As far as I know, “intellectual gymnastics” is used in a negative sense. For example, the discipline of philosophy can be belittled as “intellectual gymnastics”.
However, a university in Japan seems to be advertising one of their new academic programs by using the term in a positive sense as follows:
Advanced Education in the Liberal Arts -- Intellectual Gymnastics
Is this simply an incorrect use because of connotation? Or is there maybe some other connotation of the term that is positive?
The negative version of the phrase is similar to "jumping through hoops"; in other words, performing a physically demanding or tiring task for a small or shallow reward. In this case, it's referring to the act of wasting your mental strength thinking about something that doesn't deserve the time and effort.
The positive version takes into account the positive connotation behind exercising: the brain is a muscle, and giving it a "work out" strengthens it. In this case, doing "unnecessary work" is still of benefit.
I believe the first version is, as you said, far more common, but both seem to make sense in their own contexts.
My feeling -- strictly a personal reaction -- is that it is inherently negative. "Intellectual exercise", OTOH, has positive or negative connotations depending on context.
Scientists now tell us that intellectual exercise, e.g., solving puzzles and learning new things, can delay Alzheimer's disease seems to me to be positive.
A philosophical argument is an intellectual exercise that is generally unrelated to the world beyond the armchair and thought experiments seems to me to be negative.
It's not unheard of to try to overcome the negative connotations of a phrase. Look at how some companies tried (and a few succeeded) to turn the always negative Brand X into a positive brand name.
'Good' intellectual gymnastics:
[Emphasis mine]
The knowledge of at least rudiments of logic is recommended as one of the foundations of the true humanism of our time: excellent intellectual gymnastics, scientific logic is capable of clarifying confused thought, by banishing expressions with ambiguous meaning, by eliminating the vague "more pernicious than error." This intellectual gymanastics ... is so indispensable.
[M. Boll and J. Reinhart, "History of Logic", 1961:9-46] Quoted from Akissi, 2011, p.95
And
Fortunately for the world, metaphysics has no practical application; and the great systems which have been reared up and crumbled away again during the past few thousand years have had little effect than an educative one: furnishing intellectual gymnastics of a high order to the student.
[The Edinburgh Review: Or Critical Journal, V 207 p.112 A. Constable, 1808]
The Osaka University was right in stating on its Advanced Education page:
Advanced Education in the Liberal Arts -- Intellectual Gymnastics is a curriculum for students who have attained a certain level of specialized professional knowledge, students who will soon be entering the real world where understanding and abilities are needed in addition to their specialized professional knowledge.
