Random - a definition attempt

[Barbarian]

In the aftermath of some ugly threads about free will, I thought I would isolate the issue of randomness and attempt some clarifications via discussion here. I am not sure where this 'here' would be better located - S&S and Philosophy both were contenders. So I tossed a coin ...

First, a few definitions - feel free to propose adjustments, as no doubt you would feel anyway:

An elementary snapshot is a subset of the universe selected in such a way that both the distance between its elements and the relative speeds of its elements are negligible. Let us say that the relative speeds of the elements are less than one-hundredth of 'c' and the distance between any two elements is less than 3 meters (that's ten feet, IIRC). These numbers were chosen in a totally arbitrary manner, BTW.

A system is a collection of elementary snapshots - basically, we propose that those elementary snapshots all correspond to the same system in different states.

A state of a system is a subset of the elementary snapshots making up the system. IOW we abstract away some information regarding the elementary snapshots, and as a result some of these snapshots become indistinguishable. The equivalence classes of indistinguishable snapshots are the states of the system, then. This can be repeated with states instead of snapshots, to get more abstract states. And vice versa: it will most often turn out that what we considered elementary snapshots are already states.

A physical process is the change of the state of a system over a finite time period, starting from a certain state. As we see, the same system and initial state can give rise to different processes depending on the state abstraction.

Now: a process is random if all laws in the universe cannot restrict it to have a single final state, but they do restrict it to have only one of the possible final states. This should be the only usage of random - all others appear to me to be just less coherent approximations.

Some consequences:

A chaotic system is not random. A chaotic system evolves in a totally deterministic manner, and all problems of predictability stem from practical issues such as inexactness of measurement and limited precision of calculations. Inexactness of measurement has its limits in quantum mechanics, but then it is the nature of quantum mechanics which leads to inpredictability, not the chaotic character of the system. All other limits on measurement and computing precision may one day be overcome, and this would make any definition of random dependent on technological advancement. I do not like that.

There is no such thing as partially random or not completely random. Something is either random or it isn't, in which case it is deterministic, because the involved process either has a single next state or it has many. So a process yielding 99.99% of the time a certain result and 0.01% of the time another result is completely random, as in not restricted to a single endstate. Many people will say that random means somehow having equal probability, but they do not really mean this, e.g. they do not call a 45-55 chance non-random. (These people do have a point, though: what they are intuitively saying is that a non-uniformly random source of events has less enthropy than a uniformly random one, and thus the two cases are not equivalent, the difference of enthropy acts as extra information we have about the non-uniform case, aiding prediction somewhat. The crucial distinction is that no prediction can be made at all for any actual measurement - predictions involving the probability are not predictions.)

Maybe QM measurements are examples of true randomness. If we measure a parameter of a system not in an eigenstate of the operator corresponding to the parameter, we know the probabilities of the outcomes but not the actual outcome. I am not sure QM is really random, I keep hearing about attempts to make it deterministic, but if it were truly random, it would be a prime example of what I mean by random here.

There was some attempt to identify random sequences by their Kolmogorov complexity. E.g. a heads-heads-heads-... sequence is not random while a more complex one is more random. It should be clear that both the uniform sequence and the complex one can come from a random source. Moreover, it seems that one can learn to spot real random sequences among artificially chaotic ones (you can make up sequences you think are random-looking, while someone else can put together sequences by tossing a fair coin, then some third person can be told exactly which ones are really random and which ones are artificial. This third person can learn to distinguish with fairly high confidence between truly-random and artificially-random sequences, although it is allegedly very difficult to explain the difference, apart from the fact that truly-random sequences are not as chaotic as artificial ones, the latter displaying an unnecessary variety. I suspect that the distribution of homogeneous sequences is the key, but that is just a suspicion.)

Justification:
(What justification? http://www.iidb.org/ubb/biggrin.gif )

People typically identify random with unpredictable. I propose to use theoretical unpredictability, because practical unpredictability is a moving target, and any definition based on it would be moving too - IOW I do not like the idea of processes suddenly becoming deterministic as technology and science advances.

All predictions are based on and limited by (physical) laws. Therefore, the theoretical limit of predictability is provided by the availability of physical laws. The fact that we do not know them all and some of their currently known forms are probably false should not matter here. It would appear that all physical laws can be formulated in a form of restriction, taking a set of hypothetical outcomes and pointing out the possible ones. When we are done with applying all laws, we either restricted the process to one outcome, or we did not.

What do you think?

[llanitedave]

Now: a process is random if all laws in the universe cannot restrict it to have a single final state, but they do restrict it to have only one of the possible final states. This should be the only usage of random - all others appear to me to be just less coherent approximations.

There's a recursive component to that definition which prevents it from being rigorous:
It's trivial that the laws of the universe would restrict a process to one of the possible final states, because the laws of the universe are what define those possibilities.

Also, I get what you're trying to say, but I can't agree with it: Your definition could apply to a chaotic outcome as well.

A true definition of random is indeed hard, because there is no theoretical way to discern the difference between a random result and a well-calculated pseudorandom one.

"Random" simply implies a process that outputs a range of possible results with no discernable pattern. I won't try to pass that off as a formal definition, though.

[epepke]

Well, "random" is a word that it used in a couple of ways. The most common use of the term "random" is statistical. I presume you mean the other definition of random, that is, in informal terms, unpredictable.

There's a simpler definition. I think this is from Knuth, but I can't seem to find it. Consider a sequence S of n numbers, S(n). Define a number s(k) as random if it is a member of the sequence S(k), where k >= 1 and k < n. Say that p(k) is the probability of guessing the number s(k) with no information. Consider all algorithms A that can predict s(k) with a probability of P(k). S(k) is a random sequence, and s(k) is a random number, if there exists no algorithm A that can predict s(k) with P(k) greater than p(k) for arbitrarily large k with less information than S(1..k).

Proving that no such algorithm exists is rather difficult, and it's also provably impossible for arbitrary sequences S.

[Barbarian]

There's a recursive component to that definition which prevents it from being rigorous:
It's trivial that the laws of the universe would restrict a process to one of the possible final states, because the laws of the universe are what define those possibilities.
Exactly. I start feeling that my attempts to define process and final states were, in fact, misdirected - sounds like I tried to grab too much at once. But nonetheless, what I am trying to say is that such processes are the sole source of true randomness.

Actually, scrap the definition attempts I gave for snapshot, state and process, use whatever intuitive definition you feel appropriate.
Also, I get what you're trying to say, but I can't agree with it: Your definition could apply to a chaotic outcome as well.
Disagreement here. If a chaotic system is deterministic, then for every starting state it has a single, well-determined final state - our issue is that we do not know the starting state to the necessary precision, and that we are, perhaps, unable to perform the predictive simulation with sufficient precision. But those are practical issues, not theoretical ones. For any particular chaotic system we may be able someday to perform the feat of sufficient simulation, nothing in principle prevents us from doing so.

It is often said that QM principially prevents us from knowing the initial state. This is true, but let us have again a deterministic chaotic system. Part of the input state is the energy of a particle, and let us assume that initially the particle is not in an energy eigenstate. Depending on the QM interpretation you choose, you can say that the particle jumps into a certain energy eigenstate in a random manner per my definition and then the measurement proceeds deterministically as above. Or you can say that the superposition state of the particle extends to the apparatus and therefore the chaotic system will end up in a superposition of multiple deterministically defined states, each one based on deterministic evolution starting from an energy eigenvalue, and then only the choice of the observed outcome will be the random - multiple outcome - event. In other words, the unpredictability, the random component is entirely due to the quantum measurement and has nothing to do with the chaotic character of the system.
"Random" simply implies a process that outputs a range of possible results with no discernable pattern. I won't try to pass that off as a formal definition, though.
Exactly. However, to me the "no discernable pattern" means "cannot see the deterministic mechanism generating the outcomes, even if such a mechanism exists", combined with "hence we cannot predict the outcomes". But this has an implicit element of "at our current level of science and technology" in it, which I wish to remove.

[Barbarian]

Well, "random" is a word that it used in a couple of ways. The most common use of the term "random" is statistical. I presume you mean the other definition of random, that is, in informal terms, unpredictable.
I do not think there are such definitions of random which do not fall back to individual outcomes being unpredictable. The statistical usage is drawn from observing real-world unpredictable events, and I am not aware of any purely mathematical source of randomness. The math I have learned starts with the assumption that there are random events, with multiple outcomes, realizations of actual values of a random variable, providing absolutely no hint as to the origin of such an entity. Statistical usage is a bit wider, though, and can be used to describe pseudorandom stuff as well.

I think the difference is between a subjective definition of random (we cannot predict it based on our knowledge, technology etc.) and an objective one (it is theoretically impossible to predict it or rather: there is no such thing as predicting it). I posit that subjective definitions are exposed to erosion due to technological advancements, and not suitable for a metaphysical approach. The snag is that we only know if something is subjectively random, I do not see any way to prove that something is objectively random, and doing so may, in fact, be a theoretical impossibility - nonetheless, in a metaphysically oriented discussion the cases we have to consider are objectively random vs. deterministic, not subjectively random vs. the already understood.
There's a simpler definition. I think this is from Knuth, but I can't seem to find it.
I have not encountered that definition yet, and unfortunately I do not understand the way you have put it. I have only found some hints at it in the ACM index (Knuth about the identification of pseudorandom sequences), but did not actually see the definition. Let me go through it, to show where my understanding problems lie:
Consider a sequence S of n numbers, S(n). Define a number s(k) as random if it is a member of the sequence S(k), where k >= 1 and k < n.
I am not sure what this means. Are we going to elaborate upon S(.) later, but declare beforehand that its members are random numbers if S(.) conforms to some criteria subsequently provided?
Say that p(k) is the probability of guessing the number s(k) with no information.
Some information is needed, or else we do not know the range of the number, and must assume all (natural?) numbers are equally probable, which is undesirable. I assume you mean no information about previous members of the sequence currently being generated. This does not preclude that we know, for instance, the distribution of s(k).
Consider all algorithms A that can predict s(k) with a probability of P(k).
I assume you are not proposing algorithm classes per P(k) value, only that the P(k) is a shorthand for P(A,k). This appears to be confirmed later.

I also assume that predicting the value of s(k) with probability P(k) means that the algorithm proposes a value v and we find that the probability of s(k) being equal to v is P(k).
S(k) is a random sequence, and s(k) is a random number, if there exists no algorithm A that can predict s(k) with P(k) greater than p(k) for arbitrarily large k with less information than S(1..k).
Does having S(1..k) as information mean that we know s(1), s(2), ... s(k)? I cannot interpret it in any other way, and knowing the last element, s(k), does indeed help predicting s(k). I assume you meant S(1..k-1).

With those assumptions, the definition appears to mean that a random sequence is characterized by the need to use all previous values to predict an actual next value better than sheer chance. But in order to comment on it, I would like to ask you to confirm my clarifying guesses above.

