What is randomness?

[Ruskin]

Ruskin:

Those "random" characters come almost exclusively from the left side of the keyboard.

Are you left handed?

No I just hold the mouse in my right hand and type with the left. I have enough dexterity in my left hand to push keys just not to write or draw/paint or anything.

[EMTe]

1. Common sense approach. Randomness is false premonition that chaos can exist. The assumption that chaos exists is based on our incapability of understanding and counting everything at once. For example weather - there are so many factors involved that weather apperas to be extrememly chaotic and random issue.

2. Mathematical approach. Randomness in mathematics is not important to discuss, because it's like many other mathematical terms (including uncertainty, infinity etc.) abstract and doesn't relate to everyday issues. It's a term like many other, useful for calculations, but not meaning anything particular.

3. Physicists approach. There may appear with time that on lowest level there exists something like randomness understood as uncalculable appearance of events (see quantum fluctuations). Free of causal relationships etc. But we don't know them, however it's fun to debate such possibilities. It's science fiction though.

[enegue]

enegue said:[br][/br]However, philosophically, you still hold out for the possibility of spontaneous uncaused events, whereas I don't.

I don't believe anything can be uncaused. There is no causeless effect. I only offer that it is impossible for Humans to be able to comprehend all of the causes involved, and that's enough for something to be considered random. Apparent randomness is sufficient for the qualification of the concept.

Then we are on the same page, entirely.

On what basis do you establish, to your satisfaction, the existence of these causes as distinct from "mere" correlations?

What are "mere correlation"?

Cheers,
enegue

[Spiral Out]

Spiral Out:

I don't believe anything can be uncaused. There is no causeless effect.

On what basis do you establish, to your satisfaction, the existence of these causes as distinct from "mere" correlations?

Correlations would be apparent between independent effects, causes apparent to an effect. Separate causes and their associated effects would be correlates to each other, not necessarily causes of each other.

[Keen]

Me and Steve have been debating about randomness which is a two-parter. First part. With a number such as π, let's say the sequence has been computed to a billion places. Now let's say the digits are known to be uniformly distributed. I'm saying that based on the study of π which is known to have a random sequence, it has a uniform distribution of its digits (up to a billion places). Steve is arguing that suppose for the next billion places, the distribution becomes weighted towards some of the digits which could happen. Then what does randomness mean?

If you take a sequence, that you can calculate, it's never random, but pseudo-random at best. Random is something, which you can't predict, but it still follows some precise distribution.

Second part: can it be proven for any irrational number that the distribution of the digits can be equally weighted?

Actually it's not true. I think that pi has its digits uniformly distributed, but it's definitely not true for every irrational number. Let's just consider a number that you put more and more 0 between two 1: i.e 1,01001000100001000001.... it's definitely irrational, because the number of zeros increases constantly, so you can't find a repeating pattern in it. However I'm pretty sure, there are more zeros than ones.

[Steve3007]

Keen: The definition of an irrational number is not that it has no repeating patterns. It is that it cannot be expressed as the ratio of two finite integers. But I'm not a good enough mathematician to know whether the two are always associated with each other. Interesting question?

[Keen]

@Steve3007 You can trust me on this: A number is rational if and only if at some point the sequence of its digits is periodic i.e repeats itself. The proof of that statement is not completely obvious and quite technical, because you have to realize what notations like 0.133333333.... (rational) and 0.0123456789101112.... (irrational) ad infinity exactly mean, but yeah it's possible to prove it.

So for instance 0.142142142.... is rational. It's actually equal to 0.142*(1/(1-1/1000))=142/999. While 1.010010001.... I mentioned earlier isn't.

[Steve3007]

OK, that's interesting. Thanks! I'll have to do some research on that.

The only proof I know of in relation to irrational/rational numbers is that famous one discovered by Pythagoras of the irrationality of sqrt(2). That seems to me to be a classic example of a beautifully simple but profound proof. And I've read that it was suppressed by Pythagoras because the idea that numbers could be irrational didn't fit with the general philosophical idea of mathematics being "perfect". A pity, as the proof seems to be pretty close to demonstrating just that.

One more point:

The example you gave of adding successively more and more zeros between the ones. As you pointed out, that number doesn't have a repeating pattern. But I guess it can be expressed as an algorithm (as you just did by describing it). So presumably that means that it is not transcendental?

[Philosophy Explorer]

Me and Steve have been debating about randomness which is a two-parter. First part. With a number such as π, let's say the sequence has been computed to a billion places. Now let's say the digits are known to be uniformly distributed. I'm saying that based on the study of π which is known to have a random sequence, it has a uniform distribution of its digits (up to a billion places). Steve is arguing that suppose for the next billion places, the distribution becomes weighted towards some of the digits which could happen. Then what does randomness mean?

If you take a sequence, that you can calculate, it's never random, but pseudo-random at best. Random is something, which you can't predict, but it still follows some precise distribution.

Second part: can it be proven for any irrational number that the distribution of the digits can be equally weighted?

Actually it's not true. I think that pi has its digits uniformly distributed, but it's definitely not true for every irrational number. Let's just consider a number that you put more and more 0 between two 1: i.e 1,01001000100001000001.... it's definitely irrational, because the number of zeros increases constantly, so you can't find a repeating pattern in it. However I'm pretty sure, there are more zeros than ones.

