An early stage of the search for a general formalism of such systems might require the following basic questions to be addressed.

1. Given that a class of 'random' event is truly 'causeless', would we expect the chance of the event occuring to remain the same over time?

2. If a class of event is not truly causeless but it could nevertheless be proved that the system was so complex that the cause for any particular event could never be ascertained, then, could such a system exhibit the same behavior as a truly causeless system?

That is a question I always wondered myself, and always end up with the same philosophical question : are prime numbers random? I believe that is the basis of any random process.

Dear Christopher,

You have stated the problem accurately. Nevertheless, I think that the question can be simplified by focusing our attention on cause-effect chains instead isolated phenomena. To this end, denote by A

Interesting.

Dear Costa,

Dear,

In this paper of mine,

http://scholar.google.es/scholar?cluster=762709012915259345&hl=es&as_sdt=0,5

you have an exact deterministic distribution of primes. The distribution is picewise defined, that is to say, there is no finite formula denoting the whole function. Nevertheless, for every finite prime sequence the corresponding expression is also finite.

In any case, since every exact distrubution cannot be finitely defined, the prime sequence seems to be of random nature, but it is an optical delusion. Notice, that there are effective algorithms to discern whether or not a given number is prime. There are also algorithms determining the smaller prime greater than a given integer. By contrast, I do not know any algorithm determining the result of tossing a die.

In any case, your claim could be true depending on which is the inference between relative frequency and probability. That is to say, which of the following alternatives you prefer?

a) Relative frequency depends on probability.

b) Probability depends on relative frequency.

I think that both alternatives are true in some kind of phenomena.

For instance, a) is true in random experiments like tossing a coin.

By contrast, the probability of remembering sentence depends on the frequency in which it occurs. This is why, to remember some sentence, any student repeat it again and again, in order to increase its occurrence probability.

Dear Jean Pierre,

The title of this thread is not exclusive. You have stated another possibility. Take into account that those phenomena induced by an endless set of causes are handled as if they were of random nature. In other words, both situations are similar in practice: Causeless and endless set of causes. In the first case there is no cause, while in the second one, it is not possible to know their causes, because of its infiniteness.

Dear Professor Juan-Esteban

Can we say the that (all?) events which are not random cannot be be causeless?

Dear Issam,

It depends on the assumed criteria. For example, if A is causeless, but A causes B, then B is caused; however the couple (A,B) regarded as a whole is also causeless.

Perhaps there are neither pure causeless phenomena nor pure caused ones. For example, if you toss a die, the toss cause is your will, but the result is at random. Suppose that in a given toss the result is 5. You can say, that you are the cause of this result, however the result is at random.

In fact, there are people who think that randomness cannot exist, because every thing is caused by God. I do not think so. When there are different possible universes and there is no reason to choice this one, the God's choice must be at random. Mainly, because it is very difficult to believe that this is the best possible universe. Thus, the assumption of God as the first cause does not avoid the randomness nature of our Universe structure.

But our universe is not random, there is ample evidence it is fine-tuned for life.

Dear Issam,

It is the structure and configuration what is of random nature. Do you know any reason compelling the light speed to be 300000 km instead of 310000km?

Is there any reason to be fine-tuned for life? When there is no determinig reason for a choice, the choice is causeless, therefore at random. Suppose that I ask you to choose a number less than 100. Suppose that your choice is 23. If you have no reason to this choice, it is at random. I can figure out a Universe better than this one, therefore the choice of this Universe structure, in my opinion, cannot be induced by any optimal criterion.

Juan-Esteban,

There actually may be an underlying cause why the speed of light is what it is. Current studies focus on the effect that the possibly quantized nature of spacetime (the medium through which light propagates) would have on the speed of light.

Also - any psychologist would probably uncover the reasons why you'd come up with 23 - there would be subconscious triggers in your brain and memory - which would bring the number 23 to the fore of your conscious awareness ahead of any other number. True randomness is likely very elusive here.

I concur with you that it's relatively easy to conceptualize a world, or at least a biosphere, which would be 'better' than the one we experience. In particular, predation is the chief ordering principle in the biosphere, whereas it is easy to conceptualize a world where collaboration would be the overarching ordering principle. I sometimes get the impression that those in the US who speak of intelligent design have never seen the quite haunting - and seemingly pointless and avoidable - horrors of the Serengeti, for instance.

This being said, there is no evidence that ours is the only world (or for that matter the only universe), and that there are not other, more perfect worlds, so a number of arguments are void.

Finally, explaining things by the agency of God does not begin to explain anything at all. As Paul Davies once forcefully argued, positing a God merely displaces all of the existential questions by one step without resolving any.

Dear Ransford,

My claim about light speed is stated as an illustration, not as a theorem or axiom. Notice than in a previous message I have synthesized the question in a more abstract formalism. It is very simple, and I try to repeat it.