[llanitedave]

Disagreement here. If a chaotic system is deterministic, then for every starting state it has a single, well-determined final state - our issue is that we do not know the starting state to the necessary precision, and that we are, perhaps, unable to perform the predictive simulation with sufficient precision. But those are practical issues, not theoretical ones. For any particular chaotic system we may be able someday to perform the feat of sufficient simulation, nothing in principle prevents us from doing so.

It is often said that QM principially prevents us from knowing the initial state. This is true, but let us have again a deterministic chaotic system. Part of the input state is the energy of a particle, and let us assume that initially the particle is not in an energy eigenstate. Depending on the QM interpretation you choose, you can say that the particle jumps into a certain energy eigenstate in a random manner per my definition and then the measurement proceeds deterministically as above. Or you can say that the superposition state of the particle extends to the apparatus and therefore the chaotic system will end up in a superposition of multiple deterministically defined states, each one based on deterministic evolution starting from an energy eigenvalue, and then only the choice of the observed outcome will be the random - multiple outcome - event. In other words, the unpredictability, the random component is entirely due to the quantum measurement and has nothing to do with the chaotic character of the system.


You start out saying you disagree, and then you spend the whole paragraph practically copyrighting my logic!

Except for one or two points: I disagree that " those are practical issues, not theoretical ones". They are very much theoretical. They utilize ideas from measurement theory, which states essentially that all measurements are approximations. Therefore, you cannot in principle *prove* that "for every starting state it has a single, well-determined final state", because you never know precisely what that starting state is. The unpredictability is a matter of principle. What *IS* predictable is that the final state will lie somewhere within the bounds of whatever "attractor" the chaotic system defines. But that's really the only difference between chaos and randomness.

I'm no expert, of course, and you very well may be, but I think we're approaching a blurring of boundaries here that will satisfy neither one of us.

[Schneibster]

I've been mulling this over all day, and I don't really have anything definitive to say, but I do have a couple of observations.

I had thought to name the digits of irrational or transcendental numbers as random. However, I found that irrational numbers had been constructed by associating digits after the decimal point, in the decimal system, with the whole numbers, and then setting every one that corresponded to a perfect factorial to one and all others to zero. This is certainly an irrational number, and it might even be transcendental (IIRC, it is). This was proven (whichever it is) by Liouville, the same one who wrote Liouville's Theorem, back in the nineteenth century. So here is an irrational or perhaps even a transcendental number whose value is not random. <Sigh.>

I think that I see where the arguments about measurement theory and chaos are coming from, and I have an observation that, taken charitably, makes both llanitedave and Barbarian right (and taken uncharitably makes you both wrong, but I prefer the charitable interpretation). The difference between you is a difference in approach. llanitedave likes real systems; Barbarian is thinking of mathematical systems.

Real systems have a bottom level, where they cannot become simpler (or at least as far as we can tell, they cannot do so without completely changing character). I am speaking of the level of quarks, leptons, and force-exchange particles (there unfortunately does not appear to be a collective name for this last group). There may be something smaller than quarks and leptons that makes them up, but if there is, it is not merely some smaller particle- it will have to be inherently of a different character than quarks or leptons to give them the characteristics they must have to exist and explain our macroscopic world in detail. As a result of this, there is a fixed base level of complexity below which the real world (as we know it now- I will not speculate footlessly here) does not go. Eventually, you will always come down to a requirement for a precise definition of both position and momentum, and you will always be prevented from determining both with unlimited accuracy. Not merely due to any inability to measure, either; experiments such as Aspect's realization of the Einstein-Podolsky-Rosen experiment, when combined with Bell's Theorem, make it clear that it actually doesn't have any value. And in the face of this, it is ridiculous, ultimately, to state that any real process can ever be said to be "deterministic."

Mathematically speaking, however, there is no such "bottom end" limit. In fact, it is a characteristic of systems that display non-linear dynamics that the boundaries between various strange attraction zones become literally infinitely complex. No matter how closely you examine such a boundary, there is always detail that fades out of sight, becoming smaller and smaller until it is lost beyond your ability to make it out. Magnify that, and you will find yet another area where the same thing happens. And this is a known feature of the graphs of many if not most such functions. For instance, given a mathematical method (specifically, Newton's method) for calculating the roots of polynomials with a certain number of answers (and of course, a certain degree of polynomial which determines it), you will find experimentally that there are as many regions of strange attractors as there are possible roots (which of course is the same as the degree of the polynomial equation). Now, in the areas between those attractor basins, you will find regions generally halfway between the centers of the basins where any two points that appear adjacent and go to the two closest attractors are always separated by at least as many points as there are other attractors. And the deeper you look, the more complex this boundary becomes, just as irrational numbers multiply whenever you try to enumerate all of the irrationals between two rationals, making the reals uncountable and inenumerable. I do have to point out, however, that every single point where an exact value is defined results in one of the attractors; however, in the regions of confusion midway between two basins, one cannot be sure that the result will wind up in either adjacent attractor.

I'm sure you both can see how this makes both of you right (and both of you wrong, but we're not concentrating on that). I'll also point out that there are no answers more correct than your two, and many answers that are much less correct. Which makes both of you just about as right as it is possible to be in respect to this question.

I'll point out just a few pitfalls to consider:
1. Who's to say there isn't a level of reality deeper than particle-based quantum field theory? Not me, and not anyone at this time. There are at least two theories that involve deeper structures that could turn out to be the basis on which quantum field theory rests.
2. It is unlikely that we will ever develop simulations for most of the types of hard problems that yield complexity (chaos). When the generation of shapes of the types seen in graphs of functions at the hearts of such problems are made, there may be no faster way to compute the points than the fractal function itself. I know of at least one class of such problems for which this has been proven.
3. Picking any truly random output, if you wait long enough, you will always see any finite collection of values that appears non-random; just as if you wait long enough, you will get as many heads in a row as you care to name. While it is unlikely that any extremely large number of heads in sequence will happen in many, many times the expected lifetime of the universe, it is equally certain that given long enough, such an outcome is inevitable. So the incredibly improbable becomes the unavoidable. For instance, somewhere in the decimal value of pi, there is a string of ten thousand fives in sequence.

HTH, although I suspect it won't. ;)

[epepke]

I do not think there are such definitions of random which do not fall back to individual outcomes being unpredictable.

From dictionary.com:

1. Having no specific pattern, purpose, or objective: random movements. See Synonyms at chance.
2. Mathematics & Statistics. Of or relating to a type of circumstance or event that is described by a probability distribution.
3. Of or relating to an event in which all outcomes are equally likely, as in the testing of a blood sample for the presence of a substance.

I am not sure what this means. Are we going to elaborate upon S(.) later, but declare beforehand that its members are random numbers if S(.) conforms to some criteria subsequently provided?

Yes, but you already figured that out.

Some information is needed, or else we do not know the range of the number, and must assume all (natural?) numbers are equally probable, which is undesirable.

OK, then no more than O(1) information.

Does having S(1..k) as information mean that we know s(1), s(2), ... s(k)? I cannot interpret it in any other way, and knowing the last element, s(k), does indeed help predicting s(k). I assume you meant S(1..k-1).

No; I mean including s(k). Otherwise, you could construct a sequence where s(k) = sum(s(j)) with j = [1, k - 1], which would be quite deterministic.

But it doesn't have to be the exact set S(1..k). One could try to predict s(k) from another sequence R or set M. But there cannot exist such sequences R or sets M with less information in the Shannon sense than S(1..k), or else the sequence is not random.

[Preno]

I think we already have a sufficient definition of randomness - a string (of bits) is random if it cannot be compressed any more (i.e. if there is no program able to produce the string shorter than the string itself). I think that definition is generally better, since it does not need to use the notion of "it couldn't have been otherwise".
I had thought to name the digits of irrational or transcendental numbers as random.Computable numbers cannot be random, since they can be easily defined using a finite number of signs. Now, as for uncomputable transcendental numbers... erm, I don't know. Guess they can be called random in a certain way.

[Barbarian]

You start out saying you disagree, and then you spend the whole paragraph practically copyrighting my logic!
That may well be the case - misunderstandings abound, and I am always unsure I understood what I reply to. I only disagreed with the statement that chaotic systems are random, and the second paragraph in the quote was not directly linked to our disagreement.
Except for one or two points: I disagree that " those are practical issues, not theoretical ones". They are very much theoretical. They utilize ideas from measurement theory, which states essentially that all measurements are approximations.
Measurements indeed are always approximations, especially if we set out to measure something that by definition has no set value - QM teems with such examples. I was not explicit in stating that I think there is a definite state - characterized by whatever form of value, be it a complete wavefunction or something simpler, like total energy - only we cannot access it. Is that assumption incorrect?
Therefore, you cannot in principle *prove* that "for every starting state it has a single, well-determined final state", because you never know precisely what that starting state is.
I may be misunderstanding you again. Are you saying that if I happen to know that a system is deterministic, but do not exactly know the initial state and am therefore unable to predict the final state, then there is no definite final state?
The unpredictability is a matter of principle. What *IS* predictable is that the final state will lie somewhere within the bounds of whatever "attractor" the chaotic system defines. But that's really the only difference between chaos and randomness.
That is where we differ. I do not think there is any principial unpredictability at all in the case of a deterministic system, chaotic or not. There is a not-satisfactory-precision-of-knowledge-for-initial state problem instead, but that does not matter here, even if for some systems it proves fatal in practice. Of course, we are not talking about practical predictability here.

In retrospect, it was a poor choice from my part to use "predictability", because that word is too intrinsically linked to practical considerations. Should have used "having a determined, mechanically reached final state".
I'm no expert, of course, and you very well may be, but I think we're approaching a blurring of boundaries here that will satisfy neither one of us.
I am not an expert either, far from it, and I did not attempt to create that illusion.

