Proposed computer experiment for determining the age of the universe

[Gustafson, S]

This post describes a computer experiment that could be carried out and that could produce a more accurate value for the age (or radius) of the universe than the currently estimated 13.7 billion years (or 13.7 billion light years). It refers to the post of 14 Feb 07 in the Speculations Forum, but the current post considers only factual material and realizable experiments.

 

1. Computer experiments which used an approximate entropy metric for randomness have shown that the digits of √2 are more random than the digits of e. This result is “mainstream” and has been reported in a leading peer-reviewed journal: S. Pincus and R. E. Kalman, “Not all (possibly) ‘random’ sequences are created equal”, Proc. Nat. Acad. Sci., vol. 94, pp. 3513 – 3518, April, 1997.

 

2. There is reason to believe that the randomness of pi expressed in any base (i.e., 3.14159…in base 10) changes near a precision of one part in 10 to the 121st power and that the precision at which this change (if any) occurs could be determined accurately using the following computer experiment. Find the randomness of the sequence of the digits of pi from digit 1 to digit n and from digit n + 1 to digit 2n using approximate entropy, find the fractional change in randomness, repeat for n = 2, …, 242, and plot the fractional change in randomness versus n. Examine the plot for changes in level, slope, curvature, etc., near n = 121. Generate plots using other measures of randomness and using pi expressed in other bases to characterize the changes (if any).

 

3. If changes in randomness are found for many measures of randomness and for many bases and if they can be interpolated to the same fractional digit location (e.g., n = 121.327), then a more accurate value for the radius of the universe (currently estimated at 13.7 billion light years) could be proposed. The proposed value would be such that the precision at which the change in randomness occurs equals three times the square of the ratio of the Planck radius (4.05 10E-35 meters) to the radius of the universe.

 

4. Someone fluent in Matlab, Mathematica, etc., could carry out this computer experiment. Is anyone interested?

[insane_alien]

can i just ask what abstract mathematical concepts have to do with the age of the universe? it seems to be a bit irrelevant.

[John Cuthber]

How do you propose to measure the radius of the universe?

Last time I heard most people thought it was infinite.

If there's an "edge" wouldn't dividing that radius by c give at least as good an estimate?

Whenever you do the maths the value for "The proposed value would be such that the precision at which the change in randomness occurs "

will be the same because one of the nice things about maths is that it is constant.

Planck's constant is also a constant.

The radius of the universe is either expanding or infinite. If it's infinite the expression you give makes no sense.

If it is expanding then you are saying that something constant is a function of something that is changing.

[Gustafson, S]

To "Insane alien": I agree that abstract math concepts such as pi may seem to have little to do with measureable quantities such as the radius of the universe. However the Platonic view of mathematics (perhaps the most widley accepted view among mathematicians) is that mathematical results are discovered---not invented or formally synthesized---in the same way that results are discovered in the physical sciences.

 

To "John Cuthber": If space is curved, as is generally accepted, then the curvature may be said to correspond to a radius of curvature, which may be identified with the radius of the universe. Also, the value of pi is constant because, as you indicate, it is a mathematical construct, but some measues of the randomness of its digits may change with location of the digits. Finaly, it is generally accepted that the universe is expanding, but over times that are much smaller than its 13.7 billion year age we may regard it as stationary.

[John Cuthber]

Pi can be calculated, if you tell me what digit you want, I can (in principle) tell you what it is. It's not random. The 3rd digit of pi (in denary) is always going to be 4. Nothing random about it and the same goes for any other digit in any base.

 

You cannot have a changing value of the radius of the universe if it is calculated from a collection of constants (Planck's constant and some parameter calculated from the (randomness of the) digits of pi). Whenever you calculate it it will give the same answer. 6 times 7 will still be 42 in 13.7 billion years and it was just the same as close back to the big bang as you like. Whather or not pi was still about 3 before the big bang is a matter of philosphy and I'm not sure the question has a meaning. If you take the view that maths was discovered (ie like America, it was already there for Columbus (or whoever) to find) then pi was 3 and a bit before anyone anyone noticed it and it will still be 3 and a bit forever.

 

Since as you say, the radius is changing, there is a problem with your theory.

