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Old 2004-12-02, 00:45   #34
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There's an algorithm to compute the value of PI to any desired precision. Therefore it's determined and predictable and cannot be random.

A property of randomness is that there's no algorithm that can reproduce the sequence.
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Old 2004-12-04, 12:46   #35
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Another cross reference that may be causing confusion is the debate whether "in the digits of pi base ten, the digits 0-9 are randomly distributed".

The particulars of that debate cannot be used in this one. Quite the opposite: such a debate couldn't even be framed unless the digits in question were determined and predictable.

Last fiddled with by Maybeso on 2004-12-04 at 12:46
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Old 2004-12-05, 04:21   #36
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I've been thinking about this issue for a few days, bugged by the feeling that with Jindyu's defintion, pi could be considered 100% order. But I think it's impossible. I don't think anything in this universe is 0% or 100% order.

Quote:
Originally Posted by jinydu
Well, what does it mean for something to be random? When we say that something is random, we mean that it is:

1) Undetermined
2) There is more than one possible outcome
In irrational space, Jindyu, Luigi, Mally, and Cold Fury think pi is constant. Shifting my argument a tad, I'll now argue that in RATIONAL space pi is part random.

I'll concede that in irrational space pi may be determined in the sense that it's constant (only one possible outcome), but, now, I'll argue that to determine something means to bring it into rational space. And in so doing, pi becomes part random.

But instead of defending my position by re-interpreting 'undetermined', which you guys disagree with, I'll hide behind definition number 2. In the process of determining pi, there is more then one possible outcome: 3, 3.1, 3.14, 3.141, etc. each of which is, in real and rational number space, a different outcome.

Which is the basis of my real random number scheme: it is the determination of one of the two pseudo-random number generating schemes and/or one of the two irrational numbers that creates a truly random number. 100% random? No, there's no such thing as 0% order.

Well, maybe in irrational space.

Last fiddled with by shaxper on 2004-12-05 at 04:26
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Old 2004-12-05, 05:04   #37
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A property of randomness is that there's no algorithm that can reproduce the sequence.

I would have to expand this a bit by saying that a random event cannot be predicted. Conversely, that which can be predicted cannot be random.

This entire thread seems to be focusing into a disagreement over the old and proven wrong position that "if all possible variables are known, the end may be predicted from the beginning". In other words, if all possible interactions are known, then the universe's destiny is predictable from the big bang.

I would then point to the uncertainty principle and show a single event that is unpredictable and therefore the entire argument falls apart.

By definition, Pi is not random. It is precisely defined but cannot be represented in a decimal expansion.

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Old 2004-12-05, 17:02   #38
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Quote:
Originally Posted by shaxper
I've been thinking about this issue for a few days, bugged by the feeling that with Jindyu's defintion, pi could be considered 100% order. But I think it's impossible. I don't think anything in this universe is 0% or 100% order.
Actually, I've been trying to argue that pi is a mathematical constant whose existence and value can be derived deductively from mathematical axioms. Thus, its existence and value is independent of anything that happens in the physical universe. We can conceive of a universe with no stars, but a universe without pi is unthinkable. Thus, in some sense, pi goes beyond this universe.

Quote:
Originally Posted by shaxper
But instead of defending my position by re-interpreting 'undetermined', which you guys disagree with, I'll hide behind definition number 2. In the process of determining pi, there is more then one possible outcome: 3, 3.1, 3.14, 3.141, etc. each of which is, in real and rational number space, a different outcome.
I think what you mean is that you can vary the outcome by varying the number of digits you calculate. But this is easy to control, just program your computer to compute a certain number of digits, and then halt. For example:

[algorithm to compute pi]
[end at 1 million digits]

That computation would have only one possible outcome: The value of pi to a million digits.

I don't really understand what you mean by Rational Space and Irrational Space. Could you explain that more please?

