Probability is an illusion
Probability is the mathematical study of chance which basically considers events that are uncertain. We can't make definite claims in probability.
The usual way the subject is introduced to the novice is with dice, coins and cards. This is "probably" because the mathematics was invented to study gambling.
Anyway, let's take a 6-sided dice and see how the math works:
6-sided dice is numbered: 1, 2, 3, 4, 5, and 6
A basic definition of probability:
Probability of an event E = (Number of ways event E can be realized) ÷ (Total number of possible events)
So what is the probability of getting odd numbers with one roll of a dice?
O = number of ways we can get an odd number (1, 3, 5) = 3
T = total number of possibilities = 6
So, probability of getting an odd number with one roll of a dice = O ÷ T = 3/6 = 1/2 = 50%
It has been verified that if you actually do roll a dice, say x number of times, the probabilty of getting an odd number with one roll approaches 50%. That is to say the system (one roll of a dice) behaves exactly as if it was probabilistic. You can try it with different conditions, say rolling a number less than 3, etc., and the actual results of rolling the dice will match the probability calculated. Just to make the point clear the system behaves as if it was truly probabilistic.
That said let's consider the system (one roll of a dice) mechanistically. We know from basic physics that given all the information (force, direction, mass, etc.) of the system (one roll of a dice) we can predict the outcome with perfect accuracy. In other words the system (one roll of a dice) is, well, deterministic (certainty assured).
So, even though the system (one roll of a dice) is positively deterministic (certainty assured) it's behavior is probabilistic.
A few questions:
1. Is probability an illusion?
2. What causes an unequivocally deterministic system to exhibit probabilistic behavior?
The usual way the subject is introduced to the novice is with dice, coins and cards. This is "probably" because the mathematics was invented to study gambling.
Anyway, let's take a 6-sided dice and see how the math works:
6-sided dice is numbered: 1, 2, 3, 4, 5, and 6
A basic definition of probability:
Probability of an event E = (Number of ways event E can be realized) ÷ (Total number of possible events)
So what is the probability of getting odd numbers with one roll of a dice?
O = number of ways we can get an odd number (1, 3, 5) = 3
T = total number of possibilities = 6
So, probability of getting an odd number with one roll of a dice = O ÷ T = 3/6 = 1/2 = 50%
It has been verified that if you actually do roll a dice, say x number of times, the probabilty of getting an odd number with one roll approaches 50%. That is to say the system (one roll of a dice) behaves exactly as if it was probabilistic. You can try it with different conditions, say rolling a number less than 3, etc., and the actual results of rolling the dice will match the probability calculated. Just to make the point clear the system behaves as if it was truly probabilistic.
That said let's consider the system (one roll of a dice) mechanistically. We know from basic physics that given all the information (force, direction, mass, etc.) of the system (one roll of a dice) we can predict the outcome with perfect accuracy. In other words the system (one roll of a dice) is, well, deterministic (certainty assured).
So, even though the system (one roll of a dice) is positively deterministic (certainty assured) it's behavior is probabilistic.
A few questions:
1. Is probability an illusion?
2. What causes an unequivocally deterministic system to exhibit probabilistic behavior?
Comments (146)
How do you figure this? Practically speaking, physical science is always subject to some degree of inaccuracy.
Physics/mechanics???!!! We've put men on the moon. Surely a humble dice is within its reach.
To quote Regis, Is that your final answer?
It doesn’t take 100% accuracy to put men on the moon. Also modeling gravitation in space is much easier than modeling all frictions on a dice thrown in the air and bouncing on a surface: the dice will bounce differently depending on the hardness of the surface at the precise point where it bounces, and a tiny change in the angle at which the dice bounces will totally change how it bounces and its subsequent motion, so it’s a chaotic system, a tiny difference in initial conditions will change the final state of the dice and in most cases we can’t measure all relevant variables with sufficient accuracy. Also, the guys going to the moon could control their trajectory to some extent during the flight, whereas we don’t have little guys controlling and stabilizing the dice while it flies and bounces :wink:
Exhibit is the keyword here. What is exhibited is in the eye of the beholder - it is not just an objective property of a system. A system may hypothetically be perfectly deterministic (although how could you know that with certainty?), but if you don't know enough about its behavior, then you can, at best, predict it probabilistically. Typically, when a die is thrown, there is no practical way to predict its exact trajectory (even if there was a fact of the matter about what that trajectory would be), nor even which side it is more likely to land on. Lacking such information, the prudent bet is to distribute probability equally between each of the six sides. (If you systematically fail to do this, then a competing player could exploit your bias to gain an advantage.)
Quoting TheMadFool
Calling it an "illusion" implies that you know better. But you don't - hence probability. Probability is a function of our uncertainty.
Is there any other answer? Look, I'm willing to accept that there is an issue of scale when it comes to laws of nature. For instance the quantum world is claimed to be fundamentally different from the world of suns, planets and galaxies and coincidentally what sets them apart is that the quantum is probabilistic; not so suns, planets and galaxies.
However, a dice and a planet or you or me seem to be within the range of the mechanistic physical laws of Newton. Measures of force, mass, angles, and other relevant data are all we need to precisely predict what the outcome of dice-roll will be. Kindly read my reply to Leo below:
Quoting leo
Ok. Let's suppose that a normal-sized dice is a "chaotic system" and is actually probabilistic. Just blow-up the dice - increase its size to that of a room or house even. Such a dice, despite its size, would continue to behave in a probabilistic manner despite our ability to predict the outcomes accurately. After all you do accept that rocket trajectories can be predicted and therefore controlled.
Quoting SophistiCat
Yes. My issue is that a deterministic system performs like a probabilistic one. It's not an issue of incomplete data for computation as an enlarged dice, something we can have adequate data on, will continue to be probabilistic. We can even calculate the probabilities of every possible outcome which will match the experimental results. Yet, such a system (enlarged dice rolling) is in fact deterministic.
To All
Why does a system whose outcomes we can actually predict behave as if we can't do that? That's what bothers me. Thanks.
For quantum events as far as I am informed they are understood to be probabilistic in nature. However the using of bigger scales doesn't negate this it only creates a state where events thought of as inpossible in the mechanistic view have a very low probability.
However lets ignore that considering a chaotic system the scale of the system shouldn't have an influence on it being chaotic or not. Chaotic means as others have stated before a sensitivity to input/relevant factors.
This means in this example that if you role a dice with a specific force that leads to a certain result the increasing of the force by 0,00...1 can lead to a completley different result. Same for angle and all the other factors. I don't see how this should change based on scale.
However lets ignore this case aswell and assume a completley deterministic system.
If you have a set of factors (force, angle, ect) and you throw the dice it gives a specific result. If you reproduce the same initial conditions you create exactly the same output. This is technically still under the scope of probability by it being modeld as 100 % chance to get the same result. However if you just role the dice again without fixing the intial conditions it should be logical to conclude that different intial conditions can produce a different outcome.
Therefore roling a dice with not fixed initial conditions is not clearly determined and thus probabilistic. Theres a chance that you use initial conditions I which lead to specific outcome O.
So somehow I don't get what your asking.
If you imagine a scenario of throwing a rock that is soley dependant on force and not chaotic at all (more force=flies further) you seem to be asking in my understanding if I throw a rock multiple times (with different force) why doesn't it always lead to the same result. (Excluding chaos). However as soon as you include chaos (different hights different density of air, relationship air particles with object) the clear predictability is gone.
So you seem to be investigating a chaotic system however meanwhile refusing to view it as chaotic and postulating it to be clearly predictable.
Did I missread you somewhere?
Yes we can. The ‘definite’ claim is probabilistic though. Mathematics operates within abstract hypotheticals it is not based experimentation.
If we set up a mathematic problem where the outcome is A or B we can never ever get an answer other than A or B - the probability of getting A or B is 100%.
If you’re simply stating that mathematical models do not map 100% onto reality then why are you bothering to state this? It’s obvious. I guess some mathematical model would map onto the entire universe but I don’t see how we’d have anyway of knowing this even if we happen to stumble across it by pure fluke.
What is the difference between probability and certainty in your view?
Imagine two 6-sided dice A and B
Imagine A is rolled in our world by a person like you or me. The outcome of the dice is predictable given initial conditions. You can't disagree on that.
Now imagine dice B exists in a non-deterministic world where probability is real.
If you test dices A and B you'll observe that both yield the same results as if both resided in a non-deterministic world.
Can you explain why?
Mathematical probability isn’t based on observation/experimentation. It is used to interpret experimentation and observation thought aided my measurements.
Don’t conflate the abstract with the concrete when talking about mathematical models and reality.
Well you said Quoting I like sushi
The word "definite" is usually associated with certainty which is the antithesis of probability. I thought you had an interesting take on the subject.
Actually I think we can calculate "exact" probabilities e.g. in the chance of getting a heads on a single coin-flip is "exactly" 50%. No more, no less.
However, I'd like you to take a step back into the nature, as generally understood, of probability which is basically an uncertainty about a given event. For instance we have a "definite" number, say 70%, when meteorologists predict the weather. Yes, I'm sure about 70% - it's a definite quantity - but are you certain that it'll rain or not?
Nope. As good as 50% but not exact. For starters force of coin flip, wind factor and the weight distribution of the coin are all physical factors in the real world - not to mention the rare occasions where a coin lands on its edge.
Granted, in day-to-day speech we refer to a coin toss as being 50-50.
Quoting TheMadFool
If you cannot answer that question yourself I don’t think I know what you’re trying to talk about? ‘Definite quantity’? This is just word play isn’t it. Giving an ‘exact’ number doesn’t mean anything out of the context it is given in.
Honestly, I’ve no idea what you’re talking about. Sorry :(
No, you still don't understand. Suppose the die already landed and came to rest, but you can't see it - it rolled under a couch. The outcome is not just deterministic - it's already determined. Does that fact help you with guessing the answer? It doesn't, and you still don't have anything better than probability. So would you say that a die lying on the floor under your couch "exhibits probabilistic properties?" That would be stupid (but what else is new?)
Probability is about what you don't know. Whether what you don't know is inherently indeterministic or whether it is a matter of fact doesn't make a difference to you. You still don't know what you don't know. So what do you do? You could say "I don't know," but if you have to make decisions based on incomplete knowledge, you use probability.
If we somehow launch the house-sized dice fast enough it isn’t clear that we could predict individual outcomes accurately, because again an extremely tiny change in how the dice is thrown or in the wind or in exactly where and at what angle it bounces would change the outcome. While if we launch it so slowly that the dice doesn’t rotate, we would predict the outcome but then it wouldn’t be probabilistic anymore.
Rocket trajectories aren’t predicted with 100% accuracy, in the atmosphere their trajectories are constantly corrected and stabilized in order to stay on a given course, and in space they are also corrected every now and then, because they encounter accelerations (due to various phenomena: wind, dust, radiation pressure, ...) that aren’t predicted beforehand with 100% accuracy.
However it is possible to have systems where overall the outcomes are overall probabilistic even though we can predict each outcome individually. Consider a simple system where you throw the dice so slowly and at such low height that it doesn’t rotate while in the air and it doesn’t rotate after bouncing. Then you can always predict what the outcome will be, but in order for the outcomes to be probabilistic you have to change the initial conditions probabilistically (namely the initial orientation of the dice). So if 1/6 of the time you start with 1 up, 1/6 of the time 2 up ... and so on, you know that the dice will land 1/6 of the time with 1 up, 1/6 of the time with 2 up and so on.
It is also possible to have systems where you don’t control the initial conditions (contrary to the above example), where you can predict each individual outcome and where the outcomes are still probabilistic overall. For instance let’s say you pick 1000 people randomly (from various places without looking at them) and each of them writes their age on a piece of paper, so you have 1000 pieces of paper with some number on them. Now say you pick 10 pieces randomly, you add the 10 numbers together and you divide by 10, that gives you the average of these 10 numbers, let’s call it N1. You put the 10 pieces back with all the other and again you pick 10 pieces randomly, you carry out the same process and you get another number, N2, and so on and so forth, you do that many times. Each time you pick 10 pieces, the outcome number can be predicted (you just have to take the sum and divide by 10), however it can be shown mathematically that the numbers N1, N2, N3, ... follow a probabilistic distribution, a so-called normal (or Gaussian) distribution, that’s to say that most N values will be gathered in a tight range, and the further away from that range the less and less values there are, and you can compute how probable it is for a random N value to fall into such and such range. So before you pick 10 pieces of paper again, you know how likely it is that their average will fall into such and such range.
Going back to the example of the dice, is it possible to not control the initial conditions, to be able to predict each individual outcome and yet the overall outcomes are probabilistic? It is possible in some specific cases:
What has an influence on the dice is the initial conditions (the orientation of the dice when it is thrown and the velocity and angle at which it is thrown), and the forces acting on the dice during its motion: air friction that depends on the initial conditions, on air density, on wind; bouncing force provided by the surface on which the dice bounces that depends on the initial conditions, on the local hardness/elasticity of the surface, on the shape of the surface as well as that of the dice; and surface friction when the dice is rolling on the surface, which also depends on initial conditions and on the surface itself as well as that of the dice
If the air density, air wind, and the properties of the surfaces didn’t change, then that means that if you launched the dice several times with the same initial conditions, you would always get the same outcome. So if these properties didn’t change, in order for the outcomes to be overall probabilistic, the initial conditions would have to be probabilistic (so you would have to always throw the dice differently). Since the forces acting on the motion of the dice depend on the initial conditions in a complex enough way, a tiny change in initial conditions will be enough to usually change the outcome. And since the dice can only land in 6 possible ways, we can expect to get each side about 1/6 of the time.
Otherwise, if you always launched the dice with the same initial conditions, in order for the outcomes to be probabilistic the air density/wind and/or the properties of the surfaces would have to change between each throw, but then it would be impractical to accurately measure these evolving conditions and so it would be impractical to predict individual outcomes.
So in practice, if you conduct the experiment in a closed system where the air density is constant, where there is no wind besides that generated by the dice, and where the surfaces are hard and regular enough (both the surface of the dice and the surface on which the dice bounces), then it is possible to predict individual outcomes and to have overall probabilistic outcomes only if you always change the way you throw the dice (otherwise if you throw it in the exact same way you are guaranteed to get the same outcome and then the overall outcomes won’t be probabilistic).