[Ndi]

2. There is reason to believe that the randomness of pi expressed in any base (i.e., 3.14159…in base 10)

 

a) Pi is not random.

b) If it had ANY "randomness" it would be used for generating random numbers. It's not because it's predictable

c) Pi in base Pi is 10. Throwing "any base" is unlikely to be helpful, especially since you don't seem to see where the "magic" of pi is coming from. Pi is what happens when you try to measure things that were not in the original blueprints.

 

A meter is a meter because we said so. A foot is the foot size of some king. 1 meter = 3.2808399 feet. Does that help me calculate the radius of the universe? no. What it does is tell me that if I want to measure king feet I'd better switch to a more suitable system. We work in round numbers that we arbitrarily chose, in a base that is arbitrarily chosen. It's bound to be unsuitable in places, such as square roots.

 

If you look at Pi computation/approximation algorithms you'll see it all boils down to "bad" operations for our usual bases. Try a Google search on Gauss-Legendre or Borwein's 4-th order convergent algorithm.

 

As for 1) and 3) in your post, I'm at a loss. As pointed out in other posts, the "randomness" of such numbers is humanly perceived only because of base limitations. They are constant through time and specific to our base and system. Switch to radians and a circle becomes 2 base Pi. Half a circle becomes 1 base Pi. Conversion between bases is 3.1415 just as conversion between feet and meters is 3.2808.

 

Just because you can't measure a wheel with a stick doesn't make the wheel holy. It makes the stick bad.

[John Cuthber]

Err, I think "Pi in base Pi is 10. " is misleading. Non- integer bases run into problems.

In base 10 I can use all the digits 0 to 9; there's no need for anything bigger than 9. In binary I only have 1 and 0.

What digits can I use in base 0.5 ?

 

And, at the risk of arguing against myself, pi is random in a rather obscure sense. Given that a particular digit is, for example, 3 does not give any indication of what the next digit is. Similarly the sequence of digits 25356 is just as likely to be followed by any digit as any other. It is, in fact, used in random number generation and, as such, it was tested for randomness (of this type) a long time ago to a lot of digits. It passed so there's not going to be anything interesting in the first couple of hundred digits.

http://news.uns.purdue.edu/html4ever/2005/050426.Fischbach.pi.html

[Ndi]

[1]Err, I think "Pi in base Pi is 10. " is misleading. Non- integer bases run into problems.

In base 10 I can use all the digits 0 to 9; there's no need for anything bigger than 9. In binary I only have 1 and 0.

What digits can I use in base 0.5 ?

 

[2]And, at the risk of arguing against myself, pi is random in a rather obscure sense. Given that a particular digit is, for example, 3 does not give any indication of what the next digit is. Similarly the sequence of digits 25356 is just as likely to be followed by any digit as any other. It is, in fact, used in random number generation and, as such, it was tested for randomness (of this type) a long time ago to a lot of digits. It passed so there's not going to be anything interesting in the first couple of hundred digits.

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Yes, you run into problems because of the integer base way of thinking. Rules are simple. Start counting, when you reach base increment next digit and reset first. Rinse and repeat.

 

I can have numbers A B and C, in base D. A, B, C, AA, AB, AC, BA, etc. You can have non-integer numbers in a base, so in base 0.5 I can count in non-ints. 0.1, 0.2, 0.3, 0.4, 0.499999, 0.1|0. This is a notation problem, not a numeration issue. We write numbers with no spacing so itis human-readable with ints alone. Yet we have this issue for ints as well. In hex, we use A B C D E and F for that reason alone. 0xFF is actually 1515 but that raises confusion. So to keep glued-figure-format, we use letters. At the risk of repeating myself, this is not a number/math issue, but a human readable format issue. It has nothing to do with impossibility. We do run into problems, yes. This does not disprove the truth.

 

IMO, "What digits can I use in base 0.5 ?" is misleading because you try to apply int numbering to a non-int base. We can use powers that are non-round, roots with non-round numbers too. Just because we devised ways to cast them to an int to do on paper is a different story. Computers don't use int casting or paper tricks to do float divisions.

 

As for [2], I still fail to see your reasoning. Nobody can tell you what follows 3 because you have insufficient data. 2 might be after 3. Or 4 or 72. Now, what follows the first 3 in PI, well, that's not random. It's 1. then 4,1,5,9,2,6 etc. NOT RANDOM. Covering your eyes doesn't make it random.