Consider any phenomenon or fact X0. If X0 is causeless, then it occurs at random. If X0 is caused by X1, then recursively you can suppose that either X1 is causeless or caused by X2. In the first case the couple (X0,X1) is causeless. Repeating the process we have a chain X0,X1,X2....Xj. If the chain is finite, then Xj is causeless and the whole chain occurs at random, although for every n < j, the phenomenon Xn is caused.

Now consider the light speed C. If you know the reason X0 of its value, then you can build a chain of causes C, X0, X1....if the chain is finite, it is causeless. By contrast, if the chain is infinite it is also causeless; because of an infinite sequence is, by definition, an endless one; consequently its primitive cause cannot be reached.

As you have said, "a psychologist would probably uncover the reasons why you'd come up with 23". Denote this reason by X0. Is X0 causeless? If the answer is yes, then 23 occurs at random, because the primitive cause X0 is itself causeless. If the answer is not, then there is a reason X1 for X0 and the procedure can be recursively applied until a causeless reason can be found.

In fact, the random nature of a phenomenon depends on its primitive cause. For instance, suppose that you have two chickens A and B. Tossing a coin you choose which of them will be killed for diner. Suppose, that the toss chooses B. The cause of the dead of B is your action with a knife, but its dead occurs at random if you consider the choice procedure.

Finally, you have said: positing a God merely displaces all of the existential questions by one step without resolving any.

This sentence agrees with my claim. What I have said is that by using God's hypothesis the random nature of our universe does not vanish.

In any case, the question depends on the meaning you assign to the term God.

The concept defined by Spinoza states a different situation, since he is the essence of our Universe. Perhaps it is only a semantic question.

Finally, I prefer a formal treatment of the question. Let us see;

Denote by A ≤ B the relation "A is caused by B". It is not difficult to see that it is transitive, therefore a pre-ordering. Every complete chain A1 ≤ A2 ≤… is causeless, provided that it is finite. If it is infinite, its cause cannot be reached, that is a similar situation.

I like your formal treatment ....

Fully agreeing with the Ur-cause of anything being elusive, however once the chain has been set in motion you have a phenomenon of ordering emergence where formal cause and effect links do occur.

So the whole argument has to do with the ultimate cause or causes of anything - the universe, life itself, etc.

Either the set of ur-causes is unfathomable (for example, if time is linear, which is often an implicit assumption. Time cannot however just be 'ohne weiteres' linear, or delayed choice experiments would not work), or - there is a possibly non-nil chance that we have a partially bootstrapping universe, in which because of local time loops some phenomena could bootstrap themselves into existence.

Is randomness only locally perceived because of our distinctive and differentiated view of the universe of being either causal (relativistic view) or non-causal (quantum physics) and our model of our universe as one that has both interconnected and independent parts? However, if we dismiss this perception of "independence" as only being "local" and consider everything as interconnected to everything, then doesn't the phenomenon of randomness cease to exist?

The total traffic intensity in a network, S, at a given busy hour, given weekday and given month is random (normal distribution).

It can be divided into traffic from N origins i=1..N, and we find that

each of these components Oi at a given busy hour, given weekday and given month is also random (normal distribution)!

Otherwise if we consider the N destinations, j=1..N, we also find that

each of these components Tj at a given busy hour, given weekday and given month is also random (normal distribution)!

Furthermore, we (Kennedy, Berry 1995) showed that if we look at each of the pairs i,j, for i,j=1..N, we surprisingly also find that each of these sub-components Nij, at a given busy hour, given weekday and given month is also random !! (normal distribution)

So the appearance of the total traffic S being random can be seen as a consequence of its components and their components being random.

Looking at things the other way around: I am seemingly in control of what second and minute I am sending this post to you. "I cause the traffic." Many other people ACTING INDEPENDENTLY of me

are in the same position. In large numbers this causes the traffic we generate between any two places, Nij, to be random.

[Kennedy, I.G., and Berry, L.T.M., Origin-destination traffic variance and skewness, March 1995 Trans SAIEE 86:1 32-37]

Dear Juan-Esteban,

I like your demonstration of the causeless of a finite chain [Xn,Xn-1,Xn-2...Xi...X0] while any given event Xi is caused by Xi-1 (sorry, but I have changed the notation for a easier discussion, to my opinion):

Consider any event Xi. If Xi is causeless, then it occurs at random. If Xi is caused by Xi-1, then recursively you can suppose that either Xi-1 is causeless or caused by Xi-2. In the first case the couple (Xi;Xi-1) is causeless. Repeating the process we have the chain [Xn,Xn-1,Xn-2....Xi...X0].

As the chain is finite, then X0 is causeless and the whole chain occurs at random and the final event Xn is also causeless, although for every i >0, the event Xi is caused (i.e., by Xi-1).

1) I see however a contradiction in the above reasoning: logically speaking, is it possible to say both "Xi is caused by Xi-1" and "Xi is causeless"?

If I understand your reasoning correctly you assert that any Xi is finally causeless because the whole chain [Xn,Xn-1,Xn-2....Xi...X0] is causeless, am I correct?

