Universe is not deterministic.

Guv,
Your post about my name, and my reply is in another area of chit chat, I can't rmember which, I will go look later.
Firts on fractals. You are wrong about this. On the open university, I saw them demonstatre that a fractal can be generated by random processe. The method was to have asquare, with a point out side it, at any point (a random point if you like) and mov it to within th squar, and then mov it random distances within the square, and drawing a line betwwen the points each time it moved. This lead to a fractal patteren contaning traingles. I assume that the open university would not delibratly misinform me, so if they are mistaken, then you can take it up with them.
As for th erandom number, it would be possible to say that a random number, as I said befor, is one made by a random process (which I have yet to define! Maybe I'll work it ot later.) Anyway, if a random sequence can be made by a random process, so can a random number (singular), allowing a random number, not sequence (if there are such tings as random proccses, if not then no, there are no random numbers!)
As for the other argument, it is silly. Inth e question you defineid an algorithim for that sequenc that is far shorter than the 'print digits'!! I f you hadn't, you would have had to write the whole dam thing out, rather than give a precise method with instructons for creating it, in a shortert way then writting the whole lot out! To make even shorter,
PrimeNumber to power of 200, split into 5 digit groups, leave remander in own group.
Nowhere in that definiton does it say the algorithim must be in mathimatical terms, and I have mearly used the most convienint language to express it! Anyway, my disscussion of random was to do with the idea of lack of order being an idae of random. If there was little oredr in the 5 digit blocks, they could well be random! YOu say, ah, they are not, look how I created them, but I can say, if I define random as a lack of order, the method of creation is irrelevent in a definiton of random. If I define it as a result of a random process, the of course they would not be random (again, assuming that random processes exist!)
You missed the main thread of my idea there, Guv, which was to do with the lack of order being a definition of random.
Anyway, the point of this forum was to discuss determinism, not randomness, so I will do that for a while.
As I said before, weher a process is probablistic or not, gives no proof either way of the determinisitic nature of the universe. Deterministic, or predictable if all rules and initial conditions are know, given enough time and computional power (brainds, computers, whatever). If an event defies predction, like 'Will it come down heads or tails?' or 'When will this specific atom decay' can either be unpredicitable either beacuse the conditions are not known (The power, spin of the coin, the position and momentum of atoms in ths air) or the rules governing the process are not known (Radioactive decay). You cannot prove it either way just by argument. Let me say that again. YOU CAN NOT PROVE IT EITHER WAY BY ARGUMENT!!! The deterministic/probalisticviewpoints do not allow for thier proof with the evidence presented, the definitoin of random being irelevant to the converstion, thoug I myself got sidetracked. Anyway, I'll leave that post there, just telling Guv that the name question is at the Strange Dice forum, check it out there.

I would think that there is very little that you can prove about one model in opposition to another purely by argument. The whole idea of physics is to provide a mathematical model for the real world, and in general we use the one that seems to describe the world around us most accurately. It's not really a case of whether you can prove it's true or not, just whether it's the closest model you've got.

I hope that makes sense to someone.

It does make sense, and I know that, but the argument here isn't 'Which provides a better model of the world', but 'which is right?'
Anyway, they both, with the evidence available provide equally good models, so a determination on this basis is impossible. Popper would have therefore said, if one cannot be disproved, it is not a godd hypothesis, With determinism, you cannot disprove it, as you can always say 'There are rules/factors you don't know about', and with probability, you can't disprove it, as saying something has a probability is not a straight disporvable concept. So, neither are good, useful hyptothisis, according to Popper!

Outside of this forum is a world of activities which often interferes with the critically important issues being decided here. I apologize for allowing a faulty sense of priorities to cause me to ignore this thread for about a week.

Before going into detail, let me state my basic belief about the way the universe functions.

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In previous posts, I claimed that restarting the universe (not really possible) would result in some (or many) events being different the second time around. This statement cannot be backed up by any logical argument. It is the way I believe a probabilistic universe would behave if restarted. What I think can be backed up by logical arguments is the statement that the universe is not deterministic. In principle, accurate predictions cannot be made.

