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Old 2004-11-29, 09:47   #23
shaxper
 
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Quote:
Originally Posted by jinydu
Just because you can't write out the whole decimal expansion doesn't mean its undetermined. Here's the classic example:

1/3 = 0.33333333333333333333333333333333333333333333...
...
Your counterexample doesn't refute my argument because the number can be written as 1/3. It's finite, determined.

My point is that an infinite series is undetermined.

Quote:
Originally Posted by jinydu
Morever, I would argue that the same is true of irrational numbers that do follow a pattern, for instance:

0.1234567891011121314151617181920212223242526272829303132333435
If the pattern can be expressed in a finite number of terms, then yes, I agree with you. But if it's an infinite series of terms, then I disagree. It's only determined to the term that you express it to. Past that, it's undetermined.
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Old 2004-11-30, 05:09   #24
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But if you say that pi's decimal expansion is only determined to the digit that someone calculates it to, this would imply that in Newton's time, pi was only determined to only 35 digits. Yet I am certain that, had Newton calculated and doublechecked more than 35 digits, his results would have been identical to those of today's computers.

My point is that pi is as determined as 0 or 1 and it can be written in a finite number of "terms". Its just that these terms are not expressible in terms of a decimal expansion.

Last fiddled with by jinydu on 2004-11-30 at 05:10
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Old 2004-11-30, 14:41   #25
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Quote:
Originally Posted by jinydu
But if you say that pi's decimal expansion is only determined to the digit that someone calculates it to, this would imply that in Newton's time, pi was only determined to only 35 digits.
Yep.

Quote:
Originally Posted by jinydu
Yet I am certain that, had Newton calculated and doublechecked more than 35 digits, his results would have been identical to those of today's computers.
Yes, that's true. But he didn't.

Quote:
Originally Posted by jinydu
My point is that pi is as determined as 0 or 1 and it can be written in a finite number of "terms".
Why do you put terms in quotes?

Quote:
Originally Posted by jinydu
Its just that these terms are not expressible in terms of a decimal expansion.
Or as a finite series.
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Old 2004-11-30, 15:41   #26
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Quote:
Originally Posted by shaxper
Your counterexample doesn't refute my argument because the number can be written as 1/3. It's finite, determined.

My point is that an infinite series is undetermined.
If the pattern can be expressed in a finite number of terms, then yes, I agree with you. But if it's an infinite series of terms, then I disagree. It's only determined to the term that you express it to. Past that, it's undetermined.

Shaxper allow me to tell you that you have adopted the views of Leopold Kronecker 1823-1891. which fell out of favour in the latter half of the 19th century.
The primitive intuitionism which Kronecker advocated was founded upon 4 precepts.
Of relevance here is the 4th and I will quote
"One cannot consider actual, or completed infinities. We may contemplate the construction of a set that can be added to without limit as a 'potential infinity', but we may not view this operation at any stage as having produced a member of some 'completed infinite' collection. The production of this completion would have required the illegal notion of an infinite number of operations in its production"
'However Kroneckers tendencies were soon to come into conflict with those of a far more intense paranoid personality whose name is remembered where Kroneckers is now largely forgotten.
The man was georg Cantor 1845-1918
Cantors early work was of a sort that violated all of Kroneckers key
precepts. His first paper on sets appeared in 1874. It was non constructive. It used actual infinities: it made liberal use of the reductio ad absurdum. It was also fundamentally new and innovative And Kronecker presented it as a siren call to the younger generation of mathematicians luring them away from the straight and narrow road defined by the natural no.s into a mad house of meaningless infinite entities'. Source book "Pi in the Sky" by J.D. Barrow.
Mally
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Old 2004-12-01, 03:47   #27
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Quote:
Originally Posted by mfgoode
Shaxper allow me to tell you that you have adopted the views of Leopold Kronecker 1823-1891. which fell out of favour in the latter half of the 19th century.
The primitive intuitionism which Kronecker advocated was founded upon 4 precepts.
Of relevance here is the 4th and I will quote
"One cannot consider actual, or completed infinities. We may contemplate the construction of a set that can be added to without limit as a 'potential infinity', but we may not view this operation at any stage as having produced a member of some 'completed infinite' collection. The production of this completion would have required the illegal notion of an infinite number of operations in its production"
'However Kroneckers tendencies were soon to come into conflict with those of a far more intense paranoid personality whose name is remembered where Kroneckers is now largely forgotten.
The man was georg Cantor 1845-1918
Cantors early work was of a sort that violated all of Kroneckers key
precepts. His first paper on sets appeared in 1874. It was non constructive. It used actual infinities: it made liberal use of the reductio ad absurdum. It was also fundamentally new and innovative And Kronecker presented it as a siren call to the younger generation of mathematicians luring them away from the straight and narrow road defined by the natural no.s into a mad house of meaningless infinite entities'. Source book "Pi in the Sky" by J.D. Barrow.
Mally
I've read Charles Seife's "Zero" and so I know that being associated with Kronecker certainly isn't a complement.

