As far as I can tell, the logic goes something like this:

- pi is an irrational number
- since it is irrational, it's decimal expansion never repeats
- since it never repeats and it goes on forever, every possible sequence must eventually come up somewhere.

That third step might sound reasonable, but it is wrong. Let me illustrate with a quick example, I will provide an irrational number in which every sequence does not come up:

.10110111011110111110...If we continue this in the obvious way, we get an irrational number in which many sequences of numbers never show up, 00 is an example. So what do we have here? It is a sequence that never repeats, that goes off to infinity, and yet many finite sequences never show up inside. This may seem counter intuitive, but the example is before you.

Let's return to the statement made in Person of Interest. The statement remains that every number shows up within the digits of pi. I have provided an example of an irrational number that does not have this property, which I would argue at least puts a bit of doubt on this statement, but it doesn't prove it wrong. It demonstrates that the above logic is faulty, it can't be true simply because pi is irrational, but perhaps there is something else about pi that makes this true. I couldn't think of a way to prove this either true or false, time to consult the internet! It turns out that this is actually an unsolved problem. Nobody knows whether or not what Finch said is true or not (which I why I put mistake in quotation marks in the first paragraph). That is how hard these problems are to solve.

One last thought here, if the decimal expansion to pi has every possible finite sequence contained within it, that means that somewhere we will find a billion zeroes in a row. Put in those terms, this seems obviously false...and yet, it could be true. No one knows. Infinities are weird!

**Edit**

I was thinking of another angle to the billion zeroes thing this morning. It will take me a couple of steps to get there.

First, think of the 10 available digits, we would probably expect that each one would show up at some point. No single digit seems to have a greater chance to show up than any other. There's no reason to think that a 5 has a better chance of turning up than a 7 right?

Now, the digits that we have are somewhat arbitrary, we have 0,1,2,3,4,5,6,7,8,9 because we count in base 10, and there's nothing really special about 10 except that is the number of fingers that we have. So what if we counted in a different base? We would need a bunch of new symbols, but fundamentally nothing would really change. What if we instead counted in base 100? Every 2 digits of our normal decimal expansion of pi would be a single digit in the other base. I don't want to come up with 90 new symbols, but I could write the decimal expansion of pi in base 100 like this:

(03).(14)(15)(92)(65)(35)...

In base 100 this is 6 digits. Given this view of the decimal expansion, I see no reason to think (00) would have any less chance to come up as any other number. By thinking in base 100=10^2, I can argue that there is no particular reason to think 2 zeroes won't show up at some point.

Now, if I want to think about getting a 1,000,000,000 zeroes instead of 2, why can't I use the same idea and write the digits of pi in base 10^(1,000,000,000)? I would argue that there is no reason to think the zero of this base has any less chance of showing up compared to any other number in that base.

I want to stress that this is absolutely not an argument that we will definitely get a billion zeroes somewhere. It's more an argument that it is much less impossible sounding than it might initially seem. A little change of perspective can really screw with your intuition.

I admit I had the same reaction. In the same show Finch told the kid he needed to use atomic variables (talking about a new compression algorithm that was ground breaking in its power and elegance). I think the words rudimentary and ridiculous fell out of my mouth before I knew what was happening :-)

ReplyDeleteBack to the subject at hand. Apologetics and infinity. AFAIK the uses of infinity in apologetics has to do with resolving the argument of the origin of the universe (does it have one). There is a pretty good discussion of the difference between real infinities and things that are not (like pi which converges to a finite value) in "Reasonable Faith" by William Lane Craig.

That said my understanding is that the philosophical discussion has been rendered moot by physics. Apparently the Theists were right, the universe does have a beginning (IMO with the metaphysical implications that entails). The argument now is about what was before that beginning (and just what "before time began" may mean).

To paraphrase Pilate, what is proof?

David

That atomic variables stuck out to me as well. I don't know what they are, but it seemed odd that such a genius programmer would not think to use the right type of variable. If it was so perfect for his purpose, I would think it would have occurred to him previously. I guess it is hard to write a proper eureka moment.

DeleteAs to WLC, I would disagree with the argument that the discussion here has been rendered moot. It is true that we have a beginning of the universe, but as you said we don't know what happened before (or if that even is a coherent question to ask). But one possibility is that our universe was born out of a previous universe, perhaps through a black hole bounce or a big crunch or something. And perhaps that universe was born from a previous one, etc. And we are back to infinity. Now I'm obviously not going to argue that this is the way it is, but I see no reason to rule it out as a possibility.

As to the reasonable faith discussion. A quick google search landed me to the following pages: Omniscience and actual infinity, counting down from infinity. I don't have time to look at them at this moment, but now that it is on my mind I want to get back to this in the next few days. Are either of those pages what you are thinking of, or is there another you have in mind?

Too funny that you ended your post with the "billion zeroes" scenario

ReplyDeleteHausdorff, because a similar thought was going through my mind as I was reading about sequences of zeroes in pi being extremely unlikely given that it is a ratio.Along the lines of

David's injection, I always found it amusing to think of God sitting around in nothingness for an infinite amount of time before He decided to create something. Given that God allegedly had no beginning, but the universe did, God was around forever before us doing absolutely nothing. ;-)Yeah, the billion zeros throws me off too, it really doesn't seem right. But then, think about doing long division. If things line up just right you can get a random string of zeroes in the middle, I don't really see why that can't happen somewhere down the line in the digits of pi. If I had to guess I would say no, I don't think it would happen...but maybe.

