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 intuition 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.
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.
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:
- 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.
- 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.
- 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.
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.
