numerical methods (was a question about infinity / countability)

[kirby urner]

On Sat, Jul 16, 2016 at 8:05 AM, Joseph Austin <drtec...@gmail.com> wrote:

A company I worked for once hired Carl Sagan to help us pitch a super-computer. This was in the 70's.

At that time, he was predicting an imminent ice age!  So much for global warming.

Yes. 

Al Gore's movie ('Inconvenient Truth' -- directed by someone else) also predicts an ice age, but only after we finish with the global warming part.  Like a roller coaster. 

What goes down must have gone up.

 

But on to the irrational.

So one cannot diagonalize a square! 

The transcendental numbers escaped being solutions to any polynomial or really any algebraic expression.

A Ramanujan generator that spits out the digits of pi, or the one we had here in Python, don't count as algebraic expressions.

The 2nd root of 2 as expressed as the diagonal of a square (might be the edge of a 2-area triangle in Martian units), is at least algebraic and not transcendental.

We have to remember the real numbers do not cover all the roots of polynomials either i.e. algebraic numbers may include the roots of -1 i.e. i to even integer powers.

Complex numbers are the crowning achievement.  The Reals are a mere subset of those.

With Reals we're stuck on the number line.  With Complex Numbers we get the Argand Plane.

But then lets remember we've learned how number lines may be intersected at the origin. 

We disallow them to be joined end-to-end i.e. three number lines in a triangle, each edge with double arrows <---------> would not make sense to us.

I tried using number lines as edges of a tetrahedron, which is just not done:

http://coffeeshopsnet.blogspot.com/2016/05/4d-meme.html

(see below -- "three-vector zig-zags")

 

If that is true, to "square" a building, such as a back-yard deck, or even the pyramids,

one cannot use a "square" but must use a rectangle, such as a 3x4x5 triangle.

You mean if you want perfect integers for each side?  I get it.

But when it comes to energetic phenomena we have no theory as to what we'd be measuring after a few significant digits.

We come up against Heisenberg and Planck before long.
 

(Which reminds me, my hometown's most famous modern structures, RBF's Climatron and Eero Saarinen's arch, are both based on triangles, not squares.)

Yes, RBF is appropriately commemorated at the Climatron, along with the engineers who actually did the designing and building.

Fuller gets a lot of flak for taking the 10,000 foot view and not actually getting his hands dirty on these domes.

Here's another dome he gave a speech at when finished, but was not chief engineer for:

http://www.treehugger.com/culture/a-necessary-ruin-the-story-of-the-loss-of-bucky-fullers-union-tank-car-dome.html

He held the patent.  On the octet-truss as well.  That's the thing A.J. Bell was working on, an inventor of the telephone (great minds think alike eh?).

http://breakfornews.com/treecn/images/geniusatwork.gif

Natural philosophy had fragmented into other sub-disciplines before Bucky. 

The academic breed that remains and uses the word "philosophy" in its masthead, is neither expected nor encouraged to patent anything.  Publish or perish, not patent or perish.

Fuller is not a philosopher in the modern sense because he patented stuff.  Kant didn't, Nietzsche didn't.  QED.

Wittgenstein was interested in propellers and might have come up with a patent in due time, but helping Bertrand Russell lay the foundations of mathematics became an all-consuming passion. 

A book on logic comes under copyright, and is proprietary in that sense.

Had Wittgenstein gone into computer languages, he might've availed of GNU's GPL had it existed in his day (science fiction).  He was never in it for the money.

In the US there's a concept of software patents but not so much in the EU. 

https://en.wikipedia.org/wiki/Software_patents_under_the_European_Patent_Convention

I'm thinking of RSA as a good example of an algorithm that enjoyed a patent for awhile.

http://archive.oreilly.com/pub/a/linux/2000/09/08/rsa.html  (about when it ended)

This article says RSA was not patented outside the US because published there before a patent was sought by MIT.

"MIT decided to patent the algorithm, but because it had been published before the patent was applied for, it couldn't get foreign rights to it." (page 2)

But would the EU have allowed it a patent in the first place, even if not?

The Article 52 cited by Wikipedia bans patents on "discoveries, scientific theories and mathematical methods". 

I'd say RSA is a mathematical method.

