<big snip, most of which suggest we agree on some things>
> > But the rule-based paradigm is fundamental here. >Formal proof in mathematics is rule-based.
Not necessarily. There's a substantial literature that shows how to prove theorems by algebraic construction. For example, geometry theorems are often convertible to algebraic system solution.
I think the reality of rules is that too many of them are confusing as a programming paradigm. Few people have the knowledge and discipline to write rule sets which are defined without overlaps, in a more-or-less unstructured problem domain, where the solution is, as you say, "emergent".
> Mathematica implements a formalist vision of mathematics filtered > through a physicist's pragmatism. That's why it's so good at mathematics >, especially the applied mathematics of science and engineering. That's partly why it fails so spectacularly on the mathematics examples which I've pointed out again and again.
In >Mathematica, programming is secondary: partly emergent and partly added on. > If programming is your primary focus, Mathematica should probably not be your language of choice.
And so we agree, that if programming a solution to a problem is the key to reaching your goal, perhaps you should not be using Mathematica.