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All I was able to find through Google regarding the subject is from The Blind Spot, where it says, in part,

The question at stake is the nature of mathematical knowledge and the difference between a question and an answer, i.e., the implicit and the explicit. The problem is delicate mathematically and philosophically as well: the relation between a question and its answer is a sort of equality where one side is “more equal than the other”: one thus discovers essentialist blind spots.

Can you explain/elaborate what they're saying and what it means?

An issue that occurred to me is there are more questions than answers, e.g., we can ask about the halting problem, but can't answer the question in general. Yet both questions and answers are representable as finite strings of symbols (from a finite alphabet). So let S denote all strings, and q,a in S denote a question,answer string. Then there are relations Q,A subset SxS where qAa iff a is an answer to q (i.e., q may be a question with many different answers), and where aQq iff q is a question for which a is an answer (i.e., a may be an answer for many different questions).

I was trying to proceed along these lines, to expand on "more questions than answers" with respect to the cardinality of these Q,A relations, etc. But couldn't really get anywhere, so then tried Google, but only found the preceding quote, which just furthered the confusion. So (a) what's that quote saying, and (b) how should I proceed?

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    You start a formal process to analyze the issue, but if you want to proceed that way, you need to continue on, and define a proof system that will allow to analyze deductions, then establish whether all strings can be reached in your formal system. Then maybe look at what Goedel did.
    – Frank
    Mar 19 at 14:29
  • "The brown one, I want the brown one" said Zelda. "I intelligo ma'am. Fine taste you have. Perhaps you would like to see more of my collection?" "No, that'll be do!"
    – Agent Smith
    Mar 19 at 16:23
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    On Stack Exchange, questions are at the top, answers at the bottom. That was easy.
    – Barmar
    Mar 19 at 21:40
  • @Frank Thanks. Yeah, Godel indeed sounds like a good approach. See my comment to Marco, below, expanding on that idea a bit. You think that's a good Godel-based approach, along the lines you had in mind? Or, if not, then in more detail, what kind of Godel-based approach are you suggesting? As per proof system, maybe the question string should itself enumerate the rules of inference, etc.
    – John Forkosh
    Mar 20 at 6:11
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    It used to be said that there was a question in the Cambridge Philosophy exam that asked "Is this a question?", to which someone apparently got full marks by writing "yes, and this is the answer".
    – Michael Kay
    Mar 20 at 16:05

4 Answers 4

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If a question-answer pair (Q, A) is compared as to the amount of information each encodes, we might say that A > Q insofar as A "fills in the blanks" in Q. For example, 2x + 11.7 = 449,304 might be seen as having three "chunks" of given information (the fixed numbers, maybe including the implied indexes for the arithmetical operators (the + having an index of 1 in the basic hyperoperator sequence, the 11.7 having an implicit moment of division in the representation of its fractional component and then division being a negative operation of index 2). Decoding what x is might be thought of as providing you with more information, then, and so any question that can be formulated in such a way (as a matter of solving for variables, so to say) would be one where, if it has a uniquely paired-up answer, has less information in it than the answer.

However, consider the difference between:

  1. Is 1123 a prime number?
  2. Is 1123 a balanced prime number?

Or:

  1. What number is next after 2?
  2. What prime number is next after 2?

Either pair of questions seems, internal to the pair, to have the same answer ("Yes," for the first two and, "3," for the second). But here the framing of the questions involves different amounts of specification, and so some "virtual" information, at least, that varies.

Now, if the subquestion relation is a logical example of the powerset relation (maybe via category-theoretic mappings involving "power objects"), we might say:

Take a set of questions Q1 paired to a set of answers A. Assume that A enters into the evocation relation in (one version of) erotetic logic. Now define the set of all questions evoked by A as the erotetic powerset of Q1, i.e. here Q2 (this need not indicate the numerical successor exactly, but only the successor by type). By Cantor's theorem, erotetic powersets will always be different in cardinality compared to their bases, and given sufficient well-ordering procedures (classically the generic choice axiom), we can say that an erotetic powerset is more specifically larger than its base (in the same overall well-order) (in an unorderly or outright disorderly world of sets, it could be more precise to speak of powersets as incommensurable with their bases, at least in the zone of infinity).

