First of all, some context about this article. Vladimir Arnold was a great mathematician, but his opinions were often extreme, and expressed in a way that drew

First of all, some context about this article. Vladimir Arnold was a great mathematician, but his opinions were often extreme, and expressed in a way that drew many antagonisms. In other words: he was respected for his technical works, but his philosophical positions were often decried. See for instance https://hsm.stackexchange.com/questions/12322/some-references-for-vladimir-arnolds-thesis-mathematics-is-a-part-of-physics https://hsm.stackexchange.com/questions/12322/some-reference... What Arnold criticizes is "Maths modernes", a way of teaching mathematics from theory down to practice. It was introduced in France by the Bourbaki group, and it was a great failure! Though "Maths modernes" have been removed from schools, I believe its influence is still strong at higher levels: I remember an elite student (from ENS, which is the top post-graduate for this in France) at an oral fro the national exam that recruited teachers. He clearly had no practical experience on the subject, but he could get through thanks to a higher-level theory. The jury was impressed, though that was inappropriate since the theory he used was not in the program of the exam. As Arnold expresses in this article, anyone interested in mathematics should build a personal understanding of the concepts. And the stress on modelling is also right. But saying that everything should be rooted in the physical world is more dubious. That same argument was used, centuries ago, to reject √-1 and complex numbers. By the way, the fact that theory sometimes goes far from the initial modelling of reality is not restricted to mathematics, see Feynman famous quote "Nobody understands quantum mechanics." Now your question: "books that teach mathematics" is too broad a category. For instance, I liked the (French) books on geometry (differential or classical) by Michel Berger, and I think they were prioritizing the practice and introducing concepts very gradually. But I had years of practice and solid bases in mathematics before wandering in these texts, so what was simple and evocative to the past me could be abstruse and theoretical to the present me. My own experience is that introducing mathematical concepts through physical representations is useful but not sufficient. To build a personal representation, I often needed practice. After many exercises and questions, even if some of it was repetitive and calculatory, my inner understanding grew and I had a mental image of "how it works".