Mathematics - the language of a deterministic Universe?

[quanta'namo nay!]

 

That is true. They are chaotic, which means that even though they behave deterministically they are unpredictable. So all you can do is predict probabilities.

 

 

Quantum theory is inherently random, probabilistic and (sometimes) acausal. Even though it can be described (very accurately) by mathematics.

 

So what do you mean by probabilistic exactly? What qualifies something to be probabilistic?

[studiot]

Thank you for your references, I will look at them shortly.

 

Meanwhile please look at this discussion about the word 'random'

 

http://www.scienceforums.net/topic/84215-chance-vs-probability/page-2

 

I extract my post#29 for convenience.

 

 

So let us continue to examine the word 'random'.

 

A random sequence is a sequence that cannot be expressed more compactly than a complete list by any algorithm.

 

(after Chaitin and Solmonoff. What Wikipedia call the Kolmogorov definition if you care to look it up)

 

Let my sequence, drawn from a binary system i.e. 1 or 0 for simplicity, be {A} where A is either 0 or 1.

 

Now the question arises:- Is this sequence random?

 

Well, mathematically it conforms to the above definition so it is random.

 

But a physicist might well wish to distinguish between circumstances as to how I arrive at this sequence.

 

For instance if I always calculate A = 4/4 I will always arrive at the sequence {1} and if I always calculate A = (4-4) I will always arrive at the sequence {0}..

 

So my result is predeterminate

 

But if I flip a coin and choose A = 1 for heads and A=0 for tails then which sequence I arrive at is indeterminate or at the behest of chance.

 

Final point let me thank you on conducting an adult and professional discussion without the rancour too often seen here.

You are certainly proving up on your original statement. and your points are not easy to answer.

 

 

[quanta'namo nay!]

 

studiot just answered this. What do you mean by deterministic?

 

I can't see how he did exactly. Something being deterministic means that the state of an entity ends up from a specific state #1 in a specific state #2 after a specific operation or transformation (i.e. in reality a physical event).

 

edited:

Ok. So as per studiot's post above about "random" it seems that the term could be understood differently depending on who you ask. I think I will look into that.

Edited February 2, 2015 by quanta'namo nay!

[swansont]

I can't see how he did exactly. Something being deterministic means that a specific state #1 of an entity ends up in a specific state #2 after a specific operation or transformation (i.e. in reality a physical event).

I thought the answer was quite straightforward. What precisely is the problem? A fair die has an equal probability of showing you each value after a roll. On a d6, the probability of rolling a 6 is 1/6, and the probability of not rolling a 6 is 5/6.

 

 

What if the specific state can't be determined? You may have a 1:1 mapping of the input to the output, but what of physical limitations on the precision of the input? e.g. what if the input needs the simultaneous value of position and momentum, to perfect precision? Is such a system deterministic (and is impossible according to the Heisenberg Uncertainty Principle)? Does that help us in any way?

[quanta'namo nay!]

I thought the answer was quite straightforward. What precisely is the problem? A fair die has an equal probability of showing you each value after a roll. On a d6, the probability of rolling a 6 is 1/6, and the probability of not rolling a 6 is 5/6.

 

 

What if the specific state can't be determined? You may have a 1:1 mapping of the input to the output, but what of physical limitations on the precision of the input? e.g. what if the input needs the simultaneous value of position and momentum, to perfect precision? Is such a system deterministic (and is impossible according to the Heisenberg Uncertainty Principle)? Does that help us in any way?

 

Well I don't have a problem with this "probabilistic" concept. It means that by definition everything is probabilistic. We can use calculations of probability if the deterministic model is too complex to handle or there are too many unknowns.

 

Something being probabilistic in this sense is compatible with a deterministic universe. So probabilistic and deterministic are two good friends

[Strange]

 

So what do you mean by probabilistic exactly? What qualifies something to be probabilistic?

 

It is not deterministic; it is random; it cannot be predicted, even in principle.

Edited February 2, 2015 by Strange

[studiot]

You seem to only have considered two possibilities. Probability v Determinism

 

Nature is more diverse than that, both swansont and I have mentioned situations other than probability where the mathematical equation does not have a nice tidy deterministic output.

