To review, in May of 2003, I asked my
Pickover Discussion Group to consider a concept that
had been on my mind for many years.
Eric says:
Hey, Dr. Cliff. I was just reading your "infinite happiness Pi" theory,
and if an infinite Pi encodes for all possible realities, then, being a bit of a pessimist,
am I not also in an infinite variety of hells? For example, isn't there encoded in Pi
a reality in which I'm forced to service Roseanne Barr a number of times a day?
-Eric
From: "Pete B" :
I have been very amused by this entire thread. Cliff wondered why his original question was met with such skepticism. Well,
consider...
We talk of "information" being "encoded" in the digits of Pi, without actually ever providing explanation of precisely what those terms
mean, nor any possible mechanism (other than science-fantasy stuff) to make any use of this info, if it does exist.
I would like to know exactly how, with very specific, practical, technological detail, any such "sequence" of digits in Pi could ever, in
any even slightly plausible, humanly-possible scenario, be used to accomplish anything of significance whatsoever (beyond
calculating properties of circles and such, that is).
What does everyone mean by "coding"? Do they mean some kind of binary string that can be used as a "program" of some sort in
some kind of super-intelligent computer? Or is it perhaps some kind of binary-controlled super machine, sort of like a real-time
process control system code? Or what? But OK, let's assume, just for purposes of sheer speculative fantasy, that you have
somehow devised a scheme to discover such a meaningful sequence of digits (a topic for exploration later). Voila, here is the human
genome, all 3 billion DNA base-pairs. (We will ignore the fact that we already have found that little piece of information) So... now
what?
Do we take this "program" and then plug it into our Ajax Amazing Human Body Production Machine? Press the Enter key and the
conveyor belt starts, out pops our very own human? Shall we name him/her/it perhaps Pi The First? Or perhaps we twist a few dials
and so forth, throw in a little genetic variation and a bit of fur, and produce Pi The First's pet dog, a female called, oh, heck, maybe
April? What exactly do we do with this wonderous new source of information, what good is it if it did exist? Most important of all,
how do we market it so we can make a fortune before the patent expires on our AAHBPM?
And who gets to take care of all those freaks, anyway? The only sure thing is that Congress will tax them, the PETA people will
scream if you touch them, the Democrats will pass laws to allow them to marry your daughters, the Republicans will demand that the
voting districts be realigned to avoid any possibility the freaks might become Democrats, and most of all, the neighborhood will go to
hell if they move in next door to you.
Then, of course, when that left-hand veeblefetzer hangs up in our machine and produces fourteen new bodies with one gene horribly
mutated, causing the body to be forever a vegetable, who gets to keep them? Hey, it's your machine and your scatter-brained idea,
don't come looking to me for a handout to help you. And will it be legal to kill them? Wow, another field day for lawyers, the thought
of the fees to be made boggles the imagination...
This whole idea of some kind of "information" being "encoded" in the digits of Pi is IMHO just a good basis for a sci-fantasy novel,
and not much else. In order to even decipher any such code in an infinite series would require a device with an infinite processor
capability, so it is forever beyond our reach, just like immortality and that secret formula that turns cabbage into gold. So why
bother? Even if such information did somehow "exist" in that series of digits, it would require an intelligence of infinite capability to
recognize it, but the last time I looked, Hootie Johnson was busy keeping guard on his golf course to keep out the riff-raff.
Let's imagine your own self (or anything else that you consider significant enough to be worthwhile looking for) was encoded in Pi,
just for the sake of discussion. How would you know it? How would you recognize it? And assuming you found it, what then?
Oh, wait. I know: "Digits Of Pi: The Movie". Yes, of course, silly me, how could I have forgotten Hollywood? But... who gets the
movie rights?
:=) :=)
Talk about needing a reality check....
Pete B
From: "Nadilo" : Nick, I said to be finite just for sake of the argument which is that we are imagining pi to be infinitive but it cannot be infinitive really if necessary condition for exsitence of everything is being in some form of matter ,and we consider matter to be finite in quantity even measured and counted on smallest possible level.Therefore,we cannot used pi as universal code as a whole because it can exist only partially in moment of time. Also,this idea that all which existed and exist can be codeed in pi does not answer how to distingush real entities from those who are not. Let assume that Cliff is coded in pi as he said that he thinks it is possible. How to distinguish real Cliff from Cliff very similar but not genuine,let assume one who does not like sushi?So it is problem of real indentity. Anton
From: "Pete B": Mark, what you say may be true, but you are putting the cart before the horse. Suppose the entire movie that I called "Digits Of Pi" is indeed "encoded" somewhere in the digits of Pi. OK, now what? It's in there someplace, so what do we do? The only way we would know that we had found it is if we knew ahead of time what that string was, with perfect precision and accuracy. So we have to find what the movie itself is before we discover its' code so that we can make the movie which then gives us the code to make the movie so that.... See what I mean? Any such info would have to be known ahead of time, unless one comes up with some kind of ultra-supercomputer to spit out the required search target strings on request -- oops, wait a minute, that means that just like above we still do not need to look in Pi, we already have the info we need. The only possible way such "information" could be extracted from strings contained in the digits of PI would be if we knew what to look for, and if we know that, I'll venture a bet that it is far cheaper to just use what we already know than to find the strings in Pi for the same purpose. I actually would rank this type of speculation right up there with the questions about angels dancing on pinheads, or, to phrase it slightly differently, angels dancing on the heads of pins. Hilarious material for a comedy show, good fodder for a sci-fantasy novel, nice exercises for theoretical mathematicians and prisoners held in solitary confinement for life, but that is about the extent of it. It is about as useful as speculating on whether the moons of planets orbiting the farthest known quasar really are made of green cheese. Look at it another way: if we take an infinite number of computers, capable of generating perfectly random strings (which by definition must be infinitely long) and let them run for an infinite time, they will output an infinte number of strings of characters containing all the information that is possible to be discovered. So what? Are you going to start at that end of the pile while I start at the other end, and see what we find? I have better things to do with my short lifespan. As to Cliff's comment, I agree it is a thread worth preserving somehow... but I doubt if it is done anytime soon. :=) Pete B
Pete, The fact that all possible finite numerical sequences exist in an infinite number of locations within pi might or might not have any practical use for humanity, but it's interesting stuff to speculate about. Someone (ollyhardy I believe?) in one of these threads had the idea of using pi as a universal database of information. We'd just point to the starting address within the expansion of pi instead of sending the actual information. For example, the new Matrix movie can be encoded as an .mpg file. This same sequence exists an infinite number of times within the expansion of pi. If we had the starting address of one of these sequences and a computer capable of spitting out the stream at that address, we could all just watch the movie by going to that address. Unfortunately, I think the number representing the starting address of the sequence would be much larger than the actual sequence itself, thereby negating any practical savings. On the other hand, we can use exponents to represent very large numbers, such as 10^12^(10^12). So, if the Matrix movie or some other useful set of data, was at, say 10^12^8 - 2^3^4 + 72 * 3 - 7, for example, this concept might one day prove useful.
From: "Tufrmone" Dear Pete: I'm no engineer nor mathematician - But. Wouldn't a compression algorthim work in this type of situation. The algorhtim searchs for strings of commonly known numbers that must be part of any proposed problem solution or information and confines itself to examining the strings related to or near the known string until it finds the right algorythimic solution for what ever problem is being solved through Pi??????? Tufr
Pete B wrote: Mark, what you say may be true, but you are putting the cart before the horse. Suppose the entire movie that I called "Digits Of Pi" is indeed "encoded" somewhere in the digits of Pi. OK, now what? It's in there someplace, so what do we do? The only way we would know that we had found it is if we knew ahead of time what that string was, with perfect precision and accuracy. So we have to find what the movie itself is before we discover its' code so that we can make the movie which then gives us the code to make the movie so that.... See what I mean? Any such info would have to be known ahead of time, unless one comes up with some kind of ultra-supercomputer to spit out the required search target strings on request -- oops, wait a minute, that means that just like above we still do not need to look in Pi, we already have the info we need. The only possible way such "information" could be extracted from strings contained in the digits of PI would be if we knew what to look for, and if we know that, I'll venture a bet that it is far cheaper to just use what we already know than to find the strings in Pi for the same purpose. I actually would rank this type of speculation right up there with the questions about angels dancing on pinheads, or, to phrase it slightly differently, angels dancing on the heads of pins. Hilarious material for a comedy show, good fodder for a sci-fantasy novel, nice exercises for theoretical mathematicians and prisoners held in solitary confinement for life, but that is about the extent of it. It is about as useful as speculating on whether the moons of planets orbiting the farthest known quasar really are made of green cheese. Look at it another way: if we take an infinite number of computers, capable of generating perfectly random strings (which by definition must be infinitely long) and let them run for an infinite time, they will output an infinte number of strings of characters containing all the information that is possible to be discovered. So what? Are you going to start at that end of the pile while I start at the other end, and see what we find? I have better things to do with my short lifespan. As to Cliff's comment, I agree it is a thread worth preserving somehow... but I doubt if it is done anytime soon. :=) Pete B
From: "tmredden" I thought it was proven that pi is transcendental (not infinite, BTW. Infinite numbers are VERY LARGE and pi is only somewhere between 3 and 4). In order to do so, it should be necessary only to prove that there are no circles for which the circumference and the diameter are both rational numbers. How that would be done I'm not certain (been a long time since I studied math.) Of course, then, I had to search google, and easily came up with http://pi314.at/math/irrational.html which proof I don't yet understand, but I'll take their word for it, for now. Todd
From: "Pete B": Tufr The problem is not so much to do with finding the strings, it is to do with knowing what to find. Some issues that affect this are seemingly insurmountable. Consider: There has been talk from Cliff and others about, for example, "Cliff is encoded in the digits of Pi." Well, what the hell does that mean, really? What is the information being encoded? Some might say it is the complete genome sequence for Cliff, but I'm afraid that will not do, and will be useless to give us a Cliff. All that the genome sequence does is to show nature what it needs to know to produce Cliff. But how nature uses that information to produce a living, breathing, conscious and intelligent organism that writes great books and hostsa such great discussions... ahh, not a clue. So perhaps we "encode" the complete details of the organism called Cliff, down to the subatomic level, as far as possible. Well, first, that is impossible to determine on a subatomic level, thanks to that good old H's Uncertainty Principle. But if it were, we are still doomed to failure. Say we are able to do this feat, and we "encode" every single minute detail of information needed to exactly reproduce every single atom, molecule, ion, electron, proton, gene, protein, and so forth present in Cliff, with every property needed such as what molecule is next to which atom, and so on. Nope, too bad, that's no good either. First, it only includes the state of Cliff that exists for perhaps a nanosecond or two, after that, poof, that Cliff is gone and another whole "code" is needed for the Cliff existing during the succeeding nanoseconds, ad infinitum (or ad nauseum). Cliff is dynamic, if nothing else. Then there is the problem of the things we cannot precisely and accurately enumerate. Little insignificant things like his conscious mind. How do you encode a persona, a set of emotions, a feeling? Beats me! And everybody else that has ever studied it, too. Again, not a clue. Never has been, never will be. Then, of course, there are the issues arising from dealing with infinite sequences of information. For example:
*The digits of Pi are indeed an infinite sequence, but they are most definitely not random in the slightest. They are as precisely determined and preordained as is the molecular composition of salt. Every time you use some algorithm or other process to produce the sequence (as much of it as possible in human terms), it will be precisely and exactly the same as the any other attempt.
