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 — leo

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!

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! — Mark Dennis

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:

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 — leo

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.

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. — Mark Dennis

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:

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). — leo

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.

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. — Mark Dennis

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:

Those are part of the same postulates you are speaking about? It isn't based on assumptions either it is based on evidence and fact. Astronauts age differently as do the twins. This is all in line with special and general relativity unless some definitions changed? We are talking about physics here right?

How is Gravity not involved in the twin paradox? Are the Twins floating in a vacuum? How barbaric!

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? — TheMadFool

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.

here'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. — 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.

i mispelled illusion as allusion.

Those are part of the same postulates you are speaking about? It isn't based on assumptions either it is based on evidence and fact. Astronauts age differently as do the twins. This is all in line with special and general relativity unless some definitions changed? We are talking about physics here right?

How is Gravity not involved in the twin paradox? Are the Twins floating in a vacuum? How barbaric! — Mark Dennis

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.

Also it's a thought experiment, we haven't tried it in practice. — leo

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.

it is possible that the will is fundamentally not deterministic, not determined by deterministic laws. — leo

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.

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. — Mark Dennis

Yep I think so too :up:

For example; a lot of us here can probably notice with some quick introspection that they have probably been motivated to act at times by thoughts about the future.

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, 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. — the mad fool

[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.]

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. — the mad fool

[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.]

.

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. — TheMadFool

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?

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? — TheMadFool

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.

Let’s go baby steps:

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.

Thanks for your patience. I don't think I've "discovered" anything. My question is more a reflection of my fundamental misconceptions on a subject, here probability.

Anyway you said:

1. The behavior of the die is caused by its symmetry and then you said

you throw it quite randomly, — 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’).

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. — 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).

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. — 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.

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. — TheMadFool

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.

Because of the nature of number generating algorithms, so long as the original seed is ignored, the rest of the values that the algorithm generates will follow probability distribution in a pseudorandom manner. — Random seed

Yes, but randomly does not imply non-deterministically. — leo

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"?

3. Theoretical probability calculations has as a fundamental assumption that what is being calculated is random. — TheMadFool

Not exactly. In your OP you correctly said:

Probability of an event E = (Number of ways event E can be realized) ÷ (Total number of possible events) — 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:

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. — 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.

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 — 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.

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"? — 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).

The outcome of die throw can be calculated probabilistically e.g. probability of getting a 3 is 1/6 — TheMadFool

Isn't it probable that you roll the dice six times and never get a 3?

The outcomes of a set of 1000 die-throws can be predicted probabilistically e.g. 3 will appear approx. 166 times — 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?

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. — 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.