Your comments?

2) Let's take the example of our universe.

Our universe was clearly incredibly fine-tuned to have been able to produce the structures it has produced. Thus, it is difficult to think that it was itself the product of randomness if our hypothesis is that there is only one universe because the probability for its emergence would have been incredibly low.

By contrast if our hypothesis is the existence of a multiverse with the possible emergence (at random) of many universes then the incredible fine-tuned appearance of our universe is much more understandable: what do you think about it?

Dear Marc,

You have said:

"Our universe was clearly incredibly fine-tuned to have been able to produce the structures it has produced"

I agree with you.

In any case, take into account that an ovule is a collection of atoms and so is a spermatozoid. Both are ruled by the laws of quantum mechanics, which are of random nature. Can you believe that the fact of puting all together give rise to a human being with a brain? It is not easy to believe, but very easy to test.

Assuming that all phenomena in our universe are equiprobable, it is not possible. But probability depends on structure. For instance, suppose that the planet trajectories are chosen at random. Those planets with an open trajectory no longer will be found in any system. By contrast, those planets with a periodic movement (circle or ellipsis) will be always found. If at the very beginning all trajectories were possible, after billions of years only the periodic ones possess a high probability of existence.

Notice, that the foundational laws of physics are conservative. The only condition imposed by Einstein to his gravitational law is the covariant derivative of the Ricci-Einstein's tensor R to be null. In other words: R cannot change. The probability of existence of a property or magnitude that cannot change is 1.0.

Starting with a chaotic configuration, through billions and billons of changes, every system tends to stable configurations or attractors.

I do not know what is the proper answer, but if the probability of each phenomenon can be deduced from its structure, its existence is explained, simply, by its structure.

This is why the proper question should be: Can the structure of our universe be explained by it self or need some external explanation?

Our universe is highly improbable, but we cannot argue against the evidence for existence of it!

Dear Ian,

Chosen at random the number 345568584723654256954654056 it is also highly improbable to happen. Nevertheless in this occasion it has happened.

345568584723654256954654056 SNAP!

Dear Juan-Esteban,

Thank you for your answer to my point 2) about the emergence of our universe.

However I am not sure to understand you correctly.

For instance I don't really agree with your comment "In any case, take into account that an ovule is a collection of atoms and so is a spermatozoid. Both are ruled by the laws of quantum mechanics, which are of random nature. Can you believe that the fact of puting all together give rise to a human being with a brain? It is not easy to believe, but very easy to test. Assuming that all phenomena in our universe are equiprobable, it is not possible".

Well, first I don't see the analogy with the issue of the emergence of our universe.

Second, I think the problem of the apparent miracle when an ovule and a spermatozoid merge is analog to the more general problem of the emergence of all the organisms on Earth, our species included. To me this is the consequence of about 4 billion years of Darwinian evolution (and thus of natural selection).

Besides, from your speech "For instance, suppose that the planet trajectories are chosen at random. Those planets with an open trajectory no longer will be found in any system. By contrast, those planets with a periodic movement (circle or ellipsis) will be always found. If at the very beginning all trajectories were possible, after billions of years only the periodic ones possess a high probability of existence" you clearly infer there has been a selection. If so, I totally agree with you.

You didn't comment my point 1) about the contradiction I see when you assert both "Xi is caused by Xi-1" and "Xi is causeless" (?).

Dear Marc,

The example of the ovule only is exposed to illustrate that, even being a random phenomena, if the probability of developing a human being is high comparing with any other configuration, the process can seem to be deterministic. Nevertheless, it is not deterministic and can be unsuccessful, for instance by some virus contamination, by abortion and so on.

To answer the other question, see another illustration. Suppose that you decide to kill or not to kill a chicken depending of the result obtained by tossing a coin. Denote as X1 the toss and as X2 the fact of killing a chicken. The fact X2 is not at random. You take a knive and cut the chicken neck without any randomness. However, X1 is at random. The concatenation of both facts X1+X2 is also at random, because its result cannot be predicted. Nevertheless, once the coin is tossed, the result of X2 can be predicted and, therefore, it is deterministic.

Dear Juan-Esteban,

I agree with your reply to my point 1).

However I propose another illustration: take the emergence of our universe. Under the hypothesis of the existence of a multiverse it is very likely that our universe has emerged at random. Let call this event X0. Many events following X0 were caused by X0. This is the reason why contemporary cosmologists are able to describe the evolution of our universe from its very beginning: what do you think about it?

Regarding your point about the fusion of an ovum with a sperm, I agree that:

1. The fertilisation of the ovum is at random as the specific sperm which will fecund the ovum is not predictable a priori;

2. Then a chain of deterministic events follows. However the final product (i.e., the baby) is not totally predictable as non-predictable events can occur (e.g., virus contamination).

In addition I think there are deterministic chaos processes involved in the development of the embryo. Hence, such processes, as you know, imply that the precise evolution of the system is not predictable after some time.