*Since the digits of Pi are not random, but the sequence is indeed of infinite length, that means that there are indeed an infinite number of finite-length strings within the sequence. But that definitely does NOT mean that every possible string is included in this sequence. In fact, the number of possible finite-length strings of all random kinds NOT included in the Pi sequence is infinite.
*The total number of perfectly random infinite sequences possible is infinite, and as such, those infinite sequences will contain absolutely every possible finite-length string that can ever exist or possibly exist, including the strings that somehow may "encode" Cliff, whatever that means.
*Since the total number of infinite random sequences is infinite and contains every possible infinite sequence of digits, that random set of infinite sequences therefore would contain the sequence that is Pi as one of its' members, as well as an infinite number of other random sequences as members.
*If you were to start to examine the infinite string of digits that is the string we call Pi, then for any finite time you examine the sequence, you will find a finite number of meaningful strings, but there will be an infinite length of the sequence remaining to be examined, which will contain an infinite number of other undiscovered meaningful strings, and which would take an infinite amount of time to examine.
*Since the number of known strings from any such examination is finite, and the number of unknown, undiscovered strings is always infinite, the chance that the unknown string that encodes Cliff is in the finite number of strings you have already examined, rather than in the remaining infinite number yet to be determined, is the ratio of known strings to unknown strings, or precisely zero. And it will always be zero, no matter how many strings you examine. It will, furthermore, always be unknown because we already know that it is not in the strings we have already examined; if it was, we would not have to search any further.
*Even worse, since the infinite-length random sequence, containing the finite-length string that represents Cliff, is only one of the infinite number of possible random infinite sequences, and the infinite sequence that is Pi is precisely only one of those sequences in that set, the chance that the member of that set we are examining, the Pi infinite sequence, is the sequence that contains the string that is Cliff is one of an infinite number of possibilities, which again is precisely zero. So, the chance that Cliff is in the Pi sequence, as opposed to some other of an infinite number of perfectly random sequences, is zilch..*Even if there are an infinite number of infinite-length random sequences containing the string that encodes Cliff, there will always be an infinite number of other such sequences that do not contain that string. So we will never find the Cliff string. Unless we live forever. Ain't infinity fun? You can demonstrate anything!! :=) :=) (Now I eagerly await all the posts showing the errors of my statements.) Pete B
. From: "Pete B" Nobody said Pi was infinite. Pi is the ratio of the circumference of a circle to its' diameter. That ratio is transcendental, meaning it can only be expressed as an infinite series of digits, but Pi itself is definitely not infinite. Pete B
From: "Nadilo" Todd, Graham used argument that pi is not infinitive because it is smaller than 4 so I will answer you in same manner as to him. It is not infinitively big but it is infinitively long. A subject of discussion was is it possible that pi is information from which is possible to extract everything which existed and exist and possibly will exist because every enitiy might be coded in finite string of numbers and infinitely number of finite strings of different sequences of numbers are existing in pi. First,pi is infinite but that is imagened conception which cannot be manifested in known universe in which necessary precondition for existence is to be form of matter and considering that mattter is finite in quantity a pi cannot be bigger than limitation of imposed by quantity of matter which is on disposal to represent it. Now why is that significant? It is significant because if ,for example,a object which we want to extract from pi is placed after 10 ^1 000 000 000 digits of pi ,we have to calculate pi until that digits and to calculate it until that point will need enorm amount of time and we will not be able to store information of calculated number because it will be too big even all matter in known universe is used. Second,it is problem of indentity.Let assume that we found what we were looking for (sound like U2) how we will know that is a real thing? Perhaps because it correspond with what was searched. But if we had information what we are seaching why search another time. So,it must be that we do not know what we are searching and therefore we will not know what we will found,if we will found. Anton
From: "mwganson": Cliff's assertion that he is encoded somewhere in pi is not dependent to any degree whatsoever that we have a clue as to what that encoding might look like. Now, if we are to make any practical use of this information, then clearly we must have an in-depth knowledge of it. But, I don't think Cliff is suggesting that humanity (or some other entity) will someday take the digits of pi that encode him and use this blueprint to reincarnate another Cliff. So, let's not get bogged down in the notion that there muse exist a practical use for the information in order for it to exist. > Some issues that affect this are seemingly insurmountable. > Consider:
> There has been talk from Cliff and others about, for example, "Cliff is encoded in the digits of Pi." Well, what the hell does that mean, really? What is the information being encoded? Some might say it is the complete genome sequence for Cliff, but I'm afraid that will not do, and will be useless to give us a Cliff. All that the genome sequence does is to show nature what it needs to know to produce Cliff. But how nature uses that information to produce a living, breathing, conscious and intelligent organism that writes great books and hostsa such great discussions... ahh, not a clue.
> So perhaps we "encode" the complete details of the organism called Cliff, down to the subatomic level, as far as possible. Well, first, that is impossible to determine on a subatomic level, thanks to that good old H's Uncertainty Principle. But if it were, we are still doomed to failure. Say we are able to do this feat, and we "encode" every single minute detail of information needed to exactly reproduce every single atom, molecule, ion, electron, proton, gene, protein, and so forth present in Cliff, with every property needed such as what molecule is next to which atom, and so on.
> Nope, too bad, that's no good either. First, it only includes the state of Cliff that exists for perhaps a nanosecond or two, after that, poof, that Cliff is gone and another whole "code" is needed for the Cliff existing during the succeeding nanoseconds, ad infinitum (or ad nauseum). Cliff is dynamic, if nothing else.
> Then there is the problem of the things we cannot precisely and accurately enumerate. Little insignificant things like his conscious mind. How do you encode a persona, a set of emotions, a feeling? Beats me! And everybody else that has ever studied it, too. Again, not a clue. Never has been, never will be. Okay, so there is no way possible for any type of scanning mechanism to possibly examine Cliff in the highest level of detail, down to the smallest subatomic particle of his makeup, so what? That we are not able to create the code ourselves, does not prevent its existence within the digits of pi. Imagine a planet orbiting a star in a galaxy that lies beyond the observable universe. That is, the light emitted by that star has not yet had the time to reach us. Now, given this scenario, there is no telescope in the world that is capable of viewing that planet. But, there is a stream of digits.somewhere within the infinite stream of digits we know as pi, that can be taken and interpreted as a 1024x768x32bit jpeg image of that planet from 1,000 miles above it. There is no way for us to take this picture, and encode it into jpeg code, but it still exists somewhere within pi. There is no way for us to validate any particular stream of digits is an accurate depiction of this planet, either, but that goes to practical usage rather than the existence in and of itself of the jpeg stream. Similarly, we don't have a copy machine capable of taking Cliff's sequence within pi, but that does not preclude its existence within pi. You suggest that even if we could create an exact duplicate of Cliff, down to the tiniest subatomic particles, that it would go poof within a nanosecond or two. That could be, but I'm not willing to concede that point. We just don't know what would happen. We might have a living, breathing Cliff who, apparent to that Cliff, has just been teleported to a new location in space-time, with all of his memories, etc., up to that point in his life perfectly intact. On the other hand, we might have a dead body. This all goes to the question of whether one's soul is separate and apart or whether it is contained within one's physical embodiment. This debate, however, is a whole new can of worms, worthy of a separate thread.
> Then, of course, there are the issues arising from dealing with infinite sequences of information. For example:
> *The digits of Pi are indeed an infinite sequence, but they are most definitely not random in the slightest. They are as precisely determined and preordained as is the molecular composition of salt. Every time you use some algorithm or other process to produce the sequence (as much of it as possible in human terms), it will be precisely and exactly the same as the any other attempt. This very fact is beneficial to us if we are ever going to be able to use pi as a sort of universal database of information. I envision a day when everybody could carry around his own personal pi machine, capable of producing the digits of pi at any starting point, up to some finite number high enough to make it usable, and interpret these digits as a variety of encoding types, such as .mp3, .mpg, .txt, .html, etc. The only problem, aside from the awesome computer power that would be needed, is that the starting address for useful strings will probably take more storage space than it would take to simply store the .mp3, .mpg, etc. data streams. Just a sci-fi pipe dream at this point, I know.
> *Since the digits of Pi are not random, but the sequence is indeed of infinite length, that means that there are indeed an infinite number of finite-length strings within the sequence. But that definitely does NOT mean that every possible string is included in this sequence. In fact, the number of possible finite-length strings of all random kinds NOT included in the Pi sequence is infinite. Name me one single finite-length string that is not in pi. I don't even need the name of the string, just the length of it. If it is of n length, whatever value n is -- so long as it is finite -- we can expect to find 10^9 such strings within the first 10^9 * 10^n digits.
> *The total number of perfectly random infinite sequences possible is infinite, and as such, those infinite sequences will contain absolutely every possible finite-length string that can ever exist or possibly exist, including the strings that somehow may "encode" Cliff, whatever that means.
> *Since the total number of infinite random sequences is infinite and contains every possible infinite sequence of digits, that random set of infinite sequences therefore would contain the sequence that is Pi as one of its' members, as well as an infinite number of other random sequences as members.
> *If you were to start to examine the infinite string of digits that is the string we call Pi, then for any finite time you examine the sequence, you will find a finite number of meaningful strings, but there will be an infinite length of the sequence remaining to be examined, which will contain an infinite number of other undiscovered meaningful strings, and which would take an infinite amount of time to examine. But, as I've shown, we don't need to examine all of the infinite strings in order to find any particular finite length string. If n is the length of the sought after string, then we should expect to find it 10 times within the first 10*10^n digits.