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Various posters have requested a definition of random. Let me try to come up with a definition, and then further discuss the issue of the universe not being deterministic. As mentioned elsewhere, this and various other concepts cannot be understood via a dictionary-like definition. Some pertinent background is required.

I am sorry that I ever used the word random in this thread, because the issue being discussed could be dealt with using terms like probability (or probabilistic) and deterministic. Since it was used and has become an issue, a discussion of what is meant by random seems necessary.

Random suggests unpredictability, lack of pattern, unknown causes, et cetera. These terms are often used in definitions of random, but should be considered characteristics of random processes, not defining terms. It is analogous to considering the word nitrogen. At room temperature, it is a colorless, odorless gas. You could not define it precisely using terms like colorless, odorless, and gaseous. A given isotope of it can readily be defined unambiguously by its atomic number and atomic weight, but the other words are merely associated with properties of nitrogen. Unfortunately there is no definition of random which is as simple as defining elements by the particles in their nuclei,

Random is a term from the mathematical discipline which deals with statistics and probability. To understand its meaning, some background knowledge of probability is required. First, probabilities are almost always expressed as values between zero (no chance) and one (certainty). An exhaustive set of probabilities always add up to one. For example, when throwing dice, there are eleven possible totals (2 through 12, inclusive). If you correctly calculated the eleven probabilities and added them up, the sum would be one.

Now starting with fairly simple assumptions, mathematicians have come up with formulae or algorithms for computing various probabilities. For example: If P+Q =1, P is the probability of a win, and Q is the probability of a loss, then it has been proven that expanding (P+Q)^n provides the probabilities of 0, 1, 2, 3 wins in n plays. For six plays, the expansion would be

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P^6 + 6*P^5*Q + 15*P^4*Q^2 + 20*P^3*Q^3 + 15*P^2*Q^4 + 6*P*Q^5 + Q^6

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I f you want to know the probability of 4 wins and 2 losses, evaluate: 15*P^4*Q^2. The above is called a binomial probability distribution. I think that the Normal (or Bell-shaped) curve is the limit of the binomial distribution when n approaches infinity.

There is a multinomial distribution analogous to the above when there are more than 2 possibilities. Eg: Expand (P+Q+R)^n for a situation with three possibilities. There are Poisson probabilities applicable to situations for which an average is known. Eg: If you expect something to happen 5 times in an hour, Poisson probabilities will calculate the probability of its occurring 0, 1, 2, 3. . . times in a particular hour (at least I think this is a Poisson situation). The formulae for Poisson probabilities is P = e^-a*a^k/k!, where a is the average number of occurrences, e is 2.71828..., and P is the probability of k occurrences.

When developing formulae or algorithms for dealing with probabilistic situations, a mathematician always starts with assumptions stating that no deterministic mechanism is involved. For example, when considering the roll of a single die, it is assumed that the probability of rolling each number is the same, namely 1/6 (Id est: It is assumed that nothing is creating a bias favoring one number over another). A random process is defined as a process to which the above type of mathematics is applicable. "Random" in this context relates to the assumptions about lack of bias or lack of a deterministic mechanism. A mathematicians might say that such and so is a random process with binomial probabilities.

Note that the above is a discussion of mathematical concepts, not laws of physics. The mathematicians are saying that if a given process is probabilistic (or random), the given probability formula is applicable. They are not making claims about the world of physics. They are not claiming that such processes exist outside the world of mathematics.

Now, moving to the world of physics. When physicists discover that data and predictions for a given process are modeled by deterministic equations (or formulae), they believe it is a deterministic process. When data and predictions about a process are modeled by probabilistic (or statistical) formulae, they believe it is a probabilistic (or random) process.

