"question theory"?
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"question theory"?
Is there a mathematical discipline for constructing and analysing questions, something like "question theory" maybe?
I mean, if logic is the discipline for constructing, analysing and proving propositions (=answers!), there should be a discipline that deals with questions, too.
I have tried to find out, whether such a discipline exists, but couldn't find anything yet. Does someone have an idea?
Actually I would expect it to be a branch of logic, if it exists at all...
PS: Sorry for any grammatical errors, English is not my native language. Thanks for your help!
I mean, if logic is the discipline for constructing, analysing and proving propositions (=answers!), there should be a discipline that deals with questions, too.
I have tried to find out, whether such a discipline exists, but couldn't find anything yet. Does someone have an idea?
Actually I would expect it to be a branch of logic, if it exists at all...
PS: Sorry for any grammatical errors, English is not my native language. Thanks for your help!
Re: "question theory"?
Do mean, something that answers such as "What constitutes a 'good' question?" If that's the case, it's kind of settled on a casebycase basis. For example, http://people.math.jussieu.fr/~leila/gr ... geConj.pdf It doesn't seem like it would be a very fruitful area of study to me. Then again, I may not understand what you are trying to say.
 gmalivuk
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Re: "question theory"?
Is there really more to a question than a proposition with something missing?
I mean, sure, there's some interesting logic and linguistics there, but it's not like questions are some kind of entirely different beast that would need a whole new branch of mathematics to deal with it.
I mean, sure, there's some interesting logic and linguistics there, but it's not like questions are some kind of entirely different beast that would need a whole new branch of mathematics to deal with it.

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Re: "question theory"?
gmalivuk wrote:Is there really more to a question than a proposition with something missing?
I don't know, but I would guess a question is simply a proposition with "gaps". The simplest example would be an equation with an unknown variable.
In any symbolically formulated proposition you could replace one or several symbols by questionsmarks (or something else) and thereby turn it into a question.
Still, I don't see why one couldn't build a whole theory around that:
 Defining equivalence for questions (e.g. "5+[?]=8" ~ "3+[?]=6")
 Does the set/class/category of questions (is there such a thing??) have a certain algebraic structure? Is it a group, field, algebra, module,...?
 What is the most reasonable / most obvious / most interesting question to ask, given a certain set of knowledge?
> teaching computers to pose reasonable questions
 Are there "impossible questions"? Perhaps something like "the question about all questions, wich don't ask about itself", in analogy to Gödel's incompleteness?
 ...
What do you think?
 gmalivuk
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Re: "question theory"?
I'd say these are no longer questions, but propositions, despite the presence of "question words" for them in English: "What you add 5 to in order to get 8 is what you add 3 to in order to get 6."thefreiburger wrote: Defining equivalence for questions (e.g. "5+[?]=8" ~ "3+[?]=6")
How could it have such a structure? Propositions don't, so why should questions? Does the set/class/category of questions (is there such a thing??) have a certain algebraic structure? Is it a group, field, algebra, module,...?
I think this is pretty much computational linguistics, rather than a field of mathematics. What is the most reasonable / most obvious / most interesting question to ask, given a certain set of knowledge?
> teaching computers to pose reasonable questions