Moreover, there are some reasons to believe that the powerset relation might inflate its bases to the size of the entire universe of sets. This is trivially so in a pocket-sized set theory where the universe has size ℵ1 (if well-ordered, anyway) regardless, but less trivially, see Storer[10], esp. ch. 3 and 4 for the predicativist equation of an unrestricted subset collection over the naturals with an unrestricted set simpliciter, or closer to the mainline Matthew[21], e.g. pg. 34 for forcing class-many Cohen reals to inhabit the natural powerset. And Conway's surreal infinitesimals occur for every transfinite ordinal, so a Continuum made up of surreal infinitesimals is divisible to absolute infinity, i.e. to a degree equivalent (as an ordinal) to the cardinality of a proper class. So perhaps the erotetic powerset relation can take any infinite set of questions and any infinite set of answers to those questions, and extrapolate a cosmic-scale question-set.

It does seem epistemological, though (c.f. Ted Wrigley's answer to this OP), in the sense that we can imagine a proper class of questions and a proper class of answers, or one set (even a set) of all possible questions and answers together, but it is not as if we know even all the questions themselves, much less their answers. "Long ago," Aristotle thought to carve nature at some finite number of overarching joints, characteristically reflected from kinds of questions, but such theories about categorizing objects always seem to branch off indefinitely or loop in on themselves, or to have trivial or excessively obscure primary conditions of identification, in the end. So one imagines that of the domain of questions there will be no end, and the horizon of our knowledge will always hold the edge of question-space at least a little farther out than the edge of answer-space (or inchoately superimposed upon each other, and changing, so that definitely the space of settled questions and solved problems appears subsumed at the edges in a greater space of open questions and unproven solutions).

As far as the quoted passage goes, you can take "more equal than the other" in the sense that, e.g., "What is x?" can be read for, "What = x?" or even, then, "W = x?" The variable that grounds the process of determining the other variable has something "more to it," something to do with equivalence relations/equality (note that those phrases "equivalence relations" and "equality" are not the same in meaning), so there is some propriety in saying that W or x is "more equal" to the other, than the other is to it. Suppose, if you will, that there is an equivalence relation, we'll use a repeated equals sign to mark it out, which is not commutative. So: W = x, W == x, but ~(x == W), perhaps. (I realize that equivalence relations are defined as symmetric, but this phrase "more equal than" might then be construed as a degenerate such relation, or we would expand to some broader context involving also pseudo-equivalence, etc. and so there's probably a more canonical possible explanation of the quoted phrase than what I'm offering here.)

ADDENDUM: there is a theory, going back as far as I know to one Lennart Åqvist, according to which questions can be interpreted as epistemic imperatives. I used to think that George Lakoff's thing about "don't think of an elephant" meant that even normal assertoric functions encoded for imperatives, too, but I later thought that we could distinguish "don't think of a general unicorn" from "don't think of a particular one" and so now I'm not so sure. So maybe questions can be differentiated from answers in part by reference to the former having imperative counterparts that the latter lack.