Do physicists know how gravity works? Do they know what "causes" gravity? Not really. Do they know why gravity is attractive instead of repulsive? No they do not. In fact, they are now considering experiments to determine if antimatter attracts or repels ordinary matter. A hollow sphere exerts no gravitational force on objects inside it. An intelligent species which evolved inside a hollow planet (far removed from any other planets or stars) would not be able to deduce the existence of a gravitational force. In the absence of the data, physicists would not know that gravity existed. Will physicists ever know why there is gravity and understand the basic causes underlying it? No they will not. What they do know is that Newton's differential equations fit the observational data as accurately as can be determined experimentally for most circumstances. They also know that Einstein's equations fit the data better in certain situations. I know that physicists used to claim that gravity is deterministic. I think they still do, although I am not sure about the implications of the search for a theory of "quantum gravity."

Do physicists really understand quantum processes? No they do not. They do not understand these processes any better or worse than they understand gravity. They do claim that quantum behavior is contrary to our intuitive concepts of how the universe functions. They are more aware of their lack of "real understanding." They say that this is because our intuition is based on classical world experiences/perceptions which have no counterpart in the quantum world.

Why do physicists refer to some processes (Eg: Gravity) as deterministic and refer to others (Eg: radioactive decay) as random? They use those terms because of the mathematics which seem to model the processes and/or predict numerical results. Is the nature of the mathematics proof that the processes are deterministic or random? I do not think so. Will we ever have proof one way or the other? I doubt it. Absolute proof is something mathematicians manage to do. At least they claim to, and their claims are generally believed. Physicists are stuck with theories which model the data very well and provide the ability to make predictions and build things like lasers.

Now, when faced with a process which is accurately modeled by probabilistic mathematics, should you believe that is it deterministic or should you believe that it is probabilistic (Id est, random)? I think it is more reasonable to assume that it is probabilistic. Will the data associated with radioactive decay always match a (Poisson, I think) probability distribution? Yes it will. It matches as accurately as current technology is capable of measuring the data, and it is ridiculous to believe that more accurate future measurements will show that it does not match such a distribution. No matter what we learn in the future, the data associated with radioactive decay (and other quantum world phenomena) will still look like probabilistic (or random) data.

Guv

I want to start by saying I sincerely hope you do not see what I am about to say as quibbling or nit-picking in any way... but I feel there are some fundamental assumptions that have been made which cannot be validated.

Please... I truely hope you look at this rationally and logically and consider them as valid because that is the nature in which they are going to be presented. If you believe I am in error then please by all means provide serious and logical reasons as to where I am in error so that I may learn from those mistakes.

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When developing formulae or algorithms for dealing with probabilistic situations, a mathematician always starts with assumptions stating that no deterministic mechanism is involved.

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This is a pretty big assumption. How can the fact that they cannot "determine" a deterministic mechanism mean there is NOT a deterministic mechanism? Simply the lack of finding anything does not mean it doesn't exist.

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For example, when considering the roll of a single die, it is assumed that the probability of rolling each number is the same, namely 1/6

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That is another very large assumption. Are you actually talking about 2 dice here or are you talking about some modified flat and non-faced cubical objects under SPECIFIC conditions?

If we are talking about "normal" dice being rolled in a "normal" situation then you would be very much in error if you assumed the probability of rolling each number was 1/6. I could state for you at least 3 if not more factors that would influence the outcome towards one number or the other.

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Note that the above is a discussion of mathematical concepts, not laws of physics. The mathematicians are saying that if a given process is probabilistic (or random), the given probability formula is applicable

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This is the bit I am failing to understand. Yes I agree it is probabilistic... but I don't see where it is then automatically considered "random".

Perhaps it is my definition of random that is therefor in error... but you did state yourself that "Random suggests unpredictability, lack of pattern, unknown causes, et cetera".

Taking the example of the dice you could predict EXACTLY what you believe the outcome would be (a bell distribution of results between 2 and 12 with the majority of results totalling 7), and that it "appears" to lack a pattern does not actually mean it DOES lack a pattern.

You also say "unknown causes". That indicates that causes exist but that they are not currently known. Why then is it at its very CORE random even though you as much as suggest that a "cause" could at some stage be found.