And now to finally answer your questions, notice that in both this last example and in the example with the numbers on pieces of paper, any individual outcome is deterministic, but the way these outcomes are distributed is probabilistic. In the last example of the dice, in all the ways that the dice can be launched, in about 1/6 of the case the dice lands on a given face, because the dice can only land in 6 different ways, and the forces acting on the dice are complex enough that they don’t lead to one side being more likely than the other, whereas if the forces were much simpler (like in the example where the dice never rotates during its motion) then one side would be preferred. In the example of the pieces of paper, it can be shown mathematically that the outcomes are distributed in a probabilistic way, following a Gaussian probability distribution: that’s called the “central limit theorem”.
In deterministic systems probability is not fundamental in the sense that if we can’t predict an individual outcome it’s only because of incomplete knowledge. As to your second question, the deterministic system does not exhibit fundamental probabilistic behavior, a more correct way to phrase it is that it exhibits statistics, due to the configuration of the system itself. In the dice example it is possible to configure the system in a way that some outcome is preferred, for instance always starting with the same initial conditions. Or we could start with different initial conditions but make one side of the dice more sticky than the others so that the dice will land more often on the sticky side.
In deterministic systems probabilities are not fundamental, rather we compute statistics that depend on the configuration of the system, which can be interpreted as a characteristic or a property of the system. The probabilities in deterministic systems refer to incomplete knowledge, so in the example of the dice where the outcome can be predicted, we would say the dice has a given probability of landing on a given side when we have incomplete knowledge of the initial conditions or of how the system reacts to these initial conditions.
Say there are N different possible configurations of initial conditions, and we know the system always reacts the same to a given initial condition (because in that particular system when we precisely measured the initial condition and we kept it the same we always got the same outcome), and we know that in about N/6 configurations the dice lands on a given side (because we have thrown the dice a great number of times in that system with different initial conditions and that’s what we have noticed), but this time when we throw the dice we don’t measure the initial condition (the initial velocity/angle/orientation of the dice), then we say that the dice has about 1/6 probability of landing on a given side, but that’s only because we don’t know in which deterministic configuration we are once the dice is thrown, because we haven’t carried out the necessary measurements that would provide us with that knowledge.
And depending on the system the number doesn’t have to be exactly N/6 for each side, in most deterministic systems it wouldn’t be exactly N/6, depending on the dice and on how the system reacts to it.
That turned out longer than I expected, hope that helps.
If we keep the exact same initial conditions A will always produce the same result unlike B that will produce differenig results due to its indeterminsim.
So I disagree that they produce the same result.
There is no justification for certainty in our perception of the real world. Any claims for certainty are either a subjective convenience or an illusion.
In an ideal world, human knowledge would contain complete understanding of how the universe works.
Lacking that, we rely on models, representations, and approximations to the physical behavior of the world. The idea that knowing the current state of the universe, by knowing the positions and motions of every element, is an impossibility, since that awareness is always historical.
The rules (approximations of laws) for tossing dice are incomplete. Predictions are based on statistics, a history of past events.
I apologize for lumping you all in a group but all of you deny that there is a problem here. Maybe as Sophisticat said I've misunderstood but I hope to express my thoughts more clearly in this post.
Firstly we must agree that we can predict, using physics, the outcomes of events at the human scale. Set aside the complexity of the issue for the moment and consider that given all initial values of a system the outcome pathway is fully determined. leo said that this isn't possible with 100% accuracy which I disagree to. Take the simple example of a space probe. Using rockets in the right locations and fired for the correct durations we can and do predict that the probe lands right side up. This is a 100% prediction accuracy and if there are any inaccuracies they are due to unforseen cotingencies like wind or instrument malfunction.
I hope we can agree now that physical systems at the human scale are deterministic and that includes a fair 6-sided dice which I want to use for this thought experiment.
Imagine there are two people A and B with one fair 6-sided dice. A throws the dice but is oblivious of the initial state of the dice. B has complete knowledge of the initial state of the dice. As you see A can't predict the outcome but B can.
The theoretical probability that the dice will land on a number less than 5 is 4/6 = 2/3 i.e the dice should show a number less than 5 two-thirds of the time (66.66%)
We are now going to calculate the experimental probability. A throws the dice 100 times and checks how the dice lands.
Remember that each time A throws the dice, B can predict the outcome of the dice. So...
The result of the experiment of a 100 throws of the dice will show that the dice lands on a number less than 5 approximately 66 times. This result is in agreement with the theoretical probability calculated (4/6 = 2/3 = 66.66%). In other words the system (person A and the dice) behaves like a probabilistic system as if the system is truly non-determinsitic/probabilistic.
However, B knows, since he knows the initial states of each dice throw, that the system (person A and the dice) is deterministic/non-probabilistic and that each outcome is predictable.
1. We know that the system (person A and the dice) is deterministic because person B can predict every single outcome.
2. We know that the system (person A and the dice) is probabilistic because the experimental probability agrees with the theoretical probability which assumes the system is non-deterministic.
There is a contradiction is there not?
Quoting A Seagull
We can't say that because we know the system (person A and the dice) is deterministic because person B can predict the outcomes of every single throw of the dice.
As a wise man once said: " The world is is not only queerer than we suppose it is queerer than we can suppose.
To try to impose one's pre-supposed ideas and assumptions about the way the world 'should' be is naive.
If you still have a problem with probability and tumbling dice, I suggest you re-visit your assumptions regarding the way you think the world should be.
I wouldn't say that theoretical probability assumes the system is non-deterministic. Rather, it assumes that the system has certain regularities that enable us to calculate frequencies of possible outcomes. In the example of throwing the dice, the regularities are (deterministic) laws of physics that transform initial conditions to outcomes.
If there is a contradiction, it is only in how the system is being represented. In this scenario, person B has complete information about the system whereas person A has only partial information. The difference is not in the system but in the information that each person has.
Which is to say, you can represent the system (person A and the dice) from either person A's point-of-view or person B's point-of-view, which avoids contradiction.
And if A threw a hundred sixes in a row it wouldn't be behaving like a probablilistic system?
I said it’s possible in very specific cases, most of the time it isn’t.
Quoting TheMadFool
It seems you missed the part where I said that the rockets are constantly controlled and stabilized during their flight, so as to remain on the desired trajectory. That control and stabilization is not predicted in advance, it dynamically adjusts to the conditions that the rocket encounters through a negative feedback mechanism.
Quoting TheMadFool
Without the dynamical control/stabilization the probe would never reach its destination. The unforeseen conditions that the rocket/probe encounters are precisely what makes the outcome unpredictable from the initial conditions alone. Which is why I said that the outcome is determined from the initial conditions alone only in simple cases where everything that happens is predictable, if you read my long post carefully you will find the answers to your questions.
Quoting TheMadFool
Then consider the experiments where the outcome of each individual throw can actually be predicted, that is in simple situations with no wind except that caused by the dice, constant air density, constant temperature, smooth and hard surfaces. As I explained to you in real experiments the outcome usually won’t be exactly 1/6 for each side.
If the experiment is deterministic then there are a finite different number of ways to throw the dice, a finite number of initial conditions and outcomes, let’s call it N. If you measure all the outcomes, usually each side won’t appear exactly N/6 times. As I explained, if one side of the dice is slightly more sticky than the others, that side will show up quite more often. And even with the same stickiness, consider that a dice is not exactly symmetrical due to the dots or numbers printed or engraved on the surface, so in deterministic experiments the outcomes wouldn’t have exactly equal probability.
If instead you consider experiments where there is wind, or changing temperature, or irregular and soft surfaces, you can’t predict individual outcomes, and the various forces act together in such a complex way that they don’t prefer any particular side of the dice, even starting from the exact same initial conditions.
So for your thought experiment to make sense, consider that in deterministic systems where the outcome can actually be predicted in practice, starting from the same initial condition leads to the same outcome, and in such systems the dice does not land exactly as many times on each side.
Quoting TheMadFool
No, because again, as I and others have explained, the theoretical probability does not assume the system is non-deterministic. In deterministic systems probability is not fundamental, if it confuses you use the term statistics instead.
You can come up with deterministic systems in which the dice will always land on the same side, or twice more often on some sides than on some the others. In some deterministic systems the dice lands about equally often on each side, in some other deterministic systems that’s not the case at all. Consider these latter systems and that will help you understand your error.
In deterministic systems the outcome is about 1/6 for each side only when the configuration of the system is such that it does not prefer any particular side. Some deterministic systems are like that. Some aren’t.
The problem is you assume in any deterministic system where each individual outcome can be predicted the dice will always land about equally often on each side. That’s wrong. Also in most situations where we throw a dice we can’t predict each individual outcome. Try to build a deterministic system in which you can predict each individual outcome, that will help you understand your error too. In your reasoning you assume that each individual outcome can be individually predicted, then consider real systems where that’s actually the case, where that’s actually done in practice, otherwise your thought experiment isn’t connected to reality.
In some deterministic systems the dice lands about equally often on each side because in such systems the symmetry of the dice doesn’t lead the system to prefer a particular side. How often an object lands on each side depends on the shape of the object, on its symmetries. You get symmetrical outcomes when you are dealing with a symmetrical object and when the system doesn’t break that symmetry. This doesn’t mean that the deterministic system is exhibiting non-deterministic behavior, there is no mystery here.
A coin is partially symmetrical, such that it lands most often heads or tails, but it can also rarely land on its edge. Yet you can construct a deterministic system in which it always or almost always lands on its edge (make a system in which the coin bounces or slides on inclined surfaces so that it ends up in a groove the same width as the edge of the coin). In this case the deterministic system prefers a particular symmetry of the object. In that system the coin wouldn’t land heads 50% of the time and tails 50% of the time. And just because you can say that in this system the coin lands on its edge about 100% of the time, using probability jargon, that doesn’t mean that the deterministic system exhibits non-deterministic behavior, just like when a coin lands heads or tails about 50% of the time that doesn’t mean that the deterministic system exhibits non-deterministic behavior, just like when a dice lands about 1/6 of the time on a given side that doesn’t mean that the deterministic system exhibits non-deterministic behavior, with the dice too the system can be configured so that some particular side/sides is/are preferred...
It seems that if your goal is to put a man on the moon and you put a man on the moon, your knowledge was 100% accurate. Now, if you wanted to put a man on a certain area of the moon that is only 50 meters in diameter, then that would be a more difficult stunt to pull off. That would require more specific/relevant knowledge to accomplish.
It's interesting to note that the difficulty of some task seems to coincide with it's probability of being accomplished. The more difficult the task, the lower the probability. How difficult a task is is dependent upon how experienced we are with that task - how many times we've done it and worked out all of the kinks in our understanding of the process that it takes to accomplish the goal. It seems to me that these indicate some kind of subjective skewed view of the world where we are imposing our probabilities and level of difficulty out onto the world that isn't probabilistic or difficult/easy. It just is a certain way, which includes the amount of knowledge we have about it.
Our knowledge isn't accurate because we aren't omniscient. We don't have direct access to the entire universe at every moment of our life. What gives us the power to get close enough, if not all the way, is mathematics. Mathematics allows us to summarize our knowledge into simple formulas. It is why scientists are searching for a theory of everything - a formula that explains reality with 100% accuracy and can make virtually any prediction. What human beings would do with this knowledge is a topic for another thread.
The fact that we can load dice so that they increase the chances of rolling a 6 to almost 100% means that we must know something about dice-rolling. Maybe certainty and knowledge come in degrees rather than in bits. If we had enough information about the dice, the dice-roller, and the environment, we'd be able to predict what the outcome would be.
Notice that if we increase the likely outcomes by giving the dice-roller a 20 sided die (for all you pen & paper RPGers out there) then we increase the amount of information we'd need, increase the difficulty, and lower the probability of a particular outcome. The probability of some outcome is constrained by the possible outcomes in a given process, like dice-rolling with different sided-dice. Rolling a 4-sided dice increases the probability of all the outcomes, and makes it easier to predict the outcome.
Information is the relationship between cause and effect. The more causal relationships we are talking about in any given causal process, the more information in that system. Dice rolling has numerous causal processes involved meaning that it would require for some mind to have access to all of that information to predict the outcome. The more information in that system, the more information we need to make predictions about that system.
We also need to take into account that each particular moment of dice-rolling is different. The conditions of each dice-roll aren't the same. The prediction made about one particular dice roll won't apply to the next because there may be different conditions, like how the dice-roller is holding the dice, or the weather changes, etc. So in each instance, the information changes and we need to update our information in order to make a proper prediction. Because some prediction worked and then doesn't work in the next instance isn't to say that our knowledge is inaccurate, or can't be certain. It is to say that the process is different and so the information we have is not relevant to the current situation. So we would be making a category error in applying information to a situation that it doesn't apply, not that our knowledge can't be accurate.
Quoting Pantagruel
It's not just the world, but my own mind. I have reasons for behaving the way I do, or for the conclusions I come to. That is how reasoning works. You use reasons to support your conclusion. Your reasons are usually observations. Reasoning is causal, and can be predictable when you have access to the information in another person's mind - like when you know how they think because you have the experience of having lived with them for 25 years.
"B knows the initial states" But he cannot know the future with certainty. Some factor that will intervene causing variation. The Neil Armstrong moon landing, a classic example of human intervention, when needed. There is an issue in quality control methods that repeated operations don't produce 'exactly' the same results every cycle. The variations are classified by a level of acceptance/significance.
This problem has its roots in an ideal world of absolute values. Current research denies this.
Re:quantum physics, 'It's not the physics that is strange, but the expectations of the researchers'.
Great advice. Thanks.
Quoting litewave
It assumes the principle of indifference in this case - that all dice outcomes have an equal probability of 1/6. This assumption helps us calculate the probabilities and the experimental results match the predictions of this assumption. This is an assumption that the system is probabilistic.
Quoting Andrew M
That makes sense but the issue is that the system (person A and the dice) behaves probabilistically as if B's knowledge amounts to nought. B knows what will happen with each throw of the dice BUT the system behaves as if that knowledge is irrelevant.
Quoting Dawnstorm
Good point. Anything's possible in a game of chance. However, the issue is of predictability. Person B, given he knows the initial state of the system (person A and the dice) is able to predict every outcome; implying that the system is deterministic. However, the system behaves as if that (deterministic character) isn't the case.
Quoting leo
:chin: Kindly read my reply to litewave
Quoting Pantagruel
How did you come to know that?
Quoting sandman
Then sports or life, as we live it, would be impossible. Granted that not every pool player can predict the path of the black ball but experts do it as a matter of routine.
To All
It appears that there is an "explanation" for the situation.
The initial state of the system (person A and the dice) is, as assumed, random. Thus producing the results that are probabilistic.
Yes, person B can predict the outcome of each dice throw but he's oblivious about what these initial states will be. In other words B can predict the outcome of the initial state of the system but can't predict what these initial states will be.
