# Explaining Probabilities in a deterministic world

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1. Apr 23, 2015

### asmani

Let's assume we're living in a mechanical deterministic world. Now do you agree that any uncertainty is a result of lack of knowledge?

We flip a symmetric coin. The equations of movement are deterministic, but the outcome is uncertain, with probability 50% tail or head. Thus, it's the initial state that is unknown and makes the uncertainty. How is it that the initial state is in 50% of times in favor of tail and 50% of time in favor of head?

Now, this initial state which comprises of the initial velocity vector, initial position, air turbulence, etc... is itself an outcome of the past deterministic processes that has happened through deterministic equations, but with uncertain 2nd level initial state that has been with probability 50% in favor of initial state that leads to head, and 50% in favor of initial state that leads to tail.

We can continue this questions infinitely or go back to the beginning of the world. my question is, where does this 50-50 distribution (or any other PDF in general) comes from?

2. Apr 23, 2015

### gleem

Actually the initial state is not know. The initial state of the flipped coin is not just its starting orientation , but also its initial translational velocity, its rotational velocity , its height above the final resting place, and the conditions (geometry, roughness, elasticity, etc ) of the final resting place are all unknown and variable.

3. Apr 23, 2015

### paisiello2

If we live in a deterministic world, as you said we must assume, and if we have perfect knowledge then by definition there is no 50/50 probability.

4. Apr 23, 2015

### Stephen Tashi

Since you posted the question in a section about mathematics, you need to give a mathematical definition of "uncertainty" in order to pose a specific question. If you are speaking of "uncertainty" and "knowledge" as a mental states, then your question isn't about mathematics.

5. Apr 24, 2015

### asmani

Of course, and as we know from chaos theory, almost the whole world is involved. So you can suppose by initial state, i mean the initial state of the whole world at the moment of flipping.
You're right. I didn't find a philosophy forum in here, so I figured this is the most relevant section to post it. I'm not talking about "uncertainty" and "knowledge" as a mental states, but after all, don't probability distributions depends on the point of view of whose person we're looking at the problem? I mean, probability of tail-head is 50-50 for me, but for a physicist who has enough data and computational capacity, it's not 50-50.

Okay now let me try to be more specific. Uncertainty is when the probability of an event in the future is neither 0 or 1 (in the POV of person X). And knowledge is how much information the person X has about the initial state.

I know I'm still probably way off from academic formulations. I have an electrical engineering background, so I know just a little about probability theories. Isn't there anything as POV (point of view) in probability theories? Because I think probability distributions must be different to people with different knowledge.

6. Apr 24, 2015

### gleem

Sure knowledge affects the probability of events. Think of counting cards in Black Jack

7. Apr 24, 2015

### Stephen Tashi

That isn't question about the mathematical theory of probability. It's a question about the outlook of a person applying the mathematical theory of probability. Some people do take the view that probability is not an objective property (i.e. that it isn't a property like mass or temperature).

Well, you're defining "uncertainty" in terms of "probability", so you either have to define "probability" or take "probability" as an undefined term. Neither course looks promising for sorting out the relation between "probability" and deterministic situations.

You'll find that trying to discuss any type of uncertainty in a mathematical context is difficult. Logic and mathematics deal with statements. Statements (by their formal definition) are either true or false and they are not both true or false. So a fundamental difficulty is that a concept of "uncertainty" (such as "probability") is an attempt to attach a property to a statement (such as "the coin lands heads") that does not correspond to "true" or "false".

The way to handle this difficulty is to make statements about probabilities. For example, let A = "the probability that the coin lands heads is 0.5". But, from that given information, all we can deduce is statements about other probabilities. Attempts to deduce non-probabilistic conclusions from statements about probabilities don't work. (For example, "If the probability that the coin lands heads is 0.5 then in 100 tosses it will land heads 50 times" isn't a theorem.) Given a statement about probability, the closest one can come to getting probability out of the picture is to make a statement about a limit of probabilities. When applying probability, people often interpret the fact that a limit of a probability is 0 or 1 to mean that some event is impossible or happens with certainty. However, that is an interpretation, not a mathematical result. (For example, in logic there is a principle of deduction called "modus ponens" that has the pattern: Given A implies B and given A , we may conclude B. However, there is no principle of deduction that has the pattern' Given A implies B and given A happens with probability 1, we may conclude B.)