That's essentially the same dilemma I've been wrestling with in saying one can't create a sphere, 

But then ping pong balls exist, as do bowling balls, so what is it you're having trouble with exactly?  We see bubbles everywhere.

Philosophy woulda been different if we'd regarded the actually existent energetic special cases to have the winning hand just by virtue of energetic existence.

A real sphere in the wild is more perfect than anything dreamed of in the imagination, regardless of "precision" because imaginary / idealized / theoretical spheres (squares, circles, lines, points, planes) don't exist.  That makes them secondary, not as interesting.

What concerns us is "the real deal" i.e. phenomena we actually encounter experimentally.  We have no examples of spheres for which "pi to a billion places" makes any real sense. 

Ergo our mathematics is only approximate and has a tendency to carry us away tangents. We get distracted by the useless features (they're not really useless though, as we develop computational muscles by extending Pi -- it's an engineering challenge, even just storing all those digits).

However we decided a long time ago to value the imaginary / idealized above the special case energetic.  It's sort of a religious thing, that "the world" is fallen and that what we imagine we can imagine in our "mind's eye" is somehow closer to God.

But what if "made in God's image" means like "on reality TV"? 

We're all in this image we call Universe, on God's TV as it were. A real energetic sphere, one of God's, is no less admirable than one we can only think of, but not actually create.

We're proud of the fact that we're able to generate Pi to way more digits than any physical theory has need of or could explain having any need of.

The best we might say, to distinguish the road not taken (the philosophy we discarded) is:

"Nature is not using Pi"

All that means is, despite Pi's occurrence everywhere in expressions of physical laws, we have no persuasive argument for nature making any use of our "infinitely precise Pi" of mathematical invention.

The efficacy of our real number mathematics does not prove atoms and molecules and other quantum phenomena have any use for pure continua or "infinitely precise" anything.

The physical process of computation requires only discrete / digital events, let us posit.

 

that is, a 3D closed symmetrical object of uniform curvature and density.  

One arrow defines a ray from a dot (dot = like a star).

Two arrows from the same starting point make the hands of a clock (any angle).

Close the gap between the tips to make a triangle (as seen from somewhere else).

Three arrows make a tripod, an apex, which, when set on the ground makes a tetrahedron.

Four arrows make an origin with four quadrants enclosing it symmetrically.

Three arrows inevitably define the six lines of interrelationship.

I can see why an ethnicity would go with three arrows (tripod) as defining volume.

Three is half of six. 

Each 3-segment zig-zag of a tetrahedron defines the other one:

http://www.rwgrayprojects.com/synergetics/s01/figs/f0801.html

As for why these three arrows need to be at 90 degrees to one another, that seems an unnecessary restriction / stipulation.  Why not go with 60 if we wanna be canonical?

 

I've also claimed that the Babylonians were wrong: one cannot create a 360º circle 

(120º is ok, but you can't divide 3º into thirds, at least not with Euclid, and I doubt the Babylonians had a more sophisticated method).

You're talking about "with only a straight edge and compass" I guess.

Fortunately, we have other tools in inventory. 

We don't use Turing Machines either.  They're theoretical machines, not practical.

 

I'm now suspecting the root of the entire irrational (and hence uncountability) dilemma is Euclid's 3rd Postulate.  

But what is geometry without circles?

What are circles without their Euclidean spin?  Do we need Euclid to have our circles?

Vinyl records and paper plates are in no need of Euclid's postulates to exist.

 

Can we really have both atomism and the continuum, both numbers and circles, in the same consistent mathematics?

Why did you need "the continuum" again?  What's it good for?
 

Or put another way, can we really prove theorems about the continuum and "infinities" by considering strings of discrete symbols?

Might we have entire branches of math that are not concerned with such proofs, and still be considered maths?

I'd say we do already.

 

And is there perhaps a middle way--what I intuitively call a "uniform density" surface in which all "points" are "regularly" spaced?

When we create an XYZ apparatus and turn / twist molecules inside its scope, we're positing a frequency i.e. XYZ cubes might have nano-meter or pico-meter edges.

XYZ is a "uniform density" of reference points. 

IVM is the same (fills space with a different scaffolding, same idea though:  reference points).
 