 

swansont mentioned the uncertainty principle which is a physics principle that the product of momentum and position or energy and time cannot be known exactly.

As a result the more deterministic your calculation about one the less deterministic is your calculation about the other.

This is nothing to do with mathematics, but to do with the physics of reality.

 

I mentioned that many equations have no solution in closed form. That means that it is impossible to arrive at a perfect deterministic value for the output Y (given an input X) that the equation is calculating. Bessel's equation is a simple(?) example. Of course we can get as accurate as we wish (unlike with the uncertainty principle). This is inherent in the nature of the mathematics and nothing to do with the physical world that such equations are used to represent.

Edited February 3, 2015 by studiot

[ajb]

An equation (or function) may have multiple inputs and the output may be an equation or a numeric value. If you use sets then you are talking about multiple instances of usage of the function I suppose. If you have a different view point to this I'd ask you to present a simple example with some simple set(s).

I guess you should be more careful with what you mean by inputs and outputs. Where do they live?

 

A function, depending on the context, I would understand as a mapping f from one set to another such that it is many-to-one. In practice the domain and range may well be the real line, but you can have much more general things here if you want. (And there are things that are called functions that are not really functions.)

 

An equation should be understood as a particular kind of equivalence relation, it states that things on the left and things on the right are the same. Again, to make sense of that one would usually assume the objects in question are from a set. So, if we have a function you would like to write f(x) =y, where f(x) and y belong to the same set. By a solution of an equation you are looking for all x in the other set (could be the same set of course) such that when acted on by f you get y.

 

If your definition of deterministic is that given a many-to-one mapping f: S1 -> S2, then f(x)=y is unique for all x in the domain of f, then your definition is a little empty.

 

Anyway, as already stated by others, mathematics also encompasses probability including stochastics. People have developed tools for dealing with random variables, systems whose states vary randomly, random walks, the spectral theory of random matrices and so on. Often one will end up at best describing the systems in terms of probability/statistics like mean values and so on, which is common in physics also when dealing with statistical mechanics and quantum theory.

[quanta'namo nay!]

 

It is not deterministic; it is random; it cannot be predicted, even in principle.

 

Deterministic is not probabilistic? Are you suggesting that we can not calculate probabilities for anything deterministic?

[Strange]

 

Deterministic is not probabilistic? Are you suggesting that we can not calculate probabilities for anything deterministic?

 

I didn't say that. I sadi that for non-deterministic systems (i.e. those that are truly random) the only thing we can calculate are probabilities.

[quanta'namo nay!]

You seem to only have considered two possibilities. Probability v Determinism

 

Nature is more diverse than that, both swansont and I have mentioned situations other than probability where the mathematical equation does not have a nice tidy deterministic output.

 

swansont mentioned the uncertainty principle which is a physics principle that the product of momentum and position or energy and time cannot be known exactly.

As a result the more deterministic your calculation about one the less deterministic is your calculation about the other.

This is nothing to do with mathematics, but to do with the physics of reality.

 

I mentioned that many equations have no solution in closed form. That means that it is impossible to arrive at a perfect deterministic value for the output Y (given an input X) that the equation is calculating. Bessel's equation is a simple(?) example. Of course we can get as accurate as we wish (unlike with the uncertainty principle). This is inherent in the nature of the mathematics and nothing to do with the physical world that such equations are used to represent.

 

Well I can find only the two possibilities for making physical predictions, these are physically probabilistic calculations and physically deterministic calculations. These two appear to be compatible with each other. Neither of them is random though.

 

I didn't say that. I sadi that for non-deterministic systems (i.e. those that are truly random) the only thing we can calculate are probabilities.

 

Would not truly random systems follow no patterns at all and thus also produce random frequencies of occurrences (random probabilities)?

[Strange]

Would not truly random systems follow no patterns at all and thus also produce random frequencies of occurrences (random probabilities)?

 

No.

 

Well I can find only the two possibilities for making physical predictions, these are physically probabilistic calculations and physically deterministic calculations. These two appear to be compatible with each other. Neither of them is random though.

