On Mon, Dec 12, 2011 at 11:21 AM, Dan Christensen <email@example.com> wrote:
>> >> B = [ f(x) for x in A ] >> > > B is not the range of f. The range of f is as subset of B in this case. >
Yes, and for my part, B would not be a set after that last assignment, but a list, another object type.
Your contention that mathematics boils down to axioms and theorems is of course the classical presentation. However, there's also a more ethnographic approach, combined with geography, which spends a lot of time looking at the practices and skills of a people.
These often seem like waste-of-time side bars, like who cares how the Incas did brain surgery or accounted wheat stores, but that's not exclusively the kind of narrative we're investing in.
The skills in question may involve donning a backpack with an immersive-style 11-lense camera views, stitched together by software to give those Google Street view like 360 bubbles. Taking those on hikes, bringing back to the school, adding to the server. Consult the school to get a sense of insect counts over time.
Or consult the Dzong (another type of institution with partially overlapping responsibilities).
I'm not saying to avoid proofs, not by any means. I'm into Fermat's Little Theorem as a stepping stone to RSA. Euclid's Extended Algorithm, modulo arithmetic... lots of number theory stuff to prove, using totatives and totients (all pre-college along the DM track we're doing).
> > There is a Set predicate in my system. Only if you declare an object to be a set can you apply the various axioms of set theory on it, e.g. selecting an arbitrary subset of it. This setup avoids a number of theoretical difficlties that makes the system easier to use and more "mathematical." >
We can use sets to purge dupes out of lists. That's one of the exercises I score. There's a distance education component. This stuff about the print( ) function is going out there, but not with as much math as it could.
In the wings: modules to generate new kind of science and Game of Life style arrays, a Mandelbrot Set generator, lots more fun and games with permutations. If we become certified to offer something more like DM towards a post- high school degree, then we might tap more of this.
In the mean time, I own the material under my own name and can dribble it out to other sources. The material becomes more standard that way, which advantages my home faculty in Sebastopol (the base) if and when.
>> I'm from the >> Wittgensteinian camp on >> matters mathematical, which nets me some ridicule >> from Hansen, but I >> say it saves me time. >> >> I'm not as tempted to waste so many hours on the >> union and >> intersection of sets when we could also be playing >> with lists, tuples >> etc. Lists of lists, or multi-dimensional lists, are >> worth relatively >> more time, sets relatively less. > > I don't spend any time on unions and intersections in my tutorial, but if you want to make up lessons based on them, I do have built-in notation to handle them. All of the rest is also possible. >
I've been looking over the documentation at your site.
I haven't done a whole lot with truth tables in my writings here.
The geometry I develop is spatial to begin with and only gets planar later. Polyhedrons (wholes, shapes) before polygons (facets, fields).
Our geometry links to geodesy (GIS / GPS).
This is a STEM curriculum, not trying to "keep it pure" (deliberately bouncing around between S, T, E and M at high frequency, a pedagogical / andragogical strategy).
> Write to me with your ideas, and maybe we can come up with something. But do have a look at the tutorial. I think you will find that it is a nice introduction to logic and proof. It takes you all the way from proving if A and B is true then B and A is also true to proof by induction and elementary number theory in 10 lessons. >
I haven't found the tutorial yet. Is it a PDF?
I'm interesting in the segments linking boolean expressions to logic gates. A few "over their heads" documentaries on integrated circuit design, fresh from Intel, go here.