Comments. Thanks.
Thats like saying you can predict what someone will conclude without knowing their premises. Its nonsense. You would not be predicting. You'd be guessing.
What then is the correct explanation?
Probability, in my understanding, is the presence of multiple outcomes, each with its own weightage in terms of likelihood.
The opposite of probability, determinism, is that there is only one outcome given the initial state of a system.
In my example the system (person A and the dice) can have initial states that are probabilistic in nature. Even person B who can accurately calculate the outcome of the system doesn't have access to what initial states will obtain. It's here that probability creeps into what is actually a deterministic system.
Interesting. You are kind of making the case for a 'cumulative empirical intuition'. Even though the knowledge you are talking about doesn't achieve the level of formal conceptualization, it is possible to have a complete-enough knowledge of a system to apprehend it as deterministic. Performance knowledge preceding conceptual awareness. That approach does hold water I think.
He just can't specify which individuals. Statistics is relative to group behavior.
Predicting die toss outcomes relies on the same type of statistics.
I have answered your questions, if you don't want to bother reading/understanding what I've explained at great length in order to help you understand, I won't bother any more.
Physics still relies upon a margin of error, significant digits, and laws, in order for events to operate. The throwing of dice takes on multiple potential outcomes that are significantly sensitive to a multitude of initial conditions. Sending a rocket to the moon, while full of risks is more consistent in its initial conditions and there are known states that rocket must be in various time slices to successfully and safely land on the moon.
1.) No, probability is not an illusion.
2.) Mediating variables/forces alter outcomes from deterministic systems.
What about a quantum coin or a quantum dice? Get ready for the deep mindfuck that is the quantum world of "subjective facts"
That can be explained in an intuitive way though, without giving up an objective reality, when we take into account the fact that on small scales the instrument of measurement interacts with what is being measured in a non-negligible way, thus two different observers (or instruments of measurement) can make different observations/measurements :smile:
Your response doesn't answer the fundamental question; How does a subjective microverse create an objective macroverse?
QM is not subjective.
Lack of effort... cute.
You have brought argumentation you claim is evidence, but is not any sort of evidence. Quantum mechanics follows laws and then there are also 6 postulates of quantum mechanics as well. Wave-particle duality represents that depending upon the experiment or measurement being employed, one or the other will show up, but this hardly subjective; furthermore there is causality and determinism within QM, but it is counter-intuitive to our experiences in classical physics, (the macro-world).
To very simply and accurately answer the question of subjectivity, if QM were not objective than the correspondence principle would not hold where small particles and forces give rise to what we see in macro-physics, but rather, we would see all sorts of non-sensical changes.
http://dailynous.com/2019/03/21/philosophers-physics-experiment-suggests-theres-no-thing-objective-reality/
However I'm thinking more of narrow subjectivism within paramaters of probability. I should have been clearer.
It still doesn't take away from the fact that if you read the link I shared and do some thorough research of the concepts contained within, you'd find that the conceptualization of a Subjective Fact within QM is entirely consistent with experimental data we currently have.
Some other good material to read up on; Quantum Eraser experiments and their variants plus the myriad of different double slit variations. Then you have the curious behaviour of light somehow bending around a galaxy only on one side only to be viewed later as bending around the opposite side with no explanation of how the light could have made it from one side of the galaxy to the other faster than the speed of light.
I don't claim to make any assumptions about your familiarity with QM but I'm very aware of my own and I'm up to date enough to know a curveball to my own understanding of QM when I read it and what I shared is definitely a curveball. What do I know though? Just relaying what the experts in QM are currently saying and matching it with the things I've already learned over the years.
Let's start here since it is fundamental mathematically:
http://vergil.chemistry.gatech.edu/notes/quantrev/node20.html
Without the postulates, no math can be discussed regarding QM.
I'm trying to figure out what you think a "probabilistic system" should look like. "The initial state of the system" is different for A and B. For A, it's simply a game of dice. For B, it's the current state of the universe. For A probability only allows six outcomes. B could know that A will die of a heart attack before he ever gets to throw the die (and his hand cramps, so the die doesn't even drop). In my view you're comparing apples and oranges. A asks "What are the odds?" and B asks "What will happen?"
B uses the chain of causality to compute the outcome. A uses probability to compute the odds. Take the following example:
A bag contains only red balls. You draw one of them in the hopes of it being red.
A will use probability theory and know immediately that given that he'll successfully draw a ball it will be red (because there's only one option).
B will have to go through multiple computations to figure out which ball A will draw and then check its colour. B will know, though this process, if A will successfully draw a ball, if so which one, and by implication its colour.
In this limited case, A and B will come to the same conclusion. Why? Because the probability to draw a red ball from a bag that only contains red balls is 100 %. B has a lot more information that pertains to the situation, though, including whether A will draw a ball at all.
I'm not sure I understood you correctly, though. I'm right in assuming that B follows the chain of causality (taking into account all data he has) and doesn't encounter a truly random process (which would contradict determinism)?
Of course, given perfect knowledge in a deterministic system, the question "What are the odds?" is superfluous, because it's always 100 %. But A has very limited knowledge.
A and B have different perspectives: A's tends to be more efficient (but he'll have to contend with risk), and B's tends to be more accurate (but he'd probably die of old age before he finishes the computions).
But the presence of multiple outcomes are in your head, not out in the world. It wouldn't be correct to call them multiple outcomes. Only one outcome occurs, not multiple ones. Possibilities are not outcomes. They are ideas in the head in the present that can change your behavior to be more in tune with the reality of the situation, or the imagined situation that you call a possible outcome.
The weightage of some outcome is dependent upon the information you have about present conditions and the effects they leave in some future moment, and the amount of outcomes we are talking about - like rolling a six-sided dice vs a 20-sided dice. Because the 20-sided dice has more "possible outcomes" than the six-sided one, the probability of any particular side being on top decreases. This is all the result of our ignorance. If we weren't ignorant of the facts of the present conditions and the effects they lead to in the future, then there would be no such thing as possible outcomes. The one and only outcome would be known.
The initial states aren't probabilistic in nature. If someone doesn't know the initial state, then how can they know some future state?
Experiments are typically done in an isolated environment to eliminate outside influences. Even with a mechanical tossing arm, at a microscopic level, it doesn't impart exactly the same impulse to a die each toss. Thus you don't have complete knowledge of the die state,unless you monitor the complete process, which itself introduces extraneous factors. You can only know the past!
Glad to find somebody who gets the math! I hope you understand it better than I do because it still crunches my head sometimes; before we carry on though, are you familiar with the concept Supervenience? X Supervenes Y; meaning X can change when Y has changed?
What makes me say QM is subjective is probably me mispeaking or not being specific enough. QM is a field of Objective human inquiry. However the Quantum Realm is subjective and as everything outside of the quantum Realm supervenes on the quantum realm; coupled with the new quantum coin inquiries leads me to some questions. Mainly how is a subjective quantum realm creating what seems to be a physically objective reality?
The reason I asked about Supervenience is that the link you shared reads as if it has forgotten Supervenience as it orders things strangely through the linear human discovery as opposed to the real orders of Supervenience.
Its kind of like using Atom to write code. Imagining atom as the universe, Sure the code you are writing is physics, but atom is written in Java (Quantum Mechanics). physics as we knew it before has kind of had a hole blown in it if you keep attempting to declare its dominance over QM.
So please tell me you are aware of Supervenience?
reality only happens one way, and can only happen one way
possibilities and probabilities only exist in the ignorant mind that doesn't know which way its going to happen. if you had more knowledge and knew for-sure then you would see there was no other possibility or probability.
there is no potential in reality
"Do or do not, there is no try" - yoda
How else can the 'factual' deterministic properties of a system be defended or even described, except in terms of the behavioural frequencies of ensembles of similar systems, to which a limit argument is then applied to produce a statement that has no correspondence to reality.
Determinism is neither a factual nor a logical concept; logic refers to statements that we treat as being identical. But identicality isn't a factual statement. All we have is 'factual similarity judgements' that refers to distinguishable facts which 'share' a set of preconditions and consequences in the sense that we have put them into a rough correspondence. Identicality is our treat of 'similar facts' as synonyms.
Thanks for your comments. It's been some time so you might have lost the train of thought.
Probability, in my opinion, has to be objective or real. By that I mean it is a property of nature just as mass or volume. So, when I say the probability of an atom of Plutonium to decay is 30% then this isn't because I lack information the acquisition of which will cause me to know exactly which atom will decay or not. Rather, radioactivity is objectively/really probabilistic.
If you agree with me so far let's go to my example: person A who doesn't have knowledge of the initial states of each dice throw and person B who has.
The fact of the matter is that, experimental probability? the outcomes of a throw of a dice, say done a 100 times, will be an almost perfect match with the calculated theoretical probability. For instance the probability of a dice throw with outcomes that are odd numbers is (3/6) or 50% and if you do throw the dice 100 times there will be 50 times the dice shows the numbers 1, 3, 5 (odd numbers).
This match between theoretical probability and experimental probability is "evidence" that the system (person A and the dice) is objectively/really probabilistic.
However, person B knows each initial state of the dice and can predict the exact outcome each time.
So, we have an "apparent" conundrum on our hands. The system (person A and the dice) is deterministic for B but it is also probabilistic (for A and B) in an objective sense.
At some level of the experiment, probability has creeped into the system (person A and the dice). The outcome of the dice is determined by the initial state of the dice. In essence we can replace the outcome of the dice with its initial state since the latter determines the former.
If, as the experiment reveals, the outcomes are indicating the system (person A and the dice) is objectively probabilistic, then it must be that the initial states are probabilistic. After all the outcomes are determined by the initial states.
What do you think?
I never said that probability wasn't real. I said it is imaginary. Our imaginations can cause us to do things - like behave as if some other possibility was real in the sense that it exists as something other than an imagining.
I don't know what you are talking about when you are saying that "the probability of an atom of Plutonium to decay is 30%". Radioactive decay is a lawful process:
https://www.nuclear-power.net/nuclear-power/reactor-physics/atomic-nuclear-physics/radioactive-decay/
Quoting TheMadFool
Like I said, if someone doesn't have knowledge of the initial state, then how can they have knowledge of some future state which is just another initial state to some other future state further down the causal chain?
Quoting TheMadFool
I don't see outcomeS. I see an outcome. There is only one outcome, but in the eyes of the ignorant there are multiple outcomes. How do you reconcile the fact that you have multiple outcomes in your head but only one outcome occurs - and maybe one that you didn't have in your head. If you didn't predict the outcome then what happened to all those outcomes you did predict in your head? They weren't really outcomes then, were they?
Quoting Harry Hindu
What is your explanation for why the system (person A with the dice) is behaving probabilistically?
You mentioned an important element in the system - ignorance. Person A is ignorant of the initial state of each throw of the dice and person B is ignorant of which initial state becomes a reality even though he knows the outcome after any particular initial state is selected.
So you think probability is an illusion and is just a symptom of ignorance?
There's one issue here that bothers me. If probability is an illusion/imaginary how is it that, in a simple game of dice, the principle of indifference - a feature of true/non-imaginary probability - helps us calculate probabilities that match experimental results? This isn't about ignorance is it? A deterministic system is conforming to a principle that applies only to objective probability. That would be like, in essence, being able to predict random numbers. There's something wrong. Care to take a shot at this. Thank you.
Sorry I missed your reply. Consider that on very small scales, when you try to measure a tiny thing (for instance by sending electrons or light towards it and measuring what's reflected), the act of measurement itself (which includes bombarding whatever you're measuring with electrons or photons) changes the position/trajectory of what you're measuring sufficiently that by the time you get the measurement you don't really know where the thing you have measured is anymore, you simply know approximately where it was and how fast it was going, but you don't know where it is and how fast it is going.
On the macroscale, the photons that for instance your body emits have a negligible influence on what you observe for instance with your eyes, the presence of your body does not change the position of a rock or of a wall if you're simply looking at it.
The popular interpretation of quantum mechanics is that probability is fundamental, which leads to all sorts of confusion and apparent paradoxes, but it's not necessary to see it that way, it can be interpreted in an intuitive way (as above). There's a similar situation in relativity, special relativity can be interpreted in a way that doesn't lead to all kinds of incomprehensible paradoxes (like the twin paradox).
Quoting TheMadFool
I already explained in my previous posts everything you need to understand your confusion, if only you made the effort to read and understand. I won't repeat everything obviously as you would probably again not care to read the whole thing.
The fact of the matter is that, sometimes, you will throw the dice a hundred times and you will get the same outcome a hundred times. It is possible that you throw the dice a hundred thousand times and that it never lands on some specific number.
Would you count that as evidence that the system is not really probabilistic?
Quoting TheMadFool
Yes, in a deterministic system the outcome is determined by the initial state, which includes the initial position/orientation/angle/velocity of the dice, the air density, wind, shape/hardness of the ground and so on. And again, as I explained several times, make one side of the dice more sticky than the others and the dice will land more often on that side, changing the probability distribution. Always start with the same initial state and the dice will always land on the same side, changing the probability distribution (100% for that side, 0% for the other sides). Would you say that in a system where the dice always lands on the same side, the system is inherently probabilistic?
Reflect on that, in order to understand the problem in your reasoning.
(hint: a dice is symmetric, if you don't break that symmetry then the rest of the system has no reason to break it either)
(hint2: if you rotate the dice in your hand for a while in arbitrary directions without looking at it and then you look at which side is up, you will get each side about equally as often, does that mean that the dice is inherently probabilistic? That the system dice+hand is inherently probabilistic? Or neither?)
What does intuition mean to you though? I've got what my answer is or what I think it might be but I'm curious to know yours.
Do you think it might be possible that what is intuitive to you isn't intuitive to me? If so, why? Thank you for the constructive points and I am looking forward to hearing more :)
I'm never sure where I stand on probability but whether the universe is entirely probabilistic I cannot say. What I'm sure of however, is we are ignorant of what all the probabilities are. Unfortunately due to the very nature of human knowledge, there is an insurmountable wall between individual knowledge and collective knowledge. While technology and science are lowering the height of this wall it is still there. The wall of ignorance is made of time. For all we know, in the vastness of human inquiry and knowledge we may in fact collectively know all there is to know or at least all probabilities have been guessed... But how could the individual ever know what the collective knows in its full scope? You would have to know what everyone has said of everything and recall it all at once.
Sorry if this is too off topic for the OP. I'll let the mods decide if this should be moved elsewhere.
I meant intuitive as in particles can be seen as having a definite trajectory even when they aren't observed, as in one particle doesn't follow several trajectories simultaneously, as in things do not behave in a fundamentally different way than what we're used to observe.