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Re: "question theory"?
gmalivuk wrote:thefreiburger wrote: Defining equivalence for questions (e.g. "5+[?]=8" ~ "3+[?]=6")
I'd say these are no longer questions, but propositions, despite the presence of "question words" for them in English: "What you add 5 to in order to get 8 is what you add 3 to in order to get 6."
I absolutely disagree
"5+[?]=8" is definitely a question! The answer to that question would be "5+[3]=8" or maybe just "3".
"5+[?]=8" and "3+[?]=6" are equivalent because they have the same answer. The proposition that they are equivalent, would be a proposition, of course. What do you think?
gmalivuk wrote: What is the most reasonable / most obvious / most interesting question to ask, given a certain set of knowledge? > teaching computers to pose reasonable questions
I think this is pretty much computational linguistics, rather than a field of mathematics.
Actually I didn't mean questions in human language and "knowledge" about concrete everyday situations, at least not primarily. What I meant was: give a computer a certain set of propositions about a certain (mathematically formalized!) topic/problem. Then let the computer ask a question. Then add the answer to that question (which is a proposition!) to the set of given propositions and let the computer ask the next question.
The questions should somehow "make sense" instead of being just random. Something an interested math student would ask his teacher, if he were totally unable to make his own propositions.
Example:
Let's say the computer knows natural numbers, nothing more. Then we give it the proposition "5 is a prime number". Then the computer should find an interesting question:
"Is 6 a prime number?"No, "Is 7 a prime number?"Yes, "Is 8... ...would be a bad solution.
"Which of the natural numbers is/are prime?" would be better.
"How many prime numbers are there?" would also be a good question.
"What makes a natural number prime?", "How can I find prime numbers?" would also be interesting questions to ask.
All these propositions and questions would be formulated in a formal language, of course. So this would have nothing to do with (human) linguistics predominantly, but could perhaps someday be useful to computational linguistics, when transferred to other, less purely mathematical problems.
I believe that asking (good) questions is one of the essential skills of a mathematician, at least as important as finding answers/solutions and proofing them. But while one can study for years to learn how to find answers, we never learn to ask the right questions systematically. We just ask them intuitively, but perhaps this intuition can be investigated, formalized, improved somehow?
As far as I can see, there seems to be no such branch of math yet. Why?
 Talith
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Re: "question theory"?
It sounds to me like your 'question theory' is simply a propositial calculus but with a model proposed, a valuation given to a set of variables, and then you want to ask, for what values of other variables do certain propositions evaluate to T. So essentially, you have nothing more than a special case of a formal logic system. With regard to your computer question: why would we want a computer to give us questions to answer when it can already answer the questions it's posing?
Re: "question theory"?
Talith wrote:why would we want a computer to give us questions to answer when it can already answer the questions it's posing?
Because coming up with answers is easy. It's coming up with good questions that's difficult.
 Talith
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Re: "question theory"?
If we were talking philosophy, maybe you'd be right. But any question posed by a computer in the form of a finite sentence from some language with a given valuation for a set of variables can be proven either true or false by the computer itself, so why would we want the computer to give us the question?
Re: "question theory"?
Talith wrote:If we were talking philosophy, maybe you'd be right. But any question posed by a computer in the form of a finite sentence from some language with a given valuation for a set of variables can be proven either true or false by the computer itself, so why would we want the computer to give us the question?
Yes of course my answer is philosophical. But so is the question. The OP asked: "Is there a mathematical discipline for constructing and analysing questions ...?"
And the answer is: No, because it's too hard a problem. In terms of tests for artificial intelligence, computers are getting pretty good at the Turing test. They are better than the best human chess player. Computers are even making progress in automated theorem proving.
But nobody has any idea how to program a computer to ask good questions. That's a very high human art, one that we currently have no way to model or study mathematically. Because if we could ... we would!
As you point out, propositions are simply finite strings from some language that are evaluated according to rules. A machine can analyze a string to see if its wellformed. In some situations, a machine can determine if a proposition is true. But no machine currently imagined can analyze a string to see if it's interesting. Artificial intelligence researchers should give that some thought. It's an interesting ... question.

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Re: "question theory"?
Talith wrote:why would we want a computer to give us questions to answer when it can already answer the questions it's posing?
Wouldn't that be a substantial progress?
Today we can give a computer a certain welldefined problem and with some luck, make it solve it automatically.
If a computer could ask questions, we could just tell it: "Here you have a set of axioms and definitions. Go ahead, find some interesting propositions!". We could have a computer writing a math thesis. I think that would be great!
You can build up a theory about almost everything, step by step, by using old propositions to prove new ones. And usually one proven proposition makes a lot of new ones possible. But if you can't ask questions, how can you decide what to propose next? How do you avoid getting lost in all those possibilities?
Thanks for your answers

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Re: "question theory"?
Today I talked to a math professor who teaches logic, and she told me she didn't know about any such thing as "question theory", at least not within mathematics. If she doesn't know about it, then probably it really doesn't exist.
So now there are three possibilities:
1. No one has ever thought about it.
2. Someone has thought about it, but decided that the topic is so trivial and boring, that there is no point in doing any research.
3. People have been thinking about it, but no one has ever accomplished anything noteworthy, because it's really "too hard a problem".
I cannot see why 2) would be the case, because then we would already have questionposing computers, right?
So it has to be either 1) or 3). In this case, we have a whole unexplored field with a lot of research left to do.
Anyway, I'm all the more interested now!
So now there are three possibilities:
1. No one has ever thought about it.
2. Someone has thought about it, but decided that the topic is so trivial and boring, that there is no point in doing any research.
3. People have been thinking about it, but no one has ever accomplished anything noteworthy, because it's really "too hard a problem".
I cannot see why 2) would be the case, because then we would already have questionposing computers, right?
So it has to be either 1) or 3). In this case, we have a whole unexplored field with a lot of research left to do.
Anyway, I'm all the more interested now!
Re: "question theory"?
thefreiburger wrote:If a computer could ask questions, we could just tell it: "Here you have a set of axioms and definitions. Go ahead, find some interesting propositions!". We could have a computer writing a math thesis. I think that would be great!
This is already being done, but the outcome is not exactly what you apparently think.
The reason for this is that in any sufficiently strong axiomatic system (i.e. one that is "interesting" regarding its complexity), there are uncountably (okay, mathematically speaking, countably  but you get the idea!) many theorems, and the incredibly vast majority of them is boring and, very often, trivial for humans to see. That is why typical mathematics is (to my knowledge) rarely studied in this form (though in some specialzied areas especially in CS, automatic proof generators are used).
Now, if you actually refer to defining "interesting" in a sense a computer could handle, that's a different matter and, I'd argue, not really possible.
And as usual, the allknowing Wikipedia can tell you much more about this. It's a huge area of research.
I even recall seeing a page that had a convenient button that would guarantee you your own "unique" theorem (from basic first order logic) randomly generated and proven on pressing a button, but I can't find it right now.
Edit to add: Not sure if Theory Mine was what I had in mind (if so, then in some free version), but there you go. This is a commercial (!) provider, though.
Re: "question theory"?
Desiato wrote:Now, if you actually refer to defining "interesting" in a sense a computer could handle, that's a different matter and, I'd argue, not really possible.
If formulating interesting questions is truly something that humans can do and machine's can't do, then we have finally put an end to the dream of strong AI  the belief that we can eventually program machines to display humanlike intelligence.
http://en.wikipedia.org/wiki/Strong_AI
I doubt the proponents of strong AI would accept that the game's over and they've lost.
That's why the OP's question is genuinely interesting. It is clearly MUCH harder to formulate a good question than it is to answer a question once it's been formulated. [Aside: Does this remind anyone of P versus NP?]
Have we at long last identified a stringprocessing task that humans can do easily, and that machines can never do, even in theory? If so, then how do humans do it? What do our mushy brains do that electronic circuits, even ones of arbitrary complexity, could never do? Are humans really qualitatively different than machines? If so, by what mechanism?
Yes this is philosophy. But it's also computer science and logic. What is it about the human brain that lets it immediately identify a string as interesting or not? And if a brain made of neurons can do it, why can't a machine?

