For Arnold, mathematics is rooted in physical intuition and experimental inquiry. Can you name some math that is completely disconnected from that intuition?
Mathematics may be rooted in that, in a historical or pedagogical sense, but areas of math can certainly be disconnected from physical intuition. Non-measurable sets (e.g. those in Banach-Tarski) and transfinite numbers cone to mind.
Non-measurable sets are precisely the kinds of things Arnold wanted marginalized in mathematical pedagogy, instead of placed front and center. They are necessary auxiliaries to the main theory, that of measures and integration but auxiliary nonetheless.
I disagree that transfinite numbers are detached from physical intuition since most of the ones you or I could write down can be easily visualized with a few ellipses here or there. But i do think Arnold would consider them marginal players. Perhaps he thought set theory was a formalist distraction from the main of mathematics!
E.g. logic? A pretty important part of mathematics that is hard to marginalize, but that can't be observed experimentally. Rather scientific observation presupposes logic ability.