[PenultimateManStanding]

In Carl Sagan's book Contact on which they based the movie with Jodie Foster, he presented a beautiful conception: at one point he gets around to discussing the idea of pi calculated in the base 11 number system. After calculating for a huge number of digits a startling pattern emerges with all kinds of amazing symmetry revealing the signature of God. I think it was the alien race that revealed that to humans. But getting back to pi as we know it: one can find on the internet pi calculated to rather large numbers of digits. This number which cannot be written exactly nonetheless has the property of being the multiple of the diameter of a circle which gives the circumference. Not a trivial property. Take any random interposition of two digits and you no longer have pi (provided they are not the same digit). Obviously, there is no limit to the number of such interpositions. Do these new numbers have any interesting properties? do they have any meaning? Pi certainly has a meaning, what about all these others? This is a notion that makes me think that science and mathematics, as known, is merely a drop in the bucket compared to what is in principle knowable, that not only are numbers infinite, but knowledge is too. We humans are going to run out of gas, pardon the pun, before we come even close to unraveling all the mysteries.

[Zentric]

$this->bbcode_second_pass_quote('PenultimateManStanding', 'I')n Carl Sagan's book Contact on which they based the movie with Jodie Foster, he presented a beautiful conception: at one point he gets around to discussing the idea of pi calculated in the base 11 number system. After calculating for a huge number of digits a startling pattern emerges with all kinds of amazing symmetry revealing the signature of God. I think it was the alien race that revealed that to humans. But getting back to pi as we know it: one can find on the internet pi calculated to rather large numbers of digits. This number which cannot be written exactly nonetheless has the property of being the multiple of the diameter of a circle which gives the circumference. Not a trivial property. Take any random interposition of two digits and you no longer have pi (provided they are not the same digit). Obviously, there is no limit to the number of such interpositions. Do these new numbers have any interesting properties? do they have any meaning? Pi certainly has a meaning, what about all these others? This is a notion that makes me think that science and mathematics, as known, is merely a drop in the bucket compared to what is in principle knowable, that not only are numbers infinite, but knowledge is too. We humans are going to run out of gas, pardon the pun, before we come even close to unraveling all the mysteries.

[PenultimateManStanding]

$this->bbcode_second_pass_quote('Zentric', ' ')why should we even expect the relationship to be "rational"?

[PenultimateManStanding]

if people want to use "rational" to mean western bias or philosophical myopia, that's fine. I'm using the word to mean a ratio of two discrete numbers, period. have a nice day.

[Specop_007]

Its well understood, although rarely mentioned, that there are certain problems our current system of math is either poor or downright impossible to answer.

[PenultimateManStanding]

$this->bbcode_second_pass_quote('Specop_007', 'I')ts well understood, although rarely mentioned, that there are certain problems our current system of math is either poor or downright impossible to answer.

[Zentric]

$this->bbcode_second_pass_quote('PenultimateManStanding', 'I') would guess that this discovery was for them something analogous to what the Heisenberg Uncertainty Principle was in modern times: something really strange and unexpected. And it isn't just a circle that can be constructed but not measured: if one constructs a right triangle with sides of 1 and 2 units, the hypotenuse is then radical 5 and it can't be measured either.

[PenultimateManStanding]

This way of looking at it: humans are "irrational" to expect "tidiness" in numbers doesn't intrigue me. All this does is to muddy up the word and it is definately something that Raph loves to do. There are all kinds of amazing properties of integers themselves. Mysteries, unknowns, unexpected surprises, etc., that come up in number theory without having to go to irrational numbers to find "awe". What I was saying before, and will repeat: one can't measure pi with number systems. same for radical 5. Some were saying, "sure you can", and I said, "how?" I have not received my answer. I suggest that is because I am correct that one uses discrete number system to measure and that pi and radical five are convenient abstractions for unmeasurable quantities. valuable abstractions, to be sure, but abstractions nonetheless.

[Zentric]

$this->bbcode_second_pass_quote('PenultimateManStanding', 'T')his way of looking at it: humans are "irrational" to expect "tidiness" in numbers doesn't intrigue me. All this does is to muddy up the word and it is definately something that Raph loves to do. There are all kinds of amazing properties of integers themselves. Mysteries, unknowns, unexpected surprises, etc., that come up in number theory without having to go to irrational numbers to find "awe". What I was saying before, and will repeat: one can't measure pi with number systems. same for radical 5. Some were saying, "sure you can", and I said, "how?" I have not received my answer. I suggest that is because I am correct that one uses discrete number system to measure and that pi and radical five are convenient abstractions for unmeasurable quantities. valuable abstractions, to be sure, but abstractions nonetheless.

[PenultimateManStanding]

$this->bbcode_second_pass_quote('Zentric', '
')You lost me on what you mean by "convenient abstractions for unmeasurable quantities." What, after all, is so abstract about either pi or sqrt(5)? I mean, look, I've just now expressed them both precisely.

[PenultimateManStanding]

from wiki: $this->bbcode_second_pass_quote('', 'T')he word number has no generally agreed upon mathematical meaning, nor does the word number system. Instead, we have many examples. Thus there is no rule to say what is a number and what is not. Some of the more interesting examples of abstractions that can be considered numbers include the quaternions, the octonions, and the transfinite numbers.

[Free]

Well I didn't read the thread completely so I don't know if it has been mentioned, but there is indeed the concept of countable and uncountable infinity, the cardinality, of a set.

For example the natural numbers have the cardinality of countable infinity,"Aleph0".

So have the rational numbers, which can be shown with a tricky but easy to understand proof (a kind of grid where you can count through).

Similarly you can show that the set of the real numbers has the cardinality of uncountable infinity, that they are "more infinite" than the natural numbers - basically the proof shows that how many real numbers you ever get, you can always construct a new one with the old ones...

[PenultimateManStanding]

There was a mathematician who used to post here a ways back, I don't remember his name, but I recall he was very haughty and condescending. My level of understanding of math is rather limited. Not that I couldn't have gone a lot further if I'd chosen to. I took a course in number theory about 5 years ago which was the hardest thing I ever did but it was fascinating. The endeavor is too rigidly left-brain. I'd almost prefer to think like Raphael than devote myself to math. So unless someone says something requiring a reply, here's PMS signing out on this one which may now go down the memory hole!

[PenultimateManStanding]

$this->bbcode_second_pass_quote('Free', ' ')
For example the natural numbers have the cardinality of countable infinity,"Aleph0".