On 7/5/07, Terrence Brannon <[EMAIL PROTECTED]> wrote:
Question 1 - how does one learn mathematics?
Mathematics is an extremely broad topic area. That said, several elements seem critical. First, of course, you need an interest in the topic, and sufficient drive to draw connections, and that sort of thing. But I imagine that's not exactly what you are asking, but that instead you are asking about techniques for learning, which include things like picking up vocabulary (which the dictionary can help with), distinguishing truth from fiction (which experiments can help with), associating previously unrelated ideas together (which is something exercises in general can help with), and finding an appropriate approach and rate of progress (which requires judgement on your part). Etc. Anyways, to properly understand rank you need a decent understanding of array operations. And vice versa. And if someone has any of the concepts wrong, the only reliable mechanism for resolving those issues is to get them to test their knowledge against something they can trust.
Playing around with examples is OK, but a Ph.D in math made it clear to me that in all cases, the definition is God. It's OK to get an idea by playing around with things, but that is just an idea, the definition has final say-so.
I don't think anyone would disagree with you on that. In the realm of "trying things out", I think that means that you should try and predict what results you'll be getting rather than just trying things out blind. And, of course, there's plenty of math which can be intractable numerically. But that's beyond the scope of J.
So, likewise, learning J should be like following a proof in a textbook, right?
What kind of proof? If I recall correctly, we tend to ask students to study a variety of mathematical concepts for years (and try to ensure they get a robust practical background in some of those topics) before we ask them to understand the simplest of proofs.
But then again, math is limited by Goedel's Incompleteness theorem whereas computers can do more?
That does not match my understanding, of Goedel's Incompleteness theorem, nor of the character of computers. As I understand the incompleteness theorem: a mathematical system of sufficient complexity will allow an infinite number of statements which are not resolvable simply by recourse to previous axioms (definitions). For example, you might have a mathematical system which allowed you to assert "Bob is Green" where you could have that be a true statement or a false statement without conflicting with the rest of the system. And the incompleteness theorem says that even after you pick an truth value for this statement, there could still be an infinity of other statements whose truth values have not been determined. by the system (X=2, "Rose is 314159"). Anyways, a critical feature of Godel's theorem is that the mathematical system has to be complex enough to allow such statements before this becomes an issue. Another aspect of Godel's theorem is that it involves infinities (and, thus, extends well beyond the scope of computer implementations which are always finite). -- Raul ---------------------------------------------------------------------- For information about J forums see http://www.jsoftware.com/forums.htm