Last fiddled with by jinydu on 2004-12-05 at 17:03
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Old 2004-12-05, 18:12   #39
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Quote:
Originally Posted by jinydu
I don't really understand what you mean by Rational Space and Irrational Space. Could you explain that more please?
And tell us in which space do you think to live...

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Old 2004-12-06, 16:50   #40
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Cool To create a real random number

[QUOTE=Fusion_power]A property of randomness is that there's no algorithm that can reproduce the sequence.

I would have to expand this a bit by saying that a random event cannot be predicted. Conversely, that which can be predicted cannot be random.

This entire thread seems to be focusing into a disagreement over the old and proven wrong position that "if all possible variables are known, the end may be predicted from the beginning". In other words, if all possible interactions are known, then the universe's destiny is predictable from the big bang.

I would then point to the uncertainty principle and show a single event that is unpredictable and therefore the entire argument falls apart.

By definition, Pi is not random. It is precisely defined but cannot be
represented in a decimal expansion.] / Unquote

I perfectly agree with your definition. Its the only one that makes any sense of mathematical randomness.
Please refer to Xyzz's new Thread 'Measuring randomness' and the website given for other related 'randomness's'. It is very instructive and educational.
To the last line I would like to add ' in a finite length of time ' But Pi is a complete concept and equal to 'Aleph null' in Cantors theory of transfinite cardinals.
Mally

Jinydu : Quote/ Actually, I've been trying to argue that pi is a mathematical constant whose existence and value can be derived deductively from mathematical axioms. Thus, its existence and value is independent of anything that happens in the physical universe. We can conceive of a universe with no stars, but a universe without pi is unthinkable. Thus, in some sense, pi goes beyond this universe./ Unquote.

Very true. Pi does not even require a universe to exist. It is an abstract concept in the MIND OF THE CREATOR .
Kronecker was against the irrational, leave alone the transcendental numbers.
His famous dictum in response to Cantors theory of the transfinite cardinal numbers was ' God only made the whole numbers. The rest is the work of man (menshenwerk). However Cantor proved him very wrong and his theories (Cantors) were accepted by the math fraternity but many years after his bitter controversy with Kronecker.
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Old 2004-12-09, 10:54   #41
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Quote:
Originally Posted by Fusion_power
I would have to expand this a bit by saying that a random event cannot be predicted.
I disagree with this change because many things that were once thought unpredictable have become predictable. Therefore, we shouldn't include this definition unless you want to add "at this time" to your definition.

Quote:
Originally Posted by Fusion_power
This entire thread seems to be focusing into a disagreement over the old and proven wrong position that "if all possible variables are known, the end may be predicted from the beginning". In other words, if all possible interactions are known, then the universe's destiny is predictable from the big bang.
I disagree again. I don't think I'm intending to extrapolate to that conclusion, but if you can point to something specific I said, I'll reconsider.

Quote:
Originally Posted by jinydu
Actually, I've been trying to argue that pi is a mathematical constant whose existence and value can be derived deductively from mathematical axioms. Thus, its existence and value is independent of anything that happens in the physical universe. We can conceive of a universe with no stars, but a universe without pi is unthinkable. Thus, in some sense, pi goes beyond this universe.
Can you expand on this? Why is a universe without pi unthinkable? Is it just pi or all constants such as the speed of light? Or the laws of the universe such as the laws of thermodynamics?

Quote:
Originally Posted by jinydu
I think what you mean is that you can vary the outcome by varying the number of digits you calculate. But this is easy to control, just program your computer to compute a certain number of digits, and then halt.
Then that would be the same outcome. 3.1 and 3.1 are the same outcome, but 3.1 and 3.14 are different outcomes.

Quote:
Originally Posted by jinydu
I don't really understand what you mean by Rational Space and Irrational Space. Could you explain that more please?
We live in real space which is, I think, the sum of all other spaces. One of those spaces is rational space, which is best described with numbers. Another space is irrational space, which is best described with pictures (fractals).

