# Are intuition-based questions off-topic?

I was perusing the "Close Questions" queue just now and noticed a common theme. It seems like a large number of questions involve gaining intuition about some result; most of them use the word "intuitively" and/or have the tag.

Is this a change in site policy? Some of the questions thus tagged seemed like pretty good ones to me, but I'm admittedly not one of the most active or longest-term users of this stack.

• – user223391
Nov 9 '17 at 18:34
• What is intuitive to one asker, is not the same as what is intuitive for another. What counts as "intuition", any way? Nov 9 '17 at 18:40
• I wish I could see the list you saw. Personally I wouldn't vote to close such a question out of hand. I would have to read it and decide if the thing the OP was had some hope of a good answer. I can't think of a really good "bad" example of something I would reject. Nov 9 '17 at 18:43
• @rschwieb check the history of the close review queue; various recent items have the word in the title. Or you could even check the reviews of OP on their activity tab.
– quid Mod
Nov 9 '17 at 18:45
• @quid Thanks: I never noticed that menu before. Nothing on the first page seemed like a poor intuition-type question to me. Looks like perhaps there are a user or more who don't like such questions... Nov 9 '17 at 18:47
• @amWhy I'm not positive about your intentions with the comment above, but I hope it wasn't to be taken as discouragement for questions about intuition. If you just search for "intuition" and sort by votes you can find (IMO) several well-asked and highly regarded questions of the sort. Intuition is a core skill in mathematical thinking, I should say, so it's natural to seek it. That's not to say it can be abused by askers, of course, but then again, a lot of phrases can be abused that way. Nov 9 '17 at 18:52
• My guess is that analogies and some mneumonic devices would be good material for an answer providing intuition, and perhaps other right-brain oriented explanations. For example, I've always wanted an intuition for Hua's identity, but the explanations I've found are always trivial computations. The computations strike me as totally left-brain material. Nov 9 '17 at 18:56
• @rschwieb your hope is correct. My comment was not intended to be a dismissal of any/all posts that include the word "intuitive". I get concerned when there is no context given, to gauge the level the asker is at, or to have at least a hint of what kind of answer would meet the given users request for an intuitive explanation, etc. But with context, much more can be gleaned as to how best to explain or answer a question "intuitively". Nov 9 '17 at 18:57
• @amWhy: So you're intuitively ignoring the word intuitively?
– Asaf Karagila Mod
Nov 11 '17 at 19:52
• @AsafKaragila =P Nov 11 '17 at 21:22
• Best case scenario: We should create a chat room named "Intuition". So those seeking intuition, and those wanting to share it, can meet and interact. Nov 11 '17 at 22:01

Intuition is extremely important in mathematics and not always explicitly taught in math courses. As a working mathematician, I need "hard" knowledge to prove and define things in detail, but "soft" knowledge is an indispensable guide that steers my work. We would lose much if we were to ban intuitive questions here. It's not easy to apply the same standards to hard and soft questions, but I very much want intuitive questions asking for professional opinions of experts to be on-topic.

A hard math question can be very concise but still clear. An intuition question needs more explanation as to what is actually sought for. They are different in nature, but all questions on the site should be sufficiently clear and have enough context. This just happens to mean different things in practice for different kinds of questions.

• You can't teach intuition. You can only learn it.
– Asaf Karagila Mod
Nov 10 '17 at 8:00
• @AsafKaragila I disagree. Intuition is different to teach than what is usually known as math, but not impossible. You can certainly make it easier for your students to learn it, and I call "making it easier to learn" teaching. Nov 10 '17 at 9:07
• You can only obtain intuition by practice. The teacher's role is to help you get there faster with the right exercises, solutions, and as much feedback as possible. But it's still hard to impossible to impart intuition onto the student "just like that".
– Asaf Karagila Mod
Nov 10 '17 at 9:14
• @AsafKaragila True, it doesn't happen "just like that". The student has to go the distance and the teacher is there to support it, be it intuition or "real math". It's often hard to impart hard facts "just like that", too, especially for younger students. Nov 10 '17 at 9:25
• @AsafKaragila I knew the fundamental theorem of calculus for a long time before I grasped the basic intuition that we are summing up a bunch of tiny changes to obtain the total change. That kind of intuitive understanding is invaluable, and often math.stackexchange is the best place to learn it. Many math books fail to explain the intuition; but someone who has managed to grok the intuition can then explain it clearly to someone else. Nov 11 '17 at 0:34
• @littleO: Using hype ideas (like "someone who has managed to grok the intuition can then explain it clearly to someone else", or the horrible horrible thing "you don't truly understand something until you can explain it to your mother/grandmother/cashier at your local supermarket/druggie homeless living in your dumpster") is not conducive to a conversation about anything other than about silly things like that...
– Asaf Karagila Mod
Nov 11 '17 at 0:36
• @AsafKaragila I am just saying that for me personally, I have often found these intuitive explanations to be extremely valuable. Nov 11 '17 at 0:37
• @littleO: But it's not an "intuitive" explanation any more than "the determinant of an n-by-n matrix is the volume of the image of the unit cube" is an intuition about the definition of determinant. It is just recasting the definition in a different way. Intuition is when you just feel that a Cohen real cannot destroy a ground model tower, because it's a Cohen real and there's no way a ccc forcing could do that, but you can't quite put your finger on that. (Later you email someone and they send you a sketch of a proof, which reaffirms your intuition as correct and builds it a bit more.)
– Asaf Karagila Mod
Nov 11 '17 at 0:40
• @littleO: Or when someone asks you what is the intuition behind the definition of a limit step of an iteration of symmetric extensions, and you reply that is is sorta like taking "everything definable from finite fragments of the generic", stressing that this is not really the definition nor it corresponds to the actual definitions, but that it is a good approximation as to how one should think about it at most times, even though ultimately one should work with the proper definition. That's intuition, sure, but that is meaningless to someone who hasn't developed that intuition already.
– Asaf Karagila Mod
Nov 11 '17 at 0:42
• @AsafKaragila I think maybe the disagreement over "intuition" on math.stackexchange simply stems from different people having different definitions of the term "intuition". Nov 11 '17 at 0:50
• @littleO: I think it should be quite intuitive as to what "intuition" means. And if that confuses you, that's exactly my point.
– Asaf Karagila Mod
Nov 11 '17 at 0:52
• Comments are not for extended discussion; this conversation has been moved to chat.
– quid Mod
Nov 12 '17 at 18:28