 

I would say there are three categories:

 

Deterministic systems that are simple enough to be predictable.

Deterministic systems that are so complex and/or chaotic that they are not predictable (except in terms of probabilities).

Random systems that are not predictable (except in terms of probabilities).

 

We know all three types of systems exist. Although the first probably only exists as simplified idealizations.

[quanta'namo nay!]

If truly random systems follow some patterns of probabilities, why would you say they are truly random? Is there a difference between a random system and a truly random system or are you just making a clear distinction between random systems and too complex to predict systems?

[Strange]

If truly random systems follow some patterns of probabilities, why would you say they are truly random? Is there a difference between a random system and a truly random system or are you just making a clear distinction between random systems and too complex to predict systems?

 

That is pretty much it. For example, the decay of atoms or fundamental particles is no unpredicatle in the sense that the weather; they are unpredictable because they are random and without cause.

[quanta'namo nay!]

 

That is pretty much it. For example, the decay of atoms or fundamental particles is no unpredicatle in the sense that the weather; they are unpredictable because they are random and without cause.

 

But you don't know if in reality anything is without a cause.

[studiot]

 

Well I can find only the two possibilities for making physical predictions, these are physically probabilistic calculations and physically deterministic calculations. These two appear to be compatible with each other. Neither of them is random though.

 

 

Whilst I am happy to discuss A or B type considerations with you, the real world, both of Nature and Mathematics admits of types C, D .....

 

If you cannot see these surely the logically correct course of action is to ask for further detail, rather than pretending they don't exist.

 

Do you understand what 'no solution in closed form' means, for instance?

 

If I am wasting my time addressing your points whilst you ignore mine please tell me as I have many other things to do.

 

If you wish to continue the discussion I have shown you two additional situations, and there are yet more examples of non determinism and ways to cope with it.

 

The next one would be limit state design philosophy.

 

Do you know what this means?

Edited February 3, 2015 by studiot

[Strange]

 

But you don't know if in reality anything is without a cause.

 

Well, no. The nature of science is that you can never know anything for sure. But there are things for which there is no known possible cause and for which current theory rules out a cause. (You can always say, "ah but there must be a cause" but at that point you have abandoned science.)

[quanta'namo nay!]

 

Whilst I am happy to discuss A or B type considerations with you, the real world, both of Nature and Mathematics admits of types C, D .....

 

If you cannot see these surely the logically correct course of action is to ask for further detail, rather than pretending they don't exist.

 

Do you understand what 'no solution in closed form' means, for instance?

 

If I am wasting my time addressing your points whilst you ignore mine please tell me as I have many other things to do.

 

If you wish to continue the discussion I have shown you two additional situations, and there are yet more examples of non determinism and ways to cope with it.

 

The next one would be limit state design philosophy.

 

Do you know what this means?

 

You seem to be talking about the nature of an individual result of an operation on a mathematical equation. Perhaps I have not managed to express clearly what I mean. I meant that all mathematical equations are deterministic in the way that if you take an equation which takes any inputs say x, y, z and t, and then see what the outcome is with some specific selections for these inputs you will always deterministically get the same result if you repeat and don't change the input. Your inputs could be any kind of equations, numeric values or a mixture of both.

 

When you solve for your mathematical equation with your input, that input always determines your result. Your result will only change when you change your input, that's the determinism I am talking about. I am not talking about the specific results themselves just that they only change when the input changes.

 

Well, no. The nature of science is that you can never know anything for sure. But there are things for which there is no known possible cause and for which current theory rules out a cause. (You can always say, "ah but there must be a cause" but at that point you have abandoned science.)

 

I think science is about observable evidence. In your previous post you stated:

 

I would say there are three categories:

 

Deterministic systems that are simple enough to be predictable.

Deterministic systems that are so complex and/or chaotic that they are not predictable (except in terms of probabilities).

Random systems that are not predictable (except in terms of probabilities).

 

We know all three types of systems exist. Although the first probably only exists as simplified idealizations.

 

It seems that we can not know that all those three types of systems exists, I think you are saying that too above in the first quote. We only know that the first two categories exist.