> *Since the number of known strings from any such examination is finite, and the number of unknown, undiscovered strings is always infinite, the chance that the unknown string that encodes Cliff is in the finite number of strings you have already examined, rather than in the remaining infinite number yet to be determined, is the ratio of known strings to unknown strings, or precisely zero. And it will always be zero, no matter how many strings you examine. It will, furthermore, always be unknown because we already know that it is not in the strings we have already examined; if it was, we would not have to search any further. Let's say we are looking for a particular string of digits of length n, where n is finite. Now, I calculate that we will find a copy of this string within the first 10^n digits. It might turn out, however, that this particular string does not exist within that limit. We can expect to find 10 such strings within the first 10*10^n digits of pi. It could be that there is no such copy even within that span, but the probability is small, only 1/10. We can view this generally as the probability of *not* finding a string of length n within the first x * 10^n digits of pi as 1/x. Thus, the higher x, the lower the probability that we will not find our string. If we make x a very large value, such as, say, 10^9 (one billion), then the probability of *not* finding our string within x * 10^n digits is 1/x or one in a billion. There are an infinite number of copies of our string within pi, but we are really only interested in finding the first such copy. So, the ratio of found copies to the ratio of total copies will always be finite versus infinite, which approximates to zero, I suppose. But, this has nothing to do with the probability of finding a particular string as I have illustrated above.
> *Even worse, since the infinite-length random sequence, containing the finite-length string that represents Cliff, is only one of the infinite number of possible random infinite sequences, and the infinite sequence that is Pi is precisely only one of those sequences in that set, the chance that the member of that set we are examining, the Pi infinite sequence, is the sequence that contains the string that is Cliff is one of an infinite number of possibilities, which again is precisely zero. So, the chance that Cliff is in the Pi sequence, as opposed to some other of an infinite number of perfectly random sequences, is zilch. I disagree. There are an infinite number of possible infinite-length random sequences, but this does not imply that one and only one of them can contain a particular finite-length sequence, such as Cliff, or 1234. It means that *every* one of them will contain Cliff and 1234. Consider these multiple infinite length sequences: the set of all real numbers, the set of all integers, and the set of all whole numbers. Now, these are 3 different infinite-length sequences, yet all 3 contain the number 1 as a member. (I believe 1.0 is a real number, is it not?) Just because there are multiple (indeed infinite) infinite sequences, does not prove (or even suggest) that a finite sequence can exist within one and only one of them. This is like saying that the string 1234 exists within pi, but not within e or Sqrt[2].
> *Even if there are an infinite number of infinite-length random sequences containing the string that encodes Cliff, there will always be an infinite number of other such sequences that do not contain that string.
> So we will never find the Cliff string. Unless we live forever. We *might* not ever find the Cliff string. We *definitely* will not recognize it if we do. It could well be that the very first 10^12 digits of pi contain the Cliff string. Probably not, but possible. .It could be that we are staring Cliff right in the figurative digital face and without even knowing it.
> Ain't infinity fun? You can demonstrate anything!! > > :=) :=)
> (Now I eagerly await all the posts showing the errors of my statements.) > > Pete B
In summary, I would suggest that Cliff and every other finite length sequence exists within pi. Whether we feeble humans are able to recognize the pattern or not, bears no impact whatsoever upon the patterns existence within pi. The practical implications, whether any exist or not, are irrelevant to the very existence of all finite length strings somewhere within the digits of pi. The prospect of *not* finding a finite length string is readily calculable as 1/x where we search for an n length sequence within a span of x * 10^n digits. (Well, actually this formula needs a little bit of work, since 1/x, where x == 1, implies a probability of 1, which is not accurate. I should probably qualify it by requiring that x > 1.) Pi is of infinite length, but we need *never* search its entire infinite length (which would be impossible anyway) to find the first occurrence of any particular finite length string. The entire sum of all human knowledge could be encoded within a string of digits to be interpreted, for example, as an .mpg movie stream. This movie, even if entirely comprehensive, is still finite, since the depth of human knowledge is finite, and hence, encoded somewhere within pi. The final point, though probably not the most significant, is that even though there exist infinite sequences other than pi, these have no bearing upon which finite sequences can or cannot exist within pi. There is no mutual exclusivity among infinite sequences. --Mark
From:
From: "Graham Cleverley" For my money Pete B was right on the button, but I'll add a couple of responses in support. > I'll mix my thoughts in where appropriate.
So will I. > > > But, I don't think Cliff is suggesting that humanity (or some other > entity) will someday take the digits of pi that encode him and use > this blueprint to reincarnate another Cliff. So, let's not get > bogged down in the notion that there muse exist a practical use for > the information in order for it to exist. OK. Leave that problem aside. > > > > Okay, so there is no way possible for any type of scanning > mechanism to possibly examine Cliff in the highest level of detail, > down to the smallest subatomic particle of his makeup, so what? > That we are not able to create the code ourselves, does not prevent > its existence within the digits of pi. Again, OK, concentrate on whether the code is there, not whether we could create it. > Imagine a planet orbiting a star in a galaxy that lies beyond the > observable universe. That is, the light emitted by that star has > not yet had the time to reach us. Now, > given this scenario, there is no telescope in the world that is > capable of viewing that planet. But, there is a stream of digits > somewhere within the infinite stream of digits we know as pi, that > can be taken and interpreted as a 1024x768x32bit jpeg image of that > planet from 1,000 miles above it. Granted that there is such a stream of digits, whether it exists of not is pi is not guaranteed. It may do, but it may not. You're making the assumption that pi contains all finite length strings of integers, which is not proven. It admittedly contains a countably infinite number of such strings, but somehow you have to prove a bijection (a one-to-one mapping) of that infinite set to the set of all possible finite strings. This is so far missing. > > > > *Since the digits of Pi are not random, but the sequence is > > indeed of infinite length, that means that there are indeed an > > infinite number of finite-length strings within the sequence. > > But that definitely does NOT mean that every possible string is > > included in this sequence. In fact, the number of possible > > finite-length strings of all random kinds NOT included in the Pi > > sequence is infinite. > Pete is correct here. > Name me one single finite-length string that is not in pi. I can't, because I don't know all the digits of pi. What I believe I know is that there are more possible finite-length strings than the number of such strings present in pi. It's possible to know that.without identifying any particular string. > I don't even need the name of the string, just the length of it. > If it is of n length, whatever value n is -- so long as it is > finite -- we can expect to find 10^9 such strings within the first > 10^9 * 10^n digits.
From: "Pete B" : I will comment on some of the claims and statements you put forth. Some, however, are sheer unsupported speculation on your part, with no proof. I will merely point those out without comment. Interspersed below... Pete B
From: mwganson > Tufr > > The problem is not so much to do with finding the strings, it is to do with knowing what to find. Cliff's assertion that he is encoded somewhere in pi is not dependent to any degree whatsoever that we have a clue as to what that encoding might look like. I beg to differ. It is without any shadow of a doubt totally and unqualifiedly dependent upon the prior knowledge of exactly what that enoded string is. If nobody knows the composition of the string, there is no way it can be proven to exist anywhere, let alone as a sequence of digits in Pi. If you claim otherwise, you are merely guessing or offering your opinion, which is totally unsupported by proof and therefore is purely imaginary speculation. That is, it is in the same class as those pin-headed angels I mentioned earlier. I will also assert that although you cannot prove it exists without actually knowing what it is, I can and have offered proof that it is highly doubtful that it exists at the very least. Now, if we are to make any practical use of this information, then clearly we must have an in-depth knowledge of it. But, I don't think Cliff is suggesting that humanity (or some other entity) will someday take the digits of pi that encode him and use this blueprint to reincarnate another Cliff. So, let's not get bogged down in the notion that there muse exist a practical use for the information in order for it to exist. It may not require practicality, but it must be proven to be a reality, ie that such a string exists. So that statement is answered same as the previous one. at least until you offer some such proof. > Some issues that affect this are seemingly insurmountable.
Consider: > > There has been talk from Cliff and others about, for example, "Cliff is encoded in the digits of Pi." Well, what the hell does that mean, really? What is the information being encoded? Some might say it is the complete genome sequence for Cliff, but I'm afraid that will not do, and will be useless to give us a Cliff. All that the genome sequence does is to show nature what it needs to know to produce Cliff. But how nature uses that information to produce a living, breathing, conscious and intelligent organism that writes great books and hostsa such great discussions... ahh, not a clue.
> So perhaps we "encode" the complete details of the organism called Cliff, down to the subatomic level, as far as possible. Well, first, that is impossible to determine on a subatomic level, thanks to that good old H's Uncertainty Principle. But if it were, we are still doomed to failure. Say we are able to do this feat, and we "encode" every single minute detail of information needed to exactly reproduce every single atom, molecule, ion, electron, proton, gene, protein, and so forth present in Cliff, with every property needed such as what molecule is next to which atom, and so on.
> Nope, too bad, that's no good either. First, it only includes the state of Cliff that exists for perhaps a nanosecond or two, after that, poof, that Cliff is gone and another whole "code" is needed for the Cliff existing during the succeeding nanoseconds, ad infinitum (or ad nauseum). Cliff is dynamic, if nothing else.
> Then there is the problem of the things we cannot precisely and accurately enumerate. Little insignificant things like his conscious mind. How do you encode a persona, a set of emotions, a feeling? Beats me! And everybody else that has ever studied it, too. Again, not a clue. Never has been, never will be. Okay, so there is no way possible for any type of scanning mechanism to possibly examine Cliff in the highest level of detail, down to the smallest subatomic particle of his makeup, so what? That we are not able to create the code ourselves, does not prevent its existence within the digits of pi. Indeed it does not. But yet again, the string must be proven to exist for anything else to be valid in talking about it. Imagine a planet orbiting a star in a galaxy that lies beyond the observable universe. That is, the light emitted by that star has not yet had the time to reach us. Now, given this scenario, there is no telescope in the world that is capable of viewing that planet. But, there is a stream of digits somewhere within the infinite stream of digits we know as pi, that can be taken and interpreted as a 1024x768x32bit jpeg image of that planet from 1,000 miles above it. There is no way for us to take this picture, and encode it into jpeg code, but it still exists somewhere within pi. There is no way for us to validate any particular stream of digits is an accurate depiction of this planet, either, but that goes to practical usage rather than the existence in and of itself of the jpeg stream. Once again, I do not agree. All of your arguments do not offer one shred of proof that such a string exists, they merely speculate as to possibilities IF the string exists. Similarly, we don't have a copy machine capable of taking Cliff's sequence within pi, but that does not preclude its existence within pi. I never claimed that such a string was precluded from existence. I did show that the likelihood/probability of it existing within the digits of Pi is zero, however, but that is because we are dealing with an infinitely-long sequence. You suggest that even if we could create an exact duplicate of Cliff, down to the tiniest subatomic particles, that it would go poof within a nanosecond or two. That could be, but I'm not willing to concede that point. We just don't know what would happen. To the contrary: Cliff and all people are living organisms. Body chemistry and biological processes are continuously altering the composition of our bodies every second we live. In order for that to occur, nano-second duration subatomic processes are required to always be present and ongoing. In fact, I will venture the claim that, if it were not so, if those processes all stopped for even a second or so, the body would die. We might have a living, breathing Cliff who, apparent to that Cliff, has just been teleported to a new location in space-time, with all of his memories, etc., up to that point in his life perfectly intact. On the other hand, we might have a dead body. Sheer speculation.... This all goes to the question of whether one's soul is separate and apart or whether it is contained within one's physical embodiment. This debate, however, is a whole new can of worms, worthy of a separate thread.