[Barbarian]

I had thought to name the digits of irrational or transcendental numbers as random. However, I found that irrational numbers had been constructed by associating digits after the decimal point, in the decimal system, with the whole numbers, and then setting every one that corresponded to a perfect factorial to one and all others to zero. This is certainly an irrational number, and it might even be transcendental (IIRC, it is). This was proven (whichever it is) by Liouville, the same one who wrote Liouville's Theorem, back in the nineteenth century. So here is an irrational or perhaps even a transcendental number whose value is not random. <Sigh.>
It is transcendental as well, if you mean Sum[1/2^(n!), {n,0,Infinity}]. To my horror, I realized I do not remember the proof ... As another example we have a pretty simple algorithm to calculate the nth hexadecimal digit of Pi, so the sequence of hexa digits of Pi is also not really random.
I think that I see where the arguments about measurement theory and chaos are coming from, and I have an observation that, taken charitably, makes both llanitedave and Barbarian right (and taken uncharitably makes you both wrong, but I prefer the charitable interpretation). The difference between you is a difference in approach. llanitedave likes real systems; Barbarian is thinking of mathematical systems.
It would be cleaner to say that I am thinking of certain abstractions of physical systems, where the abstraction consists of accepting the existence of exact initial states, respecting known physics but disrespecting practical difficulties, including difficulties potentially taking an infinity of time to solve. I feel this is a valid way of thinking about real systems and not just a fatally flawed attempt of modeling reality with math - unless we go a bit positivist and say that a real system has a state only as far as it can be measured exactly, in which case no system has a state.
Real systems have a bottom level, where they cannot become simpler (or at least as far as we can tell, they cannot do so without completely changing character). I am speaking of the level of quarks, leptons, and force-exchange particles (there unfortunately does not appear to be a collective name for this last group). There may be something smaller than quarks and leptons that makes them up, but if there is, it is not merely some smaller particle- it will have to be inherently of a different character than quarks or leptons to give them the characteristics they must have to exist and explain our macroscopic world in detail. As a result of this, there is a fixed base level of complexity below which the real world (as we know it now- I will not speculate footlessly here) does not go. Eventually, you will always come down to a requirement for a precise definition of both position and momentum, and you will always be prevented from determining both with unlimited accuracy. Not merely due to any inability to measure, either; experiments such as Aspect's realization of the Einstein-Podolsky-Rosen experiment, when combined with Bell's Theorem, make it clear that it actually doesn't have any value. And in the face of this, it is ridiculous, ultimately, to state that any real process can ever be said to be "deterministic."
Question here: why would I be reduced to the need to know the position and the momentum at the same time? Now that is an impossibility I would respect. The total wavefunction of the system at the beginning of the experiment is just as a valid initial state "value", and it evolves deterministically, at least up to the first measurement or even all the way to the end if we consider relative-state interpretation. We probably cannot know the initial wavefunction either, but that is a restriction I do not respect, because as far as I know the wavefunction still exists down there, even if we cannot measure it. If QM gets replaced with something really weird, we will still have a notion of state and deterministic evolution: all mathematically supported models of reality we had so far had them. So in such a theory the random-equals-multiple-next-states definition would keep working.
Mathematically speaking, however, there is no such "bottom end" limit. In fact, it is a characteristic of systems that display non-linear dynamics that the boundaries between various strange attraction zones become literally infinitely complex. No matter how closely you examine such a boundary, there is always detail that fades out of sight, becoming smaller and smaller until it is lost beyond your ability to make it out. Magnify that, and you will find yet another area where the same thing happens. And this is a known feature of the graphs of many if not most such functions. For instance, given a mathematical method (specifically, Newton's method) for calculating the roots of polynomials with a certain number of answers (and of course, a certain degree of polynomial which determines it), you will find experimentally that there are as many regions of strange attractors as there are possible roots (which of course is the same as the degree of the polynomial equation). Now, in the areas between those attractor basins, you will find regions generally halfway between the centers of the basins where any two points that appear adjacent and go to the two closest attractors are always separated by at least as many points as there are other attractors. And the deeper you look, the more complex this boundary becomes, just as irrational numbers multiply whenever you try to enumerate all of the irrationals between two rationals, making the reals uncountable and inenumerable. I do have to point out, however, that every single point where an exact value is defined results in one of the attractors; however, in the regions of confusion midway between two basins, one cannot be sure that the result will wind up in either adjacent attractor.
The issue is that some people will then claim that since we cannot be sure of this, the brain can generate free will, and God can instrument miracles, to name just the two most outstanding claims. But that is just it: we cannot be sure of the outcome, so what? The final state is still written in the book of fate, even if we cannot read it - you get my drift.

[jcsd]

I had thought to name the digits of irrational or transcendental numbers as random. However, I found that irrational numbers had been constructed by associating digits after the decimal point, in the decimal system, with the whole numbers, and then setting every one that corresponded to a perfect factorial to one and all others to zero. This is certainly an irrational number, and it might even be transcendental (IIRC, it is). This was proven (whichever it is) by Liouville, the same one who wrote Liouville's Theorem, back in the nineteenth century. So here is an irrational or perhaps even a transcendental number whose value is not random. <Sigh.>

What is being got at is the cocnept of a normal number which is basically what epeeke is getting at i.e. it's a numebr whose digits shows the same distribution a if they were randomly generated (Liouville's constant is trivially non-normal in base-10 as it contians only two numbers in it's digits, as are all rational numebrs as they necvessarily contian recurring sequences).

However normal numbers cannot be described as truly random, as random simply means that something is govenred by probabilty and normal numebrs cna be generated without the recourse to probabilty.

For example the following number is transcendental and normal in base-10:

Champernowne's constant : 0.1234567891011121314........

[Barbarian]

I think we already have a sufficient definition of randomness - a string (of bits) is random if it cannot be compressed any more (i.e. if there is no program able to produce the string shorter than the string itself). I think that definition is generally better, since it does not need to use the notion of "it couldn't have been otherwise".
This definition simply says that we can use such a sequence as a random bit generator (by returning the bits in order one by one) if it is maximally difficult to predict them. As such, it does not deal with the randomness in real systems at all, and this is my first issue with it.

My second issue is that the string is completely predictable (even if by a long algorithm), hence non-random. We can note, though, how much related this definition is to the proposed one, which postulates principial unpredictability due to insufficient restrictions within the structure of the world, while this one emphasizes maximal difficulty - but not impossibility - for prediction.

My last issue is, how do you apply this definition to a single event? Tossing a coin once yields a random measurement. How do we determine if it was random or not?

In a world where algorithms and finite strings of bits are the only reality, this definition would indeed be the closest thing to random.

[Schneibster]

It is transcendental as well, if you mean Sum[1/2^(n!), {n,0,Infinity}]. To my horror, I realized I do not remember the proof ... As another example we have a pretty simple algorithm to calculate the nth hexadecimal digit of Pi, so the sequence of hexa digits of Pi is also not really random.Well, technically, pi was originally discovered as the ratio of a circle's diameter to its circumference on a plane. So, technically, it has to have an exact value; but when we represent it as a decimal number, the sequence of digits is random. "Random" does not mean "incomputable;" it's a slipperier concept than that.

It would be cleaner to say that I am thinking of certain abstractions of physical systems, where the abstraction consists of accepting the existence of exact initial states, respecting known physics but disrespecting practical difficulties, including difficulties potentially taking an infinity of time to solve. I feel this is a valid way of thinking about real systems and not just a fatally flawed attempt of modeling reality with math - unless we go a bit positivist and say that a real system has a state only as far as it can be measured exactly, in which case no system has a state.My thinking was that you were concentrating more on the math than on the reality. In math, of course, there can be true randomness, and it is embodied in the math that generates chaotic results.

Question here: why would I be reduced to the need to know the position and the momentum at the same time? Because if you do not, then either the position or momentum must have a random value, and measurements in such circumstances indicate that they actually do have such a random value. Your point was that chaotic systems are nevertheless deterministic; my point was that real systems are never deterministic, ultimately. At the bottom, their behavior is and must be probabilistic. And, as I said above, "probabilistic" is the definition of "random" in mathematics.

Now that is an impossibility I would respect. The total wavefunction of the system at the beginning of the experiment is just as a valid initial state "value", and it evolves deterministically, at least up to the first measurement or even all the way to the end if we consider relative-state interpretation. We probably cannot know the initial wavefunction either, but that is a restriction I do not respect, because as far as I know the wavefunction still exists down there, even if we cannot measure it. If QM gets replaced with something really weird, we will still have a notion of state and deterministic evolution: all mathematically supported models of reality we had so far had them. So in such a theory the random-equals-multiple-next-states definition would keep working.Hmmm, wavefunction determinism. Strictly speaking, the definition of "random" in mathematics is "probabilistic." And determinism states that all future states can be determined from the current state. So what we're saying here is that QM provides "deterministic randomness." Which is true. But from the viewpoint of classical determinism, there is no such thing. And that was my point.

Another point worth noting is that the wavefunction does not tell you what you will measure; and science is about what you measure. When you measure, you get a value- but the wavefunction can only tell you about the eigenvalues of the function. I would argue that the wavefunction does not actually tell about state- only about eigenstates. Einstein argued that it does not constitute a complete description of reality for precisely this reason. The only thing he failed to take into account was that it is possible that there is no complete description of reality possible on the terms that we are used to thinking of such things, biased as we are by our day-to-day experience. As far as we can tell, that is indeed the case.

The issue is that some people will then claim that since we cannot be sure of this, the brain can generate free will, and God can instrument miracles, to name just the two most outstanding claims. But that is just it: we cannot be sure of the outcome, so what? The final state is still written in the book of fate, even if we cannot read it - you get my drift.No, there is no such writing. The probabilities of all possible outcomes might be written there, but reality does not contain a description that allows even the theoretical determination of the outcome; it is not deterministic.

On the other hand, each and every point on the graph of solutions by Newton's method to the polynomial has a real value, and results in one and only one of the possible roots. So the mathematics of chaos is, as you maintain, deterministic.

My point was about recognizing this difference between the natures of mathematics and of physical descriptions of the real world.

[Schneibster]

What is being got at is the cocnept of a normal number which is basically what epeeke is getting at i.e. it's a numebr whose digits shows the same distribution a if they were randomly generated (Liouville's constant is trivially non-normal in base-10 as it contians only two numbers in it's digits, as are all rational numebrs as they necvessarily contian recurring sequences).

However normal numbers cannot be described as truly random, as random simply means that something is govenred by probabilty and normal numebrs cna be generated without the recourse to probabilty.

For example the following number is transcendental and normal in base-10:

Champernowne's constant : 0.1234567891011121314........Yeah, another good example.

But I'd also like to point out that while computable, the sequence of digits that represents pi is also random, in that you cannot predict the next digit knowing all the previous ones; the only guide is the computational method.

The first part of Preno's point that uses the information theory definition of random is strong; however, I'd argue that "computable" and "random" need not be mutually exclusive.

[jcsd]

Yeah, another good example.

But I'd also like to point out that while computable, the sequence of digits that represents pi is also random, in that you cannot predict the next digit knowing all the previous ones; the only guide is the computational method.

The first part of Preno's point that uses the information theory definition of random is strong; however, I'd argue that "computable" and "random" need not be mutually exclusive.

Pi is a bad example as we don't even know if it is a normal number or not.

The defintion of random is simply something governed by probailty, it's that simple. A random sequence is one where each memebr is generated randomly, but soemtimes we talk about 'randomness' loosely as for example in sequenecs which share properties with random sequences. A sequence where all terms are independent (and hence cannot be compressesd), must be generated randomly.