Sure it's possible that what is intuitive to me isn't intuitive to you, however it seems to me that most people find it unintuitive to imagine a single particle following two different trajectories at the same time, or to imagine two twins each aging more quickly than the other when they are in relative motion and yet when they reunite one has aged more than the other, actually I believe I have yet to find one person who finds that stuff intuitive :)
Well, what I percieve to be intuition; is right now telling me to point out that obviously two twins age differently when apart. Time is relative. If one spends time in a mountainous region or is an astronaut that has done a round trip to the moon what did you think was going to happen?
I think this is where we are getting into something really fascinating! Join me in an intuition thread later!
But it's not that the two twins age differently that's necessarily unintuitive, it's that at every moment each twin is aging more quickly than the other, twin A ages more quickly than twin B and twin B ages more quickly than twin A, yet when they reunite only one has aged more than the other, if you find that intuitive then indeed hats off to you and I want to hear more :grin:
How repeatable was this observation? How consistent? I'm only vaguely aware of the summaries of a few of the studies but I'd need to go deeper to determine any stance on the matter yet. I'll share my thoughts on intuition and what it is with you in my discussion soon. Working on it now. It will make sense then why current use of the words intuitive and unintuitive is probably misleading us from the nature of the phenomenon of Intuition itself.
To put the matter succinctly, imagine that at every moment during your trip the theory tells you that your twin is aging more slowly than you, yet when you reunite with him he has aged more, and the theory explains why in a convoluted way. It is unintuitive to me and to many that at every moment your twin ages more slowly than you yet when you reunite with him he has aged more. That's a paradox, yet the popular interpretation of special relativity is that it is what really happens.
A less popular but more intuitive interpretation is that during the trip the other twin does age more quickly. Technically it's not an interpretation of special relativity as it doesn't start from the same postulates as special relativity, but it is experimentally equivalent (in the sense that the two theories make the same observable predictions, but they give different explanations as to what is really going on behind the scenes).
The idea that the other twin ages more slowly is not something that can be directly observed/tested since we don't have instantaneous signals that can tell us how fast the other twin is really aging at every moment, we only infer that from the theory. But that's precisely the point, we are not forced to use an unintuitive theory to explain what we do observe, we can explain the same observations in an intuitive way.
Looking forward to your thread :up:
I agree this is my intuition on the matter as well. Simply due to the knowledge that gravity stretches time. As for the Postulates; I don't like to assume anything. Physicists and mathematicians can assume what they want. We shouldn't conflate scientific facts and evidence with the opinions on them.
Well the idea that "gravity stretches time" is based on assumptions, it's not something we observe directly. Any prediction is necessarily based on assumptions. And gravity isn't involved in the twin paradox, but all of that would be better suited for another thread :wink:
How is Gravity not involved in the twin paradox? Are the Twins floating in a vacuum? How barbaric!
Yes.
The only possible outcomes of rolling a six-sided dice is rolling a one, two, three, four, five or a six. Both and person A and B know this and we don't need to know the initial states to know this because we are confining the outcomes to the die only. No matter how you roll the dice, or what the weather conditions are, there will only be an outcome of 1-6 on the die roll because that is what we are focused on.
Quoting TheMadFool
The principle of indifference is based on our ignorance of the facts. When you don't know the facts, every possibility is equally possible.
I agree however i would like to note that just because something is an allusion does not make it completely useless. If I plan to go to walmart for low prices (walmart usually has low prices), it just might so happen i will be mowed down by a serial killer. Probability is used to increase a persons chances of making the right decision. We all will likely make bad decisions the longer we live.
Again this is better suited for another thread as this is really far away from what the OP is about, so this will be my last post about that in this thread:
Observations are not assumption-free, how you interpret the evidence is theory-laden (depends on your implicit assumptions that you haven't necessarily uncovered): https://en.wikipedia.org/wiki/Theory-ladenness . As a simple example you may interpret some observation as showing you something real, or as it being an illusion, hallucination, imagination ...
Astronauts haven't been observed to age differently, it's what the theory predicts (relativity and some others). Relativity is based on postulates, obviously these postulates were chosen so as to account for many observations and experimental results, but you can't predict anything if you don't start from any assumption. Even if you somehow believe that your observations are assumption-free, there is still the problem of induction, how do you know that the universe is going to keep behaving the way it did in the past? That's an assumption.
What has been detected is that some clocks run at a different rate depending on their location and velocity. It takes assumptions to move from that to saying that "astronauts age differently".
Gravity can be neglected in the twin paradox, the paradox arises due to relative velocity, gravity doesn't have to be involved, you can have the twins in a vacuum and the paradox still applies. Also it's a thought experiment, we haven't tried it in practice.
We have though; there are countless twin studies on aging that have been done. Admittedly most of these studies are usually in the fields of geriatric care; however, with a bit of logic skills and cross referencing with the physics material on the subject matter, you can identify the evidence from those studies that pertains and shows premises with which to make arguments in this area.
You are right though, we are getting too far away from the OP. However our line of discussion does have meaning which contributes to the probability argument.
For example; the dice and coin toss that has been discussed here is very curious. Probabilities for what could happen there are endless. A six sided die will only land on a face if it is not thrown at an escape velocity relative to the strength of gravity of the object it is thrown from.
Now, if say I threw a dice on the moon with enough force for it to escape the moons and earth gravity well, we might never know which side the die will land on. Especially if the vacuum of space washes away the numerical markings. It might never even land on anything or it might land somewhere which will denature the die like jupiter or the sun.
So in the argument of probability is an illusion; I say that we do not yet know enough to say whether or not the universe is entirely probabilistic; we do know enough to know that we do not know what all the probabilities are. So our current understanding is an illusion but that doesn't have to mean that Probability itself is an illusion. It could still be or not be.
As far as I know no study has been done regarding the influence of relative velocity on the relative aging of twins.
Regarding probabilities, we know that usually probability refers to incomplete knowledge, and even in quantum mechanics where probabilities are said to be fundamental it's possible to interpret observations in a way that doesn't involve fundamental probabilities. However it isn't clear that the whole universe is a deterministic system, it is possible that the will is fundamentally not deterministic, not determined by deterministic laws.
I'll give you another hint: it's possible to prove mathematically that in a deterministic system, if your 6-sided dice is perfectly symmetrical then each side will show up 1/6 of the time, without invoking probabilities at any point. It's not a mystery, it's a consequence of the symmetries of the dice.
As you mentioned, the outcome is completely determined by the initial state. So you have to prove that 1/6 of all the initial states lead to outcome "1", 1/6 of all initial states lead to outcome "2", and so on. In order to do that you have to enumerate all the initial states.
As I mentioned, the initial state is described by various parameters: initial orientation of the dice, initial position of the dice, initial velocity the dice, initial direction of motion of the dice, initial air density at each point of the system, initial air velocity at each point of the system, initial shape of the ground, initial hardness of the ground at each point, ... and so on. Let's call these parameters p1, p2, p3, ... pn, where n is the total number of parameters.
Each of these parameters can take many different values. For instance the parameter "initial velocity of the dice" can take as many values as there are initial velocities that the dice can have. Let's say that the parameter p1 can take v1 values, the parameter p2 can take v2 values, the parameter p3 can take v3 values and so on. Then the total number of initial states is v1*v2*v3*...*vn
The key thing to use is the symmetries of the dice. These symmetries will play a role in the parameter p1 (the initial orientation of the dice).
If you keep the initial parameters p2, p3, ..., pn constant and only vary the initial parameter p1, consider how you can use the symmetries of the dice to prove that in 1/6 of all initial states the outcome will be "1", in 1/6 of all initial states the outcome will be "2", and so on.
This is my feelings on the matter too. I feel the universe has it's own dichotomy of control. Determinism is one side of that dichotomy. Will of Life seems to play by different rules in my opinion.
Yep I think so too :up:
I often turn to the Piano to explain my thoughts here; now before its creation, you could probably only determine one thing about the future, that new musical instruments will be created.
Could you have predicted the piano in all its complexity and nuance? What it would be made out of, what it could be made out of, what it could inspire, what the first key press was going to be and which note, was it the right note or was it out of tune the first time, what songs were going to be made, what books from the creative inspiration, stories, narratives, paintings, marriages, killings?
It's easy to look at the past and with 20/20 hindsight to boldly claim that everything is deterministic. How easy is it to do that from the past though?
I think if we brought a person from the past to New York city or Tokyo or Hong Kong they would probably say this was all beyond all their wildest dreams.
[Probability is a human procedure based on statistics of past events, to predict future events. The mechanism (physical laws regulating behavior) of radioactive decay is not fully understood. We could speculate that space, full of radiation, is a factor. Probability compensates for lack of knowledge, by giving the most expected outcome.
In weather forecasting, there are so many variables, it isn't possible to know their current state, in such a dynamic system. This results in weather forecasts being very local and short term.]
[No he can't. If B could predict the exact outcome, there would be no reason for probabilities, and there would be no 'game of chance'. To clarify the issue: in the process of throwing a die (singular), B does not KNOW the microscopic processes affecting the 'throw'. He hasn't refined his analysis to include factors he omits as insignificant, or there are factors he is not aware of (weather people only recently be came aware of ocean currents affecting weather patterns), or he can't monitor known factors fast enough to revise his initial prediction. Knowing the initial state does not determine an outcome with certainty. The outcomes of die tosses does not cluster around the 50% value, but has a range of +/- 49%. I see Leo has touched on this.
Let's focus on the fair coin toss, with H or T. If a coin is tossed 100 times, the outcome can be 100H or 100T, or any combination of the two totaling 100. Randomness requires that all the factors affecting the outcome are present to approx. the same degree, no bias, no dominate factor. Then the essential factor that must NOT be present, memory.
Each toss is independent of the others, and is independent of time. That means you can't predict when an H or a T will occur. This also allows for 100H in a row, with the popular response, 'but that can't happen if the odds are 50% for H'. The protester arguing 'it's such a rare event', he doesn't expect to see it. If it can't occur in his life time, or that of people before or after him, when can it happen? Time is not a factor, so a 'rare event' can happen anytime.]
I don't know whether I agree or disagree. I'm not sure what - in terms of the real world - it would mean for "probability to be real". Probability is maths, and like all maths it's applied to the real world, and so the question is whether it's useful or not rather than whether it's real or not.
A operates with a very "small" probability system, and B with a very large one. A can expand to B, and B can conflate to A. When A expands, the likelihood for throwing a particular number increases until it drops to either zero or hits 1. That's just conditional probability. A's probability table would have to exhaust all probabilities.
What if the universe doesn't have an initial state, just a string of causality that breaks at some point in the past, because stuff like frequency stops working? You could only approximately describe this with a mathematical system, right? Assuming mutliple possible initial states would work, but only if we can describe all those states and their relations such as that they are mutually exclusive.
So, yeah, what does it mean for probability to be real?
That is a consequence of how you've defined the system. It seems to me that what you're pointing out is just that a predetermined initial state is incompatible with objective probabilities.
A simpler example would be a computer simulation of a hundred dice throws. The results appear random to an observer, but they are merely the outcome of a complex deterministic algorithm. With the same seed, the results are repeatable on subsequent runs.
The only issue then is how the initial state (the seed) is set - whether to a predetermined value or to a random external input.
Sorry for the long delay in my response but I was waiting for an epiphany of sorts. My mind just drew a blank so I'm going to work at this problem from scratch if you don't mind.
1. There is, more or less, an agreement that a die throw is deterministic.
2. The outcome of die throw can be calculated probabilistically e.g. probability of getting a 3 is 1/6
3. Each outcome of a die-throw can be calculated deterministically i.e. given the initial state of the die we can accurately predict each outcome
4. The outcomes of a set of 1000 die-throws can be predicted probabilistically e.g. 3 will appear approx. 166 times
5. point 2 agrees with point 4. In other words the die-throw is behaving as if determinism is false for the die
The problem for me is 1 and 5 contradict each other.
Some (@Harry Hindu) have said that probability = ignorance but that would mean that there is no such thing as actual chance and what we perceive as chance is a manifestation of our ignorance.
However, if that's the case 2, and 4 should be false but they are true and indicate the die is behaving as if determinism is false.
Statistics is a tool used for people to make decisions, it does not typically always take into account all the laws of physics and chemistry (not all of these laws are known and also to calculate them is really hard anyway). A Brief History of Time by Stephen Hawkings addresses the problem that even if all known laws are known it is still very hard to calculate particle interactions (quantum mechanics for example).
You can help predict who will win a billairds game using statistics but that doesn't mean the statistician knows the laws of physics. Statistics is a very sloppy form of mathematics and is very different from Physics and Chemistry.
So basically you have ignored again what I have taken the time to explain to you in details. I have shown you where your error lies, and you keep ignoring it and keep restating your error again and again as if no one had addressed it. That’s not respectful. You assume that you have noticed something that other people haven’t noticed, but the reality is that some people understand why you’re wrong and explain it to you but you keep ignoring what they say, you keep assuming that you know better while you don’t.
I have shown you that you can prove mathematically that in a deterministic system, as long as the dice is perfectly symmetrical, and if there are n different ways to throw the dice, then there are n/6 ways in which the dice lands on number x (where x is 1, 2, 3, 4, 5, 6). It’s not magic, it’s not a deterministic system behaving probabilistically, it’s a consequence of the symmetries of the dice.
If you always threw the dice in the exact same way while the rest of the system remains the same, the dice would always land on the same number. But in practice you don’t throw the dice in the exact same way, you throw it quite randomly, and since there are n/6 ways in which the dice lands on number x, then in practice after many throws the dice lands on number x about 1/6 of the time. If you have 6 numbered balls in a box and you pick one randomly then put it back, after many picks you will have picked each ball about 1/6 of the time. Same principle.
If there is something you don’t understand about that, then ask. Otherwise stay ignorant if that’s what you want.
Say there are 6 numbered balls in front of you, they aren’t even in a box they are in front of you, one ball has number 1, one ball has number 2, one ball has number 3, one ball has number 4, one ball has number 5, one ball has number 6.
You pick ball number 1. Then you pick ball number 2. Then you pick ball number 3. Then you pick ball number 4. Then you pick ball number 5. Then you pick ball number 6.
You have picked each ball one time, right? You have picked each ball 1/6 of the time, each ball has been picked 16.666666.. % of the time. Does that mean the deterministic system is behaving probabilistically? No!
Just because you have picked a given ball one time out of six, just because you can express with a percentage how often a given outcome has been realized, this doesn’t imply that the system was behaving probabilistically.