 

What I don't understand is how could one distinguish between the two last categories you mention here. You discover some system you find unpredictable, how could you tell if it is a random system and not just a deterministic system too complex for you to understand?

Edited February 3, 2015 by quanta'namo nay!

[Strange]

It seems that we can not know that all those three types of systems exists, I think you are saying that too above in the first quote. We only know that the first two categories exist.

 

We observe all three categories.

 

What I don't understand is how could one distinguish between the two last categories you mention here. You discover some system you find unpredictable, how could you tell if it is a random system and not just a deterministic system too complex for you to understand?

 

Because the theory that describes it is purely probabilistic, not deterministic.

 

We have models based on, say, ballistics which describe the paths of projectiles in a deterministic and predictable way.

 

We have models of weather systems which are deterministic but can only make predictions within certain bounds and with a certain probability.

 

And we have models of quantum behaviour which are purely probabilistic. There is no "mechanism" or deterministic-but-too-complex description. And there are good theoretical reasons to think that no such description can exist.

 

All of these are, as you say, based on observation. (They must be; they are all scientific models.)

[studiot]

 

 

quanta'namo nay!

Perhaps I have not managed to express clearly what I mean.

 

Yes indeed not only I, but others as well have fully understood your points and responded showing that they do.

 

Understanding has not been so fortunate the direction.

 

 

quanta'namo nay!

You seem to be talking about the nature of an individual result of an operation on a mathematical equation.

 

I have shown you two examples where it is impossible to obtain any repeatable result or output from a mathematical equation.

One of these was for reasons of Physics, One for reasons of Mathematics.

 

You keep avoiding these for some reason of your own.

 

Why is this?

Edited February 3, 2015 by studiot

[quanta'namo nay!]

Deterministic systems that are so complex and/or chaotic that they are not predictable (except in terms of probabilities).

Random systems that are not predictable (except in terms of probabilities).

 

These are the two last categories of the three you listed. I was trying to ask you how you could distinguish between them.

 

Do you think it is possible that absolutely all events (macro and quantum) in the universe fall into the domain of causality?

 

Yes indeed not only I, but others as well have fully understood your points and responded showing that they do.

 

Understanding has not been so fortunate the direction.

 

 

I have shown you two examples where it is impossible to obtain any repeatable result or output from a mathematical equation.

One of these was for reasons of Physics, One for reasons of Mathematics.

 

You keep avoiding these for some reason of your own.

 

Why is this?

 

I don't quite get what you are actually going for.

 

Mathematical proof requires verification by testing by others. Are you with me on this requirement?

[studiot]

 

Are you with me on this requirement?

 

 

No it does not.

 

But that is totally beside the points I made, which you are still avoiding.

[quanta'namo nay!]

 

No it does not.

 

But that is totally beside the points I made, which you are still avoiding.

 

It seems that our viewpoints are just somehow different here and that's preventing us from coming to an agreement.

 

If someone manages to prove for some specific equations f(x) and g(x) that g(x) equals f( f(x) ), that proof has to always hold (i.e. is deterministic). Can we agree on this?

[studiot]

 

It seems that our viewpoints are just somehow different here and that's preventing us from coming to an agreement.

 

 

 

I have already expressed my opinion as to what impedes agreement.

 

We have understood you , but you are making no effort to understand us.

 

 

 

If someone manages to prove for some specific equations f(x) and g(x) that g(x) equals f( f(x) ), that proof has to always hold (i.e. is deterministic). Can we agree on this?

 

 

You would have to be more specific as to what you mean by f(x) and g(x) and what you mean by the equality of two functions.

[quanta'namo nay!]

You would have to be more specific as to what you mean by f(x) and g(x) and what you mean by the equality of two functions.

 

Actually that's the beauty of mathematics, there could be millions of function pairs f(x) and g(x) for which g(x) equals f( f(x) ). For any of the pair if it is proven that g(x) equals f( f(x) ) then that proof must of course always hold for those function pairs.

 

All mathematical proofs if valid must always hold, otherwise they're not valid. I think here we can agree?