[Barbarian]

Because if you do not, then either the position or momentum must have a random value, and measurements in such circumstances indicate that they actually do have such a random value. Your point was that chaotic systems are nevertheless deterministic; my point was that real systems are never deterministic, ultimately. At the bottom, their behavior is and must be probabilistic. And, as I said above, "probabilistic" is the definition of "random" in mathematics.
I do not claim to be an expert on probability theory, and it is quite possible that I am behind the curve on this one, but I got the distinct feeling that math is carefully avoiding any reference to the source of randomness, and that probability theory is not a theory of randomness but a subset of measure theory. So the definition of random as probabilistic is the way of mathematics to avoid the issue of the origin of real randomness altogether.
Another point worth noting is that the wavefunction does not tell you what you will measure; and science is about what you measure. When you measure, you get a value- but the wavefunction can only tell you about the eigenvalues of the function. I would argue that the wavefunction does not actually tell about state- only about eigenstates. Einstein argued that it does not constitute a complete description of reality for precisely this reason. The only thing he failed to take into account was that it is possible that there is no complete description of reality possible on the terms that we are used to thinking of such things, biased as we are by our day-to-day experience. As far as we can tell, that is indeed the case.
There is always the caveat that we may not know some basic fact or misunderstand the world completely, trying to force it into a model that does not fit - in fact, this is more likely than not to be the case. But there is absolutely nothing we can do about this, so we may as well resign at this thought and make bold statements based on our current knowledge with the implicit caveat that all this may be completely otherwise. IOW I am pretty happy to indulge in mental exercises considering our current knowledge as the final one, while on a parallel track I am aware of the historic precedents of such sets of knowledge proving themselves false, approximative or incomplete. We can simply co-opt absolute statements, themselves impossible to verify, to act in the place of conditional statements with the implicit caveat of "if certain parts of our knowledge are exactly right, then ...". As far as I can tell, this is a pretty common approach, albeit not a very explicit one.

With that qualification, the way I remember the QM texts I had access to - granted, all of an introductory level -, it seems to me that the wavefunction is the state, and using positions and momentums is a misdirected attempt, caused by their apparent validity in our natural range of sizes, speeds, durations and energies. So a system description using actual position and momentum at the same time is simply wrong in the light of our current knowledge.
No, there is no such writing. The probabilities of all possible outcomes might be written there, but reality does not contain a description that allows even the theoretical determination of the outcome; it is not deterministic.
My bad - in the paragraph you replied to, I was already operating under the implicit context of talking about a deterministic mechanism, because that was the issue being dealt with before. A non-deterministic mechanism (which I am pretty obviously trying to equate with random mechanism) will appear in the "book of fate", with all branches plus their respective probabilities, just as you said.

[Barbarian]

But I'd also like to point out that while computable, the sequence of digits that represents pi is also random, in that you cannot predict the next digit knowing all the previous ones; the only guide is the computational method.
But the computational method can be restarted at each digit from scratch, just knowing what position you are aiming at, thus predicting the next digit just from knowing the number of previous digits. That sounds a stronger computability property to me than having to know the values of previous digits too.
The first part of Preno's point that uses the information theory definition of random is strong; however, I'd argue that "computable" and "random" need not be mutually exclusive.
I would like to see an example of random and computable, because otherwise I would argue for the opposite, namely, that random and computable (or predictable in any theoretical way) are mutually exclusive, although not a complete partition of all cases. Computable series are called random but what the definitions behind their randomness really mean is that they rather look like random sequences, to the extent possible under the circumstances. This says nothing about the nature of their source, and I am trying to pinpoint the source of randomness.

[llanitedave]

I think that I see where the arguments about measurement theory and chaos are coming from, and I have an observation that, taken charitably, makes both llanitedave and Barbarian right (and taken uncharitably makes you both wrong, but I prefer the charitable interpretation). The difference between you is a difference in approach. llanitedave likes real systems; Barbarian is thinking of mathematical systems.



Within the caveats and clarifications that Barbarian has mentioned, I think your impression is the right one. Some would argue that there is no deep difference between the real and the mathematical, and they may be deeply right, in which case our difference is probably one of scale.

[Barbarian]

epepke - thanks for the definitions. I was going to bring them up myself, but you spared me the trouble. Let me just quote your mail - this is not arguing with you but exemplifying how definition attempts converge to my proposal.
1. Having no specific pattern, purpose, or objective: random movements. See Synonyms at chance.
This definition is a bit fuzzy but promising. Does it mean that we cannot discern the governing law selecting the outcome, or that it really does not exist? In the second case, it is synonymous with my proposal; in the first case, it incorporates things that look random because of our ignorance. But I would argue against including these cases into random, because their randomness is contingent upon our ignorance, and as we learn, the randomness dissipates. A truly fundamental, metaphysical characteristic does not behave this way.
2. Mathematics & Statistics. Of or relating to a type of circumstance or event that is described by a probability distribution.
This is tricky, as math appears to completely bypass the issue of whence random is coming from. There is no discussion, notion or idea in mathematics that I know of referring to the origin of randomness. If the above definition means that all we can know is the probability distribution, then this is equivalent to my proposal. If it means that we may have other information besides the distribution but we chose to deal with the case based on the distribution, then we have a wider definition than my proposed one, and the extra cases allowed under the umbrella of "random" are the ones which can be studied via the tools originally developed for random events. But this is just a predictable property of those tools, not a property of randomness for the underlying events. Obviously, if math does not really care about sources of randomness, it will not restrict its study to really random phenomena, will implicitly study a wider range, and the tools developed will be suitable for this wider range. What we would be looking at here would be a re-definition of random to include all phenomena manageable by the tools of probability calculus and neighbors, which did not start out to deal with random only to begin with.

It is true that there is a wide range of literature on pseudorandom numbers, i.e. how to make do best within the constraint of not really having cheap physical random generators around, and when are such pseudorandom sequences acceptable as approximating random as much as possible. But this is again not a theory of randomness: it is a theory of what can be passed instead of truly random numbers, given the severe limitations (generation from software etc.) They are the best approximation we can provide from software alone, and we may call them random because nothing more random-like can be provided; but the processes generating them are still deterministic. The rest of the program can use them as random numbers because it is unlikely that the program will accidentally have some property allowing it to predict the next number (although I have heard a hint that virtual entities in some genetic algorithm figured out the pseudorandom generator to some extent - was it Avida? got to check) but that is local ignorance. In the global scheme of things the pseudorandom generator is as deterministic as one can get.
3. Of or relating to an event in which all outcomes are equally likely, as in the testing of a blood sample for the presence of a substance.
This definition is completely unacceptable and I do not believe anyone actually employs it. I have heard it proposed myself, so I know there are people who think that this is the definition, but the same people obviously hold a very different intuitive notion of random. What we have here is mere sloppiness, people have generally pretty correct intuitive ideas but cannot formulate them into definitions, and that is what we are looking at here. Just consider the example given in the definition: presence of a substance in a blood sample has a 50-50 distribution? It is like the proverbial 50% probability of winning the lottery - I will either win or not, two cases, fifty-fifty. Same fallacy here. Or consider anyone giving you the above definition: will the claim that something having a 45% chance of happening is not random? Will he claim that the sum of points from two rolls of a die is not random (by virtue of not having a uniform distribution), although both points are? I do not think so.

[Barbarian]

Within the caveats and clarifications that Barbarian has mentioned, I think your impression is the right one. Some would argue that there is no deep difference between the real and the mathematical, and they may be deeply right, in which case our difference is probably one of scale.
Yes, the existence or lack of that difference is the core of almost all disagreements we had here. I feel a bit awed: attempting to resolve a simple - well, in retrospect not that simple - question brings us to these metaphysical heights/depths? I do not feel I can go further that way: I am already happy if I can articulate my metaphysical stance; defending it feels like too much to ask. Perhaps we can agree that if there is an exact mathematical reality down there, then the only source of randomness is the lack of complete constraining provided by natural laws?

The main thing I learned here is this: in a lousy approximation and due to lack of better words, we can say that I hold platonian views. Moreover, this is far from being a common position, as there are huge underlying assumptions I assumed to be common ones, but they apparently are not so common after all. The main one of them is that there is an exact reality deep down even if we cannot know it. That makes me a platonian. I think that meaningful parameters have exact values even if we do not know them, or even if we do not even know what the correct choice of parameters is.

I have just thought of a litmus test for holding such platonian views. You are a platonian if you agree with the following statement: it makes sense to ask whether a dimensionless universal (physical) constant is a transcendental number or not. I agree with this statement, see if you do too. (If you do, you may be up to further argument re: randomness.)

[Schneibster]

I do not claim to be an expert on probability theory, and it is quite possible that I am behind the curve on this one, but I got the distinct feeling that math is carefully avoiding any reference to the source of randomness, and that probability theory is not a theory of randomness but a subset of measure theory. So the definition of random as probabilistic is the way of mathematics to avoid the issue of the origin of real randomness altogether.Actually, this is true if and only if the sample space is not uncountably infinte, like the real numbers. If the sample space is countable, or even countably infinite, like the natural or whole numbers, then a sigma algebra can be defined on the subsets of the sample space, and measure theory must be applied; but if the sample space is uncountable, then the application of measure theory is unnecessary. There is only one sigma algebra, which becomes the entire sample space. In this second case, the source of the randomness is not measure theory, but the character of the sample space itself. You must differentiate between randomness due to the nature of the sigma algebra you use to characterize your measurements, and randomness due to the essential random character of the sample space; they are two different sources of randomness.

Mathematics in and of itself is deterministic; it is incapable of directly generating random numbers. There is no means of doing so in mathematics (unless I have forgotten something important; no doubt someone will step up to the plate if I have). Mathematics is also a language for talking about the properties of real-world systems. As such, it doesn't need to talk about any source of randomness; the randomness is presumed to be a characteristic of the system being analyzed.

There is always the caveat that we may not know some basic fact or misunderstand the world completely, trying to force it into a model that does not fit - in fact, this is more likely than not to be the case. I smell a reference to quantum mechanics- and I have to point out that because of its predictive power, quantum mechanics is unlikely to be supplanted by something that does not show pretty much all of the characteristics that quantum mechanics does with respect to randomness. Explaining the Aspect experiment without acknowledging that there is a measurable variable that does not actually have a defined value until it is measured is widely acknowledged to be impossible. But how can a variable actually not have a value in a real physical system? It is a most basic assumption, and yet Bell's inequality shows us that this absence of a value is a matter of actual physical fact. In other words, eigenstates are matters of physical reality, and they are distinguishable and therefore different from states, which are also matters of physical reality. To put this in perspective, it's as if I said that the Moon doesn't exist if no one's looking at it. (Nod to Einstein.)

But there is absolutely nothing we can do about this, so we may as well resign at this thought and make bold statements based on our current knowledge with the implicit caveat that all this may be completely otherwise. IOW I am pretty happy to indulge in mental exercises considering our current knowledge as the final one, while on a parallel track I am aware of the historic precedents of such sets of knowledge proving themselves false, approximative or incomplete. We can simply co-opt absolute statements, themselves impossible to verify, to act in the place of conditional statements with the implicit caveat of "if certain parts of our knowledge are exactly right, then ...". As far as I can tell, this is a pretty common approach, albeit not a very explicit one.You show the flexibility to follow a set of assumptions to their logical conclusion, an admirable quality. I disagree with your assessment of QM as being overturnable, at least in terms of the specific characteristics you seem most interested in; however, we could argue about it until we were both exhausted and probably not agree, and you are willing to postulate it is a final theory, which is sufficient, so let's agree to disagree pending any later discussion that might require us to examine the question again.

