would be most beneficial to have answered, they hit upon:

Q4: what’s the ordered pair (x, y), where x = the best question to ask, and y = the answer to that question?

(I will understand the goodness of a question to be measured by how much the human race would benefit from having it answered. Note that it’s unclear why Q4 should count as just one question, given that in Markosian’s story, ‘what is the best question to ask, and what is its answer?’ didn’t count as just one question. But no need to settle this matter of question counting; we can restate the puzzle: let the philosophers be granted 15 seconds in which to ask questions (in English).)

In response to Q4, the angel answers: ‘it is the ordered pair consisting of the question you just asked, and the answer I am now giving’ — that is,

A4: the ordered pair (Q4,A4)

But A4 is obviously useless; the puzzle is, as Markosian puts it, to determine what went wrong in the philosophers’ quest to learn something beneficial. We should begin the diagnosis by noting that the ‘angel’ is an imposter, for he gave the wrong answer to the philosophers’ question! Suppose otherwise— suppose A4 is the right answer to Q4. Then Q4 is in fact the best question to have asked, and A4 is the answer to that question. But that means that Q4 wasn’t the best question to have asked after all. Learning that A4 is the answer to Q4 is useless; the philosophers would have been better off asking about the best way to change a car’s oil.

Note what this does, and doesn’t, establish. It does establish that A4 isn’t the right answer to Q4; it doesn’t establish that Q4 wasn’t the best question to ask. For Q4 asks for an ordered pair; a mistaken answer to Q4 will be an ordered pair, at least one of whose members is mistaken. So one of A4’s members is mistaken. The first member of A4 says that Q4 is the best question; the second says that A4 is the answer to Q4. One of these is mistaken. We’ve already seen that the second is mistaken. But we cannot yet conclude that the first is mistaken as well.

We can, however, go on to argue that Q4 isn’t the best question. Suppose otherwise. Then it must have an answer (a question without an answer would be a very poor choice for the philosophers to ask — better to ask about changing oil). Call that answer X. X must be an ordered pair consisting of Q4 and Q4’s answer. That is, X= (Q4,X). Some would argue that there can be no such X, on the grounds that X contains itself as a member. Maybe this would be rash, since there are consistent set theories that allow such things. Anyway, we needn’t settle this question, for we can continue the argument as follows: if X is the answer to Q4, then since X is useless as an answer, Q4 couldn’t be the best question. Our reductio assumption is thus contradicted.

This is not to say that the philosophers made an awful blunder by asking Q4. True, Q4 isn’t the very best question they could have asked. (And what’s more, they could have known this, by duplicating the reasoning in the previous paragraph.) But we can’t be too hard on them for not coming up with a perfect question. Coming up with and agreeing on the very best question to ask would surely have been an unprecedented philosophical triumph.

It may be objected that, in light of the catastrophe with the angel, Q4 was worse than non-perfect: it was nearly the worst question to ask. But this wouldn’t follow from what has been said. The fault, as I’ve said, in part lies with the ‘angel’, for he gave the wrong answer to the question. For all we’ve said, Q4 may have been a perfectly reasonable question to ask. The argument that Q4 had an ‘unfounded’ answer of the form:

X = (Q4,X)

depended on the assumption that Q4 was the best question to ask. When that assumption is dropped, the answer to Q4 may very well be perfectly informative. It will have the form

X = (Q,Y)

where Q is the best question, and Y is its answer. Since X and Y are answers to different questions (Q4 and Q, respectively), they may very well be distinct, and hence X may very well be useful.

But there is a further challenge to the wisdom of the philosophers, according to which Q4 was quite a foolish question to ask. The argument that Q4 might have a useful answer depended on the assumption that Q4 is not the best question, for in that case the answer to Q4 might look something like this:

(‘What is the solution to the problem of world hunger?’,Y)

But how could ‘What is the solution to the problem of world hunger?’ be a better question than Q4, given that the answer of each gives the solution to world hunger? If anything, Q4 seems a better question than ‘What is the solution to the problem of world hunger?’, because in learning its answer we learn, not only the solution to the problem of world hunger, but also something additional: that it would have been best to ask about world hunger.