But I don't think that this discussion is applicable. I'm not saying that the value of pi is undetermined because it's an infinite series. I'm saying that pi is not 100% order because it's an infinite series, that it - like everything in the universe - is part random and part order.

Jindyu is, I think, arguing that pi is 100% order because the nth term is defined. It's not unpredictable. Yet, by his definition, which is a good one, the nth term hasn't been determined. What is the nth term? Back in Newton's day, it was 35. Today it's prolly several million and growing every day. But whatever it is, it's finite. And there will always be an n+1 term that hasn't been determined. While it hasn't been determined, n+1 is random. Once it's determined, it's order.

This changes the definition of random from being unpredictable to being undetermined.
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Old 2004-12-01, 04:44   #28
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Quote:
Originally Posted by shaxper
I've read Charles Seife's "Zero" and so I know that being associated with Kronecker certainly isn't a complement.

But I don't think that this discussion is applicable. I'm not saying that the value of pi is undetermined because it's an infinite series. I'm saying that pi is not 100% order because it's an infinite series, that it - like everything in the universe - is part random and part order.

Jindyu is, I think, arguing that pi is 100% order because the nth term is defined. It's not unpredictable. Yet, by his definition, which is a good one, the nth term hasn't been determined. What is the nth term? Back in Newton's day, it was 35. Today it's prolly several million and growing every day. But whatever it is, it's finite. And there will always be an n+1 term that hasn't been determined. While it hasn't been determined, n+1 is random. Once it's determined, it's order.

This changes the definition of random from being unpredictable to being undetermined.
I think the other point I'm trying to get across is that the (n+1)th is always computable in principle. If you claim that the (n+1)th term is random until it is computed, then it follows that the randomness of any particular term will change with time. Put another way, the determined/random status of the nth term depends on what we humans do.

Currently, the value of pi is known to roughly 1.24 trillion decimal places: http://www.sciencenews.org/articles/...4/mathtrek.asp. Thus, according to your claim, the 1.25 trillionth decimal place is currently random.

I'm going to list a series of hypothetical situations. In your opinion, which of them would result in the 1.25 trillionth decimal place becoming determined? For the sake of argument, assume that the calculations were done correctly.

a) I use a supercomputer to calculate pi to 1.25 trillion decimal places. Unfortunately, the supercomputer catches on fire before I can back up the data and the hard disk is completely destroyed.

b) I use a supercomputer to calculate pi to 1.25 trillion decimal places. Unfortunately, I have a very paranoid personality and I refuse to publish the data or show it to anyone.

c) An alien civilization in a far away galaxy computes pi to 1.25 trillion decimal places. However, we never know about it because either they don't know of our existence, they don't care about our existence, or communication is impractical.

d) Mathematicians succeed in calculating pi to 1.25 trillion decimal places, but decide not to share the results with the public because they think that the public will not be interested.