Hopefully you agree with that. So stop saying that a deterministic system is behaving probabilistically simply because you can express the outcomes in terms of percentages, in terms of ratios. If you cut a pizza in 4 equal parts, each slice of the pizza is 1/4th of the pizza, 25% of the pizza, that doesn’t mean that the pizza is behaving probabilistically...
If you can understand that basic error you keep making then maybe you can start understanding the rest.
Anyway you said:
1. The behavior of the die is caused by its symmetry and then you said Quoting leo
2. Just because the outcomes can be expressed as a percentage doesn't imply that the outcomes are probabilistic
Firstly, why did you say "you throw it quite randomly"? I would infer from it that it is necessary for randomness to enter into the system (the die) at some stage of an experiment.
Secondly it isn't the mere fact that I can express the outcomes as percentage but that these percentages agree with the theoretical probability which is possible if and only if the die is random. Yet, as you seem to agree the die outcome is deterministic in nature.
How do you reconcile the fact that the die is a deterministic system and yet behaves probabilistically? I'm as nonchalant about this as I would be if someone said s/he could predict the outcomes of random events.
Okay, sorry for reacting that way, it’s just not pleasant to take time to explain something carefully in order to help you see your misconception only to be ignored again and again.
By the way I’m not a native English speaker and I just realized that the singular of ‘dice’ is ‘die’, so I just learnt something from you (I was wondering why you were always spelling it ‘die’).
Quoting TheMadFool
Yes, but randomly does not imply non-deterministically. For instance we have random number generators that are deterministic. We might say it’s not true randomness, but practically the outcomes appear random.
The reason I talked about throwing the die randomly, is that if you repeatedly throw the die in the exact same way then the outcome will always be the same, say you will land 100% of the time on number 3. In a deterministic system, when you start with the exact same initial conditions you get the exact same outcome. So clearly, the reason that the die doesn’t always land on the same side has to do with how you throw it.
If you always start with only two different initial conditions, you would only get at most two different outcomes. In order to have each side of the die showing up, you have to throw the die in many different ways. But throwing it in many different ways is not enough to have each side showing up 1/6 of the time, because if you can predict the outcome that results from given initial conditions, you could arrange to throw it in a thousand different ways and always get the number 3.
So in order to see each side showing up 1/6 of the time, you don’t just have to throw the die in many different ways, you have to not control the initial conditions, you have to pick the initial conditions in a somewhat random way. (well you could also arrange to select initial conditions so that each side shows up 1/6 of the time, but in practice the initial conditions are selected without knowing the outcome in advance).
Quoting TheMadFool
First you have to understand that the die itself does not behave probabilistically, you think it does because of your misconception. As you can see from what I said just above, the theoretical prediction that each side shows up 1/6 of the time is not always valid, in practice it is valid if you throw the die in many different ways without knowing the outcome in advance (without selecting particular initial conditions in order to get the outcomes you want).
Then the question becomes, if we throw the die somewhat randomly (without knowing the outcome in advance), why is it that after many throws each side shows up about 1/6 of the time? This is what seems like a mystery, but once you understand there is no mystery.
Since there are many different ways to throw the die that lead to a given outcome (say number 3), it is possible to throw the die in a thousand different ways and always get the number 3, even without selecting the initial conditions so as to get the number 3, even without knowing the outcome in advance. It is rare, but it is possible, it can happen. So in fact the theoretical prediction that each side shows up 1/6 of the time doesn’t always work, but most of the time it does work, most of the time that’s approximately what we get. Why is that?
The reason is the exact same reason why if you have 6 numbered balls inside a box and you pick a ball without knowing the outcome in advance, most of the time you will get each ball about 1/6 of the time. It can happen that you pick the same ball 100 times in a row, but it’s rare. The answer doesn’t have to do with probabilities, but with statistics.
There is only one way to pick the ball number ‘1’ 100 times in a row. There are only six ways to pick the same ball 100 times in a row (you might pick always number 1, or always number 2, ...). There are many more ways to pick only 2 different balls during 100 picks. There are many more ways to pick only 3 different balls during 100 picks. There are many more ways to pick each ball at least one time during 100 picks. And there are many more ways to pick each ball about 1/6 of the time, than to pick one ball 95 times and each other ball only one time. The number of ways that exist can be calculated, maybe I’ll do that when I have the time.
But basically, the reason that most of the time each side of the die shows up about 1/6 of the time, is that there are many more combinations where each side shows up about 1/6 of the time than there are combinations where some sides show up much more frequently than the others.
And this is only true if the die is symmetrical. If one side was much more sticky than the others, the die would land more often on that side, and then one outcome would show up more often than the others, and then most of the time that’s what we would get, even though in rare cases that outcome would show up as frequently as the others.
Let me know if something isn’t clear still.
That 2 and 4 are true is completely consistent with determinism. A deterministic algorithm is able to produce pseudorandom numbers that follow a probability distribution. And the generated sequence is repeatable.
Quoting Random seed
This is what I want to discuss if you don't mind and thanks for your effort in trying to make me understand.
We have to revisit our assumptions:
1. Either a system is deterministic or it's random but not both
2. The die is a deterministic system in that with the necessary knowledge of the initial state of each throw we can predict every outcome accurately
3. Theoretical probability calculations has as a fundamental assumption that what is being calculated is random. The theoretical probability of the die showing three is 1/6
4. An experiment is done and the die is thrown 1000 times. In accordance with the theoretical probability we'll get three on the die approx. 166 times or 1/6 of 1000 throws
5. The fact that 3 and 4 agree with each other implies the assumption that the die is random is correct
Notice that 2 states the die is deterministic and 5 states the die is random/non-deterministic and this is a contradiction because of 1.
I'd like to give my own "solution" to the paradox:
A deterministic system can't be random and the die is behaving as if it is random. This implies that a random element was introduced into the system (the die) at some stage of the experiment (throwing the die 1000 times) and I think this happened when we chose the initial states of each of the 1000 die throws - all initial states were chosen randomly and so the outcomes conformed with the theoretical probability which makes the assumption that the system (the die) is random.
Do you agree with my "explanation"?
Not exactly. In your OP you correctly said:
Quoting TheMadFool
There is no randomness involved in that definition.
In an earlier post I gave hints as to how you can show that for a symmetrical die subjected to deterministic forces, the “number of ways event E can be realized” (for instance the number of ways that outcome ‘1’ is realized) is 1/6th of the total number of possible events:
Quoting leo
So with that definition of probability you can compute that it is 1/6 for each side of a perfectly symmetrical die without invoking any randomness.
Quoting TheMadFool
One important thing to keep in mind is that “probability = 1/6 for each side” does not imply that in practice that’s what we will get. It is possible to throw the die 1000 times and get the number three 0 time. Theoretically it is possible to throw the die an arbitrarily large number of times and never get three. As a simple example, if you always throw the die in exactly the same way, you will always get the same result. If you always throw the die in ways that never lead to the outcome three, you never get three.
What “probability = 1/6 for each side” means is that experimentally if you throw the die once in every possible way, each side will appear in 1/6th of all throws. And if you do that, there isn’t any randomness involved at any point.
When I talked of throwing the die randomly, I meant that if you throw the die in a specific way so as to get a particular result, that’s the result that you will get. So for instance if you can predict the outcome in advance for each way that you can throw the die, then you can throw the die in specific ways so that each side will show up with the frequency you want. You can make only one side always show up, or only two sides, or one side 10 times more frequently than the others, or whatever you want. But if you don’t attempt to throw the die in specific ways there is no fundamental randomness involved, it’s simply that you aren’t choosing a particular outcome in advance.
At that point the only apparent mystery that remains is why when you throw the die only 100 times, most of the time each side shows up about 1/6 of the time, and as I described in my previous post that can be explained with statistics, there is no need to invoke any fundamental randomness.
Quoting TheMadFool
I agree very partially. The theoretical probability does not make the assumption that the system is random. The die doesn’t behave as if it is random. Throwing the die 1000 times doesn’t introduce a random element.
Where I agree is that how we choose the initial states has an impact on the frequencies of the different outcomes. When we throw the die in various ways, as arbitrarily as possible, most of the time the experimental observations will be close to the theoretical probability, but it is important to see that in some rare cases, even if you pick the initial conditions as randomly as you can, you can still get frequencies that are totally different from the theoretical probability (for instance getting the number three 1000 times in a row even though you have thrown the die in many different ways without knowing the outcome in advance, this is very rare but it can happen).
Isn't it probable that you roll the dice six times and never get a 3?
Quoting TheMadFool
What does approximately mean? Doesn't it mean that it is possible that you are wrong? Isn't it just as likely that 3 will appear approx 150 times or 200 times?
Quoting TheMadFool
They are. Roll the dice and find out that it is possible to not roll a 3 in six rolls, or roll a 3 166 times out of 1000 die rolls. Your use of "approximately" doesn't supply some truth, only an approximation, so I don't see how you could say that it is true. Approximations can't be truths. They are guesses and we guess because we are ignorant.
Quoting leo
I just read a very simplified version of the law of large numbers which asserts that as the number of probability experiments increases, the results of the experiment approaches the calculated theoretical probability.
So you're right that "unexpected" outcomes such as 20 threes in a row can occur in a 100 throws of the die. However, as the number of experiments are increased, say to a million throws, the frequency of threes in that million will be approx. 1/6.
Quoting leo
Did you mean the law of large numbers?
If yes then that implies the die is behaving randomly. Whence this randomness?
You seem to be saying that probability = ignorance but that would imply that there is no such thing as randomness or even chance.
If that's the case then consider:
1. A theoretical probability assumes randomness in its calculations. The theoretical probability for a three is 1/6
2. The die thrown 1 million times will show a three 1/6 of the 1 million throws
2 is exactly as predicted by 1 and 1 assumes randomness.
According to your claim then our ignorance led to the random behavior of the coin? How is this possible? How can my ignorance lead to randomness?
You didn't answer those questions I had in my post.
There is such a thing as randomness and chance. They are ideas that stem from our ignorance. Like every other idea, they have causal power. It's just that you are projecting your ignorance/randomness/chance out onto the world where their only existence is in you head as ideas.
How does my ignorance cause the die to become random?
Separately, I must ask you this:
Are all random and chance events caused by our ignorance?
Your ignorance doesn't cause the dice to do anything. Your ignorance causes you to think of the world as probabilities and chances.
The outcome is determined. Your ignorance causes you to not know the outcome. You can only guess at the outcome. Your guess is educated in that you know the possible number of outcomes, but not the actual outcome, hence your educated guess is the probability that exists in your head, not in dice.
If it was all probability, then how is it not probable that you roll a 10 on a six-sided die? What constrains the possible outcomes?
If it's as you claim, all in my head, how does the die know to come up three 1/6 of the time? Is the die sentient and after finding out I don't have the necessary information to predict, it does everything in its power to ensure that it behaves randomly in such a way as to match my probability predictions?
Quoting Harry Hindu
It can easily be shown the P(10) = 0/6 = 0 = impossible. This has no bearing on why the die is behaving randomly.
It’s possible to throw the die a million times and get three a million times. It’s possible to throw it a gazillion times and get three a gazillion times. Because, again, in a deterministic system the outcome is determined from the initial conditions, so if you always throw the die in exactly the same way you always get the same result.
Now let’s say there are 1 gazillion different initial conditions that yield the outcome three. That means you can throw the die 1 gazillion times in 1 gazillion different ways and always get the outcome three. You can throw it 1 billion gazillion times in 1 gazillion different ways and always get the outcome three.
However if there are N different ways to throw the die (say 6 gazillion ways), and you throw the die once in each way, and the die is perfectly symmetrical, you will indeed get each side with frequency 1/6.
So why is it that most of the time when we throw the die only 100 or 1000 times the frequencies are close to 1/6? The law of large numbers does not explain why if we don’t explain why that law works.
To understand why, consider the following analogy: you have a box in which there are N balls (6 gazillion balls). N/6 balls have the number 1 on them, N/6 balls have the number 2 and so on. If you pick 100 balls arbitrarily, without looking at them, without knowing what their number is in advance, most of the time that you do that about 100/6 balls will have the number 1, about 100/6 balls will have the number 2, and so on. Why?
As I explained earlier, it has to do with combinations, not with randomness. Basically there are more combinations of 100 balls where the 6 numbers show up each with about the same frequency, than there are combinations of 100 balls where the numbers show up with very different frequencies. Try to understand why. If you don’t understand I’ll attempt to find a simple example to make it clear.
Quoting leo
Quoting leo
[quote=Wikipedia]
It follows from the law of large numbers that the empirical probability of success in a series of Bernoulli trials will converge to the theoretical probability.[/quote]
We agree on the point that the outcome depends on and can be predicted by the initial state of the die. The rest of what you said depends on this and we see eye to eye on it.
If you'll allow me to keep things simple and not get into gazillions and combinatorics I think we'll agree on the following:
1. A fair die has 6 sides.
2. For the sake of simplicity assume that each side of the die {1, 2, 3, 4, 5, 6} is an event determined by six initial states {a, b, c, d, e, f} such that a causes outcome 1, b causes outcome 2, c causes outcome 3, d causes outcome 4, e causes outcome 5 and f causes outcome 6.
3. The [i[theoretical probability[/i] for each possible outcome when the die is thrown is 1/6
4. We have to accept that if we knew which initial state obtains we can accurately predict the outcome.
5. Now imagine you throw the die without looking at which initial state the die achieves. You will see the familiar result that each outcome is 1/6 of the total number of times the die is thrown. This concurs with increasing accuracy the greater the number of experiments that are performed.
6. 3 and 5 together imply that the die is behaving randomly
7. We also know that each initial state yields a accurately predictable outcome i.e. each outcome can be known given which initial state the die assumed
8. 6 says the die is random and 7 says the die is not random
9. Somewhere in the chain events, randomness was introduced into the system. The only place possible is at the time you put the die in one of the six initial states and this was random. This makes complete sense when you consider what you said:
Quoting leo
I agree for the most part, except:
Quoting TheMadFool
If you don’t look at the initial state, you may pick unwittingly the same initial state every time (or a member of the set of initial states that yield the same outcome), so I don’t agree that we will always see each outcome with 1/6 frequency even with an arbitrarily large number of experiments. In some rare cases the frequencies will be very different, and in order to explain that we have to delve into combinatorics.
And since I don’t agree on this point (5.) I also disagree on your next point (6.), that the die is behaving randomly.