> Then, of course, there are the issues arising from dealing with infinite sequences of information. For example:
> *The digits of Pi are indeed an infinite sequence, but they are most definitely not random in the slightest. They are as precisely determined and preordained as is the molecular composition of salt. Every time you use some algorithm or other process to produce the sequence (as much of it as possible in human terms), it will be precisely and exactly the same as the any other attempt. This very fact is beneficial to us if we are ever going to be able to use pi as a sort of universal database of information. I envision a day when everybody could carry around his own personal pi machine, capable of producing the digits of pi at any starting point, up to some finite number high enough to make it usable, and interpret these digits as a variety of encoding types, such as .mp3, .mpg, .txt, .html, etc. The only problem, aside from the awesome computer power that would be needed, is that the starting address for useful strings will probably take more storage space than it would take to simply store the .mp3, .mpg, etc. data streams. Just a sci-fi pipe dream at this point, I know. I totally agree. With your last sentence, anyway :=)
> *Since the digits of Pi are not random, but the sequence is indeed of infinite length, that means that there are indeed an infinite number of finite-length strings within the sequence. But that definitely does NOT mean that every possible string is included in this sequence. In fact, the number of possible finite-length strings of all random kinds NOT included in the Pi sequence is infinite. Name me one single finite-length string that is not in pi. I don't even need the name of the string, just the length of it. If it is of n length, whatever value n is -- so long as it is finite -- we can expect to find 10^9 such strings within the first 10^9 * 10^n digits. I am not a mathematician, but it seems to me it is easy enough to prove that there are an infinite number of finite-length strings that do not exist in Pi. Here is one way, expressed as a thought experiment, and which assumes that one can actually work with infinite-length strings of digits like Pi, and that we can determine all of the substrings of Pi of any finite length: We can designate the length of the longest possible finite-length string contained within Pi as Lstr(Pi). Now list all of the unique such strings contained within Pi of that longest possible length. Obviously there is only the one string, the one that begins with the first digit of Pi and ends with the last digit of Pi. Now change the last digit of that string to any other value. That new string does not exist within Pi. Now enumerate and list all of the unique strings within Pi that are of length Lstr(Pi)-1. Now for each such string, change any one digit. All of the changed strings are not contained within Pi. Continue this process for all lengths of strings, from longest to shortest, listing all of the unique strings of a fixed particular length, change any one digit of each such string, the result strings are not contained within PI. Do the same process for all strings that begin at the first digit of Pi, and work up to the last digit of Pi. Yhen do the same for all combinations of starting digit and ending digit. By doing so, I believe you will have produced an infinite number of unique strings, none of which are contained within Pi. Here is a simple example: consider the string of digits in the infinite sequence of Pi that begins with the first digit (after the decimal point) and is of maximum length, in other words, includes all the digits in the infinite string that is Pi. Now change the first digit of that string to 2 instead of 1. That new string is not contained anywhere within Pi. Now go to the string that starts at the second digit, and change that starting digit from 4 to, say, 8. Lo and behold, another string that is not contained within the digits of Pi. There are an infinite number of such non-Pi strings. BUT... the set of all possible perfectly random infinite-length strings does include all of those strings, in fact, that set includes all possible strings that can exist. It is only because Pi is not random that it does not contain all of the possible finite-length strings
*The total number of perfectly random infinite sequences possible is infinite, and as such, those infinite sequences will contain absolutely every possible finite-length string that can ever exist or possibly exist, including the strings that somehow may "encode" Cliff, whatever that means.
> *Since the total number of infinite random sequences is infinite and contains every possible infinite sequence of digits, that random set of infinite sequences therefore would contain the sequence that is Pi as one of its' members, as well as an infinite number of other random sequences as members.
> *If you were to start to examine the infinite string of digits that is the string we call Pi, then for any finite time you examine the sequence, you will find a finite number of meaningful strings, but there will be an infinite length of the sequence remaining to be examined, which will contain an infinite number of other undiscovered meaningful strings, and which would take an infinite amount of time to examine. But, as I've shown, we don't need to examine all of the infinite strings in order to find any particular finite length string. If n is the length of the sought after string, then we should expect to find it 10 times within the first 10*10^n digits. Except you have not the slightest idea what that length is, or even WHETHER the string exists.
> *Since the number of known strings from any such examination is finite, and the number of unknown, undiscovered strings is always infinite, the chance that the unknown string that encodes Cliff is in the finite number of strings you have already examined, rather than in the remaining infinite number yet to be determined, is the ratio of known strings to unknown strings, or precisely zero. And it will always be zero, no matter how many strings you examine. It will, furthermore, always be unknown because we already know that it is not in the strings we have already examined; if it was, we would not have to search any further. Let's say we are looking for a particular string of digits of length n, where n is finite. Now, I calculate that we will find a copy of this string within the first 10^n digits. It might turn out, however, that this particular string does not exist within that limit. We can expect to find 10 such strings within the first 10*10^n digits of pi. It could be that there is no such copy even within that span, but the probability is small, only 1/10. We can view this generally as the probability of *not* finding a string of length n within the first x * 10^n digits of pi as 1/x. Thus, the higher x, the lower the probability that we will not find our string. If we make x a very large value, such as, say, 10^9 (one billion), then the probability of *not* finding our string within x * 10^n digits is 1/x or one in a billion. There are an infinite number of copies of our string within pi, but we are really only interested in finding the first such copy. So, the ratio of found copies to the ratio of total copies will always be finite versus infinite, which approximates to zero, I suppose. But, this has nothing to do with the probability of finding a particular string as I have illustrated above. All could be true... if the string exists and we know what the string is. If it does not exist or is forever unknown, then none of that applies, nor does it matter.
> *Even worse, since the infinite-length random sequence, containing the finite-length string that represents Cliff, is only one of the infinite number of possible random infinite sequences, and the infinite sequence that is Pi is precisely only one of those sequences in that set, the chance that the member of that set we are examining, the Pi infinite sequence, is the sequence that contains the string that is Cliff is one of an infinite number of possibilities, which again is precisely zero. So, the chance that Cliff is in the Pi sequence, as opposed to some other of an infinite number of perfectly random sequences, is zilch. I disagree. There are an infinite number of possible infinite-length random sequences, but this does not imply that one and only one of them can contain a particular finite-length sequence, such as Cliff, or 1234. It means that *every* one of them will contain Cliff and 1234. Consider these multiple infinite length sequences: the set of all real numbers, the set of all integers, and the set of all whole numbers. Now, these are 3 different infinite-length sequences, yet all 3 contain the number 1 as a member. (I believe 1.0 is a real number, is it not?) Just because there are multiple (indeed infinite) infinite sequences, does not prove (or even suggest) that a finite sequence can exist within one and only one of them. This is like saying that the string 1234 exists within pi, but not within e or Sqrt[2]. Your statement is not to the issue in question, unless the string of digits that encodes Cliff just happens to be a sequentially-ordered finite-length string of digits (1, 2, 3, 4, 5....). As I have shown above, there are an infinite number of finite sequences that do not exist wirthin Pi. If you can prove that one of your stated sequences (or any stated sequence) is the code for Cliff, i will concede the entire issue.
> *Even if there are an infinite number of infinite-length random sequences containing the string that encodes Cliff, there will always be an infinite number of other such sequences that do not contain that string.
> So we will never find the Cliff string. Unless we live forever. We *might* not ever find the Cliff string. We *definitely* will not recognize it if we do. It could well be that the very first 10^12 digits of pi contain the Cliff string. Probably not, but possible. It could be that we are staring Cliff right in the figurative digital face and without even knowing it. And, more likely, we will never be staring at any such sequence. So, back to those angels...
> Ain't infinity fun? You can demonstrate anything!!
In summary, I would suggest that Cliff and every other finite length sequence exists within pi. Whether we feeble humans are able to recognize the pattern or not, bears no impact whatsoever upon the patterns existence within pi. The practical implications, whether any exist or not, are irrelevant to the very existence of all finite length strings somewhere within the digits of pi. The prospect of *not* finding a finite length string is readily calculable as 1/x where we search for an n length sequence within a span of x * 10^n digits. (Well, actually this formula needs a little bit of work, since 1/x, where x == 1, implies a probability of 1, which is not accurate. I should probably qualify it by requiring that x > 1.) Pi is of infinite length, but we need *never* search its entire infinite length (which would be impossible anyway) to find the first occurrence of any particular finite length string. The entire sum of all human knowledge could be encoded within a string of digits to be interpreted, for example, as an .mpg movie stream. This movie, even if entirely comprehensive, is still finite, since the depth of human knowledge is finite, and hence, encoded somewhere within pi. The final point, though probably not the most significant, is that even though there exist infinite sequences other than pi, these have no bearing upon which finite sequences can or cannot exist within pi. There is no mutual exclusivity among infinite sequences. --Mark All of your "summary" is mostly a rehash of your unsupported speculations, I will not beat a dead horse. But it was great of you to reply. Like I said, ain't infinity fun? Me, since last night, I can't even find the damned remote for my TV....
From: "Danielle" Wouldn't pi be both finite *and* infinite? That is, realistically finite and theoretically infinite? Using the example of matter in the known universe, pi would be realistically finite. Even if we could etch a trillion digits of pi on a piece of matter the size of a pinhead, we'd eventually run out of matter in the known universe (realistically finite). However, if you take into consideration that space is an infinite vastness with no beginning and no end (which I believe to be the case) then one must assume within that infinite vastness exists infinite matter. With infinite matter existing within an infinite vastness, and infinite number of digits could be written (theoretically finite). Okay, so I'm not a math whiz and this whole pi thing is way above my reach, but just a thought or two... Danielle
From: "Pete B": Permit me to express what I think you actually mean: Pi is indeterminate, in that we can never know the value with perfect precision and accuracy. It is finite but unbounded. Its' value is expressed as a very real sequence of infinite length, but its' value is actually very exact, we just can never know it. Pete B
From: "rockinginthefreeworld1" : Don't mean to butt in on your little debate here, but did anyone ever see that "Outer Limits" episode where little men in blue have to redo every second of time when it arrives? At the end, the person who got stuck between seconds (or was it minutes, perhaps?) sees a blue tool that got left behind. It was pretty eerie. I thought of that episode while reading that last part.
>On the other hand, we might have > a dead body. Don't try this with Cliff!
> This all goes to the question of whether one's soul is > separate and apart or whether it is contained within one's physical > embodiment. This debate, however, is a whole new can of worms, > worthy of a separate thread.
