I don't know what you thought you did, but it seems to me that you ran into
some of the "natural attributes" of our universe, which lead you to
recognize those numbers. Take a look at
http://spaceprojects.arc.nasa.gov/Space_Projects/pioneer/PN10&11.html#plaque . We expect not just other humans to recognize them, but also extra-terrestrials competent at science and engineering - because of the way the universe works.
The Pioneer 10 spacecraft, destined to be the first man-made object to escape our solar system, carries the plaque shown at the URL above. It is designed to show scientifically educated inhabitants of some other star system-who might intercept it millions of years from now-when Pioneer was launched, from where, and by what kind of beings. The design is engraved into a gold-anodized aluminum plate, 152 by 229 millimeters (6 by 9 inches), attached to the spacecraft's antenna support struts in a position to help shield it from erosion by interstellar dust. At the far right, the bracketing bars show the height of the woman compared to the spacecraft. The upper barbell like figure indicated by represents a reverse in the direction of spin of the electron in a hydrogen atom. This transition puts out a characteristic radio wave 21 cm long, so we are indicating that 21 cm is our base length. The horizontal and vertical ticks are a representation of the number 8 in binary form. Therefore, the woman is 8 x 21 cm = 168 cm, or about 5'5" tall. The human figures represent the type of creature that created Pioneer. The man's hand is raised in a gesture of good will. The radial pattern will help other scientists locate our solar system in the galaxy. The solid bars indicate distance, with the horizontal bar, denoting the distance from the Sun to the galactic center. The shorter solid bars represent directions and distances to various pulsars from our Sun, and the ticks following them are the periods of the pulsars in binary form.
Pulsars are known to be slowing down and if the rate of slowing is constant, an other-world scientist should be able to roughly deduce the time Pioneer was launched. Thus, we have placed ourselves approximately in both space and time. The drawing at the bottom indicates our solar system. The ticks accompanying each planet are the relative distance in binary form of that planet to the Sun. Pioneer's trajectory is shown as starting from the third planet, Earth.
Now you can scream about context. This plaque was designed to create a context by using a combination of physical constants that an engineering space going race would be familiar with and ratios to conceptualize itself. Are you saying it doesn't work? Or is your argument specifically against my examples?
As for squares - draw a number line of evenly spaced real integers. Erect a square between the starting point and each integer distance marker. Bingo. Ratios can be area against distance too - and "understood" intuitively. Not something that requires much education beyond preschool I guess.
I am suggesting once again that the frame of reference for these things is the universe. The model is not more significant than the real thing. And the real thing does not change because there are people who cannot read a map. I think you are arguing that the only way to give an address to an extra-terrestrial would be to mail him a house. I am saying that any creature intelligent enough to figure space travel also is intelligent enough to understand the idea of maps. I guess you are suggesting that most of the people here will never end up participating in space travel. I would agree, but that still doesn't invalidate the use of maps. Pi is a transcendental. "Pi" is also a symbol that encodes it perfectly to whatever degree of precision is required. We have done the same thing with all of the other discrete numbers I used. There is a message here. These numbers are "special". Not the specific values I used, most of them are symbols in our reference frame, but they allow us to point to the underlying universal "truth". One of the benefits of using symbols like this is that they are always as perfect, "true", as they need to be. I am also prepared to bet that if and when we meet an extra-terrestrial space going creature, that it will have symbols that do the same job - and here is the real point - that refer to the same numbers/ratios.
Final comment - Godel proved that any ***closed*** system is either incorrect or incomplete. Like a creationist playing with entropy, the omission of a key part of the specification allows you to use it to point out things that would horrify the originator. For every system, there is also a metasystem that allows one to prove the system's characteristic... i.e. complete or correct or neither. I think I'm seeing a lot of the latter.
I gotta run.
> -----Original Message-----
> From: email@example.com
> [mailto:firstname.lastname@example.org]On Behalf
> Of Eric Boyd
> Sent: Thursday, May 13, 1999 11:40 AM
> To: email@example.com
> Subject: Re: virus: maxims and ground rules and suppositions
> TheHermit <firstname.lastname@example.org> writes:
> Well done, and thank-you, ERiC!!! As you figured out, and have
> subsequently had confirmed, I left them in odd dimensions and unitless
> in order to make the point that they are intrinsically recognizeable.
> And that the actual measurement and units are far less significant
> than the ratios that they represent. I hoped somebody would do this as
> it made the point far more completely than any words I could use.
> Actually, I refuted your point -- I provided the context ("frame of
> reference") in which your "truths" hold. Tim also demonstrated their
> inherent context by providing the wrong one -- in which your "truth"
> is no longer true.
> It is becuase of our similar education (the common framework of our
> society) that I was able to recgonize those numbers at all -- and I
> pointed that out with my first-century Roman Soldier reference.