Quoting TheMadFool
I do agree that the frequencies of the outcomes are related to the way the initial states are chosen. However I don’t agree that there is a fundamental randomness that is introduced. For instance you can cycle through all the initial states deterministically, and yet you will get the result that each outcome will appear overall with frequency 1/6. So it is not randomness that leads each side of the die to appear 1/6th of the time. In order to explain that, you have to take into account both the symmetries of the die and combinatorics.
The whole reasoning goes as follow: in a deterministic system, for a perfectly symmetrical 6-sided die, it can be shown theoretically (using the symmetries of the die) that each side shows up with probability 1/6 (where probability of an outcome is defined as the number of initial states that lead to this outcome divided by the total number of possible initial states).
Then what remains to explain is why, most of the time, the frequency of each outcome converges towards 1/6 as the number of experiments increases, and yet in some rare cases the frequencies of each outcome are very different even though the number of experiments becomes arbitrarily large. In order to explain that, we have to go into combinatorics.
The probability that this happens becomes increasingly small as the number of experiments becomes increasingly large, but for any finite number of experiments, there are situations where the frequencies of all sides will be very different, even if we pick the initial states arbitrarily without knowing the outcomes in advance. Because for instance, if there are N possible initial states, and you conduct 1000000000000*N experiments, it happens in rare cases that you haven’t gone through all initial states, or that you have picked unwittingly some initial states much more often than some others. And if you don’t take that into account you get the illusion that experimentally we will always see each outcome with frequency 1/6 as we increase the number of experiments, while this isn’t true.
And at the same time combinatorics will also explain while most of the time each outcome shows up with frequency close to 1/6, even though no randomness is introduced at any point. Picking an initial state arbitrarily does not imply that it is picked non-deterministically.
The medical profession does not have a sterling reputation for statistical studies. Sometimes this is due to multiple experiments in which outliers are given undue consideration. Sometimes the experiments are poorly designed.
However, having said this I will tell you that probability theory still leaves me a little uneasy, even though its applications have been largely very successful. Probability waves in QM? Who'd have thunk? :brow:
It doesn't imply that. A number-generating algorithm simulating 1000 throws can be completely deterministic. Given the algorithm, I could correctly predict every outcome before the simulation is run. For example, that throw 23 would produce a six. Yet all the outcomes taken together would follow a probability distribution.
Perhaps you just want to say that this is not true randomness, it only appears random. If so, the term you could use is pseudorandom.
Quoting Andrew M
Because...
we can introduce randomness or more accurately pseudo-randomness into a deterministic system.
This is exactly what bothers me. It should be possible to bias the experiment towards a particular outcome. Yet this doesn't happen and the die behaves in a completely random fashion as is evidenced by the frequency of outcomes in an experiment of large enough number. Why?
Quoting leo
:chin: The evidence for randomness is in the relative frequencies of outcomes in an experiment which perfectly or near-perfectly matches the theoretical probabilities.
:smile: :up: If there's a hidden message in there I didn't see it
Yes, so does any puzzle remain for you?
BTW here's a quick-and-dirty example. Suppose my algorithm for 1000 dice throws is to take the first 1000 prime numbers, calculate their square roots, multiply their decimal expansion by 6 and round up to the nearest natural number. The algorithm is deterministic and the sequence is pseudorandom.
The first few throws are:
3, 5, 2, 5, 2, 4, 1, 3
Correct. I wonder how one differentiates the true random from pesudorandom?
From an operational perspective, "true" randomness would come from outside the system of interest (such as from thermal noise or quantum phenomena that are used as inputs to the system), whereas pseudorandomness would be the result of computable processes within the system.
A determinist about the universe would regard them as ultimately the same thing. Randomness and chance would just be terms indicating one's ignorance of the relevant information for making correct predictions. Empirically, one could try to discover the underlying laws or causes that explain apparently random events.
Because as I keep mentioning, it does happen! It is rare but it does happen, even with an incredibly large number of experiments. There is a non-zero probability that you will throw the die a gazillion times and always get the same number, or never get some number, or get some numbers much more often than the others. The reason you believe it doesn’t happen is that most of the time it doesn’t happen, but sometimes it does happen.
Again this is explained with combinatorics. I’ll give a simpler analogy than earlier. Since there are N initial states, and N/6 initial states lead to outcome one, N/6 initial states lead to outcome two and so on, there are as many initial states that lead to each outcome. And once an initial state is picked, the outcome is already determined (even if we don’t know it in advance).
So a mathematically equivalent analogy of the throw of the die (6-sided and perfectly symmetrical) is to have 6 identical balls inside a box, numbered from one to six. Throwing the die in an arbitrary way without knowing the outcome in advance, corresponds to picking one ball arbitrarily without looking inside the box. Then you write down the outcome and you put the ball back inside the box, and you shake the box so that you have no idea which ball is which when you pick another ball after.
And if you do that many times, it is possible that you will always pick the same ball. As the number of picks increases it becomes rarer and rarer, but no matter how many picks you make it still can happen. Or it can happen that you never pick some ball, or that you pick some balls much more often than some others, or whatever.
So even though this shows that in some rare cases the observed frequencies will be very different from the theoretical probabilities, even with an extremely large number of picks, combinatorics also allow you to show that most of the time the observed frequencies will be close to 1/6 as the number of picks increases.
Because for instance if you make 100 picks, there are 6^100 possible ordered combinations of numbers from one to six (say 1-2-4-3-6-2-... or 3-1-1-1-1-6-...), and mathematically you can show that in most combinations, each number appears with about the same frequency. So most of the time the balls are picked with about the same frequency, most of the time each side of the die appears with about the same frequency, but that doesn’t always happen, because there are some combinations of 100 numbers from one to six in which the numbers have very different frequencies.
Mathematically it’s even possible to calculate how likely it is that after X picks, each number shows up with frequency less than say 2% away from 1/6 (or 1% away or 0.1% or whatever). And if you carry out the calculation (it’s not easy but it’s doable), you will see that there is a non-zero probability that after X picks the frequencies are very different from 1/6, no matter how large X is.
Is there something that still isn’t clear?
I'm not saying a given outcome(s) is/are impossible. Perhaps I don't see the relevance of what you're saying to what is a actually bothering me. Kindly read below.
Quoting Andrew M
@Harry Hindu look at the part underlined.
I've given it some thought and I think you both are correct but not in the way you think.
Imagine a deterministic system A (a fair die with 6 sides). Once we have all the information on A we can make accurate predictions of how A will evolve. Deterministic systems will have specific outcomes right? There's nothing random in A and so however A evolves, everything in A will show a pattern and there won't be any variation in the pattern.
Please note that patterns are of two types which are:
1. Deterministic patterns. A good example would be gravity - there's a force and that force acts in a predictable manner.
2. Non-deterministic or probabilistic patterns. A die throw is effectively random but any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time, an odd numbered face will appear 3/6 of the time.
Also bear in mind that a deterministic pattern will differ markedly from a non-deterministic/probabilistic pattern. The latter will exhibit multiplicity of outcomes will the former has only one determined outcome.
Imagine now that we lack information i.e. we're ignorant of factors that affect how A will evolve. We assumed A to be deterministic and given that our ignorance has no causal import as far as the system A is concerned, system A should have a deterministic pattern. However, what actually happens is system A now exhibits a non-deterministic/probabilistic pattern.
I will concede that there was a lack of information about system and that is ignorance but that has no causal import on A which should be exhibiting a deterministic pattern because system A is deterministic as we agreed. However, the actual reality when we do experiments we observe non-deterministic/probabilistic patterns.
Yes you aren't seeing the relevance, because it precisely addresses what is bothering you.
Quoting TheMadFool
That's the kind of thing I was referring to yesterday, you assume that you know better and that there is something you see that we don't see, while it is the other way around. Your misconception is making you believe that, because you're still not seeing your misconception.
Quoting TheMadFool
This is wrong. It is false that "any sufficiently large experiment will demonstrate that the outcomes have a pattern viz. that three appears 1/6 of the time". As I keep telling you again and again and again, sometimes no matter how large your experiment is, it doesn't exhibit the pattern you mention. Sometimes you might throw the die 1 billion billion billion billion billion billion billion billion times and always get the same number, or never get some number. It is extremely rare, that's the only reason why you haven't noticed it.
And to explain why most of the time there is the pattern you mention, the answer is, again: combinatorics. There is nothing non-deterministic in combinatorics. It simply says that for a series of X natural numbers between 1 and 6, there are 6^X possible combinations (6^X different series), and that in most of these series, when X is large, each number appears about as frequently as the others, and the larger X is the bigger the percentage of series in which each number has a similar frequency. But no matter how large X is, there are always series where the frequencies are totally different from 1/6. You have to understand that, otherwise you will never get it.
The outcomes that we observe experimentally can be explained fully deterministically by invoking the symmetries of the die and combinatorics, without invoking non-determinism at any point.
Quoting TheMadFool
This is wrong also, you throw the die in different ways that's why there is a multiplicity of outcomes, otherwise what you're saying would imply that if the die behaves deterministically it would always land on the same side no matter how we throw it, THAT would be the weird thing.
That unexpected events occur isn't an explanation for the issue I raised but...
Quoting leo
I underlined the part that made sense to me. It squares with my explanation. Thanks.
@Andrew M@Harry Hindu
Thanks all
That you can find patterns in a deterministic system doesn’t mean that the deterministic system behaves non-deterministically. That you can find patterns in the frequencies of outcomes when you throw a die doesn’t mean that the die behaves non-deterministically. That you can find patterns in the frequencies of letters in the English language doesn’t mean that English words behave non-deterministically. That you can find patterns in the relative sizes of pizza slices when you cut a pizza doesn’t mean that the pizza behaves non-deterministically...
You are conflating the existence of statistics (which can be expressed as ratios or percentages) with non-deterministic behavior, they aren’t the same thing at all. Both deterministic and non-deterministic systems have statistics, so when you express the statistics of a deterministic system it doesn’t mean that it behaves non-deterministically. It seems to me you believe you have understood while you haven’t really understood.
Just to be clear, deterministic system A is one single die roll. The next die roll would be deterministic system B, and so on.
Just because you know the conditions of A, doesn't necessarily mean you know B. While their might be some causal influence, predicting B is not predicting the same system because you may use your other hand to roll, so the dice will roll in the opposite direction and maybe you roll it with a little less force, and so on. While you may have the formula for the system, you don't have the numbers to plug in for each die roll, unless you know the conditions of each die roll.
Quoting TheMadFool
We may know the formula for gravity, which tells us how two massive bodies will interact via gravity, but we still need to know the mass and distance between the two objects in order to predict what will happen over time. We still need to have those conditions plugged into the formula.
So just because we have a formula for a type of system doesn't mean that we can always predict the outcome, because we still need more information because each system is unique, even though they share a common formula/pattern that we refer to as a "system". The "system" is actually a unique state-of-affairs in each moment. It's just that many states-of-affairs share similar qualities, like die rolls. Each state-of-affairs includes hands rolling the same die.
What I think is happening here is that you are confusing the pattern with the state-of-affairs. The pattern is in your mind. The state-of-affairs is out there. Your formula/pattern is only a partial representation of the state-of-affairs. There are other factors that you aren't taking into consideration when determining the outcome of any particular state-of-affairs. If you know ALL the factors, then you can predict the outcome.
Since some of the factors change in the next die roll (but not all of them because we are still using hands to roll the same die), we'd need to know what changed, and by how much, in order to make the prediction for the next state-of-affairs.
This also makes you confuse the two patterns you have. You're conflating knowing one die roll with knowing all of them. How does knowing the formula/pattern for gravity allow you to predict every gravitational event in the universe? Don't you still need to know the mass and distance of the planets or stars in each event?
Rolling dice would be like rolling planets together. How will each interaction evolve? How many times will there be collisions vs. establishing orbits around each other? So you're confusing the pattern in your head as if it were the state-of-affairs itself.
:up:
Tell me what is it that I didn't understand.
Quoting Harry Hindu
Correct. Thanks.
I’ve done that repeatedly in the previous posts, do I really need to say it again?
You believe that the die behaves non-deterministically, that’s wrong. When you say that it behaves probabilistically that’s what you mean right? That it behaves non-deterministically. But it makes no sense to say that a deterministic system behaves non-deterministically, it’s self-contradictory.
The fact that the outcomes of multiple die throws exhibit often particular frequencies does not imply that the die behaves non-deterministically at any time. Just like the fact that the letters in an English book exhibit often particular frequencies does not imply that English books behave non-deterministically.
Now tell me, what is it that you don’t understand about that?
Well, what is the best way to model a die throw in your view?
1. Probability
2. Determinism
Both right?
I think you’re conflating probability and non-determinism. When we say that “there is 1/6 probability that the die will land on three”, do you agree that you interpret it as saying that the die behaves non-deterministically?
It is fine to say that there is 1/6 probability that the die will land on a given side. But this simply expresses our ignorance of the initial conditions when we throw the die, it doesn’t express that the die behaves non-deterministically, and I think that’s where your confusion lies.
To say that the die behaves non-deterministically would be to say that the die doesn’t have a determined trajectory once it is thrown, that it will behave differently even if it is thrown in exactly the same way and even if everything else remains the same, but this isn’t true, or at the very least there is no evidence of that, and the frequencies of the observed outcomes can be explained without invoking any non-determinism.
When the system is deterministic, probabilities refer to incomplete knowledge, but that complete knowledge exists even if we don’t have access to it. Determinism is not incompatible with probabilities, with incomplete knowledge, with ignorance.
But when a system is deterministic, it cannot be non-deterministic, that would be a contradiction. In a fundamentally non-deterministic system, even if we have complete knowledge of it, there are probabilities that remain, for instance the system can behave differently even if it starts from the exact same initial conditions.
Whereas in a fundamentally deterministic system, when you have complete knowledge of it the probabilities disappear, you know exactly which outcome you’re going to get. If after many throws the frequencies of each outcome are about 1/6, you can explain why. If they are totally different from 1/6 you can explain that too. You can explain why in many cases the frequencies are about 1/6. There is no non-determinism.
If you don’t know anything about a die, you wouldn’t come up with any probability, you would say that the outcome of a die throw can be anything.
But through thinking, through making use of the symmetries of the die, you can conclude that there are as many ways to throw the die that lead to outcome ‘one’, than there are ways that lead to outcome ‘two’, than there are ways that lead to outcome ‘three’, and so on. This gives you partial knowledge, you know that if you throw the die once in every possible way, each outcome will be realized with frequency 1/6. That’s why we say that without knowledge of the initial conditions, the best we can say is that in all possible ways to throw the die, 1/6th of them lead to outcome ‘one’, 1/6th lead to outcome ‘two’ and so on, or in other words that each outcome has probability 1/6 of being realized.