What if there is no line of demarcation between the two? Like, what part is "soul" versus "body" anyways? Doesn't it depend on the observer? I mean, can you not keep dividing a number into smaller and smaller fractions, ad infinitum (or ad nauseum)? At some point, things get so small that you cannot make sense of them anymore, and so you do not even realize they exist! (like teeny tiny dimensions). If there is a "soul," then when is it a "soul" anyways? When an observer sees it? Feels it? Hypothesizes that it is probabilistic that it is there, somewhere? Can of worms is right, come to think of it. I think everything intersects at precisely the right place/time/whatever, and then you have your person, or you have that person's awareness that he/she is a conscious being in the world. It is as if our whole universe became self-aware, through conscious beings having evolved to this point where they can contemplate their own existence. Pretty cool of our universe, actually.: )
From: "Graham Cleverley": I said I wouldn't post any more on the topic of what the digits of pi may or may not contain, but this isn't breaking the promise. It's going off on what seems to me an interesting tangent. Say you're about to throw 2 dice. What's the probability of throwing a 3? 1/18, I assume everyone would agree. After you've thrown them and looked at them, they show a 7. What's the probability you threw a 3? Zero? Something else because you might be making a mistake? And when you've thrown them, but you haven't looked at them. What's the probability then that they show a three? Still 1/18? Or either 0 or 1 depending on what they do show? The digits of pi are much the same as the numbers of the face of the die in the situation where they have all been thrown, but you haven't looked at all of them. Those that haven't been looked at (calculated) are also in a similar position to Schr�dinger's cat while it is still in the box. To make it clearer, modify the classic experiment a little. The cat will now die with probability P within one hour. After that it will either stay dead or stay alive. You'll open the box sometime after one hour. Before you start the experiment, the probability the cat will be dead when you open the box is P. After the box is opened, the probability the cat is dead is either 1 or 0. After one hour, but before the box is opened, what's the probability the cat is dead? Still P? Or is it now either 1 or 0? Pi's digits are like the dice already thrown, or the cat after one hour. The situation is fixed. All that is uncertain is our knowledge of the state of affairs.
From: "Graham Cleverley" : I think Pete's arguments were correct. These are alternative arguments to the same conclusions that - I hope - will be the last thing I post to this thread. Again, as Mark asked I'm concentrating on whether the digits of pi contain all possible finite sequences. I depend on the following results from Cantor's theory of sets. (If you don't trust me, Google on 'Georg Cantor'). In particular, if one takes an infinite set X, and writes |X| for its cardinality - the number of elements in it, then the following are true: |X| + |X| = |X| and a fortiori |X| + |X| + ..... = |X| and therefore 2 * |X| and k * |X| = |X|, where k is any integer However, exponentiation is differents and 2^|X| > |X| and generally k^|X| > |X| A) First to establish the cardinality of the set of finite digit strings in pi: Write the strings in this pattern
3
1 , 31
4 , 14, 314
1 , 41, 141, 3141
5 , 15, 415, 1415, 31415
.....
The table contains duplicates, but it should be obvious that ultimately it would contain all the finite strings in pi. Now write
1
2, 3
4, 5, 6
7, 8, 9, 10
8, 9, 10, 11, 12
......
You could superimpose one table on the other. It therefore follows that the number of finite strings contained in pi is the same as the number of natural integers: they have the same cardinality, which is conventionally denoted as |N|.
B) For the cardinality of the set of all possible digit sequences. There are 10 such sequences of length 1, 100 of length 2, 1000 of length 3 and in general 10^x of length x. (If you use a different number base the constant term changes of course.) So the total number is 10+10^2+10^3+10^4......10^N Look at the last term. We know from Cantor that k^N > N - i.e. k^N.contains more elements than N. Some finite strings are therefore not present in the digits of pi.
C) With regard to the distribution of pi, and the probabiliy of finding a particular string in the first m digits. When m = 1, the probability of finding `3' is 1. the probability of finding any of the other 9 digits is 0. When m = 2 the probability of finding 3,1 or 31 is 1; the probability of any of the other 107 is 0. As you increase m, the distributions remain similar. There are m* (m+1)/2 strings where the probability is 1, out of a total of 10 + 10^2 + 10^3�+10^m possibilities. Some strings have probability 1, some probability 0. The vast majority have probability zero. For no string is the probability between 0 and 1 - they are always either certain or impossible. The probability of finding a particular string is not a continuous variable, not even a stochastic variable at all. You cannot use stochastic methods to predict it.
From: "Craig Becker":
> Don't mean to butt in on your little debate here, but did anyone ever > see that "Outer Limits" episode where little men in blue have to redo > every second of time when it arrives? At the end, the person who got > stuck between seconds (or was it minutes, perhaps?) sees a blue tool > that got left behind. It was pretty eerie. Based on an old Theodore Sturgeon short story called "Yesterday Was Monday". Stephen King also kinda-sorta ripped off the notion for _The Langoliers_. Which sucked, BTW. Craig [ > > Then there is the problem of the things we cannot precisely and > accurately enumerate. Little insignificant things like his conscious > mind. How do you encode a persona, a set of emotions, a feeling? Qualia? You'd have to be pretty darn good to encode that! > You suggest that even if we > could create an exact duplicate of Cliff, down to the tiniest > subatomic particles, that it would go poof within a nanosecond or > two. That could be, but I'm not willing to concede that point. We > just don't know what would happen. Don't mean to butt in on your little debate here, but did anyone ever see that "Outer Limits" episode where little men in blue have to redo every second of time when it arrives? At the end, the person who got stuck between seconds (or was it minutes, perhaps?) sees a blue tool that got left behind. It was pretty eerie. I thought of that episode while reading that last part. >On the other hand, we might have > a dead body. Don't try this with Cliff! > This all goes to the question of whether one's soul is > separate and apart or whether it is contained within one's physical > embodiment. This debate, however, is a whole new can of worms, > worthy of a separate thread..What if there is no line of demarcation between the two? Like, what part is "soul" versus "body" anyways? Doesn't it depend on the observer? I mean, can you not keep dividing a number into smaller and smaller fractions, ad infinitum (or ad nauseum)? At some point, things get so small that you cannot make sense of them anymore, and so you do not even realize they exist! (like teeny tiny dimensions). If there is a "soul," then when is it a "soul" anyways? When an observer sees it? Feels it? Hypothesizes that it is probabilistic that it is there, somewhere? Can of worms is right, come to think of it. I think everything intersects at precisely the right place/time/whatever, and then you have your person, or you have that person's awareness that he/she is a conscious being in the world. It is as if our whole universe became self-aware, through conscious beings having evolved to this point where they can contemplate their own existence.
From: "mwganson"
> I am not a mathematician, but it seems to me it is easy enough to prove that there are an infinite number of finite-length strings that do not exist in Pi. Here is one way, expressed as a thought experiment, and which assumes that one can actually work with infinite-length strings of digits like Pi, and that we can determine all of the substrings of Pi of any finite length:
> We can designate the length of the longest possible finite-length string contained within Pi as Lstr(Pi). Now list all of the unique such strings contained within Pi of that longest possible length. Obviously there is only the one string, the one that begins with the first digit of Pi and ends with the last digit of Pi. Now change the last digit of that string to any other value. That new string does not exist within Pi.
> Now enumerate and list all of the unique strings within Pi that are of length Lstr(Pi)-1. Now for each such string, change any one digit. All of the changed strings are not contained within Pi.
> Continue this process for all lengths of strings, from longest to shortest, listing all of the unique strings of a fixed particular length, change any one digit of each such string, the result strings are not contained within PI.
> Do the same process for all strings that begin at the first digit of Pi, and work up to the last digit of Pi. Yhen do the same for all combinations of starting digit and ending digit. By doing so, I believe you will have produced an infinite number of unique strings, none of which are contained within Pi.
> Here is a simple example: consider the string of digits in the infinite sequence of Pi that begins with the first digit (after the decimal point) and is of maximum length, in other words, includes all the digits in the infinite string that is Pi. Now change the first digit of that string to 2 instead of 1. That new string is not contained anywhere within Pi. Now go to the string that starts at the second digit, and change that starting digit from 4 to, say, 8. Lo and behold, another string that is not contained within the digits of Pi. There are an infinite number of such non-Pi strings.
> BUT... the set of all possible perfectly random infinite-length strings does include all of those strings, in fact, that set includes all possible strings that can exist. It is only because Pi is not random that it does not contain all of the possible finite-length strings
Pete, Like you, I guess we've beaten this dead horse enough. But, I'll fire one more salvo, just for kicks. I think you and I have different concepts of what infinity is. My concept of something, like pi, that has an infinite amount of somethings, is that there is no last something. We don't say there are an infinite number of digits in pi because there are a helluva lot of digits in pi. If that were the case, we'd just say there are a helluva lot of digits in pi. Whatever length you give to your Lstr (Pi) string, there will always be other strings that are longer. Your argument is akin to saying, take the largest possible natural number... There simply is no such largest possible natural number. No matter what number you give me, I can just say, okay, add one to that and now you have a bigger number. I see no point in repeating my other arguments any further. I'll conclude just by saying that for any finite length string of n.digits, the probability of *not* finding it within the infinite set of digits in pi is 1/Infinity, which is, in your words, zilch. Let's take one final look at my formula from a previous post:
p == 1/10^z * d
This formula gives us the probable number of times we can reasonably expect to find any given string of length z in pi, given a search span of d digits. A an example, let's say we are going to search for a string of 10 consecutive digits and we are going to confine our search to a span of 10^50 digits of pi.
p == 1/10^10 * 10^50
p == 10^40
In other words, we'd expect to see all possible strings of 10 consecutive digits appear in 10^40 different places within this span of digits. The probability of *not* finding a single one of these 10- digit strings is 1/10^40. Now, if we choose to "confine" our search to an infinite number of pi digits in order to find this elusive 10- digit string, the formula looks like this:
p == 1/10^10 * Infinity
p == Infinity
This tells us there are an infinite number of these 10-digit strings. The probability of *not* finding a single one of them? 1/Infinity, which is equal to 0, zilch, nada. Does this mean Cliff is in there? It does if Cliff can be encoded into a finite length string of digits. --Mark
From: ollyhardy Craig- I think the show you're thinking of was actually an episode of the 1980's version of "Twilight Zone". I actually think I even remember that the actor who played the main "time worker" was a middle aged black actor, best known for his role as the belligerent seargant in the film "A Soldier's Story" (don't remember his name though). Cliff, I think the Pi page is a great idea. May be if the people who posted on pi sent you an email with their posts it might facilitate your efforts. Just a thought....
From: "mwganson" I will agree that there are the same number of digits in pi as there are natural numbers. This is fairly obvious since we can directly map the set of natural numbers to the digits in pi, such as by.creating a funciton f(n), which produces the nth digit of pi.