Your question, is the reverse of the above situation. You want to make deductions of the form "If A then B" where A is a statement that does not involve probabilities and B is a statement that does. You'd have to invent a new approach to probablity theory to have any hope of getting mathematical results of that form. (Perhaps you could revive the Von Mises theory of "collectives"). In current mathematical probability theory, results about probabilities only exist in the context of some given "probability space".

If you attempt to proceed from a deterministic situation to a situation involving probability, you'll find that you have to force probability in to the situation. For example, you could take a deterministic situation and talk about observing part of it "at random" or looking at a "randomly chosen" piece of it. Deterministic situations can have statistical properties. For example, a class of students can be 51% male. But statistical properties don't imply probabilities unless you introduce some probabilistic context, such as "pick a student from the class at random".

8. Apr 25, 2015

### asmani

Thank you Stephen for your thorough answer. I'm not sure if I could correctly understand all of what you said, but I should say I've been considering probabilities as a purely subjective thing. In a deterministic closed system, if there's no conscious agent, then I guess probabilities are meaningless.

I'm again sorry that I'm way off from academic mathematical talking, but I want to know what a "probability distribution" means in a deterministic world? Suppose I'm living in a deterministic world and I know that the proposition "flip coin is head by probability 50%" is true. It means if I repeat flipping coins infinitely, the ratio will tend to 50%. Does it mean that the world's structure is so that there's a symmetry toward head and tail outcomes?

9. Apr 25, 2015

### micromass

10. Apr 25, 2015

### FactChecker

Much of "probability theory" is really the theory of guessing given incomplete information. After all, if a coin has already been flipped, we still calculate the "probability" of heads for something that has already happened and been determined. We just have incomplete information.

11. Apr 25, 2015

### PeroK

You need to question what you mean by "living in a deterministic world". And what you mean by "perfect information".

If I toss a coin now, you could measure very accurately everything about it as it leaves my hand. But, before I've tossed it, what kind of physics will tell you how I will toss the coin? I might throw it really high, or spin it really fast. What sort of physics will tell you that?

Also, what is a perfect measurement of, say, initial velocity? There's no such thing.

In general, you can only determine the outcome of relatively simple experiments. Determinism beyond that is practically unachievable.

12. Apr 25, 2015

### PeroK

PS: I'll tell you what I would do if you said you had such a theory. I'd ask you to tell me what it said I would do. If you said, my theory says you'll toss it really high, then I'd do the opposite. Where's your determinism then?

13. Apr 25, 2015

### FactChecker

Ha! But in his theory, you would TRY to toss it low and would accidentally toss it very high, just like he knew you would. It's hard to beat perfect knowledge.

Last edited: Apr 25, 2015
14. Apr 25, 2015

### Stephen Tashi

That is not a correct interpretation of mathematical probability. You can't conclude that the the ratio will definitely tend to 50%. You can only draw conclusions about the probability of the ratio being close to 50%. Assumptions about the probability of events lead to conclusions about the probabilities of other events, not to conclusions that other events definitely will happen.

If you assume a deterministic world, you have not specified any probability space. If you assume a deterministic world and also assume the coin flipper is taking "random" observations from the possible coin flips in that world, you are contradicting the assumption that the "world" is deterministic, because the coin flipper is part of the world, so his behavior cannot be "random".

15. Apr 26, 2015

### asmani

Thank you all for the replies, I need much more thinking and reading on the subject, I will get back to you then.

16. May 29, 2016

### raphalbatros

Let's say I create a purely deterministic universe in a box that cannot interact, in any way, with any other thing than itself. Then, I record every coin toss that happens in it, with their results. At the end, I find out 50,01% of the coin tosses had tail as a result.
Then, I repeat my experiment. I create another universe with differente initial conditions, I record all the coin tosses, and I find out 50,008% of the time, the result is head.
Let's say I proceed to repeat my experiment a lot of times. As a human, I happen to be able to recognize a pattern. And so, I realise something. In each experiment, the results are always around 50%.
My question:
How is it that a deterministic universe contains ensembles of events (like a coin toss) whose final states (the result) are always near to a distribution of 50 / 50 ?

17. May 29, 2016

### FactChecker

Most of probability theory is the theory of guessing given incomplete knowledge. Determinism doesn't help your guess if you don't know all the facts. Suppose your thousands of coin tosses have already happened, but you do not know the results. It doesn't matter if the coin toss physics were deterministic or not -- the results are already determined, but unknown. You still have to guess half are heads. There are mathematics theories (Bayesian conditional probabilities) to address how you should adjust your guess as you get more information. Whether you are given information before or after the experiment does not make any difference. So suppose you have done 5,000 coin flips and only know the results of the first thousand. If they were 600 heads and 400 tails, you should adjust your guess of the total results. I would guess 2,600 heads and 2,400 tails. There are much more complicated examples of Bayesian theory than this.