But if you have also partial knowledge of the initial conditions you can determine the probabilities even more precisely. For instance if you know that in specific ways that you throw the die you never get a ‘six’, then you know that when you throw the die in these ways there is 0% probability to get a ‘six’, and the probabilities of the other outcomes change accordingly. If you know that when you throw the die in a very specific way you always get a ‘four’, then you know that when you throw the die in that way there is 100% probability to get a ‘four’.
And when you have complete knowledge there is no more need to talk of probabilities, you know exactly which outcome you are going to get each time, each time you know there is 100% probability you are going to get a specific outcome.
As another example, through making use of the symmetries of a coin you can conclude that in about 50% of ways to toss a coin you get the outcome ‘heads’ and in about 50% of ways you get ‘tails’. But if you think more carefully you realize that there are also a few ways that lead the coin to land on neither heads nor tails but on its side. So if you take that knowledge into account, you come up with more precise probabilities, you say that there isn’t 50% probability to get ‘heads’ and 50% probability to get ‘tails’ but slightly less, because there is a non-zero probability for the coin to land on its side.
And if you have complete knowledge of the system, of what outcome you get depending on how you toss the coin, then there is no need to talk of probabilities anymore, you have complete knowledge so you know what you’re gonna get, and if you want you can toss the coin so that it always lands on its side.
Whereas in a non-deterministic system, complete knowledge doesn’t allow you to say what outcome you’re gonna get, there are still probabilities that remain, the exact same initial conditions can give rise to different outcomes. Personally I believe that such systems do not exist, that even quantum mechanics can be formulated in a deterministic way, and that probabilities always refer to incomplete knowledge rather than knowledge that doesn’t exist.
What are the following in your view?
1. Probability
2. Determinism
3. Non-determinism
1. Probability expresses incomplete knowledge that we have about a system.
2. The exact same initial states in a deterministic system lead to the exact same outcome.
3. The exact same initial states in a non-deterministic system can lead to different outcomes.
What are they in your view?
Also read my previous two posts carefully, I think eventually it will click for you. I’m taking quite a lot of time to help you understand, so it would be fair if you took at least as much time to read and attempt to understand my posts.
Since in a deterministic system the outcome depends solely on the initial state, the observed frequencies of the outcomes depend solely on the initial states that are chosen.
And the key point to understand: there are many more ways to pick initial states leading to outcomes that have similar frequency, than there are ways to pick initial states leading to outcomes with very different frequencies. This is a result arrived at through combinatorics, something I have mentioned a few times but that you have consistently ignored.
And this result implies that in most experiments where the die is thrown arbitrarily (that is where we aren’t preferring some initial states over some others), the observed outcomes have a similar frequency, close to 1/6. Which in no way implies that the die is behaving non-deterministically at any point. Nor that the initial states are chosen non-deterministically.
That's a great explanation. Thank you for your time and patience.
However...
In your definition of non-determinism you concede that there is something you don't know viz. the outcomes and then you go on to say that probability is about incomplete knowledge. So it must follow that non-determinism is just probability or are you claiming that there's a difference that depends on what you're ignorant about- only the initial states or only the outcomes - and probability would be an issue of ignorance regarding initial states but non-determinism would be ignorance about outcomes despite having knowledge of the initial states.
If that's the case you're making then non-determinism can't be understood in any way because the outcomes will not exhibit any pattern whatsoever. In other words non-determinism is true randomness with every outcome having equal probability and that brings us to where we began - that non-determinism = probability.
Long post here, I hope you will read all of it carefully in order to understand, I could have made it shorter but I wanted to answer you as clearly as possible.
1.
In a deterministic system, the outcome is a deterministic function of the initial state, let’s write it O = f(I). No matter how many times you run the system from the same initial state I, you get the same outcome O.
You can have incomplete knowledge of the initial state I, or incomplete knowledge of how the system behaves (the function f), or incomplete knowledge of the outcomes O, or any combination of the three.
1.a)
Without any knowledge about that system, we don’t know anything about the outcomes, anything is possible.
1.b)
If we know that the system involves the throw of a six-sided die numbered from one to six and the outcome is the top face of the die when the die has stopped moving, we know that the outcome can be any number in the set {1, 2, 3, 4, 5, 6}. This counts as partial knowledge of the outcomes O.
You can express that by saying that any outcome outside of this set cannot occur, that it has 0% probability of occurring. But for now you have zero knowledge of whether all outcomes in this set actually occur, in principle it is possible that the outcome is always ‘3’, so at this point you can’t assign any probability to the outcomes in the set.
1.c)
If we know that the initial state of the die is the initial position/orientation/velocity of the die, we can determine the range of possible initial states that exist, this counts as partial knowledge of the initial states I.
But we still don’t know anything about the function f, we still don’t know how any initial state transforms into any outcome, so we still can’t assign probabilities to the outcomes {1, 2, 3, 4, 5, 6}.
1.d)
If we know that the die is perfectly symmetrical, then combining that knowledge with our incomplete knowledge of the initial states and outcomes described in the previous paragraphs, we can conclude that 1/6th of the initial states lead to outcome ‘1’, 1/6th of the initial states lead to outcome ‘2’, 1/6th of the initial states lead to outcome ‘3’, and so on. This is the same as saying that each outcome has probability 1/6 of being realized, that’s the definition of probability. This result isn’t obvious but it can be proven mathematically, offering us partial knowledge of the function f.
If we knew nothing of the function f, even if we knew the initial states perfectly we couldn’t predict anything about the outcomes, we couldn’t predict how the die is going to behave while it is flying and bouncing, but the symmetries of the die and the determinism of the function f allow us to say that no matter how the die behaves, it behaves exactly the same whether in the initial state the side ‘1’ is facing upwards or any other side is facing upwards.
1.e)
Then if we have more complete knowledge of the function f (more complete knowledge of how the die behaves while it is flying and bouncing), and more complete knowledge of the initial state when the die is thrown, we can predict the outcome more accurately, and this changes the probabilities from 1/6 to something else that depends on the initial state. And if we have complete knowledge then we can predict the outcomes exactly from the initial states and we don’t need to talk of probabilities anymore.
2.
In a non-deterministic system, the outcome is a non-deterministic function of the initial state, let’s write it O = pf(I). Even if you run the system many times from the same initial state I, you don’t always get the same outcome O. You may get some outcomes more often than some others, but you don’t get only one outcome.
In such a system, even if you gain complete knowledge of the initial states and of the function pf, you still don’t know what outcome you are going to get each time you run the experiment. But you do know things, for instance you may know which outcomes are possible (have a non-zero probability of occurring) and which outcomes are impossible (have zero probability of occurring). You may know that some outcomes are more likely than others, and assign probabilities to them.
So as you can see, it is not the case that in a non-deterministic system the outcomes will not exhibit any pattern whatsoever, it isn’t the case that a non-deterministic system is totally random. You may know that starting from initial state I, the outcome will be O1 90% of the time and O2 10% of the time. But this probability is irreducible, in the sense that it cannot be removed from gaining more knowledge, because that additional knowledge doesn’t exist.
Basically in non-deterministic systems there is irreducible probability even if you have complete knowledge of the system, whereas in deterministic systems the probabilities are only a sign of incomplete knowledge, and disappear when we have complete knowledge.
Personally I believe that we shouldn’t even make a distinction, I believe that there is no such thing as non-deterministic systems, that probabilities are always due to incomplete knowledge. And when you see things that way you clearly see that formulating probabilities isn’t a sign that you’re dealing with a non-deterministic system, but merely that you’re expressing the incomplete knowledge you have of a system.
This is what I've been saying all along. Deterministic systems can behave probabilistically.
Let me get this straight.
1. In a deterministic system there's a well defined function that maps each initial state (I) to a unique outcome (O) like so: f(I) = O.
2. In a non-deterministic system there is no such function because there are more than one outcome e.g. initial state A could lead to outcomes x, y, z,...
You mentioned a "function" pf(I) = O but if memory serves a function can't have more than one output which is what's happening in non-deterministic systems according to you: one initial state and multiple outcomes.
Quoting leo
A fine point. :up:
Quoting leo
So, there's a difference between non-determinism and randomness but you have to admit that both can be described with mathematical probability.
Thanks for being so helpful.
No no this is where your confusion lies. What do you mean exactly by “behave probabilistically”? It can be interpreted in various ways:
I. Either you mean “behave non-deterministically”, but by definition a deterministic system does not behave non-deterministically. Also in order to arrive at the result that “1/6th of initial states lead to a specific outcome” we had to assume in the first place that the system behaves deterministically, so this result does not mean at all that the system behaves non-deterministically.
II. Or you mean that the behavior of the system depends on probabilities. But as we have seen, probabilities in a deterministic system are an expression of our incomplete knowledge of that system, and surely the behavior of a deterministic system does not depend on the knowledge that we have about it. So it isn’t meaningful to say that a deterministic system “behaves probabilistically” in this sense.
III. Or you mean that when we throw the die many times, the observed frequencies converge towards 1/6 for each outcome, which is similar to how a non-deterministic would behave. But you have to realize that this is false, because:
a) If we have complete knowledge of the deterministic system, we can throw the die such that we always get the outcomes we want, and then the observed frequencies can be totally different from “1/6 for each outcome”. Would you still say that the system “behaves probabilistically” then?
b) Whereas in a non-deterministic system, you might always start from the same initial state and get 6 different outcomes each with frequency 1/6, you never get that in a deterministic system.
c) In the experiment of the die the observed frequencies converge towards 1/6 only in special cases: when the initial states are chosen such that they lead to outcomes with similar frequencies. As it turns out, when we have no knowledge of the initial states we often choose them unwittingly in this manner, for the simple reason that there are many more combinations of initial states that have this property than there are combinations of initial states without this property (this result can be arrived at through combinatorics, if you understand this it will finally click for you, but you will never understand if you keep ignoring this).
Quoting TheMadFool
A function maps inputs to outputs. Deterministic functions (the ones we are used to) map one or several inputs to one output. Non-deterministic functions can map one input to several outputs.
One example of such a non-deterministic function would be: when input is I, there is 90% probability that output is O1, and 10% probability that output is O2. Each time you run the function you only get one output, either O1 or O2. But when you run it a very high number of times, you would get O1 90% of the time and O2 10% of the time.
If that makes you uneasy, you can consider like I do that non-deterministic systems fundamentally do not exist, that if we get different outcomes from the exact same initial state, it’s simply that we falsely believe that it was the exact same initial state, while in fact there was something different about it that we didn’t take into account.
Quoting TheMadFool
There is randomness involved in non-deterministic systems, sure. And randomness can be described with probabilities, sure. But this does not imply that there is randomness in deterministic systems. Because the probabilities in deterministic systems refer to incomplete knowledge, not to randomness. A deterministic system only seems to have randomness in it when we don’t fully understand it.
For instance with a poor understanding of how planets move, their motion in the sky can seem to be partially random, but apparent randomness isn’t fundamental randomness. As another example, you believe there is fundamental randomness involved in the throw of a die, because you haven’t yet understood fully how the frequencies that we observe can be explained without invoking randomness.
There is no confusion at all. A die is deterministic and it behaves probabilistically. This probably needs further clarification.
A die is a deterministic system in that each initial state has one and only one outcome but if the initial states are random then the outcomes will be random.
I explained carefully why saying that “the die behaves probabilistically” is at best meaningless and at worst a contradiction, and yet you’re saying I’m the one who is confused ...
Quoting TheMadFool
In a deterministic system where all initial states lead to the same outcome, even if the initial states are picked randomly the outcome isn’t random.
In a deterministic system where all initial states don’t lead to the same outcome, there are subsets of all initial states within which if you pick initial states randomly the outcomes aren’t random.
In the example of the die you can pick the initial states deterministically (rather than randomly) and still get outcomes with frequency 1/6.
Clearly, the underlying reason why the observed frequencies are often 1/6 is not that the initial states are picked randomly, you are still confused about that.
It is correct that picking the initial states randomly in the example of the die leads often (not always) to frequencies close to 1/6 for each outcome, but it is incorrect to believe that randomness is required to obtain such outcomes. The same frequencies can be obtained deterministically.
With your current understanding, you can’t explain why we can pick initial states deterministically and get outcomes with frequency 1/6 each. Because your understanding is incomplete. Now you can keep believing I’m the one who is confused if you want, but meanwhile you’re the one who hasn’t addressed many of the points I’ve made.
A variable has an event space, and that event space has a distribution. How you pick a value for the variable determines whether the variable is independent or dependent. An independent variable can be a random variable, and a dependent variable can depend on one or more random variables.
How we retrieve the values for the variable in an experiment (i.e. if it's a random variable or not) has no influence on the distribution of the event space of the variable, but it can introduce a bias into our results.
That the same variable with the same distribution can have its values computed or chosen at random in different mathematical contexts is no mystery. It's a question of methodology.
Quoting leo
I think I get what you mean.
Assuming that the die is a deterministic system two things are possible:
A. The usual way we throw the die - randomly - without knowing the initial state. The outcomes in this case would have a relative frequency that can be calculated in terms of the ratio between desired outcomes and total number of possible outcomes. It doesn't get more probabilistic than this does it?
B. If we have complete information about the die then we can deliberately select the initial states to produce outcomes that look exactly like A above with perfectly matching relative frequencies.
When I said "a deterministic system is behaving probabilistically" I did so on the basis of A above. The reason for this is simple: though each outcome is fully determined by the initial state of the die, the initial states were themselves randomly selected which precludes definite knowledge of outcomes. Thus we must resort to probability theory and it seems to work pretty well; too well in my opinion in that the die when thrown without knowledge of the initial states behaves in a way that matches theoretical probability.
I'm in no way saying 2 can't be done.
However, there's a major difference between A and B to wit the probabilities on a single throw of the die will be poles apart. In situation A, the probability of any outcome will be between 0 and 1 but never will it be 1 or 100% but in situation B every outcome will have a probability 1 or 100%
Quoting Dawnstorm
:chin:
I want you to focus on that, on this feeling that it seems to work too well. That feeling is telling you something, that there is something off in your understanding that you can’t quite pinpoint yet, so don’t stop now thinking that the confusion has disappeared. But we’re circling in on that confusion, and I think you aren’t far from finally seeing it.
Quoting TheMadFool
You absolutely have to understand this: the theoretical probabilities do not tell us about the relative frequencies that we will observe. They merely express the best knowledge that we have when we don’t know the initial state we’re in. And indeed as you have correctly noticed, it seems strange that the observed frequencies match the theoretical probabilities so well, why would they?
