## Saturday, September 12, 2015

### Compare and Contrast

I would agree that the language of mathematics is about making generalizations, but also making them precisely enough so that they hold universally thanks to some clear stipulations.  Minus the clear stipulations, one just gets dogmas (poopy belief piles).

In some of the Google Groups, such as mathfuture, I suggest abetting ordinary XYZ vector mathematics, an arithmetic of sorts, with what I put forward as a sidebar "IVM vector mathematics", with an apparatus some might claim is a parody of XYZ but which I'm happy to take quite seriously.

In comparing and contrasting the two approaches, we come to see what commonalities apply.  The two arithmetical games have the same operations.

"We use a Caltrop in place of a Jack" is one of the ground rules, differentiating respective language games.

The XYZ "jack" divvies space into eight octants whereas in Quadrays, the "caltrop" divides it into four.

XYZ (jack): (+ + +)(+ + -)(+ - +)(- + +)(- - +)(- + -)(+ - -)(- - -)

IVM (caltrop): (+ + + 0)(+  + 0 +)(+ 0 + +)(0 + + +)

Per any point, one direction stays passive in Caltrop arithmetic, with positivity (or zero) in the other three, sufficient to reach all points in a quadrant, with four quadrants spanning space.

Cartesian coordinates (also invented by Fermat) employ a "jack shape" of six spokes whereby space is subdivided into eight regions, designated by permutations in sign, positive or negative.

Tip-to-tail addition, 180 flip for negation, scalar multiplication (if allowed) works the same way as with "Jack vectors" such that all points have 4-tuples instead of 3-tuples, and negatives are not needed.  We have an isomorphism between them.

The Americans have apparently said "no" to any such curriculum for now, if we're to judge by Common Core, but then who says that's what to judge by?  Common Core actually forbids nothing, being an affirmative document in a "what to include" format, so lets not assume "verboten" where "not mentioned" is more the expected norm.

Common Core is like a cake with no frosting, or staircase with no carpet.  The frills are missing, leaving bare bones.  In adding spice with the IVM, in addition to XYZ, we're going beyond what's required.

In other words, if you're the teacher and you find introducing Quadrays to your classrooms catalyzes more productive thinking about "vectors" in general, then you have the right, as a freedom-loving American (or whatever) to say "yes" to the sharing of these ideas, if only for experimental purposes.

However, you may feel you need permission from your "church" or from whatever various designated religious authority sub-geniuses, however intriguing you find these ideas personally. It's not necessarily your call, where the education of innocents is concerned, whether to venture outside the lines or not.  Not every teacher is a wannabe rebel.  I encourage you then:  follow your conscience.

Besides, I've had a free hand to teach this stuff for some decades now, so I'm not about to complain about censorship.  I maybe don't always reach the most receptive age group, as vectors are usually saved until college and my Saturday Academy classes were more middle school on average.  That's a different issue.  The past does not dictate the future in any case.

No, I think the reason IVM mathematics makes so few inroads in America has more to do with complacency than with censorship, with the XYZ people thinking "why should I share the road with some johnny-come-lately, and what's a 'caltrop' anyway?"  The NIH ('not invented here') syndrome is prevalent.

Nobody recognizes "Quadrays" as a brand of anything (correction:  there's a flashlight by Nitecore), let alone as a geometric something, which means the whole language game might as well be from Mars.

In an age that rewards people for being "mainstream", anything out of the ordinary (i.e. "extraordinary") is left to "circus freaks" or whatever "out cast" of underdogs.

We're more likely to learn of Quadrays from wandering egoists "sirfessing" in Hobo Colleges [tm], than from Pearson or Springer-Verlag.

Speaking of Mars, that's a segue to Wittgenstein, the movie, wherein the young LW's alter ego and / or imaginary friend, is likewise a Martian -- played by Nabil Shaban, a Facebook friend (introduced to me by Trevor).