DeleteAnd yeah, I love the idea of God sitting around for infinity before he created us. I wonder what made him decide to take that action. It actually reminds me of the beyonder from the secret wars (Marvel universe cross over event). As I recall, he was content to stay in his own perfect little universe where he was everything, but then he somehow got a glimpse into ours and was like "what's going on over there".

I was thinking of another angle to the billion zeroes thing this morning. It will take me a couple of steps to get there.

DeleteFirst, think of the 10 available digits, we would probably expect that each one would show up at some point. No single digit seems to have a greater chance to show up than any other. There's no reason to think that a 5 has a better chance of turning up than a 7 right?

Now, the digits that we have are somewhat arbitrary, we have 0,1,2,3,4,5,6,7,8,9 because we count in base 10, and there's nothing really special about 10 except that is the number of fingers that we have. So what if we counted in a different base? We would need a bunch of new symbols, but fundamentally nothing would really change. What if we instead counted in base 100? Every 2 digits of our normal decimal expansion of pi would be a single digit in the other base. I don't want to come up with 90 new symbols, but I could write the decimal expansion of pi in base 100 like this:

(03).(14)(15)(92)(65)(35)...

In base 100 this is 6 digits. Given this view of the decimal expansion, I see no reason to think (00) would have any less chance to come up as any other number. By thinking in base 100=10^2, I can argue that there is no particular reason to think 2 zeroes won't show up at some point.

Now, if I want to think about getting a 1,000,000,000 zeroes instead of 2, why can't I use the same idea and write the digits of pi in base 10^(1,000,000,000)? I would argue that there is no reason to think the zero of this base has any less chance of showing up compared to any other number in that base.

I want to stress that this is absolutely not an argument that we will definitely get a billion zeroes somewhere. It's more an argument that it is much less impossible sounding than it might initially seem. A little change of perspective can really screw with your intuition.

If we were dealing with pure probability, I think you'd be on the right track. But what I keep coming back to is that this is a ratio; specifically that of the circumference to the diameter. That means that we are dealing with long division, as I think you mentioned above. I think that sets a practical limit on the number of consecutive zeros which can occur, because for each zero, the remainder gets multiplied by a factor of ten for the next division attempt.

DeleteGiven that there are long strings of consecutive non-zero numbers in pi, I think the likelihood of there being a remainder which can withstand a double or even triple failed division attempt is extremely low.

At least, that's what my neurons are telling me right now. :-)

I'm not sure about that. When we think of long division we are typically imagining 2 integers. In this case at least one of the terms (diameter and circumference) must be irrational. This fact will probably gum up our intuition about it a little bit. Either that, or I'm not quite understanding what you are saying :)

DeleteGood point! I think I had a brain fart thinking about the more-standard integer division. Doh! :-)

DeleteSince C/D = pi for D = 1 and C/D = pi for D = anything else, there is the same chance for a billion zeros for D = 1 as for D = anything.

ReplyDeleteA quick look at the first million digits (http://www.piday.org/million.php) of pi shows on sequence of 5 zeros (relatively early). That argues against the long division limit.

Looking a little more its interesting that for a million (1,000,000) places of pi we have a maximum of 6 consecutive digits (xxx,xxx) of all the other digits (and we have exactly one each of those). Who's to say if this behavior is linear but it does seem to imply a string of 1 billion zeros is possible.

About God sitting around waiting, remember time is not as it was imagined by the ancients (i.e. not the fourth dimension). Time is relative so before the beginning of the universe its hard to think about time (another reason why this leaves room for argument). TWF and I kicked time around a bit in other discussions because of the quandary of our choice and God's foreknowledge. Because we almost never travel at speeds approaching the speed of light time seems pretty stable to us. Its hard to get past that since we've known it from birth. I try not to think about it too much 'cause it makes my head hurt.

Great example with the first million digits of pi. I of course used a billion zeros for effect, but even 6 consecutive zeros seems strange at first thought. If those showed up so soon, why not 7 zeros a little later, and then 8, and so on. Somehow I would doubt this behavior is linear, but I might believe logarithmic. (I have absolutely no justification for that statement, it's just a feeling).

DeleteThat's also a really good point about what happened before the big bang. As you point out, the way we envision time isn't really correct, and the very notion of before the big bang might not even be a sensible thing to say. I remember hearing someone (Stephen Hawking?) compare this to asking what is north of the north pole. It's just kind of nonsense given the geometry involved. Perhaps there is something similar in space-time at the big bang.

I'm wading into pretty dangerous waters here as my knowledge of physics is fairly low, but you mentioned that our intuition about time is poor because we never travel close to the speed of light. Recall that time also changes near a gravity well, the bigger the gravity well the bigger the distortion. At the big bang, all of the mass of the universe was concentrated in one spot, it seems that this would be an enormous gravity well, and therefore time would be distorted in the extreme.

Although I guess it's not just all of the mass concentrated, but all of the...space...is too? Alright, my brain is melting a little, I'll leave this here.

The universe is a fallacy of "in" and "out". If a billion consecutive zeros occur in pi then the entire text of War and Peace also shows up, and so does the transcript of this blog, and so does everything that anyone can say on the subject. But pi is not "out there". You are reading those very digits, now.

ReplyDeleteI'm not sure what you mean by "a fallacy of in and out". But yes, that is basically what finch was saying, that since every possible finite sequence is contained within pi, then all data that we can ever imagine is encoded within the number somewhere. The claim that every sequence within seemed strange to me and in searching for whether or not it was true, I found that it is an unsolved problem.

Delete