And the “why” is what you need to understand now.
When you throw the die arbitrarily many times, what you are essentially doing is picking an arbitrary combination of initial states. Now why would this combination contain each outcome with about the same frequency? That’s what you can’t explain yet. Why is it that the bigger the combination, the more similar the frequencies of each outcome are? Once you understand that you will finally see the confusion, and you will finally get it.
If you pick an arbitrary combination of initial states, and most of the time that combination contains each outcome with about the same frequency, do you agree that either something magical is going on and guiding your hand when you pick the initial states, or it means that there are many more combinations where each outcome has about the same frequency, than combinations where the frequencies are different?
And indeed this is something that we can prove: in the example of the die, there are many more combinations of initial states where each outcome has about the same frequency, than there are combinations where the frequencies are different.
If you want we can focus on proving that, if you finally understand that this is the only way that we can make sense of what we observe, without invoking magic or randomness, without saying that our ignorance of the initial states somehow makes the die behave differently.
In situation A the probably of one outcome is also 1 or 100% once the die is thrown, it is simply our incomplete knowledge that makes us say that any outcome is possible, but the outcome that is about to be realized is already determined.
So there is no fundamental difference between A and B, the only superficial difference is that in situation A we don’t know what the outcome is going to be, the difference lies in our knowledge of the system and not in how the system behaves.
When we have no knowledge of the initial states, the frequencies of the outcomes are often similar simply because we pick the initial states arbitrarily, and there are many more combinations of initial states where outcomes have a similar frequency, so we pick such combinations much more often. That’s all there is to it.
The die behaves deterministically, there is no fundamental randomness, there is no magical force guiding our hand or the die in order to yield these frequencies. The probabilities we talk about are not a result of an underlying randomness but simply an expression of our incomplete knowledge. The observed frequencies often match these probabilities not because the system behaves probabilistically (randomly, non-deterministically), but because most combinations of initial states lead to these frequencies.
And you can see that it is a coincidence that these observed frequencies match these probabilities, it isn’t always true. For instance if the initial states are picked deliberately so that the observed frequencies do not match these probabilities, then this is what happens. And in some cases the initial states are picked arbitrarily and still the observed frequencies are very different from these probabilities, even after many throws.
The scenarios A and B in my previous post was to explain that deterministic systems can behave probabilistically and I think it accomplished its purpose.
Bear in mind though that I don't mean deterministic systems are non-deterministic. I just mean that sometimes, as when we have incomplete knowledge, we can use probability on deterministic systems.
Considering we can use probability on non-deterministic systems too, it must follow that probability theory has within its scope non-determinism and determinism,some part of which we're ignorant of.
Quoting leo
Yes, I believe I wrote something to that effect in my reply to Harry Hindu but that was because I thought he claimed ignorance had some kind of a causal connection to randomness. Later in my discussions with him/her and you, I realized that ignorance of deterministic systems is not a cause of but rather an occasion for, probability. I hope we're clear on that.
Quoting leo
This is an obvious fact and doesn't contradict anything I've said so far.
Just as long as what we are clear on is that probabilities only exist in the system of your mind, not in the system of dice being rolled. Determinism exists in both systems. The idea of probabilities are a determined outcome of ignorant minds. When you are ignorant of the facts, you can't help but to engage in the idea of probabilities, just as when you aren't ignorant of the facts, you can't think in terms of probabilities. The system is determined from your perspective, which just means that you understand the causal relationships that preceded what it is that you are observing or talking about in this moment.
Can you think of any point of your life where you were not ignorant of the facts and still thought of the system as possessing probability or indeterminism? Can you think of any point in your life where you were ignorant of the facts and you perceived the system as being deterministic? It seems to me that ignorance and probabilities aren't just a correlation, but a causal relationship.
So, probability didn't exist before there was such a thing as mind, say 9 billion years ago when the earth hadn't even formed? Everything was deterministic before minds came into being and now probability exists because there are now minds and to add, these minds can be ignorant.
Do you mean to imply that if, by some freak of nature, all minds were wiped out, probability would disappear?
Surely, you don't mean to say that do you?
If so, what exactly do you mean by "probability only exists in the system of your mind"?
I agree that the restricted domain herein, of die throwing, is ultimately deterministic and that whereof we're ignorant we can only guess and ignorance being a state of mind there is a sense in which your statement is true but if your statement means that non-determinism i.e. true randomness doesn't exist and that every instance of probabilistic behavior is simply us being forced to engage in mathematical guessing (probability theory) due to ignorance, then I need more convincing if you don't mind.
Quoting TheMadFool
Quoting TheMadFool
It's clear to me that you think scenarios A and B explain why deterministic systems "behave probabilistically", but as leo pointed out "behaving probabilistically" isn't well defined, and in any case the maths works the same in both A and B.
You use terms like "the initial state", and "complete information about the die", but those terms aren't well defined. "The initial state" is the initial state of a probabilistic system, but that's pure math and not the real world. We use math to make statements about the real world. The philosophy here is: "How does mathmatics relate to the real world?"
The mathematical system of the probability of a fair die has a single variable: the outcome of a die throw. There is no initial state of the system, you just produce random results time and again. The real world always falls short of this perfect system. You understand this, which is why you're comparing ideal dice to real dice. "Initial states" aren't initial states of ideal dice, but of real dice. (I understand you correctly so far, no?)
Now to describe a real die you need to expand the original system to include other variables. That is you expand to original ideal system into a new ideal system, but one with more variables taken into account. This ideal system will have an "initial state", but it's - again - an ideal system, and if you look at the "initial state", you'll see that the variables that make up the initial state can be described, too. This is important, because you're arriving at the phrase "complete information about the die" and you go on to say that "we can deliberately select the initial states." But there are systematic theoretical assumptions included in this in such a way that what initial states we pick is not part of the system we use to describe the die throw. (But, then, is the information really "complete"? What do you mean by "complete"?)
So now to go back to my original post:
Quoting Dawnstorm
Take a look at a die. A die has six sides, and there are numbers printed on every side, and it's those numbers we're interested in. This is what makes the event space:
1, 2, 3, 4, 5, 6
The distribution is just an assumption we make. We assume that everyone of those outcomes is equally likely. This isn't an arbitrary assumption: it's a useful baseline to which we can compare any deviation. If a real die, for example, were most likely to throw a 5 due to some physical imbalance, then it's not a fair die. The distribution changes.
In situations such as games of chance we want dice to behave as closely to a fair die as possible. Even without knowing each die's distribution, for example by simple rule: never throw the same die twice. The idea here is that we introduce a new random variable: which die to throw. Different dice are likely to have different biases, so individual biases won't have as much an effect of the outcome. In effect, we'd be using many different real dice, to simulate an ideal one.
And now we can make the assumption that biases cancel each other out, i.e. there are equally man dice that are biased towards 1 than towards 2, etc. This two is an ideal assumption with its own distribution, and maybe there's an even more complicated system which equals out the real/ideal difference for this one, too. For puny human brains this gets harder and harder every step up. But the more deterministic a system is, the easier it gets to create such descriptive systems. And with complete knowledge of the entire universe, you can calculate every proability very precisely: you don't need to realy on assumptions and the distinction between ideal and real dice disappears.
Under prefect knowledge of a deterministic system probability amounts to the frequentist description of a system of limited variables. An incomplete frequentist description of a deterministic system will always include probabilities, because of this. If, however, you follow the chain of causality for a single throw of a die, what you have isn't a frequentist description, and probability doesn't apply. They're just different perspectives: how the throw of a die relates to all the other events thus categorised, and how it came about. There's no contradiction.
OK we can agree on that. With the caveat that it would be wrong to expect that throwing the die many many times will always yield each outcome with the same frequency as the others, it would be wrong to expect that the observed frequencies will always match the theoretical probabilities we’ve come up with, it would be wrong to expect that if you throw the die a gazillion times you will always get 1/6 frequency for each outcome.
But still...
Quoting TheMadFool
It isn’t an obvious fact, it’s not easy to prove. And I would venture to say that if it was so obvious you wouldn’t have made this thread in the first place, and you still wouldn’t be saying that observed frequencies fit the probabilities too well.
Say you pick a number in the set {1, 2, 3, 4, 5, 6} and you do that 100 times. You get a combination of 100 numbers, each number being either 1 2 3 4 5 or 6. You can compute how many times each number appears in the combination, compute what its frequency is.
If it was obvious that there are many more such combinations in which each number’s frequency is about the same than there are combinations in which the frequencies are very different, then it would be obvious that when we throw the die arbitrarily 100 times we often get each number with about the same frequency, and we wouldn’t think that it’s weird even though we’re dealing with a deterministic system. If the latter isn’t obvious then the former isn’t either...
Also, it is misleading to say that the system behaves probabilistically, it creates confusion, what’s more accurate is to say that the outcomes of a deterministic system can be distributed evenly when the system is run many times. Fundamentally there is nothing mysterious about that, what appears to be mysterious is why oftentimes the outcomes of a die throw are distributed evenly, but this is explained as above.
What about the law of large numbers which says exactly the opposite of what you're saying? The law of large numbers states that the average of the values of a variable will approache the expected value of that variable as the number of experiments become larger and larger.
If the random variable x is the probability of getting an odd number in a die throw then we could conduct n experiments of T number of trials in each and get the following values: x1, x2,...xn
The average value for x = (x1+x2+...+xn)/n
The expected value for x, E(x) = P(x) * T = (3/6) * T where P(x) is the [i]theoretical probability[/i[ of event x.
The law of large numbers states that (x1+x2+...+xn)/n will approach E(x) = P(x) * T and this is only possible if the actual probabilities themselves are in the vicinity of the theoretical probability.
Note: my math may be a little off the mark. Kindly correct any errors
Your claim that we shouldn't expect that theoretical probabilities will not match observed frequencies is applicable only to small numbers of experiments.
Quoting leo
What could be more obvious than saying if there are more ways of x happening than y then x will happen more frequently if the probabilities of all outcomes are equally likely?
Your comments are basically about practical limitations and these can be safely ignored because, as actual experimentation shows, even a standard-issue die/coin behaves probabilistically.
Quoting Dawnstorm That is correct.
You have a flawed understanding of expected value, it is 1*P(1)+2*P(2)+3*P(3)*4*P(4)+5*P(5)+6*P(6) = (1+2+3+4+5+6)*1/6 = 3.5
That ‘law’ states that the average of outcomes will converge towards 3.5, not towards 1/6 times the number of trials (that wouldn’t make sense).
Quoting TheMadFool
People have come up with plenty of ‘laws’, are they always correct? Just because something is called a ‘law’ that means it’s always true? A ‘law’ has a domain of validity, the law of large numbers doesn’t always apply. Try to understand how and why people have come to formulate this law, rather than assuming it’s always true and applies everywhere.
There are two main limitations to this ‘law’ here:
1. Even if you pick the initial states randomly, it is possible that the average of outcomes will not converge towards the expected value no matter how many times you throw the die (it’s possible that you always get some outcome, or never get some outcome, or get totally different frequencies, it’s rare but possible).
2. You don’t know in the first place that you are picking initial states randomly. For instance if you unwittingly always pick initial states that never lead to outcome ‘6’, then it’s wrong to say that outcome ‘6’ has probability 1/6 of occurring, it has probability 0 of occurring, and then the average of outcomes won’t converge towards 3.5.
Quoting TheMadFool
Firstly, what’s not obvious is that there are more ways of x happening than y.
Secondly, the probabilities of all outcomes are the same only theoretically, in practice the effective probabilities depend on how you throw the die.
I was expecting that but the math works out in my explanation. The law of large numbers does say that the experimental probability will approach the theoretical probability.
Quoting leo
Are you in any way challenging the law of large numbers?
[quote=Wikpedia]A special form of the LLN (for a binary random variable) was first proved by Jacob Bernoulli.[7] It took him over 20 years to develop a sufficiently rigorous mathematical proof which was published in his Ars Conjectandi (The Art of Conjecturing) in 1713.[/quote]
Talk to Jacob Bernoulli :grin:
Then why did you claim it?
Jokes aside, the 3.5 value is obtained because the probability of each outcome is 1/6.
On the one hand, you say that practical limitations can be safely ignored, and on the other hand you wish to appeal to actual experimentation. You have to choose one. Practical limitations may not be important to the law of large numbers when it comes to an ideal die, but they're certainly vitally important to actual experimentation. That's a theoretical issue, by the way: the universe we live in is only a very small sample compared to the infite number of throws, and what any sample we throw in the real world converges to is the actual distribution of the variable, and not the ideal distribution (though the sets can and often will overlap).
More importantly, though, since you're talking about determinism, you're actually interested in practical limitations and how they relate to probability. It's me who says practical limitations are unimportant to the law of large number, because it's an entirely mathematical concept (and thus entirely logical). Not even a universe in which nothing but sixes are thrown would have anything of interest to say about the law of large numbers.
I'd say the core problem is that without a clearly defined number of elements in a set (N), you have no sense of scale. How do you answer the question whether all the die throws in the universe is a "large number" when you're talking about a totality of infinite tries? If you plot out tries (real or imagined, doesn't matter) you'll see that the curve doesn't linearily approach the expected value but goes up and down and stabilises around the value. If all the tries in the universe come up 6, this is certainly unlikely (1/6^N; N = number of dice thrown in the universe), but in the context of an ideal die thrown an infinite number of times, this is just be a tiny local devergance.
Quoting leo
The two of you work with different x's. Your x is the outcome of a die throw {1,2,3,4,5,6}. His x is the number of odd die-throws in a sample of the size of T. He's using the probability of throwing an odd number as the expected value. Explaining the particulars, here, is beyond me, as I'm out of the loop for over a decade, but he's basically using an indicator function for x (where the value = 1 for {1,3,5} and 0 for {2,4,6}).
As far as I can tell, what he's doing here is fine.
I think if one ignores practical limitations then we can rely on experiments. If it weren't that way we could never be sure of experimental results because they would never be perfect enough.
You mentioned things like changing the die with every throw and other variations to die throwing that, to me, were an attempt to make the process ideal.
However as you already know experimentation with die bought from any stall under normal non-ideal conditions yields results that agree with the law of large numbers. So, I saw no need for us to go in that direction because it was unnecessary.
I did some very basic research and you're right in that no finite large number can compare to infinity but it seems we really don't need infinity to see the trend of the sample mean/average value of outcomes approaching the expected value.
Thanks.
The uncertainty lies in the radioactive sample, not the cat.
The uncertainty lies in the dynamics of the toss, not the die.