> > B) For the cardinality of the set of all possible digit sequences. > > There are 10 such sequences of length 1, 100 of length 2, 1000 of > length 3 and in general 10^x of length x. (If you use a different > number base the constant term changes of course.) > > So the total number is 10+10^2+10^3+10^4......10^N I accept that there are 10+10^2+10^3...+10^n different possible digit sequences of length N. This is another way of stating that there are a finite number of possible digit sequences of length N. This is 1.111111 (with N of these 1's) * 10^N. Or, N 1's followed by a 0. > > Look at the last term. We know from Cantor that k^N > N - i.e. k^N > contains more elements than N. > > Some finite strings are therefore not present in the digits of pi. >This is where you lost me. I was with you all the way up to calculating the number of possible unique strings of within N digits, but I don't see how the fact that this value is expressible as a sum of powers of 10 can support the leap of logic that says this number must therefore be greater than an infinite number. Nobody is contending that pi contains infinite length strings (except for the single infinite length string that is pi itself). There will always be a finite number of possible unique strings of any finite length. This finite number will always be smaller than the infinite number of possible finite length strings contained within pi.
> C) With regard to the distribution of pi, and the probabiliy of > finding a particular string in the first m digits. > > When m = 1, the probability of finding `3' is 1. the probability of > finding any of the other 9 digits is 0. > > When m = 2 the probability of finding 3,1 or 31 is 1; the probability > of any of the other 107 is 0. > > As you increase m, the distributions remain similar. There are m* > (m+1)/2 strings where the probability is 1, out of a total of 10 + > 10^2 + 10^3�+10^m possibilities. > > Some strings have probability 1, some probability 0. The vast > majority have probability zero. > > For no string is the probability between 0 and 1 - they are always > either certain or impossible. The probability of finding a particular > string is not a continuous variable, not even a stochastic variable > at all. You cannot use stochastic methods to predict it.This is all fine and dandy, but it matters not that the total number of potential strings of m digits will always be more than the actual number of strings that can be found within m digits of pi. You see, we have the luxury of not stopping at m (or any other finite number of) digits in our search, because we have an infinite amount of territory in which this finite string can be. We can draw an analogy by looking at the set of natural numbers versus the set of prime numbers. Both are infinite, but clearly there are more natural numbers than there are prime numbers. This is the cardinality to which you refer. But consider the fact that no matter how large of a natural number you choose from this cardinally larger set, there is an infinite number of primes that are larger still. Likewise, no matter how long a stream of digits you select, so long as it is finite, it will always be duplicated an infinite number of times within pi. --Mark
From: "Nadilo" : I think that we forgot one more factor which is impossible to calculate and mathematicians will probably say it does not exist but it is real phenomenon nevertheless,a pure luck. Now,if it is enorm number of possibilities in which we have to find one from first try a probability that we will find it is almost zero, but............ With right "potato" everything is possible. On the other hand ,pi is irrational and infinite but is there possibility that some finite strings of numbers representing Cliff does not exist inside pi?Is it possible that pi contain infinite number of finite strings but not one finite string of numbers ,a "Cliff" one? I think that is possibility above zero and therefore real one. Let we call it "reversed potato". :) Anton
. From: "tmredden" : Pi cannot even be looked up on as infinitely long (perhaps in a semantical way,) but rather as a number which cannot be expressed precisely in a finite way. It's all a matter of precision, which even for the sake of most science requires only a few decimal places of accuracy. Don't forget, the way base ten numbers work is that each place holder further along to the right holds values only one tenth of the values to its left. As we go further out, the digits' importance shrinks away to near nothingness very quickly. I still don't see the value of using pi, or any random transcendental number, as a code for storing information. The effort of finding useful information within it requires far more work and time than its benefit warrants. If we could calculate the number of bits of information to accurately describe any object, then we could pretty easily calculate the chances of finding that set of bits within a perfectly randomly generated infinite string. But, finding a one-to-one correspondence between the data and a subset of the string could take eternity. What would be the point? Todd
. From: "mwganson" : That is a good question, Anton. Let's suppose the Cliff string is n digits in length. (This n could be 10^10^10^10 or whatever outrageously high number for n we could come up with, so let's just stick to calling it n.) The probability of finding Cliff within the first 10^9 * 10^n digits of pi is pretty high considering a reasonable expection of finding 10^9 different copies of them within this range. Still, there is a super small chance that there will be no Cliff's within that range of digits. The best way to look at it is to consider the chance that we will *not* find Cliff, which is 1/10^9. So, the chance of *not* finding Cliff is very close to zero even with this finite search range. But, there is no reason to limit ourselves to *any* finite range. (There are, of course, practical limits to the range that we will ever be able to actually search, but whether we can actually find a string or not does not preclude the string's very existence.) So, the chance of *not* finding Cliff within an infinite range of digits is 1/Infinity, which, by definition is not *close* to zero, it *is* zero. Or, as Pete would say, zilch. It might help to consider the analogy of a simple flip of a fair coin, which can result in either heads or tails. If you flip a coin just one time, there is a good chance that the result will not be heads (1/2 to be exact). If you flip the coin twice, there is a lower probability that neither flip will result in heads (1/2*1/2 == 1/4). But, a 1/4 probability is still greater than 0, so we must consider it possible, even if not probable, that you will not flip heads in either of the 2 flips. The question is, how many flips until we can no longer see it as reasonably feasible that we will never have heads? For any finite number of flips, the probability will be above 0, and can be readily calculated as 1/2^n where n is the number of flips. What you are, in effect, asking above is whether or not the probability is greater than 0 that we will never flip heads after flipping an infinity number of times. The answer to that is no. The probability is exactly 0. In other words, you *will* eventually flip heads. In other words, 1/2^Infinity == 0. --Mark
. From: "Graham Cleverley" I really tried to resist the temptation to reply here. But I failed. "mwganson" wrote:
> > I depend on the following results from Cantor's theory of sets. > > (If you don't trust me, Google on 'Georg Cantor'). In particular, > > if one takes an infinite set X, and writes |X| for its > > cardinality - the number of elements in it, then the following > > are true: > > > > |X| + |X| = |X| and a fortiori |X| + |X| + ..... = |X| > > and therefore 2 * |X| and k * |X| = |X|, where k is any integer > > > > However, exponentiation is differents and > > > > 2^|X| > |X| and generally k^|X| > |X| > > > > A) First to establish the cardinality of the set of finite digit > > strings in pi: > > > > Write the strings in this pattern > > > > 3 > > 1 , 31 > > 4 , 14, 314 > > 1 , 41, 141, 3141 > > 5 , 15, 415, 1415, 31415 > > ..... > > > > The table contains duplicates, but it should be obvious that > > ultimately it would contain all the finite strings in pi. > > > > Now write > > 1 > > 2, 3 > > 4, 5, 6 > > 7, 8, 9, 10 > > 8, 9, 10, 11, 12 > > ...... > > > > You could superimpose one table on the other. It therefore > > follows that the number of finite strings contained in pi is the > > same as the number of natural integers: they have the same > > cardinality, which is conventionally denoted as |N|. > > I will agree that there are the same number of digits in pi as > there are natural numbers.I agree that would be a trivial result. However look a little more closely at the table: it contains ALL the FINITE-LENGTH STRINGS contained in pi, and demonstrates that the set of all such strings is countable: i.e. there are exactly as many such STRINGS as there are natural numbers. (The proof above is rather similar to the proof that there are as many rational numbers as there are integers.)
> > > > B) For the cardinality of the set of all possible digit sequences. > > > > There are 10 such sequences of length 1, 100 of length 2, 1000 of > > length 3 and in general 10^x of length x. (If you use a different > > number base the constant term changes of course.) > > > > So the total number is 10+10^2+10^3+10^4......10^N.> > I accept that there are 10+10^2+10^3...+10^n different possible > digit sequences of length N.But by Cantor's standard results 10^N has GREATER cardinality than N. There are exactly as many finite length strings in pi are there are natural numbers (A above). There are more possible finite length strings than there are natural numbers (B above) Therefore inescapably there are more possible finite length strings than there are finite length strings in pi. > This is where you lost me. I was with you all the way up to > calculating the number of possible unique strings of within N > digits, but I don't see how the fact that this value is expressible > as a sum of powers of 10 can support the leap of logic that says > this number must therefore be greater than an infinite number. I accept that you don't see that, and I accept that I haven't proved that 10^|N| is greater than |N| myself. All I can do is refer you to any standard work on infinite sets. You write throughout as though there was only one infinity - as if one infinite set was the same size as any other infinite set. It isn't.
> > > C) With regard to the distribution of pi, and the probabiliy of > > finding a particular string in the first m digits. > > > > When m = 1, the probability of finding `3' is 1. the probability > > of finding any of the other 9 digits is 0. > > > > When m = 2 the probability of finding 3,1 or 31 is 1; the > > probability of any of the other 107 is 0. > > > > As you increase m, the distributions remain similar. There are m* > > (m+1)/2 strings where the probability is 1, out of a total of > > 10 + 10^2 + 10^3�+10^m possibilities. > > > > Some strings have probability 1, some probability 0. The vast > > majority have probability zero. > > > > For no string is the probability between 0 and 1 - they are > > always either certain or impossible. The probability of finding a > > particular string is not a continuous variable, not even a > > stochastic variable at all. You cannot use stochastic methods to > > predict it. > > This is all fine and dandy, but it matters not that the total > number of potential strings of m digits will always be more than > the actual number of strings that can be found within m digits of > pi. Yes it does matter because the comparison is still true if you let m go to infinity. If m --> |N| then the number of strings with probability 1 is m*(m+1)/2 --> |N|*|N|/2 = |N|. However, once again, the expression for the number of strings with probability 0 includes 10^m --> 10^|N|. And (same comments as before) 10^|N| > |N|. Same standard result. (Nothing magic about 10 here : k^|N| > |N| for any k. And k^|N| = j^|N| for any j,k.)
Brett H says: Immortality in Pi sounds like garbage to me, having looked at the "argument" at that page just now. I'm not a sequence of numbers (or bytes which represent me). The fungible "me" is not me! I am that organism which is perceiving what I perceive right now - and there's only one of me.
I'm no more to be identified with 'my simulation' (in a computer or in an abstract infinite series) than a perfectly simulated apple in my computer is the same apple that I am eating. They're distinct, and I can see that they are. I'm simply not 'my representation' - however 'perfect' that representation is. I'm 'here' and my representation is 'there', and as our perceptions are (therefore) necessarily different so will our thoughts be and thus are we not the same person, however much we look and act alike.
Brhall says: I'm no more to be identified with 'my simulation' (in a computer or in an abstract infinite series) than a perfectly simulated apple in my computer is the same apple that I am eating.
But Ollyhardy responds: You don't understand the argument being made.