If you can explain these to me I would appreciate them, despite whether you think they are folly or come from an ignorant person... the proof in my ignorance will be in the absolute ease with which you can explain why these are so ;)

Gen-X, I do not know how to answer you without being insulting, but I will try. Let me analyze some of my last post and your corresponding replies. For the time being (perhaps forever), I will ignore a previous post you made.

I posted the following.

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When developing formulae or algorithms for dealing with probabilistic situations, a mathematician always starts with assumptions stating that no deterministic mechanism is involved.

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Your comment was the following.

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This is a pretty big assumption. How can the fact that they cannot "determine" a deterministic mechanism mean there is NOT a deterministic mechanism? Simply the lack of finding anything does not mean it doesn't exist.

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Do you understand the meaning of "assumption" when used by a mathematician? It means that he intends to proceed on the basis of the statement being true. It does not mean that he believes the statement to be true. In some cases, he intends to arrive at a contradiction and thus prove that the assumption cannot be true. In other cases, he merely wants to determine the logical consequences of the assumption.

Now, I hope you know what "assumption" implies. In dealing with a probabilistic situation, what do you think should be assumed? Perhaps they should assume that differential equations are applicable and try to determine which differential equations. Perhaps they should assume that gremlins or lawyers are in charge of the probabilistic situation. If you were going to try to formalize a probabilistic situation, what assumptions would you use as a starting point? Perhaps you would assume nothing. If you make no assumptions, how do you start your analysis?

On to your next comment. I made the following statement relating to how a mathematician might start his analysis of dice throwing.

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For example, when considering the roll of a single die, it is assumed that the probability of rolling each number is the same, namely 1/6 (Id est: It is assumed that nothing is creating a bias favoring one number over another).

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Now, your first comment on the above was the following.

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That is another very large assumption. Are you actually talking about 2 dice here or are you talking about some modified flat and non-faced cubical objects under SPECIFIC conditions?

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This is alleged to be a serious discussion of a issue which is not simple, and requires that you read carefully. Note that I said a single die . How much plainer do I have to be? No, I was not talking about two dice! Did you really read what I said? Your reply suggests that you scanned it while really thinking about what you were going to post.

Next, in talking about dice, do I have to draw a picture? Have you ever been to a casino and seen dice? What are you trying to ask when you say "some modified flat and non-faced cubical objects"? I am talking about a mathematician's idealized concept of casino dice. What else could I possibly be talking about?

Your next comment on the above was the following.

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If we are talking about "normal" dice being rolled in a "normal" situation then you would be very much in error if you assumed the probability of rolling each number was 1/6. I could state for you at least 3 if not more factors that would influence the outcome towards one number or the other.

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Perhaps I should repeat my above discussion of what "assumption" implies, but if you did not understand, why don't you just reread it? Now, if you were a mathematician attempting to formalize the mathematics of dice tossing, what assumption would you use as a starting point? Perhaps, it would make sense to assume that the probability of each number was the reciprocal of the number? Do you like that better than 1/6? Perhaps a mathematicians should assume that the house has control over the dice and decides in advance what number should come up. Then you call in a psychiatrist and ask him to make some assumptions for you. Do you really understand the difference between a mathematician and a physicist? Perhaps, a mathematician should decide that dice throwing is totally capricious and not subject to mathematical analysis.

Your remarks suggest to me that you have never read a book about probability. They also suggest that you do not know the meaning of words like "assumption" in the context of the development of a mathematical analysis.

I do not know how to communicate with somebody like you. On several occasions I have decided not to try. It seems like such a waste of time. Then I say: "What the hell, I am retired and have time on my hands, why not waste some of it?"

Harbringer, sorry about that. You are correct, a random process can be used to create fractal images. At least it can be used to create images which are suggestive of fractals generated by deterministic processes.

I thought you were trying to say that fractals in general were created by random processes. One of the important concepts in chaos & complexity theory is the fact that extremely complex patterns can be produced by simple deterministic algorithms. Furthermore, it is interesting that the long term outcome of these simple deterministic algorithms cannot be predicted without actually doing the work. I hope you are not missing these points of interest.

Recently, I did notice your answer to the name question on another thread.