With that qualification, the way I remember the QM texts I had access to - granted, all of an introductory level -, it seems to me that the wavefunction is the state, and using positions and momentums is a misdirected attempt, caused by their apparent validity in our natural range of sizes, speeds, durations and energies. So a system description using actual position and momentum at the same time is simply wrong in the light of our current knowledge.This is true, in the sense that without measurement, the wavefunction encapsulates all we can know about a particle. However, from a more rigorous point of view, we also have to acknowledge that in the limit as an object becomes sufficiently large, and I'm still talking microscopic, just not sub-atomic, its wavelength becomes immeasurably short, and therefore its uncertainty of position and momentum becomes immeasurably small. Its wavefunction then becomes simple classical mechanics. And the wavefunction no longer describes probabilities at that point; they are all 1 or 0. Its eigenstates all become composed of single eigenvalues. Technically, the eigenstates still exist, but they are no longer differentiable from states. It is therefore difficult to argue that they are not indeed states. Thus, if we examine things that are small enough, or quick enough, or both, we can see that we enter a different regime; novel principles emerge, and must be applied in order to understand this different regime.

These principles provide a basis for randomness, and show that determinism cannot apply at this level.

Interestingly, the fluctuation theorem provides a basis for understanding how this non-deterministic "under-reality" can result in the classically deterministic everyday world. It also shows us that if we watch a system long enough, the indeterministic operations of this "under-reality" will make themselves felt, destroying the apparent determinism of the everyday macroscopic world. It has already shown us regimes on the microscopic scale in which the Second Law of Thermodynamics can be violated for periods of a few seconds.

My bad - in the paragraph you replied to, I was already operating under the implicit context of talking about a deterministic mechanism, because that was the issue being dealt with before. A non-deterministic mechanism (which I am pretty obviously trying to equate with random mechanism) will appear in the "book of fate", with all branches plus their respective probabilities, just as you said.Most gracious. We're very close to being on the same wavelength about this stuff. This should be a very interesting conversation. I certainly don't think it's done yet; I expect you'll have some interesting insights into the points I've brought up here.

[Schneibster]

Some would argue that there is no deep difference between the real and the mathematical, and they may be deeply right, in which case our difference is probably one of scale.I think this argument fails because although math can represent physical systems, it need not. In other words, math can represent systems that do not exist in the real world, and thus need not follow its rules. I'll pull up my Newton's-method example again and point out that if it were a representation of a real-world system, there would be a level of representation beyond which the graph would cease to describe the real world; but in the mathematical representation, it is infinitely complex. No real system can be infinitely complex, unless one speaks of the entire universe (and believes that the universe is infinite, a view to which I subscribe but against which there may be some arguments possible).

[Schneibster]

Yes, the existence or lack of that difference is the core of almost all disagreements we had here. I feel a bit awed: attempting to resolve a simple - well, in retrospect not that simple - question brings us to these metaphysical heights/depths? I have myself often wondered at the depth of these questions, despite their apparent simplicity.

I do not feel I can go further that way: I am already happy if I can articulate my metaphysical stance; defending it feels like too much to ask.Oh, I certainly hope not! :)

Perhaps we can agree that if there is an exact mathematical reality down there, then the only source of randomness is the lack of complete constraining provided by natural laws?I think there is not an exact mathematical reality in the sense you mean it, so I'd be more comfortable with a more hypothetical construction: if there were an exact mathematical reality underlying physics, then the only source of randomness would be the lack of our ability to completely constrain that reality by measurement. I must point out here that quantum mechanics states clearly that there is no exact mathematical reality; the only mathematical representation of reality that works is one that deals only with probabilities, not with exact values.

I think that meaningful parameters have exact values even if we do not know them, or even if we do not even know what the correct choice of parameters is.The problem with this is that experiment shows that in fact, there are meaningful parameters that cannot have exact values; if they did, the outcome of the experiment would be different.

[Preno]

This definition simply says that we can use such a sequence as a random bit generator (by returning the bits in order one by one) if it is maximally difficult to predict them. As such, it does not deal with the randomness in real systems at all, and this is my first issue with it.A valid point. But we essentially do get our information about the world in the form of bit strings (or rather, it can be converted to such a form). I can "compress" a bit string representing the development of a Newtonian system into a description of the initial state plus several basic rules. I can't do the same with quantum systems.
My second issue is that the string is completely predictable (even if by a long algorithm), hence non-random. We can note, though, how much related this definition is to the proposed one, which postulates principial unpredictability due to insufficient restrictions within the structure of the world, while this one emphasizes maximal difficulty - but not impossibility - for prediction.You're right here, but you're comparing two different notions of randomness.
My last issue is, how do you apply this definition to a single event? Tossing a coin once yields a random measurement. How do we determine if it was random or not?A single event (when considered separately) is random by definition.
"Random" does not mean "incomputable;" it's a slipperier concept than that.So what is it then? I argue pi is not random, because every digit of pi can be computed from a simple, finitely-long definition of pi.

[Schneibster]

So what is it then? I argue pi is not random, because every digit of pi can be computed from a simple, finitely-long definition of pi.OK, then show me how, simply knowing that the first seventeen digits of the decimal expansion of pi are 3.14159265358979323 and nothing else, we can find that the eighteenth is 8. Only by knowing the current state of our calculation can we know that; without the information of precisely what the current state of the calculation is, we must view that 8 as random.

There is no means that will reliably give us the next digit but a stateful calculation of the value of pi. Contrast this with the decimal expansion of 1/7, where, given any digit, we know what the next digit will be without having to use a stateful calculation of the decimal expansion; if the last digit is 2, then the next will be 8; if the last digit is 5, then the next will be 6. The value of any given digit of the decimal expansion of 1/7 is non-random; the value of any given digit of pi is random; pseudo-random at minimum.

Remember, pi itself is not random; but the value of any given digit in the decimal representation of pi is random. Computability and randomness are not mutually exclusive.

[jcsd]

OK, then show me how, simply knowing that the first seventeen digits of the decimal expansion of pi are 3.14159265358979323 and nothing else, we can find that the eighteenth is 8. Only by knowing the current state of our calculation can we know that; without the information of precisely what the current state of the calculation is, we must view that 8 as random.

There is no means that will reliably give us the next digit but a stateful calculation of the value of pi. Contrast this with the decimal expansion of 1/7, where, given any digit, we know what the next digit will be without having to use a stateful calculation of the decimal expansion; if the last digit is 2, then the next will be 8; if the last digit is 5, then the next will be 6. The value of any given digit of the decimal expansion of 1/7 is non-random; the value of any given digit of pi is random; pseudo-random at minimum.

Remember, pi itself is not random; but the value of any given digit in the decimal representation of pi is random. Computability and randomness are not mutually exclusive.

As I siad pi is a poor example as we don't know if it is a normal number (in any base) only that the distribution of it's digits are remarkably uniform for the first few million digits. Infact there is no proof that you need to no the preceding digts of pi to claauclate the nth digit, already one digit extracting formula for pi (in base-16) is known.

[Schneibster]

Really? That's very interesting. I'm repeating an argument from a math textbook; if it's wrong, I'm very curious to know about it. What is the formula?

I should point out that my contention was not that you need to know the previous digits; it was that knowing only the previous digits, you could not predict what the next one would be without actually continuing your calculation.

[jcsd]

http://mathworld.wolfram.com/BBPFormula.html

Can you predict the next digit in this sequence of digits:
0.1111111111111111111111111111111111111111111111111111111111111..?

The probelm I see with you rarguemnt is taht you are mereely stating that the digits of pi is non-cyclic which follows trivially from the fact that pi is irrational.

[Schneibster]

Fascinating! Thanks. The probelm I see with you rarguemnt is taht you are mereely stating that the digits of pi is non-cyclic which follows trivially from the fact that pi is irrational.No, actually I was arguing that there is no algorithm for finding digits. I think that your link blows it out of the water. I'll have to think about whether there are other irrational or transcendental numbers that might be computable but random.

[llanitedave]

Yes, the existence or lack of that difference is the core of almost all disagreements we had here. I feel a bit awed: attempting to resolve a simple - well, in retrospect not that simple - question brings us to these metaphysical heights/depths? I do not feel I can go further that way: I am already happy if I can articulate my metaphysical stance; defending it feels like too much to ask. Perhaps we can agree that if there is an exact mathematical reality down there, then the only source of randomness is the lack of complete constraining provided by natural laws?

The main thing I learned here is this: in a lousy approximation and due to lack of better words, we can say that I hold platonian views. Moreover, this is far from being a common position, as there are huge underlying assumptions I assumed to be common ones, but they apparently are not so common after all. The main one of them is that there is an exact reality deep down even if we cannot know it. That makes me a platonian. I think that meaningful parameters have exact values even if we do not know them, or even if we do not even know what the correct choice of parameters is.

I have just thought of a litmus test for holding such platonian views. You are a platonian if you agree with the following statement: it makes sense to ask whether a dimensionless universal (physical) constant is a transcendental number or not. I agree with this statement, see if you do too. (If you do, you may be up to further argument re: randomness.)

I would cordially dispute the idea that I am platonian in any significant way, and from what I have gathered up to now, the postulation of a deep reality is not necessarily a Platonic approach. It's a matter, as I understand it, on whether one regards reality as abstract or concrete. But then, I'm no philosopher.

Back to random, I'm at the point of wondering whether there is any such thing as a number that is "fundamentally" random. If we know that a pseudorandom sequence can be a deterministic algorithm, and we don't know whether any given sequence is truly random or pseudorandom, then how can we say that randomness can be anything beyond an observational bias?

Randomness, in other words, would be an artifact of our own limitations, not a property of the universe but an abstract concept of convenience. Like, perhaps, "good" and "evil".

[Preno]

OK, then show me how, simply knowing that the first seventeen digits of the decimal expansion of pi are 3.14159265358979323 and nothing else, we can find that the eighteenth is 8. Only by knowing the current state of our calculation can we know that; without the information of precisely what the current state of the calculation is, we must view that 8 as random.How does that have anything to do with the subject? Yes, if I know that there is a number that begins 3.1415, of course I don't know whether the next digit is 9 or 0. :huh:
There is no means that will reliably give us the next digit but a stateful calculation of the value of pi.There is the BBP formula. But even if it were true, so what?
Contrast this with the decimal expansion of 1/7, where, given any digit, we know what the next digit will be without having to use a stateful calculation of the decimal expansion; if the last digit is 2, then the next will be 8; if the last digit is 5, then the next will be 6.Again, so what?
The value of any given digit of the decimal expansion of 1/7 is non-random; the value of any given digit of pi is random; pseudo-random at minimum.So what is your definition of randomness?
Remember, pi itself is not random; but the value of any given digit in the decimal representation of pi is random. Computability and randomness are not mutually exclusive.Of course. Basically, you're saying "you're right, but if what you were saying was X, then you'd be wrong". I'm not saying that we can compute the next digit of a number based solely on its previous digits.