 

What comes after 3 is an incomplete question. What comes after 314 is better. 3141592 is even better. With enough info one can determine where you are in the string and start predicting with 100% accuracy.

 

This is why real crypto functions require you to move mouse, bash keyboard, hit your head against the power supply and so on. Random() has a list of "random" numbers inside. If you use Random() once, you get away with it. Keep using it, and you give enough info to be located in the random string and you can predict randomness because it's not random.

 

Not having enough info doesn't make something random. Coin toss is random. You can toss a coin a million times and you still can't tell what the next toss will be. It thought that computers and maths can't really generate random numbers because they are precise by nature. This is why hi-security systems turn to physics for randomness.

 

What you are saying in your post is limited to "next digit" or next "few hundred" digits. Just because it's non-intuitive in human terms doesn't make it random in any way. It has a rule, it's well defined, thus predictable at any point. It's so predictable that if you give me an index I can give you a digit (I only have 32M so don't overdo it ).

 

Perhaps it's the concept of random versus humanly-perceived random that's in question here?

[John Cuthber]

The fact that you don't have enough data to tell what follows "3" is precisely what makes it "random".

It would be perfectly possible to generate a transcendental number that never included, for example, 3.

"With enough info one can determine where you are in the string and start predicting with 100% accuracy."

No, you cannot because any finite sting of digits will occur in the decimal expansion of pi an infinite number of times and will be followed by each of the digits 0 to 9 (an infinite number of times for each digit). How often each digit will follow the sequence is "random" in that is unbiased in the long run. I accept that it's perfectly possible to calculate out pi to as many places as you need so it's perfectly deterministic. However, if you did generate the sequence of "digits that follow 4s in pi" it would look just like tossing a 10 sided dice; there's no pattern to it. If you did the same thing with the decimal expansion of 1/23 you would get a very clear pattern. 4 is always followed by a 3 or a 7.

 

That's the only sense in which pi is "random" and, in refutation of the original post, it has been shown to stay "random" for a lot more than 121 digits.

I have a vague feeling here that what is in question is the meaning of random, I think the one that makes pi look random is a lack of auto-correlation. Of course, since I can always calculate the next digit it isn't random at all and that's the point I made originally because it seem to rather pull the rug from under the original post.

[Ndi]

The fact that you don't have enough data to tell what follows "3" is precisely what makes it "random".

 

The need to quote "random" should worry you because you are giving "random" a meaning that is yours alone. Random does not mean you can't tell what's after three.

 

From the AHD

random

ADJECTIVE: 1. Having no specific pattern, purpose, or objective: random movements

 

No. Specific. Pattern. Pi has a specific pattern. It's just not repetitive. Just to pre-clear it:

 

pattern

NOUN: [...] 2. A plan, diagram, or model to be followed in making things

 

Pi has a model, a plan, it has a rule that makes it. It has a pattern, thus does not fit the description of "random". Making a new word and using it as an already existing work is bad for clarity.

 

Pi is not random. Pi has a counter-intuitive look if you read the decimals in order.

 

"With enough info one can determine where you are in the string and start predicting with 100% accuracy."

No, you cannot because any finite sting of digits will occur in the decimal expansion of pi an infinite number of times

 

I was talking about Pi as a random number generator, thus in a practical application in which Pi is obviously finite. Random() does no computations, it has a stored string of digits and uses that to return results. It is thus not a true random function (If you are not in the programming business, Random() is a notation for the Random function in a computer, that returns pseudo-random numbers in practical computer applications) so you can use that or decimals of Pi just the same.

 

Actually, Pi would be worse because once someone figures out where in the [finite] string you are, they can predict without a complete database.

 

and will be followed by each of the digits 0 to 9 (an infinite number of times for each digit). How often each digit will follow the sequence is "random" in that is unbiased in the long run.

 

a) It's not "random" (note the quotation again). It is UNKNOWN YET.

b) Only applies to decimal system

c) Pi has not been calculated, nor has it been proven to be infinite. It is considered to be infinite because we really tried and found nothing. That does not (yet) prove it's infinite, let alone random. And even if it WAS infinite, I'm sure that statistics will show some tendencies towards/against some figures, disqualifying it as random.

d) you have no proof that it will be followed by those figures at any time. You only have 200 billion decimals so if it's truly unpredictable, how do you know it simply doesn't go 77777777 at some point, having a finite number of fours?