HarryW & Harbringer, you are both correct in saying that the issue of deterministic or random universe cannot be proven in the sense of proving a mathematical theorem.

In physics, one only hopes to develop formulae and/or algorithms which model the experimental data and provide worthwhile predictions. When such algorithms are discovered, it seems impossible not to speculate on the metaphysics implied. Physicists do not "understand" any of these processes. "Understanding" is a metaphysical concept, as are statements relating to a deterministic or probabilistic universe.

When a process (Eg: gravity) is modeled by deterministic equations, it seems appropriate to consider it a deterministic process. When a process (Eg: Radioactive decay and other quantum phenomena) is modeled by probabilistic (statistical) mathematics, it seems similarly appropriate to consider it a probabilistic (not deterministic) process. These, of course, are metaphysical statements about the processes. They are not provable statements.

It should be noted that it does not seem reasonable to believe that future developments will change the nature of the quantum data. If radioactive decay data matches Poisson distributions to a high degree of accuracy, no future knowledge will change the nature of those measurements. Any future developments can only be expected to result in knowing that the probabilistic model is applicable to a lower quantum level.

At any rate, it seems far more worthwhile to interact with you fellows than with another poster who shall remain nameless here.

Did you notice a recent post in this thread in which I attempted to define random and make a few more remarks on this subject?

Guv,
When you were talking about a mathematician rolling a dice, and state that ihe assumes that is it probalistic in nature, a reasanble assumption, you would think, and he also assumes there is an equal chance of roling each number (1/6). I would disagree, and say that for each throw, taken by itself, there is only ONE POSSIBLE OUTCOME..that is to say that for this specific throw, with the air mocelues exactly were they are, and with everything (quantum states, EVERYTHING) as it is, there is only possible outcome, ie IT IS DETERMINISTIC.
This means if the same condititions were repeated (which could be impossible, as it is possible that time is a factor, so even if all physical factors are the same, the time will be different, and there could be other non resetable faetures...and antimatter and matter aprticle could have coliieded, ane the removal of these two particles, and the replacement of them by energy is enough in my opinion to constitute a difference) the same outcome would DEFINATlY OCCUR.
The probablistic nature of results can be explained, I believe, by this.
If there are an equal number of universal states(By which I mean the EXACT state that the univese is in) that cause each number to be rolled, then there will be an equal cahnce of rolling each number. However, this can be modified slightly. If we take the universe an instant before the dice is rolled, at that universl state, then, if it is deterministic(as I am assuming for this argument), the next universal state must follow on fom this one, and will be the same, no matter how many times this happens. This leads to the predictability of dice rolls.(I will get on to the impracticalities later!)
(So, on a side point, it would be possible to assign the universal state for any given time, though if time runs backward, this breaks down...the laws of physics actually run different if time runs backwards, so if time was run backwards for 10 year, and then run foward again, the outcome would be different, as the universal state achived by the 10 years reversal would be different to the universal state of 10 years ago, so given Guv's inital point, it would actually run differently, BUT in a prdictable, deterministic fashion!)
There could be a situation were there is an inequal ammount of univerasl states that cause 'probabilistic' events to occur..I will use a coin, ratherthan a dice, for simplicity (For my own poor brains sake, if no-one elses!)
Lets say that for this coin, there are twice as many unoversal states that support heads than tails. (please excuse the abbrivated grammar, it gets rather long) For this coin, looking in a probalisticmanner, it will come up with heads twice as often as tails. There will be a vali physical explanation (Say the coin is extremly weighte, and has a curved head side, so it rolls off frquently, for example). This physical explanation supports the deterministic version of events! In a probablistic explanation, if they ignore physical features (unlikley!!), they ill have no way of expalining the difference in odds betwwen heads and tails. If they accept the physical explanation, THEY MUST ACCEPT THAT THE PHYSICAL NATURE DETERMINES THE EVENT! (Of course, they could claim that the phyical nature only EFFECTS the event, and that some other process(v. Mysteriuos in function!)still randomizes the event. If this is the case, then if there is only a small change in the coin form perfect(say a microscopic difference in texture on one side, the rest of both sides being perfectly alike), then this will have no difference to the odds, because if the randomizing function can overcome strong physical disparities, then a minute one will be over powered, and even if the trials are continued to infinity, there will be no odds difference between the two side.)
I believe this shows how probablistic maths can be used to describe determinist events (I hope it does!), and I will now go on to talk about impracticalities.
Guv, you (sorry to bring it up, but neccasry!)you mention unpredictabilty. I know that I have stated that the univers is prdictable, but now I will contradict myself!(You also mention pattern, but I don't think anyone belives the universe has no pattern, and unknown causes can always be discovered later!)
The univers is (in my opinion) predictable in nature, but unpredictable in practice. It is easy to see why. If the quantum state of every particle affects events, then to predict events(with 100% accuracy!) you need to know these states. In fact, you need to know everything. You need to know the exact laws that gvern all interactions, you need to know the exact relative positions(posibly even absolute postions..)of every particle, and the state that they ar in (direction, speed ect), and quantum entaglements, ect. After you know all this (Which i believe would be impossible to store, as the univers already stores all this data..by the position, momenum ect of the actual particles in question, using the particle to store this data would change the data to be recorded, and it is clear that there is enouhg room to store all this info once(as it already does), it is not clear that it can store it twice, once in a form easily accesible by us!)
Okay, lets say you do stoe this data in a usable form, then you have to compute this, to gain predictions. The universe is clearly able to commpute this (In real time!), but we need to commpute the change of universal states FASTER than the univers does, AND with less particles than the universe uses (We can only ever use as many, and if we do that, there is nothing left to predict!) This is clearly impossible. (I think it is, if anyone thinks idea of a way to do these things, I will be suprised.)
So, within this univers, it is mposible to predict the outcome of the universe.(100% accuracy agian.) THis means that the universe is unpreditable in practice, which some people may take to mean as random, and fair enough, but I belive that as it is deterministic, though unpredictable, it should not be considered to run on random proccesses.
However, if, as some physicist belive, there is more than one universe, then maybe it would be possible to harness a few of these to make calcualtions!(For moral reasons, perhaps we should select ones unihabited by any for ofm life ;) ) This is a distinct, but in my opinion very unlikly possibilty,(D'OH! Shouldn't use that word in here!), but it does allow theoretical basis for predictablity back into the equation, thus removing the arguments that as it is compltly unpredictable, it may as well be random from our point of view!
Phew!!! I think I got all I wanted to say down there! Anyway, thats the universe as I see it, so tell me what you think!