In music, we encounter a similar challenge in "encircling" the sequence of "fifths" or 3:2 frequency ratios.

it is not possible to construct a "circle" which is simultaneously 'harmonious" (small integral ratios) and "closed",

so we must pick one or the other: equal-temperament (but not pure ratios) or "just" harmony (but not free modulation between keys.)

Joe

Why not look for definitions, concepts gaining meaning through use, that don't require so much concern about either infinity (uber-large) or 1/infinity (uber-small). 

Border both big and small extremes with a tetrahedron (containment) to define a "scope". 

Between the two is the "realm of the definite" (our scope). 

Outside these extremes is the realm of the Finite (but not yet definite).

This sense of an enclosure, of being inside something (a universe) is symbolically arrived at by remembering Descartes' Deficit: 

Finity - 720 degrees = curvature / firmament / ball of wax (res extensa).

That'd be in interesting starting place:  with conceptuality itself, and the sense of "containment" i.e. of being inside something with things inside it,

and with those things having outsides as well.  Russian Dolls.

Very Kantian, not to mention Cartesian, to begin with first principles.

Kirby

 

On Jul 13, 2016, at 1:25 PM, kirby urner <kirby...@gmail.com> wrote:

SYNERGETICS:

OCTAPHASE EVERYDAY MIND

by Kirby Urner

 

[Joseph Austin]

Kirby,

Thanks for the dome video.

In God's eyes, we must all be kids building castles on the seashore.  Come the evening tide, they will all be washed away.

But isn't our "day" on the beach fun nevertheless?

From whence is five-fold symmetry?  If you cut an apple, or a quince, along the equator instead of a longitude,

you can see the five seeds.

As for "spheres", of course there are raindrops and planets that are roughly round.  But necessarily "roughly",

because if they are all built of chemical bonds, they will not be perfectly smooth.

My heresy regarding "circles" stems from trying to draw them in pixels.

But if the universe is all finite "pixels", then nature must be able to get along without them.

But if nature can, can't our geometry?

Do we "need" infinity to model the "finite", or is it just a convenience, an approximation?

I suspect we may be stuck in the same kind of trap as the ancients,

who decided that the sphere and regular solids were "perfect" constructions,

and hence God, being perfect, necessarily used them to create the universe.

Speaking of pixels, I sometimes wonder whether it would be better to arrange pixels in hexagons instead of squares.

Then we could have "real" straight lines for isometric rectilinear diagrams.

But then perhaps some engineer would decide to run the phosphor lines horizontal instead of vertical  (the better to align text) and mess it all up!

Joe

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[kirby urner]

Hi Joe --

I don't know if you saw this "global matrix widget" in my blog, two images:

http://mybizmo.blogspot.com/2016/07/measure-97.html

The hexbug moves around inside the hexagon, which has little gates and a way to attach to additional hexagons.  

The hexbug travels through gates from cell to cell apparently.  A friend rescued this toy from a Good Will recycling bin.

I figure about 10 to the 15th power such hexagons, all pretty similar, plus 12 pentagons, would tile a sphere the size of Planet Earth.  

If you make each hexagon a pixel, just zoom out about 100-1000 feet and you've got your HDTV.  

Fine lines might be a few hexagons thick, giving that "shading" that gives them their anti-aliasing / smooth appearance.

Bradford might remind us we're always just looking at the surface, from the outside.  

However the same spherical mesh may be viewed from inside as well, plus we're free to create "layers" as deep as we like, needing fewer hexagons the deeper we go (given a shrinking ball).

Hexagons do a better job tiling a ball than than quadrilaterals, which distort all the way from squares at the equator to narrow / thin trapezoids at the poles, if doing the lat-long grid.  Not nearly as nice for pixels.

This could be like our "home base" for doing geometry.  

The hexagonal tiling is locally flatter than Death Valley or an ice rink.  We've got our planes (plains) for plane geometry.  No need for any "infinite planes", just a planet.

Here's a fun Youtube by my associate Gerald de Jong:  https://youtu.be/_II-uESToOs  (notice how desert-like the hexapent surface).

Speaking of tensegrity software...

Tim Tyler's stuff is also good:  springie.com  but I can only run java -jar springie.jar at the command line after downloading.  Modern browsers fight like hell to not run Java applets anymore.

Kirby