 

 

While it is true that (and I underline) it is ok to use Pi as a pseudo-random number generator in practical applications, it is not random, by definition, and furthermore, by having a function that defines it it is NOT related to universe, age, expansion or aliens, but SOLELY by incompatibilities between our perception system (circle) and our measurement system (decimal).

 

However, if you did generate the sequence of "digits that follow 4s in pi" it would look just like tossing a 10 sided dice;

 

It will be no such thing. The first 4 will ALWAYS be followed by 1. Aaaaalways. The first toss of a dice will always be unknown, like any other toss, aaaaalways. The closest I can come up with is "statistically speaking, the numbers that follow 4 have roughly the same distribution as the faces of a 10-sided dice". Which is roughly correct (probably, but not definitely). That does not make it random, as it was not random to begin with, it's a coincidence and, frankly, unlikely given enough figures- one is bound to be favored. Why not use something truly random?

 

If you did the same thing with the decimal expansion of 1/23 you would get a very clear pattern. 4 is always followed by a 3 or a 7.

 

I fail to see the distinction between a "clear" pattern and "no clear pattern". If you mean a human friendly pattern, then yes, you are correct. The 1/23 decimals will be so repetitive it's useless as a pseudo-random generator. Pi isn't. Right. But it's perception-deep and application oriented.

 

...since I can always calculate the next digit it isn't random at all and that's the point I made originally because it seem to rather pull the rug from under the original post.

 

It does (pull the rug), because that's actually the catch. Funny, that last millimeter of the meaning of the word "random" is what's making the calculation impossible and why the posts reflect that.

 

The first reply, by the insane alien, said "can i just ask what abstract mathematical concepts have to do with the age of the universe?". That's the sum of what has been explained here, one is a physical measurement and the other is a mathematical abstract constant devised to aid in calculation. It's not random nor indeterminate nor related, it's just not expressible in decimal as a real number.

 

Distinct notions, no relation, no calculation. And, sorry to be so verbose, I really hate numerology and I saw this as one of the subjects.

[John Cuthber]

I don't seem to be the only one who thinks autocorrelation is a test for randomness.

http://www.itl.nist.gov/div898/handbook/eda/section3/autocopl.htm

 

You seem to have forgotten that what I originally wrote was "And, at the risk of arguing against myself, pi is random in a rather obscure sense. " Did you miss the last clause there?

 

You wrote "I fail to see the distinction between a "clear" pattern and "no clear pattern". "

Apparently, however many of us would see them as distinct.

And you wrote "Making a new word and using it as an already existing work is bad for clarity."

I presume that I shouldn't use the word random for anything non random. The trouble is that (arguably) nothing is random.

 

"Pi has not been calculated, nor has it been proven to be infinite"

True, three and a bit hasn't been proved to be infinite; probably because, since it's less than 4, it is clearly finite.

It has been (I understand) proved to be transcendental so its decimal (or any other rational base) expansion is infinite. The proof is effectively the same as the impossibillity of squaring the circle. http://en.wikipedia.org/wiki/Squaring_the_circle

 

From that I deduce that if you are looking for any finite series of digits in pi and you don't find it, it's because you have not finished looking yet.

You asked

"how do you know it simply doesn't go 77777777 at some point, having a finite number of fours?"

I'm perfectly certain that it does go 77777777, not just at some point, but at an infinite number of points. It also has an infinite number of 4s (but not an infinite sequence of 4s because, to be infinite that bunch of 4s would need to be at the end, which would make pi rational which it isn't).

 

At the risk of being really petty I should probably point out that you wrote "The first 4 will ALWAYS be followed by 1." and you objected to one of my comments because it would only be true in base 10.

Dear Pot,

thank you for you comment,

best wishes

Kettle.

 

And btw, strictly, what I wrote would also be true in any finite, integer, base greater than 10.

 

Personally, I like numerology; it's one of my favourite jokes. Anyway, the original posting 's idea sucks for any or all of the reasons put forward.