I'm not sure about this, but I was under the impression that when things like dice and coins are mentioned here, in relation to probabilities, the idea that each result has a particular probability is assumed. What I mean is that these are just idealised probabilistic experiments, they are just representations of a probabilistic event. Trying to apply these experiments to a real world situation and then saying they wouldn't be perfect as they are assumed to be in theory seems a little silly to me. The idea of the probability needs an experiment to apply it to so that it can be easily described and likened to a real world situation (but not analysed in too stringent a way); the experiment is not proposed first and then the probability found for it in these theoretical situations, the probability is defined and then a suitable experiment thought of for it.

I agree that if the universe is deterministic in nature then we still cannot predict any aspect of the future. My reasoning for this was slightly different (but similar) but that is beside the point.

Well, you folks who believe that the universe is deterministic in principle, but not predictable in practice have some very intelligent people on your side, including Einstein. The computational difficulties are not the only problems with predictability. Both the uncertainty principle and the velocity of light limitation on collection of data make it impossible in principle (as well as practice) to collect all the data.

Without presenting any further arguments against determinism in principle, I have collected some quotes from various sites. They might be interesting. In particular, notice the "Many Worlds Idea." of Hugh Everett, and the "It ain't there" concept of Eugene Wigner. Compared to these guys, the mainstream concepts do not seem weird at all. These two have real credentials and are not considered to be quacks or charlatans. Personally, I have my doubts about their sanity.