I find these questions about intuition to be extremely valuable. A perfect example of an intuitive explanation appears on p. 553 of Terence Tao's book Analysis II:

Intuitively, one can think of the several variable chain rule as follows. Let $x$ be close to $x_0$. Then Newton's approximation asserts that $$f(x) - f(x_0) \approx f'(x_0)(x - x_0)$$ and in particular $f(x)$ is close to $f(x_0)$. Since $g$ is differentiable at $f(x_0)$, we see from Newton's approximation again that $$g(f(x)) - g(f(x_0)) \approx g'(f(x_0))(f(x) - f(x_0)).$$ Combining the two, we obtain $$g \circ f(x) - g \circ f(x_0) \approx g'(f(x_0)) f'(x_0)(x - x_0)$$ which then should give $(g \circ f)'(x_0) = g'(f(x_0))f'(x_0).$ This argument however is rather imprecise; to make it more precise one needs to manipulate limits rigorously; see Exercise 17.4.3.

I have often found intuitive explanations similar to this on math.stackexchange that I thought were very enlightening.

Terence Tao has also discussed three stages of mathematical education, in a blog post entitled "There's more to mathematics than rigor and proofs":

One can roughly divide mathematical education into three stages:

1. The “pre-rigorous” stage, in which mathematics is taught in an informal, intuitive manner, based on examples, fuzzy notions, and hand-waving. (For instance, calculus is usually first introduced in terms of slopes, areas, rates of change, and so forth.) The emphasis is more on computation than on theory. This stage generally lasts until the early undergraduate years.
2. The “rigorous” stage, in which one is now taught that in order to do maths “properly”, one needs to work and think in a much more precise and formal manner (e.g. re-doing calculus by using epsilons and deltas all over the place). The emphasis is now primarily on theory; and one is expected to be able to comfortably manipulate abstract mathematical objects without focusing too much on what such objects actually “mean”. This stage usually occupies the later undergraduate and early graduate years.
3. The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.

The transition from the first stage to the second is well known to be rather traumatic, with the dreaded “proof-type questions” being the bane of many a maths undergraduate. (See also “There’s more to maths than grades and exams and methods".) But the transition from the second to the third is equally important, and should not be forgotten.

Note Tao's emphasis on intuition here. I find math.stackexchange to be very helpful for developing the kind of intuition that Tao is referring to.

• Your first example is a proof and it is based on the definition of derivatives. The real example of intuition is the intermediate value theorem. Since graph of continuous function is without any breaks if the function takes positive value somewhere and a negative value somewhere else the graph must cross x-axis in between. This intuitive reasoning is just plain wrong because there are discontinuous functions which satisfy IVT.
– Paramanand Singh Mod
Nov 11 '17 at 6:38
• @ParamanandSingh: "This intuitive reasoning is just plain wrong because there are discontinuous functions which satisfy IVT." That doesn't make sense. The intuition you described says "IF the function is continuous, its graph has no breaks, and therefore intermediate values must be taken." No statement about non-continuous functions can invalidate that intuition. Nov 11 '17 at 13:23
• @celtschk: it takes a bit more than intuition to understand that the "continuity" does not really imply the "no breaks" or "no gaps" property, rather the effect is created by completeness of real numbers. Similarly the IVT for derivatives is based on completeness. Intuitive ideas look very fine and simple and impressive but only when backed by underlying rigor and evidence. After all there was a time when intuitively earth used to be flat (some may feel this way in present time also) .
– Paramanand Singh Mod
Nov 11 '17 at 17:32
• And let's not forget the transition from stage 3 to stage 1 - which happens when physicists teach math. :P Nov 19 '17 at 0:32

There was no very recent discussion about this, though the subject came up in the not too distant past (see the comment by Zachary Selk). In that sense, no there was no change in policy.

However, the general guidelines are very broad and for the most part finer points of policy just write done what is already done in practice. That is to say, it is normal that there is first a change in practice, and then a policy is introduced.

Likely, somebody just went ahead and tested if their point of view has some traction. When reviewing evaluate each question on an individual basis according to your standards.