Keen, I want to thank you for jogging my memory with that counterexample. Here's another counterexample to consider: .01040916253649....

Now the question is with irrational numbers such as π and others in its class, if they hold hidden properties that they would share besides the presumably even distribution of the digits? Also would every transcendental number have a potentially even distribution of its digits?

Just some further food for thought.

PhilX

[Keen]

People seem to take interest in transcendent numbers and I understand: they are fascinating, but it's also a very difficult subject in mathematics, which uses advanced algebra as well as advanced analysis.

The example you gave of adding successively more and more zeros between the ones. As you pointed out, that number doesn't have a repeating pattern. But I guess it can be expressed as an algorithm (as you just did by describing it). So presumably that means that it is not transcendental?

Unfortunately it's not that easy. There are tons of algorithms which calculate digits of pi to any precision you want. People like to run these to test how fast their computers are. On the other hand pi is transcendent. I found out very recently, that the example I gave is transcendent as well, but I haven't had the time to read the proof. There is a theorem claiming that if the number of 0 between two non null digits increases, then it's transcendent, so the example I gave is a particular case of that theorem.

Now the question is with irrational numbers such as π and others in its class, if they hold hidden properties that they would share besides the presumably even distribution of the digits?

I suppose so: pi is a very fascinating number and I believe it has quite a lot of properties. You can see it pretty much everywhere in nature. Circumference of circle is the most obvious one, but it appears quite a lot as well in optics, electronics acoustics or any other discipline where you try to describe waves. It appears also in Gaussian distribution, which describes how large quantities of random things like births, or grades behave.

Also would every transcendental number have a potentially even distribution of its digits?

As the example I gave earlier is transcendental, then the answer is apparently no.

[Steve3007]

Unfortunately it's not that easy. There are tons of algorithms which calculate digits of pi to any precision you want.

Yes but, as I understand it, the algorithm or equation that defines the number can't be an infinite sequence. Isn't pi transcendental because it can't be defined exactly by a finite length algorithm? It can only be approximated to greater and greater accuracy by an iterative algorithm. I don't know if that's true. It just sounds vaguely right!

P.S: I just looked up transcendental numbers again to remind myself and, of course, I'm talking nonsense here aren't I? It's being algebraic that is the key to not being transcendental. Not being algorithmic.

pi is a very fascinating number and I believe it has quite a lot of properties. You can see it pretty much everywhere in nature. Circumference of circle is the most obvious one, but it appears quite a lot as well in optics, electronics acoustics or any other discipline where you try to describe waves. It appears also in Gaussian distribution, which describes how large quantities of random things like births, or grades behave.

I guess one reason it's so ubiquitous is because Simple Harmonic Motion is so ubiquitous, and S.H.M. is intimately connected with circles and ellipses. I think the ubiquity of e, the base of natural logarithms is interesting too. It's also intimately connected to pi and the harmonic functions (via Euler's formula), circles and S.H.M.

In fact, it's interesting to consider which pieces of mathematics might be considered to have the widest applications in nature. S.H.M. is certainly a good candidate. Along with exponential decay perhaps?

[Keen]

P.S: I just looked up transcendental numbers again to remind myself and, of course, I'm talking nonsense here aren't I? It's being algebraic that is the key to not being transcendental. Not being algorithmic.

I think I should give some precision to that. Yes the definition of being transcendent is being non algebraic. However I should point out, that if any number is what you call "non algorithmic" exists, it must be transcendent, because there are algorithms to approach any algebraic number you want. Most known of them probably being the Newton's method. The interesting fact is that we know there are transcendent numbers which are not algorithmic. The reason behind this is, that the set of all possible algorithms that calculate digits of irrational numbers is countable, while it is a well known fact that the set of all numbers is not countable. As a conclusion, what you said wasn't a complete nonsense: there definitely is a link between transcendence and algorithmic. It's just not as simple as you thought.

A fascinating point I get from this is that mathematics get general results on numbers we can never possibly "know" as there are no tangible ways of calculating their digits.

[Atreyu]

Me and Steve have been debating about randomness which is a two-parter.

First part. With a number such as π, let's say the sequence has been computed to a billion places. Now let's say the digits are known to be uniformly distributed. I'm saying that based on the study of π which is known to have a random sequence, it has a uniform distribution of its digits (up to a billion places). Steve is arguing that suppose for the next billion places, the distribution becomes weighted towards some of the digits which could happen. Then what does randomness mean?

Second part: can it be proven for any irrational number that the distribution of the digits can be equally weighted?

PhilX

I'd just like to answer the first question, and just directly, as IMO your example only obscures the key issues.

'Randomness' refers to any phenomenon allegedly occurring without any conscious intention or force, either directly or even indirectly, as in the case of a boulder which just *happens* to follow a certain path after a man has consciously and intentionally rolled it down a hill. If something is 'random', it means that not only is it not directly a result of any conscious intention, i.e. nothing *does* it, nothing *made* it happen, but also that it has no chain of causation back to any conscious intention or decision. If you trace back cause and effect, you *never* arrive at any conscious action or intention. Ever. Only mechanical forces are involved, directly *or* indirectly.

[Steve3007]

Atreyu:

It's interesting that your understanding of "randomness" seems to be so intimately connected with the concept of "conscious intent". It looks like a very theist understanding, if you don't mind me saying so.

[Jklint]

...a Cause which has no prior Cause.