quanta'namo nay!

You could of course see from the mathematical system which was the previous state of the parameter, and observe in simulation how the parameter changes its state from one to another whenever there is any change (maybe program the simulation to detect any change in the given parameter). That allows you to also plot the state of the parameter with respect to simulated time. This would give you more insight to the behaviour of the parameter than just calculating probabilities. Wouldn't that be the same as just plotting two functions instead of one? It is interesting that the notion is the "real" line as opposed to something that is not real.

It often pains me that it is so difficult to generate "random" numbers. I usually just end up using the time a program is executed as an easy "random" input to the program as it is not in control of the program. That generates adequate variation but still the linear progression is seen as some kind of pattern. Interpret the meaning in what sense? As applied to the natural world for physics? I'd say that without physics mathematics is just abstractions absent any meaning. Suppose you want to model the path with a plotted curve. Can you find a function f(x) that gives you a "two way" parallel path? No worries about the edit..

Uniqueness is not a requirement of determinism. Only that the result is not random but repeatable and predictable.

Yes. One-to-many and many-to-one mappings are of course deterministic, you can not produce random results via any function. It would mean your results change even though what you substitute into the function does not.

Yes, indeed. The simplest solution with unrestricted domain is f(x)=-x and g(x)=x. And this here will hold always. This is the determinism I was referring to. As said If it was not deterministic it would be random, and if it was random it mostly would not hold (could hold extremely rarely when one is lucky). If you are referring to your f(x) = tan(x) and g(x) = sin(x) example, that was already addressed. Strange also addressed this. sin(x) = tan( tan(x) ) does not hold always, and as such is false as a statement. sin(x) = tan( tan(x) ) will hold when you say x=0. Then you have reduced your equation to 0=0. As this is also not random but deterministic, that will hold always when you let x=0. Yes. Newton's model for gravity is also known to be false. We can of course still use it for approximations. Science searches for the best models. In that sense a classical model always has the upper hand over any nonclassical models that produce logical paradoxes. Occam's razor says that the paradoxical model is the worse model.

In mathematics the default is no range unless specified. Have you seen anyone assume a range without specification? It is merely a problem without a presented solution. The problem is: Which functions f(x) and g(x) will hold true for g(x) = f( f(x) )? You might find some function pair that will always hold true for the above. No restrictions on domains, they can be any. If codomains of f( f(x) ) and g(x) are contradictory then of course the statement g(x) = f( f(x) ) is false and there is no proof for that particular function pair.

Suppose there are two competing mathematical models in physics supported by the exact same data (the evidence) and no data against. Would you use Occam's razor to decide which one of these models is better? Well determinism means predictability without exception. Mathematical proofs when valid are predictability without exception, they will always hold. No matter what numeric values or other substitutions you test them with they always hold. In mathematics if there is any x for which g(x) = f( f(x) ) does not hold, it means that there is no proof. If there is only a specific range of x for which the above holds true, then you need to specify that range. Limited to the range then again the proof may hold. In that case of course always. What I defined was not for any limited range, but any.

I explicitly referred to any pair of functions f(x) and g(x) that fulfil the requirement of g(x) equals f( f(x) ) and that being proven for that pair of functions. Why do you offer a pair of functions for which this has not been proven, but also obviously do not fulfil the requirement? For any pair that fulfils the requirement it is not nonsense, instead it would be a proof that holds always, deterministically without exception.

Here we can see that for g(x) equals f( f(x) ) you get: sin(x) = tan( tan(x) ) It it obvious that this does not hold, so there is no proof (for particular function pair). My point was that there was some proven function pair that fulfils g(x) equals f( f(x) ) If the proof is valid, it must always hold. See above.

Perhaps you misread, I wrote g(x) equals f( f(x) ) not g(x) equals f(x) It is not a statement but is a requirement for statements where the first result of f(x) is fed once more into the same function.

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?

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?

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? 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?

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

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