"Somewhere inside the digits of pi is a representation for all of us -- the atomic coordinates of all our atoms, our genetic code, all our thoughts, all our memories. Given this fact, all of us are alive, and hopefully happy, in pi. Pi makes us live forever. We all lead virtual lives in pi. We are immortal." - Cliff Pickover
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.
> 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.
> > 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
> > 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.)
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:
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.
I read the discussion that Cliff institued about Pi encoding reality:
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, Stephen
Although 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?
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?
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.
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
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?
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.]
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.
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.]
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
Mr. Pickover, I read some of the thread about your "we are in pi" observation.
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.)