The difference between the two is how they describe infinity. Personally, I think Cantor was wrong. There are not different "levels" of infinity, but different kinds of infinity. In rational space, we conceive of infinity as the numbers increasing without ending. In irrational space, in fractals, we conceive of infinity as magnification, i.e., we can expand (magnify) a fractal at any point without ending.

The distinction between "levels" and "kinds" is an important one because there are an infinite number of spaces that, I suspect, require different conceptions of infinity. We can easily (now) rationalize irrational space (with fractals), but can we rationalize all the other spaces? For example, I think sound (music) is another space. Can music be rationalized, e.g., can we turn music into numbers so that the patterns (order) becomes conceivable?

I think in time we will, but then there will be another space and another space and...maybe those are different levels, but it implies a hierarchy that I don't think exists. It just happens to be the chronological order in which we "discover" these other kinds of infinity.

Maybe someday we'll discover that these different spaces have an order but I can't, at this time, predict it when that day will occur.

Quote:
Originally Posted by ET_
And tell us in which space do you think to live...
Real space.
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Old 2004-12-09, 16:51   #42
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Quote:
Originally Posted by shaxper
Can you expand on this? Why is a universe without pi unthinkable? Is it just pi or all constants such as the speed of light? Or the laws of the universe such as the laws of thermodynamics?
In classical relativity --- that is the one largely created by Einstein and with subsequent developments which do not attempt to quantize spacetime. the speed of light is just a proportionality factor. Here is a little fable that may make the concept easier to understand.

Imagine a country where north-south distances are measured in miles and east-west distances are measured in kilometers. Now take a rod and place it east-west. Measure it, and find that it is x kilometers long. Take the rod and turn horizontally through 90 degrees. You find it measures y miles long.

Careful measurements made over many years shows that there is a proportionality constant, which scientists call "d" and which has units of kilometers per mile. Every single measurement of a fixed rods, with ever increasing accuracy and sensitivity, shows that d is completely independent of the length of the rod, of the location in the country, of its temperature, the material that it is made of, everything. It is a fundamental constant of the universe. Eventually some scientist got tired of the complications caused to their equations by the presence of d and made a radical proposal. They agreed to measure all distances in kilometres and defined d=1. Now all the equations became much simpler and no one had to worry about d any more. Of course, all the engineers, school-kids, travellers and so forth continued to use the traditional measurements and when anyone wanted to use the equations to calculate something which was actually useful, they had to remember to put an appropriate power of d back in the final result --- but you can't please all of the people all of the time.

Now move to our four-dimensional space time, where we measure north-south in metres, east-west in metres, up-down in metres and past-future in seconds. According to the view of classical relativity, there is nothing mysterious about the value of the speed of light in empty space being exactly the same for all observers no matter how, where or when they measure it. The quantity "c" has exactly the same nature as the quantity "d" in my mythical country. It is used solely to convert from measurements made in one direction to measurements made in another directlion. So, of course, when manipulating the equations of relativity, special and general, it is much easier to define c=1 and to measure past-future in metres too.


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Old 2004-12-10, 01:16   #43
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Quote:
the speed of light is just a proportionality factor
Doesn't this negate the type 1A supernova phenonmenon?
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Old 2004-12-10, 07:03   #44
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Quote:
Originally Posted by shaxper
Can you expand on this? Why is a universe without pi unthinkable? Is it just pi or all constants such as the speed of light? Or the laws of the universe such as the laws of thermodynamics?
A universe without pi is unthinkable because the existence and values of pi can be deduced solely from the axioms of mathematics, which don't even assume that the universe as we know it exists. Even if you could show that these axioms are false in our universe, this would not detract from the fact that the exact value of pi follows inevitably from those axioms.

Similarly, the validity of the argument: "It is raining. Therefore the ground is wet." does not depend on whether its actually raining.