Note that Einstein's deterministic view seems to be considered "less fashionable" rather than incorrect, so I am not claiming that every expert backs my point of view. What they do seem to agree with is the idea that the classical world is deterministic only if the quantum world is.

I have bolded some sentences. Otherwise, these are exact quotes.

Most physicists see quantum weirdness as something we have to accept in spite of ourselves, because of the overwhelming empirical evidence in its favor. Richard Feynman argued that we must simply admit that Nature makes no sense - knuckle under and accept the evidence, no matter how weird it seems

Computer-generated "random" numbers are more properly referred to as pseudorandom numbers, and pseudorandom sequences of such numbers. A variety of clever algorithms have been developed which generate sequences of numbers which pass every statistical test used to distinguish random sequences from those containing some pattern or internal order.

HotBits is an Internet resource that brings genuine random numbers, generated by a process fundamentally governed by the inherent uncertainty in the quantum mechanical laws of nature, directly to your computer in a variety of forms. HotBits are generated by timing successive pairs of radioactive decays detected by a Geiger-Müller tube interfaced to a computer.

The first problem lies in the apparent indeterminacy of Quantum phenomena. In the macroscopic world, if an experimental system is set up repeatedly in exactly the same state it yields the same results each time; indeed in simple cases these results can be calculated in advance. If we know the state of the system for values of t < t0, then in theory we have sufficient information to determine the state at values of t > t0. This is not true for experiments involving atomic sized structures, where identically prepared experiments can yield differing results. We have discovered however, that although the past history of such a system does not allow us to make exact predictions, it does furnish enough information to make statistical predictions of future states, as described by the rules of Quantum Mechanics. This has seemed to many people, including Einstein, an unsatisfactory state of affairs, suggesting that our understanding of quantum processes is fundamentally flawed.

But why do we expect the quantum world to permit exact predictions, why do we expect it to be deterministic? If indeed it were, then the macroscopic world would necessarily be deterministic also. But the converse argument is false. Because of the large number of quantum events contributing to one macroscopic event, the statistical quantum laws account for the determinism of the world of familiar experience just as readily as if they were strictly causal, and should cause us no surprise. It is merely our familiarity with the determinism of everyday life that leads us to expect the fundamental laws of nature to display the same property.

At one extreme we have the view advocated by Einstein, that quantum behavior is really deterministic, controlled by some system of "hidden variables" which we have not yet discovered. If only we knew how these operate we would find the quantum world as predictable as the world of everyday experience. This viewpoint is now less fashionable than in the past, for it fails to address some of the most pressing difficulties. In particular we still require an element of non-locality to explain the EPR experiments, and the fundamental question of the collapse of the wave function is still unanswered.

At the other extreme stand the followers of Eugene Wigner, who maintain that events do not really occur, the waveform does not collapse, until the results of an observation enter a conscious mind. It is possible that a consistent set of beliefs can be based on this interpretation, but at considerable cost. We must believe that nothing really happens in those parts of the universe where no conscious observers exist, and that there was no activity at all during the billions of years before they first emerged. To many people this view is far too anthropocentric.

Somewhere in between stand the followers of Neils Bohr, forming what we call the Copenhagen school. They do not discuss the reality of properties which are unobservable, claiming that we cannot know whether what we say is true, and that it does not matter anyway. The state vector, or waveform as we have called it, represents only the knowledge an observer has of a system, and this, they claim, solves the problem of its collapse, for it must change whenever our knowledge changes as the result of an observation.

Much more radical is the "many worlds" view put forward by Hugh Everett in 1956. Whenever a quantum system has a choice, then the universe splits into two, one displaying the first choice and one the second. A copy of every living creature is created in each universe, and both copies go on to lead independent lives, but without any possibility of communication between them. I cannot comment on the plausibility of this proposal without betraying my astonishment that intelligent scientists can subscribe to such an idea. Have they never heard of Ockham? Do they not understand the principle of argument by reductio ad absurdum? Ockham must turn in his grave at the prospect of an infinity of universes. And no-one should ever again use the reductio argument, now that this conclusion has been judged insufficiently absurd to disprove its premises.