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  • I would say that x = 2 contains "less information" than 2x + 3 = 7. I'm not sure this is an ambiguous avenue.
    – Frank
    Mar 19 at 17:17
  • @Frank if we were fully writing it out, I guess we'd start with something like ab + c = d and so the last two lines would read as a = 2, b unset, c = 11.7, and d = 449,304. Then the last one would have b set and would seem to be more informative on the surface than the preceding line?
    – Kristian Berry
    Mar 19 at 17:44
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    Very possible, but I think "informative" would deserve a more precise definition to make this more productive, that's all :-)
    – Frank
    Mar 19 at 17:54
  • @Frank yeah it's one of the words I'm most tempted to use while not quite knowing why it should be used, when it should be (if ever). I get hung up on the etymology first, thinking of it like a process of imprinting a Form, then I think about the quantum-information sense of it, etc. and so far it doesn't end up having a technical marker in my notes.
    – Kristian Berry
    Mar 19 at 17:59
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    @JohnForkosh somewhere in my notes I have some (probably bizarre) consideration about reducing wh-questions to y/n-questions. Note, I think, that reducing to y/n seems to mean reducing to a bivalent/LEM-space, which will fly more with some analysts than others. If variables are question-like in themselves, maybe "proto-erotetic" say, then they at least constitute an erotetic space that is not y/n-formatted, but this might be fixed below the fully propositional level (which might be y/n-formatted).
    – Kristian Berry
    Mar 20 at 8:32
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I would not care to guess what the quote was intended to mean, since it is too ambiguous. However, I suggest an important next step in your quest would be to clarify what constitutes a question and what constitutes an answer. For example, is 'what is the capital of green?' a question? Is 'what is the value of pi?' the same question as 'what is the ratio of the diameter of a circle to its circumference? or are they separate questions? How do you know whether an expression is an answer? For example, take the expression 'Birkenhead Park'. Is that an answer? If it answers two separate questions, does it count as two answers? Does an incorrect response count as an 'answer'? Is a question translated into another language the same question? Do you include questions about matters of opinion, which can have any number of answers? Do you include questions such as 'what is the exact time?' the answers to which are continuously variable? Etc etc. When you have a working definition of 'questions' and 'answers', you might be in a better position to pose the question about whether the former outnumber the latter.

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  • the ratio of the diameter of a circle to its circumference is 1/pi
    – Stef
    Mar 19 at 20:27
  • @Stef Ha! Well said! Feel free to call me the world's biggest idiot. Mwhaah!
    – Marco Ocram
    Mar 19 at 21:24
  • Thanks, Marco. Yeah, clarify what constitutes a question and what constitutes an answer indeed seems like the fundamental issue, since until that's (logically/mathematically) clarified no formal/rigorous analysis is really possible. Maybe a question string, e.g., "is it raining?" is a function q:S-->S that takes premise(s) string(s), e.g., "the temperature is T and the humidity is H", to conclusion(s) string(s), e,g, "it is [or isn't] raining". Then, as per Frank's comment re Godel above, q has an answer iff that function is computable. And then we'd have more questions than answers.
    – John Forkosh
    Mar 20 at 6:06
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This is an issue in epistemology. To keep things in the mathematical context of the quote, think about the nature of a mathematical proof. At the beginning of the proof we have a set of premises and axioms; at the end (after a series of symbolic manipulations) we have an entirely new statement. Clearly that end-statement was implied and contained within the original axioms and premises, but we did not have access to it until after we completed the proof. So what exactly did that process of symbolic manipulation add?

A question merely frames something implicit and unknown. It's like a gift-box: you can guess at what's inside by looking at the size and shape, or by shaking the box a bit, but you don't know what's inside until you open the box up and start rooting around in the packing peanuts. It isn't precisely correct to say there are more questions than answers. The issue is that the universe is always bigger (by far) than the language we use to describe it, and a question merely marks off some corner of the unknown universe for exploration.

Incidentally, this highlights some of the menacing aspects of question-asking. Many people use the act of question-asking to imply an answer while simultaneously cutting off any avenues for investigation or exploration. Sometimes this is innocuous moralizing, like asking a teenager why he thought it was smart to do such a stupid thing. Sometimes it's malicious and manipulative, an effort to get someone to believe the wrong thing for one's own benefit. We have to be clear whether someone is using the question format properly to frame a lack of knowledge, or improperly to create active ignorance. If we can't see that difference, we will suffer for it.

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A question is asked to clarify what piece of information is missing to solve a problem. An answer is that missing piece of information.

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