One might think that asking a question is very trivial; in mathematics, maybe in early years of learning, asking what 1+1 is might be very innocent, until to later realize that the answer can be complicated. After years of mathematical training, some students seem to get so used to be assigned exercises that they may think that asking questions (and solving them) probably is the easiest skills they need to acquire during their training. Nevertheless, Georg Cantor seems to hold a different opinion about questions; indeed, he claimed, “In mathematics, the art of proposing a question must be held of higher value than solving it.” What could be involved in asking a question that makes it so important, according to Cantor?
First, to liken proposing questions to an art seems to imply that some skills are needed to master such art. As a result, proposing questions can require as much attention and time as answering those questions. Besides time, I wonder how one could master this art and whether one needs to study specific rules to be considered an expert in proposing questions. If such rules exist, I suppose that some people, maybe mathematicians, have created them; if so, who are they? What inspired them to devise those rules? When it comes to rules by one person or a group, the question of cultural relativism may arise because rules, while they may make sense or are desirable for one cultural group, may be entirely repulsive for others; since practice of mathematics crosses several cultural boundaries, one may wonder to what extent such a unified art could be possible if all cultural peculiarities need to be considered. Also, this issue of cultural relativism may bring in a normative aspect to Cantor’s claim in that he may imply there are a good and a bad way to propose questions in mathematics.
Aside the two issues brought up above, one may wonder about the importance of not just questions in mathematics, which seems to be obvious, but also of the way those questions are posed. Concerning their importance, I have the impression that questions encourage or push mathematicians to establish new connections between areas, to create new theories, and to add to the list of solved problems; also, tackling problems seems to be a way to make some mathematicians’ work meaningful to themselves, and this justification of their work might be why some of them can spend years of their lives on few problems without necessarily solving them. But, I think that this apparent failure for a solution may still be considered as satisfying for some of them. For the way those questions are posed, I still don’t know what this could mean. Since Cantor unfortunately cannot be contacted, I would be delighted if some mathematicians can help. In Cantor’s perspective, I suppose to ask what 1 + 1 is might be either an awful way to ask such a question or the best way to the extent that it can be a model for other questions. Nevertheless, I suspect such a question might be too simple to be considered as a model since many other questions in mathematics may not be about simple computation. If such a question surprisingly is a model, maybe its simplicity and conciseness are its qualities.
There also is the question of medium; by art, would Cantor consider the medium used in the proposition of the question? Would asking the question in a book or paper better than in an informal setting during conversations, or vice-versa? In the current age, would he abhor PowerPoint slides and see them as the worst way to propose a question, or would he approve of them? What about e-mails, twits, blogs, or text-messages? Would they all be viable media for questions?
Furthermore, when one considers the amount of time and energy invested in solving mathematical problems and the apparent easiness to pose those problems (I have heard some questions sometimes are scribbled on margin of notebooks.), Cantor’s claim rather seems paradoxical. In such a case, what could this value be if it is not measured with time and mental energy? Perhaps, a reason for the priority of posing a problem over solving it is that the way a question is asked may influence people’s understanding of such question hence its answer: If someone misunderstands the question, an answer may never be possible or a solution may be hard to find just because the formulation of the problem is not clear enough. If this interpretation is true, Cantor’s claim seems to imply that some advances in mathematics are not necessarily guaranteed by solutions to problems but by the way they are presented.
Finally, many mathematics training programs seem to emphasize problem solving, where students merely are given exercises with no comments on their formulations. Could Cantor’s claim be understood as a prescription to those programs? Would students then be required to take classes that emphasize the form of a question rather than its solution? In such a case, writing courses with a focus on language probably would be needed, where, for example, nuances between meanings of a word would be discussed.
What, then, do you think Cantor’s sentence mean?