Also, suppose, for the sake of argument, that humans are the only species in the Universe intelligent enough to calculate pi. Now, imagine that the sun suddenly explodes, destroying all life on Earth and all records of computations of pi. Since there are now no computations of pi, does this mean that the value of pi is completely random?
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Old 2004-12-01, 06:42   #29
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Quote:
Originally Posted by jinydu
I think the other point I'm trying to get across is that the (n+1)th is always computable in principle. If you claim that the (n+1)th term is random until it is computed, then it follows that the randomness of any particular term will change with time.
Exactly.

Quote:
Originally Posted by jinydu
Put another way, the determined/random status of the nth term depends on what we humans do.
Is that so hard to accept? I can't remember the name, but there's a principle that states that an observation changes the outcome. Is this so different?

Quote:
Originally Posted by jinydu
Currently, the value of pi is known to roughly 1.24 trillion decimal places:
Wow!

Quote:
Originally Posted by jinydu
I'm going to list a series of hypothetical situations.
I've got one for you. If the Louvre is on fire and you have an opportunity to save either a child or the Mona Lisa, but not both, which would you save?

Quote:
Originally Posted by jinydu
In your opinion, which of them would result in the 1.25 trillionth decimal place becoming determined?
My opinion is irrelevant. This is your definition.

Quote:
Originally Posted by jinydu
Also, suppose, for the sake of argument, that humans are the only species in the Universe intelligent enough to calculate pi. Now, imagine that the sun suddenly explodes, destroying all life on Earth and all records of computations of pi. Since there are now no computations of pi, does this mean that the value of pi is completely random?
If there's no life, there's nobody around to define, re-define, interpret, re-interpret, or misinterpret "random." Problem solved. Case closed.

Last fiddled with by shaxper on 2004-12-01 at 06:44
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Old 2004-12-01, 08:00   #30
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Quote:
Originally Posted by shaxper
Is that so hard to accept? I can't remember the name, but there's a principle that states that an observation changes the outcome. Is this so different?
I think you're confusing two concepts from Quantum Mechanics. One is the the Uncertainty Principle, which has two components: 1) The minimum uncertainty of the position of a particle is inversely proportional to the minimum uncertainty of the momentum of the particle. 2) The minimum uncertainty in energy is inversely proportional to the minimum uncertainty in the time. That is position+momentum and energy+time are complementary pairs, knowing more about one element in the pair means that we must know less about the other element.

The other concept is that of Wholeness. Simply put, it means that the act of observing a quantum phenomenon changes the phenomenon itself.

In their original form, both of these concepts were designed to apply to experiments that took place on an extremely small scale (generally, atoms or smaller). Since then, further experiments have shown that they may also apply on a macroscopic scale. However, a claim that they also apply to mathematics would be outside the realm of physics, since physics deals only with the physical universe.

Quote:
Originally Posted by shaxper
I've got one for you. If the Louvre is on fire and you have an opportunity to save either a child or the Mona Lisa, but not both, which would you save?
I don't see how that example is relevant to whether or not pi's decimal expansion is determined or random.

Quote:
Originally Posted by shaxper
My opinion is irrelevant. This is your definition.
Of course your opinion is relevant. Otherwise, what would be the point of your being in this discussion?

Quote:
Originally Posted by shaxper
If there's no life, there's nobody around to define, re-define, interpret, re-interpret, or misinterpret "random." Problem solved. Case closed.
The problem with this is that science attempts to answer questions about what happened before life formed. The current consensus is that life (at least on Earth) formed about 3.5 billion years ago, while the Big Bang occured around 13.7 billion years ago. If we only studied things that happened after there was life to observe it, we would be missing out on about 10 billion years of interesting things.

According to your model, pi's value is determined to about 1.24 trillion decimal places. In 1900, pi was determined to only 527 digits. In 1800, 140 digits and in 1700, just 71 digits. The precise figures are not too important, the point is that the precision decreases as we go back in time. Eventually, as we keep looking back farther into the past, there would be some point, many millenia ago, at which somebody noticed, for the first time ever, that the ratio of the circumpherence of a circle to its diameter is always equal to the same constant. This would be the first computation of pi ever. But according to your claim, this was when the first digit of pi was determined. Before that moment, pi was 100% undetermined and random.

