Mathematics - the language of a deterministic Universe?

[ajb]

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.

It is not always possible to do that, especially of there are lots of things to take care of. It maybe the case that in principal you can do this, but it maybe impossible to actually do. You may be forced to look at statistical methods. This is not quite the notion of 'not deterministic' in the sense used in quantum mechanics, it is a problem of having too many degrees of freedom to cope with. Anyway, people have developed lots of mathematical tools here.

 

So one may have to calculate something using statistical mechanics, you may be able to get an exact result for something, but that result cannot be understood in a completely deterministic way. It may be an average value or a probability. In your sense this is 'deterministic' while the interpretation needs some care. It is also technically deterministic as it involves no randomness.

 

Wouldn't that be the same as just plotting two functions instead of one?

In essence yes, but which one would our system chose?

 

It is interesting that the notion is the "real" line as opposed to something that is not real.

 

 

There are lots of things that are not real and lots of sets that we cannot take to be the real line. However, from your examples it seems that we are just discussing functions on the real line.

 

 

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Edit: Maybe these further comments will help...

 

 

Let me consider a simple example. Let us take a fair dice. It has the outcomes S= {1,2,3,4,5,6}. To that set of outcomes I define a mapping to the real numbers which I will call the probability of that outcome occurring on a singe roll of the dice.

 

[math]P : S \rightarrow \mathbb{R}[/math],

 

which I naturally define as P(1) =P(2) = ... P(6) = 1/6.

 

Now for every element in S, I have assigned a number in a unique way. Every time I feed in a specific element x of S I will get the same number P(x), in this case what I get does not depend on what I feed in, but that is immaterial. This is deterministic, no randomness or ambiguities arise in this assignment of the probabilities. (Just as no problem arises in the example of y = f(x) for functions on the real line)

 

However, it is in the interpretation of what probability means where the randomness comes in. I can not predict the actual outcome of rolling the fair dice other than making statistical predictions such as I know the probability of rolling a 6 is 1/6. I cannot say if a given role will actually give me a 6. The system of rolling a fair dice is not deterministic, I can get different outcomes when rolling the dice.

 

Quantum mechanics is actually similar to this. It is deterministic in the sense that given an initial state at t=0, then I can calculate the state at any other t. I know how the states evolve in time and this is deterministic. The non-deterministic part comes in when I interpret what the wave function means and how I assign outcomes of experiments to a states.

Edited February 7, 2015 by ajb

[studiot]

So to recap

 

I offered you Heisenberg's Uncertainty principle which is not deterministic in our physical world

 

and you have not replied.

 

I offered you Bessel's equation, which has no solution in our mathematical world, although we can get very good approximations.

 

I also mentioned other areas.

 

The humble arch, one of the simplest structural forms that has been much used for thousands of years has no known mechanical solution, although again we have approximations.

 

The loads on a bridge are not deterministic.

So if I asked you to design an arch bridge deterministically, what what load would you specify and how would you analyse it to get 'determine' an answer?

 

If you were an electrical engineer the actual load on a cable is not predictable, you would use 'the diversity theorem' when designing wiring for say a kitchen installation.

 

If you are going to claim that every real outcome in this universe can in theory be predicted with 100% accuracy by some mathematical equation you must be prepared to offer an equation for any that ask.