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Re: "question theory"?
@Deslato:
I think we are talking about exactly the same thing, just with different words.
I've heard of computer programs that were supposed to generate proofs automatically, but simply got lost in the vast number of possible proofs. So 99.99% of what they did was meaningless and boring, because they were unable to distinguish the few interesting things. That's exactly what you are saying.
What I have in mind is a computer that can not just write a random math thesis, but that writes an interesting one.
So the computer has to focus only on possibly interesting propositions. This could be done by asking good questions. Good questions lead to good propositions. So the ability to find those good questions would be no more than a tool for eventually finding good propositions.
As far as I see it, this has nothing to do with automated theorem proving, because that requires an already given theorem that is to be proven.
@fishfry:
Yes, it really is philosophy! But from the philosophical point of view, I am totally convinced, that machines can do everything a brain can do (at least in theory), but not necessarily the other way around. If that's true, it's possible, that we build a questionposing computer someday, and then find out, that its questions are even much better thant the human ones. That humans have always posed "inferior" questions, because human curiosity is so intuitive, thus subjective and irrational. And that mathematics would have come much further by now, if we had had our questionposing computer right from the beginning.
Thank you for your interesting comments so far!
I think we are talking about exactly the same thing, just with different words.
I've heard of computer programs that were supposed to generate proofs automatically, but simply got lost in the vast number of possible proofs. So 99.99% of what they did was meaningless and boring, because they were unable to distinguish the few interesting things. That's exactly what you are saying.
What I have in mind is a computer that can not just write a random math thesis, but that writes an interesting one.
So the computer has to focus only on possibly interesting propositions. This could be done by asking good questions. Good questions lead to good propositions. So the ability to find those good questions would be no more than a tool for eventually finding good propositions.
As far as I see it, this has nothing to do with automated theorem proving, because that requires an already given theorem that is to be proven.
@fishfry:
Yes, it really is philosophy! But from the philosophical point of view, I am totally convinced, that machines can do everything a brain can do (at least in theory), but not necessarily the other way around. If that's true, it's possible, that we build a questionposing computer someday, and then find out, that its questions are even much better thant the human ones. That humans have always posed "inferior" questions, because human curiosity is so intuitive, thus subjective and irrational. And that mathematics would have come much further by now, if we had had our questionposing computer right from the beginning.
Thank you for your interesting comments so far!

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Re: "question theory"?
thefreiburger wrote:"5+[?]=8" and "3+[?]=6" are equivalent because they have the same answer. The proposition that they are equivalent, would be a proposition, of course. What do you think?
This seems to be similar to topology in relation to geometry. It looks at different properties of shapes than were noticed before. this theory would be relating answers to what sort of questions are the most useful in what situations and that answers are properties of the questions.

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Re: "question theory"?
thefreiburger wrote:@fishfry:
Yes, it really is philosophy! But from the philosophical point of view, I am totally convinced, that machines can do everything a brain can do (at least in theory), but not necessarily the other way around. If that's true, it's possible, that we build a questionposing computer someday, and then find out, that its questions are even much better thant the human ones. That humans have always posed "inferior" questions, because human curiosity is so intuitive, thus subjective and irrational. And that mathematics would have come much further by now, if we had had our questionposing computer right from the beginning.
I believe that both directions are equally plausible. Whatever a brain can do, we should be able to create a computer that works on the same principles, even if they are inherently biological. It should also be possible to genetically modify in a way that would create endresults that would work on the same principles of our modern computers. Both genetics/bioengineering and comptur science and the related fields are still in their infancy now and the possibilities are nearly unlimited what we be able to do with them.
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