On Wed, Sep 12, 2012 at 9:26 AM, Paul Tanner <email@example.com> wrote:
>> That's a telling statement. You consider the "real numbers" to >> "exist" and "to have been discovered". They're an empirical >> phenomenon in your mental geography, like Mt. Fuji. >> > > You think that a thing exists only if you name it or know its nature? > Hubris, anyone? >
Did the game of chess exist before it was invented? Of course not.
Does the game of chess exist now that it has been invented. Yes, of course.
Real numbers, same thing. It's not hubris.
People have gone on to invent the set of hyperreals which set is defined to contain infinitesimals that are smaller than any real number.
I think many students feel ripped off when they find out about the hyperreals as the reals were already billed as containing all those distances on the number line, yet here we have a number > 0 that's not real only because it's too small to be real. Since when did reals kick out the really small numbers? What gives?
"In 1960, Abraham Robinson provided an answer following the first approach. The extended set is called the hyperreals and contains numbers less in absolute value than any positive real number." [ http://en.wikipedia.org/wiki/Infinitesimal ]
> >> Those who don't >> use them, don't play the same games, are "science deniers" because >> they deny the "fact" or these "real numbers". >> > > Nonsense. I merely point out the complete inconsistency of denying the > existence of that which one uses. Everyone - and I mean everyone who > takes Algebra I and beyond - is taught to use mathematics based on the > binary function on the reals R that we name "multiplication", R x R -> > R such that the domain of argument x happens to be R or an uncountable > subset of R, this domain of x being uncountable guaranteeing that > almost all of the elements of the domain of x are not computable real > numbers, which explains why my talking about the cardinality of the > continuum being larger than the cardinality of the natural numbers is > so utterly relevant.
"Existence" is one of those funny words. I don't deny the existence of chess either, yet have no trouble seeing it as an ethnic / mathematical activity.
Along the digital math track, we still have math objects that sort into types (list, set, int, float, dict, str etc.) but we don't spend as much time talking about some "real type". We don't have those in the language and they're not needed to understand what's going on.
Does that mean we're against other sand-castles wherein the bread and butter object type is the real type? Not at all. We just don't feel confined to those.
> > No, we don't tell them this difference in cardinality and its utterly > profound set of implications until later in more advanced treatments > if they take these courses or if they educate themselves, but so what?
Yeah, so what. Awe yourself with Cantorism if that floats your boat. Many gifted at math will have no need for such arcana.
> Are we supposed to actually lie to them by implying to them the lie > that all the real numbers on which the math they are taught to use are > computable, which is what we do when we tell them that repeated > addition as computation of the product from two given factors can > model the multiplying of any two real numbers when the truth is almost > the complete opposite of that? What kind of good math education is > that? >
If I hand you a pencil and say it has length pi in the unit system I'm using, and then I put put five such pencils end to end and say that's 5 * length of pi, then I have told you no lie.
I tend to use phi a lot more. We have these U,V,W tetrahedrons that make the fat and thin hexahedrons, along with a "3D" golden spiral. They're in turn based on what we call T modules which only differ from E modules by scale factor.
The game is to fill space and/or make other polyhedra using tetrahedrons that are phi scaled relative to each other, meaning literally, i.e. all linear dimensions increased by about 1.618. As a result, volume increases by about 1.618 to the 3rd power.
This is bread and butter for some researchers. Zome Tool and vZome get used. The material is quite accessible to a high schooler and makes for good / interesting computer graphics / animations (I have examples at my web sites).
In other words, I use scaling by reals all the time, and am familiar with research along those lines. However when using computers we're actually only using rational numbers in the computations. There's no reason to lie about that. If we want to say phi( ) is not a number at all, but represents an algorithm that continues to converge, that's not a problem in some language games.
The T and E modules are in the published literature with many a bibliography going to the source. Their plane nets are on-line. Likewise we have the A and B modules, two tetrahedrons that, without scaling, may be used to assemble the tetrahedron, octahedron, rhombic dodecahedron, cube, cuboctahedron. A and B also have the same volume (as each other). A = B = T in volume.
Spatial geometry is getting new legs in STEM, thanks to a resurgence of interest in polyhedrons, which in turn traces to seeing so many under the microscope, both in the animal kingdom and in the world of crystals. Who would have thought, in Plato's day, that the icosahedron would turn out to be the shape of a virus, and that the number of viral capsomeres might be predicted using 1, 12, 42, 92... the icosahedral numbers.
> By the way, let's again look at that that model I gave of "repeated > addition" that shows that any element b in any field with > characteristic 0 and that contains the natural numbers can be written > as the sum of n elements for any natural number n: b = n(b/n) = (1_n + > ... + 1_n)(b/n) = (b/n)_1 + ... + (b/n)_n. Note that since we have > surjective multiplication, that means that the product of any two > elements in such a field can be written as the sum of n elements. >
I have other sources for my "repeated addition" meme.
You have refused to connect it with growing and shrinking quantities, such as volumes, which is the traditional arena wherein grain, fluid, sand, is added or removed.
I've wanted to continue a seamless progression from adding whole number quantities, then rational quantities, then adding volume by scaling by phi or pi.
You have claimed I am wrong or lying for wanting to provide this kind of continuity from N through R, and therefore I have no choice but to cross you off my list as a compatible teacher. We just wouldn't get along in staff meetings. I'll have to get my pedagogical ideas from thinkers I better understand.
> Then I hope you're aware that the notion of a continuum has been > around for thousands of years, even if as "a cultural artifact", "an > institution", >> "a set of conventions", "a set of practices", "a set of language >> games", and so, even if you hold to the idea that a thing exists only if someone as a notion of it, then the continuum has existed for thousands of years. >
Thanks to the hyperreals, it sounds like your "continuum" is still full of holes. All those infinitesimals fell through the cracks hah hah.
>> Anyway, it's becoming clear to me that you're more of a dogmatist than >> I'd realized > > You confuse being a dogmatist with wanting to avoid sloppiness and > wanting to have consistency.
The dogmatist believes science has it all figured out, nailed down, and the doubters are just faithless "conservatives" who can't admit they've been outclassed.
Just because cultural conventions come all dressed up in fancy theorems with lots of squiggles doesn't make them any more "true" or "existing" than chess is true or existing.
A lot of math goes unchallenged / ignored simply because it's not worth anyone's time contradicting it.
Once you've learned how to stack the deck professionally, you can win every hand. Tautologies beget tautologies.