[Barbarian]

Back to random, I'm at the point of wondering whether there is any such thing as a number that is "fundamentally" random. If we know that a pseudorandom sequence can be a deterministic algorithm, and we don't know whether any given sequence is truly random or pseudorandom, then how can we say that randomness can be anything beyond an observational bias?
I would go beyond wondering and would explicitly propose that there is no such thing as a random number. Only physical processes - regarded as sources of events - can be random. The same sequence can come both from a random and from a deterministic source. So randomness is not a property of the result (number or sequence) but a property of a source, of a process generating results = outcomes. The various definitions of random, when applied to sequences, are just trying to say that those sequences can fill in for truly random results to the widest extent imaginable.
Randomness, in other words, would be an artifact of our own limitations, not a property of the universe but an abstract concept of convenience. Like, perhaps, "good" and "evil".
This is what I try to clarify. I do not think randomness is due just to limitations. What I feel is that random is used both for real randomness and for those cases where it is an artifact of our limitations, IOW we just don't know enough to successfully deal with the situation in any other way but using tools of probability calculus. I am proposing to drop the latter cases from the definition, and reserve random for those cases where removal of all limitations (knowledge of all physical laws and all characteristics of the process) would still result in an unpredictable outcome. The cases I propose to drop are really deterministic, although we cannot make use of this fact.

A drawback would be that under no circumstances would then we be able to prove whether something is truly random. There would be two kinds of processes, the proven deterministic ones and the maybe-deterministic-maybe-random ones. Moreover, I do not see how would a process ever make it to the proven deterministic crowd. So the definition proposal is gonna be useful only for thought experiments and for logical deductions of statements having the template "no matter whether a process is deterministic or random, the following holds: ...". E.g. "If brain processes are deterministic, free will is not free; if brain processes are random, free will is no will; in either case there is no free will". (This is an example only: IPU should protect us from turning this thread into a freewill one.)

This way we would also be exempt from making a stand about whether there are random processes in the universe at all, or conversely, whether there are deterministic processes at all. I remain agnostic on the issue, remarking that if QM holds true, then QM measurement would be the prototypical random process.

[Schneibster]

I am proposing to drop the latter cases from the definition, and reserve random for those cases where removal of all limitations (knowledge of all physical laws and all characteristics of the process) would still result in an unpredictable outcome. The cases I propose to drop are really deterministic, although we cannot make use of this fact.I see what you mean- but I don't agree with your conclusion. I'll say why in a moment.

A drawback would be that under no circumstances would then we be able to prove whether something is truly random. There would be two kinds of processes, the proven deterministic ones and the maybe-deterministic-maybe-random ones. We measure random events all the time. When a particular unstable nucleus decides to decay is random; so is when a particular muon decides to decay into an electron, or when a neutron decides to decay into a proton and an electron. (Yes, I know, there are neutrinos involved- it's aside from the point.) Yet, we can state that, given a collection of muons, or whatever, after time x, half of them will have decayed. This remains true whether we observe them as a mass or as individual particles one after another. And this is truly random; there is no "internal mechanism" that determines precisely when a muon decays. Muons are awfully simple particles; there simply is no room for any such "mechanism." There isn't anything we're going to find out later that will shed any light on this, either. I'm very confident in stating that. Muons and their decay have been studied for many, many years, and no one has found anything that either modifies their decay time, or correlates to it.

Moreover, I do not see how would a process ever make it to the proven deterministic crowd. Actually, we are aware of processes that are deterministic. For instance, if there is a muon, it will decay with a half-life of x to an electron. We have never seen a muon that does not. It is interesting, is it not, that the same process can be both random and deterministic? And this is a feature of all of QM. There are many random processes that have deterministic results.

I remain agnostic on the issue, remarking that if QM holds true, then QM measurement would be the prototypical random process.Actually, once a measurement is made, then the result is no longer random; you collapse the wavefunction from eigenstates with eigenvalues to a single state with fixed values, except for any value that may be complementary with another value under uncertainty. The randomness is inherent in the particles before they are measured.

[Barbarian]

I'm back! :wave:
We measure random events all the time. When a particular unstable nucleus decides to decay is random; so is when a particular muon decides to decay into an electron, or when a neutron decides to decay into a proton and an electron. (Yes, I know, there are neutrinos involved- it's aside from the point.) Yet, we can state that, given a collection of muons, or whatever, after time x, half of them will have decayed.
Careful now: all you can say is that the proportion of muons decayed after a while is a random number, which, if we set the time period to one half-life of muons, has an expected value of 1/2 and a very small standard deviation. This only allows us to say that we can expect about half of them to have been decayed during that time, but does not allow us to say that exactly half of them will have been decayed. (And that was the second case ever I used the past-in-the-future tense in English.)

In addition, if by measuring random events you meant our capability of measuring the lifetime of muons and seeing that they appear to have a lifespan with an exponential distribution, I will disagree. Such a measurement is not indicative of randomness. For a finite number of muons - and I expect we have and always will have been experimenting with a finite number of them - mechanisms may exist which simulate exponential lifespan distribution beyond even theoretical possibilities of revealing them as deterministic but complex mechanisms. Being able to verify that the group of muons as a group appears to decay in accord with theoretical predictions (which postulate their decay as being random) only means that we have once again failed to falsify the model, and therefore we are justified to treat it as more correct. But we say so many times in other threads - mostly cre/evo ones - that in science there is no proven theory, only not disproven ones, that we must stay consistent and not be so sure of anything here. Not that I expect the theory behind muon decay to be proven false, but we cannot rely on it when we talk about "all laws in the universe", as in "all laws of the universe fail to restrict the process to one single next state".
This remains true whether we observe them as a mass or as individual particles one after another. And this is truly random; there is no "internal mechanism" that determines precisely when a muon decays. Muons are awfully simple particles; there simply is no room for any such "mechanism." There isn't anything we're going to find out later that will shed any light on this, either. I'm very confident in stating that. Muons and their decay have been studied for many, many years, and no one has found anything that either modifies their decay time, or correlates to it.
I do not expect it either, but as I said above, we must be consistent. All we know is this: the theory is beautiful, so far it explained whatever we have found and it seems to imply that muon decay is random. We may use the word "verified" or "proven" for such a theory, but that is just co-opting the word since its original meaning is not applicable to anything.
Actually, we are aware of processes that are deterministic. For instance, if there is a muon, it will decay with a half-life of x to an electron. We have never seen a muon that does not. It is interesting, is it not, that the same process can be both random and deterministic? And this is a feature of all of QM. There are many random processes that have deterministic results.
Just again, I am arguing for a position which is dangerously close to kookery. Namely, I argue that we cannot ever be sure that a process is deterministic, because there is always a chance that some random variable is at work in it, but it has a 99.9999..... (many 9-s) ... chance of having the same value, so we practically never meet the exception. Stating that there is no internal structure, no complexity to hide such a variable sounds too final to me. Please understand that I do not argue that there is such a variable, or that we should even take the possibility into consideration, but I do argue that we should not be so sure of anything specific when we argue metaphysics.

If we abstract away far enough, all random processes will have deterministic results. For instance, in your case the abstraction consists of "what will this muon decay into?", and the answer is invariably "an electron etc. of course". The possibility of this kind of abstraction prompted me to attempt in the OP the ill-fated introduction of equivalence classes for outcomes. But we can apply such an abstraction that the process stays random, e.g. we define decaying before or after a given time as the two outcomes. In that case the process appears to be random.
Actually, once a measurement is made, then the result is no longer random; you collapse the wavefunction from eigenstates with eigenvalues to a single state with fixed values, except for any value that may be complementary with another value under uncertainty. The randomness is inherent in the particles before they are measured.
That is okay with me. I used the quantum measurement process as the prototype of random, even if I am not prepared to state that quantum measurement is random. (It was not proven to be deterministic, that is all.) In a measurement made on a system in a state which can be written as a superposition of different "measurement outcomes", the state actually selected by the measurement does not appear to be determined, according to our current knowledge, by any physical law, while physics somehow demands that after the measurement we only have one of those states. This is exactly what I mean when I say that all laws of physics fail to restrict the process to one single possible outcome, but they do restrict the process to have one of the possible outcomes. The difference is, I do not think we can argue for any actual process to be random or deterministic in this manner, we can only state that these are the only two possibilities, and proceed to prove other statements which would fail if there existed a non-random, non-deterministic process.

[Barbarian]

I think I owe an explanation as to the reason of this thread.

In the OP I said that clarifying the notion of random was an urge rising in the aftermath of some free-will threads. The case is a bit more complicated than that. There were two different issues discussed in threads I was participating in, and both of them stayed apparently undecided because of fuzzy thinking surrounding randomness.

One was the free will issue. I do not believe in free will, and tried to lay out my arguments for it. The argument went like this: assuming that the mind and the consciousness are just products of brain processes, these processes are either deterministic within a time period, or random. Deterministic implies lack of free will. Randomness means that the final selection of outcome (after we have the possible outcomes allowed by physical laws) is not due to anything, so up to the determination of possible outcomes it is deterministic, after that the decision has no agent, hence no one to possess the alleged freedom. Three rebuttals were forthcoming, one saying that we may someday discover a physical process neither random nor deterministic, and anyway who am I to force the universe into my false dichotomies; the second rebuttal went along the lines of postulating strong emergence of consciousness, that is, that it does not really matter what the underlying processes are; and finally, I was declared guilty of disregarding from the start the possibility of supernatural influence. I can argue with strong emergence, for it is but an ad-hoc rationalization; I concede that in the case of supernatural influence anything, even free will, is possible; but I am just flabbergasted by the first rebuttal. It was also formulated as "external stimuli do influence but do not determine my decisions", and that is a clear postulation of magic land between determinism and randomness. If we had a clear definition, that case would not have survived much dissection.

The second (actually, first in timing) was natural selection. I was somewhat careless and stated in a thread that to my best understanding natural selection is a random process - you can imagine the rest. Actually, the crowd was trying to stay polite, in a bared fangs, repressed ferocity kind of way. Of course, I cannot actually tell whether NS is random or not, but it does look random to me, and the issue was that many of the participants operated under the 'random equals uniformly random' misunderstanding. The main counterargument was that a random process converging to some degenerate distribution of states is actually deterministic, as in tossing a fair coin, losing one dollar for each tails and not winning anything for heads would be a deterministic process leading to depletion of all of your monetary resources. Another idea - more understandable for me - was that I should not use random in this sense, even if I am correct in doing so, because creationists would abuse my statements or the agreements of my opponents - as if my statement of NS being random carried any weight and meant anything else for the creationist cause apart from clearly grasping at straws. Anyway, the idea there was that we must not consider processes with pretty high probability of one outcome as random. A clear definition would have spared me from having to argue and finally give up on that thread.