But with some knowledge of the physical universe, we know that can't possibly be true. Many theories, such as Newton's Theory of Gravitation (which provides a good enough approximation to make my point) depend on a precise value of pi. If was pi was truly random, the planets would swing around wildly, crashing into the sun or spiraling off into outer space. A stable solar system could not exist without a stable value of pi, and we wouldn't be around to discuss it.

Of course, you may counter some of my arguments by noting that I've slipped another assumption into your claim: namely that the computation of pi determines a certain number of its digits all of the Universe. Maybe, instead, the computation only determines the value of pi for those who actually receive the message. But if pi is random before it is computed, we would then have to be astounded by the fact that different cultures, continents away from each other, have independently computed the same value of pi. The probability that this would happen through chance alone is 10^-n, where n is the number of digits. Around 480 AD, Tsu Ch'ung Chi computed the value of pi as 3.1415926, which agrees perfectly with the first 8 digits of Western computations. The probability of this happening if pi was truly random is 1 in 100 million. I don't think it is a coincidence.

I think that a much more reasonable view is that pi is a well-defined number with one, and only one, possible value; and this value, determined solely through deductive logic, is constant and independent of any physical event.
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Old 2004-12-01, 08:29   #31
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I see a slight and wrong identification between "random", "undefined", "undetermined" and "unpredictable" in shaxper's argument.

If you were travelling towards London on a defined path, it doesn't mean that your path is "random until you reach London": it only means that you "still did not reach London following your path".

The path is well defined and determined even if London is far from you. Now, consider the ratio between diameter and circumference as a path: it is well defined too, even if we can't determine it completely; now, let's go back to London: you can say that the path measures 80 miles, then refine your computation and define it as 81 miles, 80.9, 80.86, 80.859, 80.8587 and so on. As long as you have a sufficiently precise instrument to measure it, you'll come up with a better approximation of the real distance.

Does it mean that the distance to Londom changes during your misuration, that is "random" until you add another decimal digit? Or, instead, that you acquire a better precision of the (well determined) distance?

I'll put it in another way: think about the Zenone's paradox: it says that Achilles will never reach the tortue because there are "infinite gaps" to close. But with infinitesimal calculus (the same you described in your thread to justify your argumentations) you can demonstrate that the gap closes after 1.11111.... steps. You don't need to actually compute all the 1s in the decimal part to be assured that the gap will close, because the final result is bounded.

Now, apply this idea to Pi. If the final result is bounded (we may say between 3.14 and 3.15) it is definitely not random, even if, as irrational and trascendent, we can't completely describe it.

Luigi

Last fiddled with by ET_ on 2004-12-01 at 08:33
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Old 2004-12-01, 12:35   #32
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Quote:
Originally Posted by ET_
I see a slight and wrong identification between "random", "undefined", "undetermined" and "unpredictable" in shaxper's argument.
Wrong. I just forgot the second part of jinydu's definition.

My bad.
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Old 2004-12-01, 16:23   #33
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Cool To create a real random number

Quote:
Originally Posted by shaxper
Wrong. I just forgot the second part of jinydu's definition.

My bad.

Shaxper your discussion has become in to a play of words!
We are discussing maths not latin grammar!
Et has taken the words out of my mouth but its worth having a say.
Jinydu is right: dont confuse math principles with Quantum theory
Take the case of a circle of unit diameter. Its circumference is pi units.
Now is pi here 'undetermined', 'an infinite series' '100% order' 'part random part order'? Has it an '(n+1) term' that hasnt been determined?
Your 'argument' takes me back to Greek times with Zeno and his paradoxes of Archilles and the hare and the flying arrow. Both have been explained away
today by the calculus.
Its the case of straining at a gnat and swallowing a camel!
Mally
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