Last edited: May 29, 2016
18. May 29, 2016

### raphalbatros

Edit:
I just realized, reading wikipedia, that a deterministic universe is not exactly what I had in mind. Instead, my question is referring to a universe respecting the laws of what is called "causal determinism". Which means that "any state is completely determined by prior states".

I don't make reference to an experiment where I have to guess the results given incomplete knowlegde.
I pretend that, if a purely deterministic universe was to be, it would contain such ensemble of events (like all coin tosses), whose final states (if you were to record all of them) would respect a certain distribution.
And I ask what causes this particular distribution to be the same (approximately 50/50) in every deterministic universe possible, regardless of its initial conditions.
I don't ask how probabilities behave, or how I should interpret them, I ask how it is that some aspects of a deterministic universe mimic what you would expect from pure randomness.

Last edited: May 29, 2016
19. May 29, 2016

### Stephen Tashi

You just said that you are going to pretend that there is an ensemble of events that has a particular distribution, so it's your imagination that has caused this to happen.

20. May 30, 2016

### raphalbatros

Stephen Tashi
Of course, as a thought experiment, my reasonning is not good for that reason, I understand that.
If a purely deterministic universe was to be, would it contain such ensemble of events (like all coin tosses), whose final states (if you were to record all of them) would respect a certain distribution ?
If the answer to that question is yes, then my interrogations remain. (where does that distribution come from)
If the answer is no, then the problem isn't one finally.

21. May 30, 2016

### Stephen Tashi

Likewise, where does any property of a deterministic world come from ? If you are hypothesizing a deterministic world then are you also hypothesizing "natural laws" that govern how that world behaves? If so then I suppose you must say that those natural laws cause all properties of the deterministic world - including the property that a particular set of events has a particular distribution.

22. May 31, 2016

### raphalbatros

Yes, I would say a deterministic world is entirely governed by its initial conditions combined with its "natural laws". And yes, I believe your last statement is true.

But imagine I read in a book that "e to the i π + 1 = 0". Then I go ask to my teacher "How is it that e to the i π + 1 = 0 ?". I would expect from him to explain it to me from the start (the axioms) to the final equation (e to the i π + 1 = 0). I would not be satisfied with an answer like "This equation is an implication of the axioms".

What I'm saying is, I would like to understand how those "natural laws" imply that distribution. I seek an answer like "From such natural law, you can deduce that and that and there you have it, there goes your distribution".

And, for the record, I am under the impression we currently live in such a universe.

23. May 31, 2016

### chiro

Hey raphalbatros.

There are convergence theorems regarding complex numbers (just like there are regarding real numbers) and you can get results regarding things like Taylor series and expressions for functions.

The graduate field of study that looks at this is known as Complex Analysis (or the Analysis of Complex Variables) and a good textbook should be available that covers these issues.

24. May 31, 2016

### FactChecker

I think I am starting to understand your original question better. Here are a couple of questions that might help answer your original question:
1) Given the natural laws, how much randomness in the initial conditions would be retained over time? And where would the randomness of the initial conditions come from?
2) Are there natural processes that balance the largest deterministic influences and leave the final result to be determined by tiny "random" influences?

Here is why I think these are relevant questions:
Question 1): Suppose we calibrate a coin toss so precisely that Heads/Tails are determined by the impact of air molecules in still air. The result might be determined by the velocity of a molecule that was on the other side of the Earth 10 years ago. Even in a completely deterministic world, the random distribution of initial conditions would still show up. Notice that "calibrating" the coin toss is intentionally bringing all initial states into balance so that the H/T result will be 50/50 and determined by tiny "random" initial conditions.

Question 2): Suppose there is a process that naturally brings all states into balance at a stable state. That might leave the final result of the process to be determined by tiny "random" conditions. There are many processes that converge to a stable state. That would be similar to "calibrating" a coin toss to give an exact 50/50 Heads/Tails result, dependent only on tiny "random" initial conditions.

25. May 31, 2016

### Stephen Tashi

Do you think there is one method of understanding that would work for each possible system of natural laws? It seems to me that each particular system of natural laws would have its own implications. If you are serious about investigating this problem (in the mathematics sections), you should formulate a simple mathematical example of your general question.