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Sure. A couple of comments only: The example you give of fields is a good one, but one counter-example of an area where playing around doesn't help much if at all, doesn't really affect the question whether you can learn maths by programming, to which the answer 'No' seems wrong to me. If one person is blind, it doesn't mean nobody can see. Sure, you can't learn all of it. But the other extreme, where you seem to be saying, you can't learn anything—which saying things like that you can't learn a particular piece of maths from programming "until you’ve already learned it" seem close to—seems equally wrong to me. In the area you're talking about, sure, it's hard to see how computers would help ones intuition. But that's very far from meaning that all areas are like that. Also, you're talking about problems, homework - answering other peoples' questions. I didn't have that in mind, but learning, exploring for oneself. And in areas where no-one knows the why, one is exploring the what, how things work. I think you're just thinking of abstract areas where that's not really possible, but in a lot of areas it absolutely is. But yes, I think we're not disagreeing so much as talking and focussing on different aspects/areas of mathematics. And probably you know muuch more maths than I do, and possibly mucking around with computers is of less use in the higher realms than in the lowly plains I inhabit. But still, I can't help thinking that the problem with thinking that knowledge is exhausted by know-that (know-how disregarded as not real/important because can't be written down explicitly, as diagnosed in Rationalism in Politics) is a part of thinking that the things that you say, correct as they are, fit together into any kind of argument that programming is of no use in learning mathematics. But it seems we have entirely different levels/areas in mind. Thanks for the comments, very interesting. |