20041129, 09:47  #23  
Nov 2004
2×7 Posts 
Quote:
My point is that an infinite series is undetermined. Quote:


20041130, 05:09  #24 
Dec 2003
Hopefully Near M48
6DE_{16} Posts 
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 20041130 at 05:10 
20041130, 14:41  #25  
Nov 2004
2·7 Posts 
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20041130, 15:41  #26  
Bronze Medalist
Jan 2004
Mumbai,India
2^{2}·3^{3}·19 Posts 
Quote:
Shaxper allow me to tell you that you have adopted the views of Leopold Kronecker 18231891. 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 18451918 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 

20041201, 03:47  #27  
Nov 2004
2·7 Posts 
Quote:
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. 

20041201, 04:44  #28  
Dec 2003
Hopefully Near M48
6DE_{16} Posts 
Quote:
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? 

20041201, 06:42  #29  
Nov 2004
2·7 Posts 
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Last fiddled with by shaxper on 20041201 at 06:44 

20041201, 08:00  #30  
Dec 2003
Hopefully Near M48
2·3·293 Posts 
Quote:
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:
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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 welldefined number with one, and only one, possible value; and this value, determined solely through deductive logic, is constant and independent of any physical event. 

20041201, 08:29  #31 
Banned
"Luigi"
Aug 2002
Team Italia
2·3·5·7·23 Posts 
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 20041201 at 08:33 
20041201, 12:35  #32  
Nov 2004
2·7 Posts 
Quote:
My bad. 

20041201, 16:23  #33  
Bronze Medalist
Jan 2004
Mumbai,India
2052_{10} Posts 
To create a real random number
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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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