> I'd concentrate on the definitions and theorems > (proved propositions), where algorithms are forms of > theorems. (If an algorithm has not proved or is not > at least provable, then how could one trust that the > algorithm will always deliver as advertised?)
We definitely remind 'em to read what's on the bottle, e.g. the Jython algorithm (pulled into Python from the JFC) promises only probable primes, with the degree of probability very high, but not 100%. The fact that RSA depends on these numbers is considered non-critical to security, although I'm too lazy to cite references in this context.
At the Pycon workshop for DM track teachers, we used an already cracked RSA number, giving some of the lore behind it (German security service earned itself $20K for all its hard labor). This was good advertising for the company, as you could just add a few more bits and it'd take a powerhouse greater than all Germany to crack her. Wow. Good to show how the mathematical properties of things actually feed advertising, give substance to claims (even if only a glimmering).
The "geometry of lumps" we use in some DM segments is in no way anti-theorem, even if the assumptions (definitions) are non-Euclidean by design. We want our students to have a sophisticated grasp of these "math pyramids" (axioms a foundation, theorems ascending to an apex -- not always the case, but a good image on which to build).
Wittgenstein used to quip that Bertrand Russell type Logic underpins a painted castle (higher order arithmetic) much as a painted foundation would (paraphrase) i.e. it *looks like* we're "proving arithmetic" (that's what "adding a foundation" means in that neck of the woods) even though most of these dons hadn't read 'On Certainty' yet (came out in notes first) and so didn't have those 'form of life' associations we take for granted today (in some circles, not saying every town is like Vienna or one of those).
I've updated the edu-sig group about some of our business here, hoping to keep our threads moving. There's nothing wrong with nostalgia for Turtle Graphics or Turtle Art. As a genre, these aren't going away. Python's newest turtle module (Standard Library) is being managed from Vienna at this point (might've mentioned) and is being used in elite academic trainings (for teachers as well?). Also, note use of Python in this investigation of Ulam's Spiral. These are skills-building experiences. If you're just listening to a teacher drone, I suggest you tune out and get with the program! Use an iPod maybe? We're podcasting some of our stuff.