Interestingly, both arguments accuse me of extending the scope of 'random' too much. And yet, in both cases there were examples thrown at me with the claim that they were random while they were clearly deterministic - chaotic behavior, for instance. So I decided to launch this thread to clarify the notions, to do away with magic lands between random and deterministic, to do away with chaotic being classified with random, to do away with uniformly random being equated to random. I am not really satisfied with the result so far, but that is in a big part due to my inability to participate in the discussion for almost a week now. On the positive note, for the following few days I may be able to put in more participation again.

I have obtained a copy of George Chaitin's Information, Randomness and Incompleteness, to shed some light on the less clear definitions of random - as in random sequence - offered here in this thread. I am far from having digested it, but so far my general feeling has not been changed compared to the one before the book: definitions of random sequences based on difficulty of predicting them algorithmically are attempts to find the notion closest to the real-world random in an artificially restricted universe where only deterministic algorithms exist. So no bearing to the real world case there, except that these attempts attest to the feeling of their proponents that random is deeply intertwined with unpredictability.

[Schneibster]

Glad you're back, Barbarian! This is an interesting discussion. I don't have time right now to respond- I might not this week. But I will- I'm putting this thread into my favorites right now. So give me a little bit and I'll get back to you.

[Schneibster]

Careful now: all you can say is that the proportion of muons decayed after a while is a random number, which, if we set the time period to one half-life of muons, has an expected value of 1/2 and a very small standard deviation. This only allows us to say that we can expect about half of them to have been decayed during that time, but does not allow us to say that exactly half of them will have been decayed. (And that was the second case ever I used the past-in-the-future tense in English.)And very nearly correct, too- this is one of the hardest parts of English, so no dis on you- the correct formulation is "will have decayed." The tensed version of "to be-" that is, "been-" would be correct- but not with that particular usage of "decayed." That usage would imply that a muon decays as a result of something else, which is not correct. It decays in and of itself. It would be correct to say that, for instance, a fallen tree in a forest "will have been decayed," because decay of a tree happens as a result of the acts of ants, and microbes, and so forth. A muon, however, decays as a result of its very nature, not because some outside force acts on it. "To be" in this formulation implies the existence of an outside agency, which exists in the case of the tree, but not in the case of the muon.

As far as your point goes, you are correct; it will be half within a standard deviation. I figured it was enough to say "half" without qualifying it as "about half." I don't think it makes a difference in my point, however, which was that the amount of time for the decay of any individual muon is random, and that that randomness is not the result of any underlying "mechanism," because a muon is too simple for there to be any such. And it even bolsters the point: it is so random that we have to add that it is not precisely half- it is within a standard deviation of half.

In addition, if by measuring random events you meant our capability of measuring the lifetime of muons and seeing that they appear to have a lifespan with an exponential distribution, I will disagree. Such a measurement is not indicative of randomness. For a finite number of muons - and I expect we have and always will have been experimenting with a finite number of them - mechanisms may exist which simulate exponential lifespan distribution beyond even theoretical possibilities of revealing them as deterministic but complex mechanisms. Being able to verify that the group of muons as a group appears to decay in accord with theoretical predictions (which postulate their decay as being random) only means that we have once again failed to falsify the model, and therefore we are justified to treat it as more correct. But we say so many times in other threads - mostly cre/evo ones - that in science there is no proven theory, only not disproven ones, that we must stay consistent and not be so sure of anything here. Not that I expect the theory behind muon decay to be proven false, but we cannot rely on it when we talk about "all laws in the universe", as in "all laws of the universe fail to restrict the process to one single next state".I respectfully disagree with this. Here is why: we know that a muon is extremely simple; it has only a very, very few characteristics. We know this because we have done many, many experiments with muons. This is not a matter of theory; it is a matter of observation. Characteristics show themselves by differences in interactions; one cannot see a muon. One can only observe its effects. Those effects include creating a trail in a bubble chamber, or cloud chamber, or photographic emulsion. If a speed measurement is made, and the path goes through an electric or magnetic field, then the mass can be determined by the deflection of the path. Through many other interactions, we can see its characteristics; and they are few, and simple:

Electric charge: -1
Mass: 105.6 MeV
Weak charge: -2
Spin: 1/2
Color charge: 0
Upness: 0
Downness: 0
Strangeness: 0
Charm: 0
Topness: 0
Bottomness: 0
Weak isospin: -1/2
Weak hypercharge: -1

And that's it. Note that the flavor variables (color, up/down/strange/charm/top/bottom) are all zero except weak charge. (Weak charge is a composite of weak isospin and weak hypercharge.) So a muon only has five characteristics. No other characteristics are possible, because if a muon had any other characteristics, it would interact due to them- and no other interactions have ever been seen. And this is over trillions upon trillions of interactions- and perhaps trillions is far too conservative; it might be many orders of magnitude beyond that. We have observed an incredibly huge number of muons, and not one single time has any interaction been seen that could not be accounted for by those five characteristics.

Now, this is a general property of quanta- they don't have very many characteristics. Think of a single germ cell- it has a truly huge number of characteristics; so many that it might exceed the number of muons in the visible universe; so many that we are justified in saying that it is not precisely the same as any other germ that has ever existed; in fact, it is not even identical to itself just a second ago. But a muon only has five. And every muon has those five, and they are all just the same. Which is another general property of quanta- they can be exactly the same as other quanta. No macroscopic object is like this; but all quanta are. And this is not a matter of theory; it is a matter of observation. We use the terminology, "the same," all the time for things that are not and cannot ever be truly "the same;" but in the case of quanta, the terminology is accurate.

Many, many different ways of describing the decay of muons into electrons using mathematics have been tried. Only one way yields the results we actually see: randomness. That's why we have to use a standard deviation on the half-life. If it were not truly random, we wouldn't need that standard deviation. But because it truly is random, we have to have it, or we're not really describing what's happening; the numbers we get from our equations will not be the numbers we get from observation. And remember: muons are very, very simple; they have only five characteristics. We know they can't have more, because if they did, they'd do things we don't see them do. The number of muons we have seen may be finite, but it is astronomical; it is a number so huge that we could never count it. So, can we be ABSOLUTELY CERTAIN? Hell no, we can NEVER be ABSOLUTELY CERTAIN. But can we be CERTAIN BEYOND A REASONABLE DOUBT? Yes indeed- and we passed that point long, long ago. Ask a physicist- and there are some who hang around here- how many characteristics a muon has. I guarantee they will give a small integer value. They just can't have any more than that; we'd have found out by now.

I do not expect it either, but as I said above, we must be consistent. All we know is this: the theory is beautiful, so far it explained whatever we have found and it seems to imply that muon decay is random. We may use the word "verified" or "proven" for such a theory, but that is just co-opting the word since its original meaning is not applicable to anything.

Just again, I am arguing for a position which is dangerously close to kookery. Namely, I argue that we cannot ever be sure that a process is deterministic, because there is always a chance that some random variable is at work in it, but it has a 99.9999..... (many 9-s) ... chance of having the same value, so we practically never meet the exception. Stating that there is no internal structure, no complexity to hide such a variable sounds too final to me. Please understand that I do not argue that there is such a variable, or that we should even take the possibility into consideration, but I do argue that we should not be so sure of anything specific when we argue metaphysics.By this rationale, we cannot ever discuss anything; nothing can be stated with this degree of certainty. Nothing ever. It is all based on what we have seen SO FAR.

I think this is your great difficulty: you have trouble with assumptions. I have to point out that it is an assumption on your part that there is actually a person writing this; it is an assumption on your part that you are reading this on a computer, that there is actually a real world "out there" that is creating the conditions that result in your sensory impressions; and it is even an assumption that the "you" that is reading THIS word here is the same "you" that is reading THIS OTHER word here. We need to agree on some set of minimal assumptions that we can use in this conversation; without that, I don't see how we can continue.

And one of those assumptions must be a reasonable standard of proof. If you require an infinite number of trials to prove any assertion, then NOTHING- and I do mean NOTHING- can ever be proven. On the other hand, if an astronomical number, but not infinite, is a REASONABLE standard of proof, then that standard of proof has been surpassed with regard to the decay of the muon. I state that this reasonable standard of proof is sufficient for me; and I encourage you to consider this matter most carefully.

Let me state that again another way: if you are going to require an absolute standard of proof, then you are inconsistent to fail to require it for what you yourself see, hear, taste, touch, and smell. And if you cannot trust that, then you cannot trust this conversation- in which case there is no point in having the conversation in the first place! So let's agree on a reasonable standard of proof, so that we can continue to chat! :D

By the way, this is not kookery; it might be from a scientific standpoint, but from a philosophical/metaphysical one, it certainly is not. Standards of proof must be discussed for this to be a complete conversation. So don't feel defensive about bringing it up. It's a valid question, in my eyes.

If we abstract away far enough, all random processes will have deterministic results. For instance, in your case the abstraction consists of "what will this muon decay into?", and the answer is invariably "an electron etc. of course". The possibility of this kind of abstraction prompted me to attempt in the OP the ill-fated introduction of equivalence classes for outcomes. But we can apply such an abstraction that the process stays random, e.g. we define decaying before or after a given time as the two outcomes. In that case the process appears to be random. But see, the fact is, it is random, if we accept a reasonable standard of proof. Basically what you're arguing here is that everything is not real; because it would be inconsistent to require this standard of proof for the muons, and not for everything else. This is a trap devised long ago by religious philosophers to trap the unwary.

That is okay with me. I used the quantum measurement process as the prototype of random, even if I am not prepared to state that quantum measurement is random. (It was not proven to be deterministic, that is all.) No, the proof is stronger than that. As I showed above.

In a measurement made on a system in a state which can be written as a superposition of different "measurement outcomes", the state actually selected by the measurement does not appear to be determined, according to our current knowledge, by any physical law, while physics somehow demands that after the measurement we only have one of those states. This is exactly what I mean when I say that all laws of physics fail to restrict the process to one single possible outcome, but they do restrict the process to have one of the possible outcomes. The difference is, I do not think we can argue for any actual process to be random or deterministic in this manner, we can only state that these are the only two possibilities, and proceed to prove other statements which would fail if there existed a non-random, non-deterministic process.I think you've gotten a bit sideways here. I'd like you to think over what I've said, and see whether this still makes sense in that light. I state again that "random" and "deterministic" are not mutually exclusive, but I also note that they are mutually exclusive when applied to a single value of a single state. It is only when we deal with eigenstates and eigenvalues that random outcomes can be deterministic.

[Barbarian]

My work prevents me from replying for such long periods that this thread could rightfully be nominated as the Glacier Thread of the year ...
And very nearly correct, too- this is one of the hardest parts of English, so no dis on you- the correct formulation is "will have decayed." The tensed version of "to be-" that is, "been-" would be correct- but not with that particular usage of "decayed." That usage would imply that a muon decays as a result of something else, which is not correct. It decays in and of itself. It would be correct to say that, for instance, a fallen tree in a forest "will have been decayed," because decay of a tree happens as a result of the acts of ants, and microbes, and so forth. A muon, however, decays as a result of its very nature, not because some outside force acts on it. "To be" in this formulation implies the existence of an outside agency, which exists in the case of the tree, but not in the case of the muon.
That would explain some weird reactions to the first case I used the tense. I am grateful for being corrected here, I can use all help I can get. The problem is that these tenses sound very exotic to me, as I cannot relate them to anything in my native tongue; in addition, I was already an adult when I started learning English, so the natural childlike ability to pick up a different language was mostly gone by then. Feel free to pick on my incorrect usages since I aim for perfection. Seriously.