First of all, the idea of a perfectly simulated apple in your computer is incoherent. But that's not what Pickover suggested.
The idea is that there is nothing-including you- that cannot be represented as a string of digits. If every finite string of digits exists somewhere in Pi, then a string of digits representing "you" exists somewhere in Pi. Now, that string of digits, lying inanimate in Pi as it were, is certainly not alive. But in principle, the information is there and thus, in theory, one can imagine a technology that could find "where you are" in Pi and "implement you"- recreate a living, breathing, in the flesh "you" from those inanimate numbers in Pi.
You should keep in mind that I don't think Pickover intended this as anything more than an entertaining speculation,which it certainly is. In any event, the flaw with Pickover's idea, in my opinion, becomes apparent when we ask how it is that we would find the string of digits associated with us in Pi.
The notion of searching for the string in Pi that represents "us" requires that we already know the digits we were looking for- otherwise how would we recognize them? If I already have the information, why should I bother searching for a second copy in the infinite string of digits of Pi? Seems rather like looking for a needle in a hay stack, when you already have the needle.
OllyHardy says: I might be able to say I am completly described (as in - able to be perfectly reconstructed) as the starting digit 2^289819329924353434545 with length 10^19. Sure beats sending 10^19 bytes of data. That's a good point; it would provide a significant advantage in terms of data compresion. Pi would be rather like a sort of universal hard drive that we all have local access to; all we would need to know is the "starting address" of the data in Pi. I made just this point as a contributer to Pickover's discussion, and my comments to this effect are posted at the website in question. We would still almost certainly have to be able to independently decode all of the requisite data for any given thing we might seek to find in Pi however- it's doubtful that we could ever use Pi as the primary source of data for any given thing we might seek to simulate, emulate or "resurrect".
Charles responds: I rather suspect, although (obviously) I can't prove this, that it would probably require rather more than 10^19 bytes to specify where in Pi to find an exact description of you (even if it only describes the positions of all the atoms in your body to within an acceptable uncertainty).
Allen M. writes: The advantage of knowing where a certain "person" or things identification is within PI is simply data compression.
Karl S. responds: er - i hate to rain on your parade - but - if a bit string N first occurs at position M in pi, then the size of N and M are roughly equal. no compression is available using this technique.
Sampo S also responds: I don't think there would be an advantage. As indices to PI constitute just a function from strings to strings (i.e. a code), a simple pigeon hole argument shows that you can't win in average. Plus, PI probably isn't very well suited for the compression of most data. You would probably do a lot better using an algorithmic description (a high order statistical compressor whose description is annexed into the coded representation) than indices into normal numbers. BTW, *has* PI been proven normal? I'm not really keeping up with this sort of thing...
From Ollyhardy: (er - i hate to rain on your parade - but - if a bit string N first occurs at position M in pi, then the size of N and M are roughly equal. no compression is available using this technique. K.>
What is written above makes no sense.
For an outrageously oversimplified example, let's consider the first 13 decimal digits of Pi, and say that the string "926" represents some data. 3.141592653589 All I need to store is the starting "address" and ending "address" of this string, which in this case would be the 5th digit after the decimal place and the 7th digit after the decimal place, respectively.
Starting address: 5d
Ending Adress: 7d
Data referenced: "926"
Nevertheless, I do agree withyou that the Pi data compression scheme, however interesting, is utterly impractical.
We would first have to map out Pi to, say, the first several trillion digits. We would then have to digitize any objects we would like to be able to recognize in Pi. We then would have to find where they are in Pi, or rather where they are in those first few trillion digits we mapped, and there is of course no guarantee that any given string should be found within the first few trillion digits. Or in the first trillion trillion trillion digits for that matter. And if we chose instead to simply search Pi indefinitely until we found any given string, some searches may go on forever!
The scope of any one of these sub projects alone could make the human genome project look like a "Hello World" program by comparison, and there is no guarantee that we would even be able to find any given finite string.
Interesting speculation, perhaps something for sci fi authors to keep mind, but totally unrealistic as a real world idea.
OllyHardy says: The advantage of knowing where a certain "person" or things identification is within PI is simply data compression. It would be much more efficient to simply send the starting digit and the length rather than send the entire data set that represents the person or thing. For example I might be able to say I am completly described (as in - able to be perfectly reconstructed) as the starting digit 2^289819329924353434545 with length 10^19. Sure beats sending 10^19 bytes of data.
Gary O. responds: I'm actually not sure of this. Seems to me like there's a reasonable probability that the index of the starting digit might be a 10^19-bit number, in which case you haven't saved anything.
From Allen M: Are you attempting to say that if the one billion bytes of data I was interested in began in Pi at position 5. Then when I send you the string formated as starting position, length "5,10^9" (which is only six bytes) you believe that this six bytes (M in your example) is the same (or even in the same order of magnitude) as the billion bytes it represents (N in your example). I could just as easily make the string 10^10000 bytes long at the cost of only 4 extra bytes.
You must of meant to say something different because even my five year old son can see that your response is obviously false.
Even the string of bytes I am interested in is deeply embedded in Pi, there will be a compact means of expressing the starting digit (granted it may take a lot of computation to generate this starting digit in the form X^Y^Z + n or even as x!^y! + n. But in all cases there is a M many orders of magnitude less than N.
From: "Nadilo" Todd, I agree wih you that idea of finding information about something in pi is not convincing from practical point of view. As I said, if we know what we are looking for then why search in pi for that, and if we do not know what we are looking for how we will know what we will find if we will find? But talking about storing an information of people so they might be "restored" in future when technology allow that process I think it is easier if we store genetic code and memory of individual. Genetic code we are able to store right now even technology is in its beginning and for memory is reasonable to expect that it could be done if artificial hippocampus will work as predicted. Anton
From: "Nadilo" : Thanks Mark, for the very clear comment. However, if pi is infinitive but regular number ,something as 123,123,123,123..... and so on , probability that we will not find Cliff inside it is 1 if Cliff string is 1234. As Pete said, pi is not perfectly random number and therefore it is possible that it is developing in its infiniteness but "avoiding" some possibility of development. Let assume that Cliff is 123456789.Let assume that string of pi "explore "all combinations of those numbers but not with digits 1.Let assume that when all combinations were presented that pi string goes further combinating 9 digits of numbers but not with digits 2.Let assume that when all combinations were presented that pi string goes further combinating 10 digits but not with digits 3,and like that toward infinity.Let also assume that "missing" digits is irregulary "chosen". In that case a pi will be irregular and infinitive but not perfectly random and therefore it is possibility that does not possess Cliff string. In shourt,infinite does not necessary mean absolute. Anton
From trklss: I know its an old thread, but I was reading it on reality carnival, and decided to say something about it. A lot of people said that it can be disproved, because it doesn't include other irrational numbers such as e or sqrt(2). But these numbers are not expressed in totality in the universe anywhere, because there aren't enough particles to store them, and therefore would not be included in pi. Also, it was pointed out that negative and complex numbers are nowhere in pi. This is true, but it is possible for the decoding scheme to calculate them. (There are no negative or complex numbers on your hard drive, just 1's and 0's, but software can decode those) I do believe that we are all encoded in pi. My location starts at digit 10^343+5^643+2^985, and is encoded as an autocad file containing the location of every particle in my body at 3:43 on April 2, 2003.
From Peter da Silva:
We don't know if any given sequence is in there. It's possible for a number to be infinitely long and complex while having infinitely many sequences NOT in it. For example, consider this number:
1.00010100001010000010000000001001...
That is, for each digit of pi, this number has a sequence of zeroes, followed by a 1. This number is precisely as complex as pi, but we know it doesn't contain a sequence of 10 zeroes anywhere in its expansion.
It is possible that the string that encodes you is the first string that never occurs anywhere in the decimal expansion of pi.
On the other hand, pi *can* contain any number of other transcendental numbers. Consider the number that you could generate by alternating the digits of pi and the square root of two. This number contains pi, and it contains the square root of two. It is possible that every Nth digit of pi, for a large enough N, encodes another transcendental number.
It can't encode all such numbers, only infinitely many ofthem.
The problem with all this reasoning, though, is that the same thing can be said of any nonrepeating pseudorandom sequence. It would be much more efficient for the Omega Point machine to spend its eternity infinitely many random universes in parallel, from start to finish, because it would eventually simulate the universe that contains us... and the one that contains it. Infinitely many times. Whoops, that would need infinite storage as well as infinite time, but the same is true of any machine that could pull us out of the dust of pi.
Speaking of dust...
I think you need to go read Greg Egan's story "Dust" or the novel "Permutation City" that was derived from it. Then figure out what's wrong with the experiment it's based on.
-- Cliff responds, but if we assume the digits of pi are normal, than I'd expect a "close enough" representation of us in pi.
Response to Cliff: I would hesitate to guess the probability, though... but if it's not
unity; that means there are infinitely many possible people who are not
encoded anywhere in pi.
From: "tmredden" :
Storing any kind of information directly is easier than finding it in pi.
Mike W says:
Regarding the previous post, probably true, but suppose you need to transmit a very long sequence to the other end of the universe? You can just send the address of the message within Pi, and let the receiver look it up, instead of sending the entire message. -- Mike W
From S. Harris:
I read the discussion that Cliff institued about Pi encoding reality:
http://sprott.physics.wisc.edu/pickover/pimatrix.html
I think that Omega, which is not very computable, is sometimes proposed as the "number" of wisdom.
Pi as many know passes all tests of randomness. But it is not considered random by the definition of Algorithm Information Theory (AIT) because the unending digits of Pi can be computed from a compact source or algorithm, so Pi is not truly random.
I don't think it is known if there is another Pi-like number, or perhaps an infinity of them, which can pass all the tests for randomness and are infinitely long, but are generated by a shorter effective procedure.
What would differentiate these different Pi-like "numbers" would be their initial segments or sequences such as 1.37... or 6.45... etc. If they are all Pi-like, that would mean that they would eventually encode all finite sequences just like Pi, but that the combinations of these finite sequences is juxtaposed internally, shuffled, in a sense. But they all have unique initial codings.
One could think of all Pi-like numbers as representing a unique universe (causal laws) within the multiverse. Or experientally, each individual's trajectory within or through a universe which could have infinite variation though still constrained by the foundational seed conditions of that universe.