The same can not be said about physical constants, such as the speed of light (measured in meters per second). That value cannot be calculated from any known physical theory, at least not without some experimental measurements. This is essentially what seperates mathematical constants from physical ones.

In addition, a universe with different physical laws is conceivable. To a very good approximation, gravitational field strength varies according to an inverse square law. But there is no logical reason why there can't be a universe where gravity follows an inverse cube law. In short, what separates physical laws from the value of pi is that physical laws make claims about the way this universe is, whereas the value of pi doesn't (at least, not without physical laws); instead it makes claims about the way a universe would be if the axioms of mathematics were satisfied.

Quote:
Originally Posted by shaxper

Then that would be the same outcome. 3.1 and 3.1 are the same outcome, but 3.1 and 3.14 are different outcomes.
But I don't see how you can call that "randomness". I can guarantee getting the value 3.1 by asking my computer to compute 2 significant figures, while I can guarantee getting 3.14 by programming for 3 significant figures.

Quote:
Originally Posted by shaxper
We live in real space which is, I think, the sum of all other spaces. One of those spaces is rational space, which is best described with numbers. Another space is irrational space, which is best described with pictures (fractals).

The difference between the two is how they describe infinity. Personally, I think Cantor was wrong. There are not different "levels" of infinity, but different kinds of infinity. In rational space, we conceive of infinity as the numbers increasing without ending. In irrational space, in fractals, we conceive of infinity as magnification, i.e., we can expand (magnify) a fractal at any point without ending.

The distinction between "levels" and "kinds" is an important one because there are an infinite number of spaces that, I suspect, require different conceptions of infinity. We can easily (now) rationalize irrational space (with fractals), but can we rationalize all the other spaces? For example, I think sound (music) is another space. Can music be rationalized, e.g., can we turn music into numbers so that the patterns (order) becomes conceivable?

I think in time we will, but then there will be another space and another space and...maybe those are different levels, but it implies a hierarchy that I don't think exists. It just happens to be the chronological order in which we "discover" these other kinds of infinity.

Maybe someday we'll discover that these different spaces have an order but I can't, at this time, predict it when that day will occur.
I think your definition of space is too vague. From what your post says, my best guess is that you define a "space" as an "area" with a different conception of infinity as all other similar "areas". So far, you have described three kinds of "spaces": numbers, pictures and music. Yet I don't think your explanation of why they should be considered different spaces is adequate.

In what you call "rational space", you say that infinity is conceived of as numbers increasing without end. But in fact, there are other ways to use numbers to think of infinity. Another way is to think of "infinity" in terms of limits, what something approaches as a variable gets arbitrarily close to a certain variable. The notion of limits is central to the entire (mathematical) theory of calculus.

In addition, pictures and music aren't really seperate from numbers. You seem to imply that fractals are the prime example of pictures. However, if you study Chaos Theory, you would know that fractals are mathematical objects that cannot be described without numbers. For a more detailed description of this, see http://pirate.shu.edu/~wachsmut/Workshops/Camden/. One example of generating fractals is to use Newton's Method on a cubic equation in the complex plane. If you color each point according to what root it converges to, and all three roots are distinct, you get a fractal. For more information, see http://mathworld.wolfram.com/NewtonsMethod.html.

I am even more mystified by your claim that music is another seperate space. Yet you have not explained how music has its own conception of infinity, which seems to be your definition of a "space". Furthermore, there is already a way to turn music into numbers so that the pattern becomes conceivable. The mathematical theory of osciallations has already been applied to the study of sound waves, and plenty of regularities in music have already been found. Furthermore, there is already a device that can convert music into numbers, at least to a good enough approximation that most people don't notice: its called a DVD recorder/player.

Finally, you claimed in an earlier post that to determine something is to bring it into rational space. First of all, this is inconsistent with your claim that nothing is 100% determined, since by this definition, anything in rational space would be completely determined. Furthermore, you haven't explained why determinism is only possible in rational space.
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