Here, in more careful form, is the argument that Q4 is quite a bad question to have asked. Suppose otherwise — suppose that Q4 is a somewhat reasonable question to have asked. Then it must have an answer, for any question that has no answer would be completely useless to ask. Then there must be such a thing as the (one and only) best question, for that question would be the first member of the ordered pair that is Q4’s answer. Call it Q. Because of the reasoning in the previous paragraph, it seems that Q4 is at least as good a question as Q. So Q isn’t the best question after all; at best, it is a best question — one of the questions such that there are no better questions.

This suggests that the philosophers might have better modified Q4 along the following lines:

Q5: What is an ordered pair consisting of one of the best questions we could ask and one of its answers?

This sort of question does not have a unique answer, but rather has many answers; since some of its answers may be hoped to be useful ordered pairs of questions and their answers, Q5 might seem to be a reasonably good question for the philosophers to have asked.

A curious fact about Q5, however, is that it generates a paradox, which for me is the real paradox of the question, and a paradox that I do not know how to resolve. Whether this is simply a variant of one of the more familiar semantic paradoxes, I do not know. Either Q5 is, or it is not, one of the best questions; but either supposition leads to contradiction. Suppose first that Q5 is one of the best questions. It cannot be the only best question, because then its only answer would be a useless, unfounded ordered pair of the form:

X = (Q5,X)

So it must be tied with other, presumably ‘first order’ questions, such as ‘What is the solution to the problem of world hunger?’. (If all the best questions were like Q5, then all their answers would be useless.) But now the problem is that there seems to be a danger in asking Q5: one of Q5’s possible answers is a useless, unfounded ordered pair. Since first order questions lack this trouble, Q5 would seem, after all, not to be one of the best questions. So let us consider the other supposition, that Q5 is not one of the best questions. Then there is no such danger: the first member of an answer to Q5 must be one of the best questions and thus could no longer be Q5, and thus there’s no danger that an answer to Q5 would be unfounded. But with this danger removed, it’s hard to see why Q5 wouldn’t be one of the best questions. An answer to Q5 would presumably give us an answer to one of the first order best questions, and thus it’s hard to see how Q5 would be inferior to that first order question.

That, then, is the paradox of the question: Q5 cannot be consistently supposed to be one of the best questions to ask, but neither can it be supposed to not be one of the best questions. It is no solution to reject the possibility of the angel, for the angel’s existence is not required to generate the paradox: in our present, angel-less state, we simply need to consider the value of having various questions answered. One could reject the notion that there are any such things as best questions: perhaps for every question there is a better. But it is hard to believe that we could be forced to accept such a conclusion by a priori means. Moreover, if we restate the paradox as I suggested above, so that the angel gives a fixed time period for the question, then there will only be finitely many questions stateable by humans in English.

What should we do if we are ever confronted by such an angel? One is tempted to simply avoid the paradoxical question Q5. This reaction seems irrational since the paradox is generated simply by the question’s existence, and not by its being asked. Nevertheless, if I were among the philosophers in the story, I would have suggested something like this:

Q6: What is the true proposition (or one of the true propositions) that would be most beneficial for us to be told?

In the early days of Sussex University a lecturer in Physics proposed the following question (Q) and sample answer (A) to the Physics Subboard:

Q: Propose a question for next year’s Physics Final Examination, and give a sample answer.

A: See above.

Alas, nobody had the imagination to give the obvious answer.

“What is the answer to this question?”

This is completely obvious: there is no difference between the question “What color was Alexander’s white horse?” and the question “What is the answer to the question ‘What color was Alexander’s white horse?’?”

This resembles eta-expansion.

The answer to why anything exists at all is the anthropic principle - self-aware structures are self aware, and tautologically are situated in contexts which are consistent with (i.e. explain) their self-awareness.

The authors mistake is in putting so much significance on the present moment - but it’s just as uncertain as the past, a reduction of sense and thought to narrative, relations of sense and cognition and memory interwoven through time. Choice of horizon is arbitrary.