Cliff comments: I think that there was some debate as to whether pi must code all finite sequences. My argument is that if we assume the sequence of digits in pi is "normal", then it is likely pi codes you sufficiently close that the coding is you. (I don't care if a few atoms are out of place of if the memory trace of your daughter has a pixel out of place.)"No intentional causation without explicit representation" J. Fodor, StephenAlthough the pi encodes you and even a time-evolution of you, can we say that you are actually "living" and conscious in pi? Perhaps the pi encoding is more like a movie film of you. Perhaps to say that you are "living" in pi, there has to be a dynamic (time) component or use of the coding. For example, I can see that a cellular automata (like the Game of Life) of you might actually be conscious, just like a computer simulation of the neurons in your brain might be conscious, because there is a dynamic component: one state giving rise to another. On the other hand, is there any way can imagine that the pi coding could generate consciousness? For example, if we found that exact location in pi in which your life were coded, and then processed this digit string so that it had a time component in which one state gave rise to another, perhaps this things would be conscious. But what would kind of processing would this be?
From Cliff:
I like people's skepticism. Here is my question. If we have a string of digits with a normal distribution, what can we say (if anything) about the probablity of finding a sequence in the digit string that is similar to a given target sequence? In this case, the target sequence is a coding of you (e.g. the position of your atoms to a certain degree of accuracy or the essential connections and weights for your neurons.) We can define "similar" to be some kind of percentage similarity. For example, one digit string may be "close" to another if 99% of their digits are the same and in the same position.
Maybe, I can be more concrete to start the discussion. Given: a trillion digits with normal distribution. Given: a target sequence of 100 digits. Question: what is the probablity of finding that target sequence within the trillion digits?
From Chris B:
I have a vague feeling that this whole dicussion is missing the point, which to me is: encoding a sequence that *represents* something does *not* cause that thing to exist in "real life" anymore than me saying "Jesus Christ was born in 1914, grew up in the midst of WWI, and was killed in a freak airbombing raid while on holiday in Europe." makes that happen.
Clearly any infinite non-repeating sequence codes for every possible sequence, including every permutation that could possibly happen and didn't. In fact, it's a good model for the "many worlds" interpretation of quantum reality where everything that can possibly happen does in fact happen.
But flipping a coin forever doesn't cause me to exist and ponder the flipping of coins.
You see, you have to have a method for "reading the code" and "applying meaning," whether that's a Universal Computing Machine or the Mind of God, and whether that code is ASCII, English, punch tape, or Morse code.
From Mark Nandor:
Given: a trillion digits with normal distribution. Given: a target sequence of 100 digits. Question: what is the probablity of finding that target sequence within the trillion digits? Assuming you mean an American Trillion and not a British Trillion, we're looking at, as Graham mentioned,
(10^12 - 100)!/(10^12)! = 1.000000004950... x 10^-1200
Pretty unlikely.
And is it not still unproven that an infinite string of normally- distributed digits contain every finite subsequence? There's no guarantee that a string of 1000 3s exist somewhere in pi, since there are an infinite number of other finite subsequences that could also be used. So doesn't that make this discussion at least somewhat moot?
Cliff responds, thanks!1. No, not moot. Recall that I don't require an exact match because I consider you coded in pi, for example, even if some atoms are out of place or some neuronal connections are out of place.]
2. Ok, now let us extrapolate your thinking above related to the 100-digit target that now must be found in an infinite sequence of digits rather than a trillion-digit sequence. Assume a normal distribution of digits. For now, assume something patternless as would be produced by a geiger counter. What's the probability of finding the target now?
From Mark Nandor:
I made a mistake in my math. That's what I get for trying to do it while watching football and working on another math problem at the same time.
I need to reformulate. I THINK that the following is true:
1) the probability of the first hundred digits do not match the target sequence is (1-(0.1)^100). The probability that digits 2-101 do not match the target sequence is also (1-(0.1)^100). And so on.
2) there are 10^12-99 such 100-digit sequences, as they overlap.
3) the probability that all of those sequences do not match, then, is (1-(0.1)^100)^(10^12-99).
4) that turns out to be about 10^-89.
5) for a number with 10^50 digits, the probability is about 10^-51.
6) for a number with a google digits, the probability is about 63.2%.
7) the probability becomes indistinguishable from 1 at a number of about 10^105 digits in Mathematica in a time less than one minute.
So to answer your question, Cliff, with an infinite sequence, the probability approaches 1. But I think we already knew that. The point is that we can't guarantee it that this sequence of 100 digits, or even one similar to it, will be included. There are still, I believe, an infinite number of finite sequences NOT included in the infinite sequence. And just because the probability approaches one does not mean that it is equal to one!
So is it likely that I am coded in Pi? Sure. But I still don't see that it is a done deal.
[Cliff comments: EXCELLENT. I am quite happy if you will allow me to say it is LIKELY I am in pi! The probablity approaches 1.]
Chuck G. says:
If people can be encoded by finite strings of digits then it doesn't matter whether or not we're encoded in pi. Each of us would certainly be encoded by some integer. Some much larger integer would encode our entire spacetime with all of our lives from beginnings to ends embedded in it. If all of pi somehow has existence then I'm pretty sure all of the integers exist too. So the real problem here is whether or not such an existence produces real consciousness. We don't really need to know how the digits of pi are distributed.
Graham C. says:
This is very like the one about numbers with a 9 in them. Somewhere in there infinity gets divided by infinity which is meaningless.
Moreover there's a cardinality problem in that there are aleph_1 infinite decimal expansions but only aleph_0 finite ones.
And the apparent anomaly that if you follow Cliff's logic through then the target string would not only be encoded once but a infinite number of times. Say the target is encoded finishing at position x. There are still just as many sequences to the right of x as there were sequences starting at position 1.
Paul C. says:
To make one conscious in Pi? Knotty problem really - isn't that where God comes in :-)?
I think the problem is one of precision - that we can't define a person to infinite precision hence we can never define them in Pi with infinite precision. It's the small differences that count.
[Cliff responds, interesting. My feeling is that we don't have to define you to infinite precision for it to "be" you for all intents and purposes. For example, I chop off a hair or change a memory slightly (e.g. you had a ripe apple to eat last night and not an unripe one) -- and it is still "you." But I understand where you are coming from.]
From Govert:
Cliff -- I am a freelance astronomy writer in the Netherlands, with only a little background in mathematics. I came across your web page http://sprott.physics.wisc.edu/pickover/pimatrix.html about the question whether every finite string of digits can be found in the decimals of pi. Apparently, the most 'convincing' counter argument was the one by Graham Cleverley about the fact that the number of finite strings in pi has a smaller cardinality than the total number of conceivable digit strings. However, I cannot judge his argument.
I wonder if there has been new developments on this topic. I realize that no one is as yet 100% sure that pi is a 'normal' irrational/transcendental number, but for the sake of argument, let's suppose it is. In that case, do mathematicians agree that every finite string, no matter how long, occurs in pi, and maybe occurs infinitely often? Or is there indeed a strong reason why this cannot be the case? In other words: is the jury on this question in?
One mathematics acquintance of me says that it is perfectly possible to have every possible string of digits occuring in a number with an infinite number of decimals. The argument goes as follows: every digit string is also an integer. So imagine the following number: 0.1234567891011121314151617181920212223... etc. This number contains every integer in its decimals, so it also includes every possible string of digits. (So at least you're alive forever in the digits of this particular number!). However, I'm not sure whether or not it also applies to pi.
If there are any new developments on this theme, I'd be grateful to learn
about them. Thanks in advance --Govert
From Ned: if my life is encoded in pi, is my afterlife also coded in pi?
From: Mike Frank
Mr. Pickover, I read some of the thread about your "we are in pi" observation.
http://sprott.physics.wisc.edu/pickover/pimatrix.html
I, too, was surprised about the amount of pushback you saw. I wanted to make a few points to counter some of the doubters.
1. First, we know that pi never repeats (falls into a repeating pattern), because that would make it rational, and it has been proven to be irrational.
2. Even if pi is not "normal" (does not contain all digits in any given base with equal frequency), it may still be the case that it contains all finite digit sequences (just not at the same frequency as each other, as it would if it were normal).
3. Even if pi is not normal, and worse, does not contain all finite digit sequences, it is anyway trivial to describe another mathematical object that does. A simple program for any Turing-complete model of computation can directly generate all finite sequences in a systematic order in dovetailed fashion. E.g., a simple enumeration of all natural numbers accomplishes this for all finite digit sequences not starting with 0 (which is not a significant restriction; the initial digit can just be ignored anyway). To define such a process (a counting program) from first principles doesn't take inordinately more work than to precisely define (as a process) the meaning of the phrase "the decimal expansion of pi." In fact I think a counting program will be shorter than a program to calculate pi in most "natural" programming languages. So, arguably, a simple enumeration of all finite sequences is already just about as fundamental an object as pi itself is (if not more fundamental).
4. There is every indication from physics that our universe is, indeed, a computable structure. Theories of quantum gravity indicate that in fact, it also only contains a finite amount of quantum information. Certainly, the information needed to replicate our conscious experience is only finite (our finite, noisy brains cannot possibly be aware of all the details of an infinite amount of information, anyway).
5. The simplest theory of existence (and therefore the one that, by Occam's Razor, is most likely to be correct) is the one that states that ONLY mathematically representable structures exist, and that indeed the only meaningful kind of existence IS mathematical existence. ANY mathematically describable object, structure, or situation therefore exists in the same sense as any other. This includes our universe as well as our conscious experience of it at any given moment. The nature of mathematical existence is eternal, in the sense of being unrelated to the passage of time that we perceive, which is itself just a subjective interpretation of the eternal pattern that comprises the history of our universe, when that pattern is viewed "from within," so to speak.
Some people call the above philosophical system "Metaversalism," and you can read more about it at http://www.metaversalism.com.
So, the essential philosophical content of your theory is correct, even if pi doesn't contain all finite sequences, and even if the basic philosophical ideas here don't necessarily have anything to do with the number pi in particular. ANY mathematical construction of a given pattern is fine for considering that pattern to be "eternal," and a construction like "the pattern of length so-and-so that starts at digit position such-and-such in the decimal expansion of pi" is not necessarily the simplest representation of a given pattern, and it is, as a mathematical representation, no more eternal than any other.
Finally, someone suggested using embeddings of sequences in pi as a data compression technique. Unfortunately, this does not work, because it is easy to prove (using a simple counting argument) that the number of digits required to specify where a given digit-sequence starts within the digits of pi will almost always be almost as large as the original digit sequence itself.
E.g., if one asks, "At what position in pi is the first occurrence of the 24-digit sequence 1111 2222 3333 4444 5555 6666" (or any other 24-digit sequence), the answer will almost certainly itself be a number that also has about 24 digits. On average for a random input sequence, no compression is achieved whatsoever. So, you can't expect pointing out a given sequence among the digits in pi to really be a particularly helpful way of defining that sequence.
Rather than saying "we all exist in pi," when pi is not really the key here, it is I think a simpler and more parsimonious statement to just say "we all exist in mathematics." I.e., the latter statement has greater verisimilitude. (Although stating this philosophy in terms of pi in particular is perhaps a little more enticing, due to the symbolism that we associate with the circle.)
Regards,
Mike Frank