OTOH, your guess is right, I did not want to imply the existence of an agent causing the decay.
As far as your point goes, you are correct; it will be half within a standard deviation. I figured it was enough to say "half" without qualifying it as "about half." I don't think it makes a difference in my point, however, which was that the amount of time for the decay of any individual muon is random, and that that randomness is not the result of any underlying "mechanism," because a muon is too simple for there to be any such. And it even bolsters the point: it is so random that we have to add that it is not precisely half- it is within a standard deviation of half.
I only asked because I have encountered opinions which would exactly correspond to the loose formulation you gave originally, and namely that statistical trends are deterministic. I was not sure if you were just trying to be succint assuming a common understanding or you were of the heretic sect of "anything with a high enough probability happens for sure". And if I may nitpick: one standard deviation is also not sure to contain the outcome.
I respectfully disagree with this. Here is why: we know that a muon is extremely simple; it has only a very, very few characteristics. We know this because we have done many, many experiments with muons. This is not a matter of theory; it is a matter of observation. Characteristics show themselves by differences in interactions; one cannot see a muon. One can only observe its effects. Those effects include creating a trail in a bubble chamber, or cloud chamber, or photographic emulsion. If a speed measurement is made, and the path goes through an electric or magnetic field, then the mass can be determined by the deflection of the path. Through many other interactions, we can see its characteristics; and they are few, and simple:

Electric charge: -1
Mass: 105.6 MeV
Weak charge: -2
Spin: 1/2
Color charge: 0
Upness: 0
Downness: 0
Strangeness: 0
Charm: 0
Topness: 0
Bottomness: 0
Weak isospin: -1/2
Weak hypercharge: -1

And that's it. Note that the flavor variables (color, up/down/strange/charm/top/bottom) are all zero except weak charge. (Weak charge is a composite of weak isospin and weak hypercharge.) So a muon only has five characteristics. No other characteristics are possible, because if a muon had any other characteristics, it would interact due to them- and no other interactions have ever been seen. And this is over trillions upon trillions of interactions- and perhaps trillions is far too conservative; it might be many orders of magnitude beyond that. We have observed an incredibly huge number of muons, and not one single time has any interaction been seen that could not be accounted for by those five characteristics.
This discussion is prone to be influenced by the minor fact that I know jack shit about muons ...

Fortunately, that is beside the point. I wandered off into the realm of what-can-we-know-for-sure just because I got that counterargument on other threads and wanted to play safe. If I make the statement that all processes in nature are deterministic, I get shot down. If I make the statement that all processes are random, I get shot down. If I say some are deterministic and some random, then one gang will ask how do I know there are deterministic ones, another gang will ask how do I know there are random ones and a third one will ask me how do I know these are the only alternatives. I really only want to elaborate on this last question. So if you think muons are really that simple as they seem to be, I can postpone arguing about that part.
Many, many different ways of describing the decay of muons into electrons using mathematics have been tried. Only one way yields the results we actually see: randomness. That's why we have to use a standard deviation on the half-life. If it were not truly random, we wouldn't need that standard deviation. But because it truly is random, we have to have it, or we're not really describing what's happening; the numbers we get from our equations will not be the numbers we get from observation. And remember: muons are very, very simple; they have only five characteristics. We know they can't have more, because if they did, they'd do things we don't see them do. The number of muons we have seen may be finite, but it is astronomical; it is a number so huge that we could never count it. So, can we be ABSOLUTELY CERTAIN? Hell no, we can NEVER be ABSOLUTELY CERTAIN. But can we be CERTAIN BEYOND A REASONABLE DOUBT? Yes indeed- and we passed that point long, long ago. Ask a physicist- and there are some who hang around here- how many characteristics a muon has. I guarantee they will give a small integer value. They just can't have any more than that; we'd have found out by now.
I get the idea that the chances of a muon having extra characteristics are competing with the theory starring Archangel Gibreel pushing muons around and ripping them off at random points in time. The only reason I descended on this scale of attempts for insane rigurosity was that I became, quite frankly, paranoid on this forum about leaving any gaps in an argument, as outlined above.
By this rationale, we cannot ever discuss anything; nothing can be stated with this degree of certainty. Nothing ever. It is all based on what we have seen SO FAR.
Perhaps I deleted in an editorial frenzy the part about co-opting absolute statements to mean "as far as we know ...". Anyway, here it is again: we use formulations like "we know that ..." and "it is true/false that ..." which cannot ever live up to their literal meaning. But exactly because they cannot ever be correct in their literal meaning, we can co-opt them to mean "as far as we know ..." or "since we have failed so far to disprove the theory that ...", because there is no danger of misunderstanding, as I cannot seriously mean using them in their literal sense.
I think this is your great difficulty: you have trouble with assumptions. I have to point out that it is an assumption on your part that there is actually a person writing this; it is an assumption on your part that you are reading this on a computer, that there is actually a real world "out there" that is creating the conditions that result in your sensory impressions; and it is even an assumption that the "you" that is reading THIS word here is the same "you" that is reading THIS OTHER word here. We need to agree on some set of minimal assumptions that we can use in this conversation; without that, I don't see how we can continue.
I assume there is a real world out there simply because this approach seems possible, and I cannot think of meaningful alternatives. I assume that there is a person writing answers to my posts because, according to my current understanding, the alternatives are too unlikely to warrant thinking about. If I misjudge the probability of the altenative, that's a risk inherent in being a sentient being, and I have no idea how to remedy that - nor does anyone else seem to be able to correct that situation.

I want to use in this particular case as few assumptions as possible. That is because assumptions like muon decays being actual cases of random processes will be perceived as attackable spots, and for instance the guys in the freewill thread will happily gnaw at such an assumption even if it is immaterial to the topic at hand, escaping thinking about the real matter completely. The only way, I thought, was to reduce the number of assumptions. For the sake of discussion, mind you.
And one of those assumptions must be a reasonable standard of proof. If you require an infinite number of trials to prove any assertion, then NOTHING- and I do mean NOTHING- can ever be proven. On the other hand, if an astronomical number, but not infinite, is a REASONABLE standard of proof, then that standard of proof has been surpassed with regard to the decay of the muon. I state that this reasonable standard of proof is sufficient for me; and I encourage you to consider this matter most carefully.

Let me state that again another way: if you are going to require an absolute standard of proof, then you are inconsistent to fail to require it for what you yourself see, hear, taste, touch, and smell. And if you cannot trust that, then you cannot trust this conversation- in which case there is no point in having the conversation in the first place! So let's agree on a reasonable standard of proof, so that we can continue to chat! :D
I do not think I hold anything to a higher or more absurd proof level than you do (well, maybe I do, but that's just a personality flaw). I trust statements "proven" by continued failure of disproof efforts as much as you do. I only reserve the right to talk about them in a nonstandard way, even if that means redefining "to know", or making the assumptions explicit. It is just unusual.

I do not "trust" my senses, I simply use them, without thinking about their trustworthiness all the time, basically because I know of no real alternative. How should I go about not trusting them? All I can think of is this: behave as if you trusted them and know that you have no basis to do so. I thought about this and came full circle back to the origin, just a little bit dizzy.

Douglas Adams does a better job at explaining this in the Restaurant at the end of the universe, chapter 29. The ruler of the Universe in that chapter does a good job staying consistent without too many assumptions.
By the way, this is not kookery; it might be from a scientific standpoint, but from a philosophical/metaphysical one, it certainly is not. Standards of proof must be discussed for this to be a complete conversation. So don't feel defensive about bringing it up. It's a valid question, in my eyes.
I would feel uncomfortable to use the muon decay example as a truly random event in the mentioned threads, because I have it only on authority, and it is not necessary. All I try to establish is that there are exactly two mutually exclusive categories of events. I try not to rely on extra assumptions because some people around here are happy to challenge them first and think about the importance of those assumptions later or not at all. I regard that as an unfair tactic, and have yet to find a good way to deal with it.
I think you've gotten a bit sideways here. I'd like you to think over what I've said, and see whether this still makes sense in that light. I state again that "random" and "deterministic" are not mutually exclusive, but I also note that they are mutually exclusive when applied to a single value of a single state. It is only when we deal with eigenstates and eigenvalues that random outcomes can be deterministic.
This is a difference in abstraction, and by now I am pretty sure I failed to get that idea through in the OP. Basically, we do not see processes but their abstractions. By abstraction here I mean the omission of some information, whereby states/outcomes which only differ in parameters described by suppressed information will merge into a single abstract state. Snapshots in the OP were meant to be basic states, with no information missing.

Assume we have a measurement apparatus which measures a certain parameter, like spin up/down, or energy level or something. Let us assume that the system to undergo measurement is not in an eigenstate of the operator corresponding to the measured parameter. Now if we ignore the information abotu which eigenvalue was selected, and focus on whether an eigenvalue was selected or something else, then we have a deterministic process. If we ignore extra information - I do not know what would be an example of a variable inconsequential to the measurement, but you probably would be able to bring up an example - and only consider some sort of index of the eigenvalue selected to be the outcome, then it is random. This could seem a bit contrived, but it is the only way to look at processes - we see only a number of parameters and cannot know of the existence and value of others, so we always must assume to look at an abstraction of a process. And two abstractions of the same process need not both be equally random or deterministic, although a deterministic process cannot be abstracted into random. Abstraction collates the allowable next states, and once they get down to a count of one, there is no way to increase that with further abstraction.

And now for going down the slope you offered: there is no law of physics forcing the measurement to yield one of the possible values. Laws of physics do demand the measurement to yield one of the eigenvalues but there is no law that would demand a certain one of these eigenvalues to appear in the measurement. At least that is what you get when you are sure we know something. I would probably only say that if there are random events, they would look very much the way a quantum measurement looks like.

So I guess I should have said: there are no ways to observe a process. We only observe abstractions of processes. Abstractions of processes are either deterministic or random.

[Schneibster]

My work prevents me from replying for such long periods that this thread could rightfully be nominated as the Glacier Thread of the year ... Heh, looks more like "glacier thread of the decade," being as how the New Year has come and gone. I had to think about this a long time to see some of the points you are making, and frankly I dropped in on the floor, not through lack of interest, but through lack of gumption. Your post in a related thread came to my attention, and brought this back to mind; I realized when I saw it that we had never finished the conversation.

I only asked because I have encountered opinions which would exactly correspond to the loose formulation you gave originally, and namely that statistical trends are deterministic. I was not sure if you were just trying to be succint assuming a common understanding or you were of the heretic sect of "anything with a high enough probability happens for sure". And if I may nitpick: one standard devi