Determinism: mathematical models

In summary, determinism in physics has been traditionally understood through the use of differential equations, but there are other mathematical descriptions that cannot be reduced to DEs. These include cellular automata, random processes with probability one, and systems with memory effects. The question remains whether these other forms of determinism can be shown to be equivalent to DEs or not.
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Auto-Didact
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Ever since Newton, determinism - i.e. the concept that the prior state of a system determines the next state of the system - has become an integral part of science, in particular of physics. This is particularly because Newton invented a mathematization of the concept of change: the calculus; in doing so he did much more.

Applying this new form of mathematics to study changes of physical phenomenon, Newton discovered that the concept of motion follows very strict mathematical laws; this essentially was the discovery of the laws of physics. The laws of physics have since then always come to us in the form of differential equations (or generalizations thereof).

To make a long story short, differential equations - or any abstractions thereof or of descriptive functions where the input and outputs reside in the same space - are the prototypical mathematical implementation (or 'mathematization') of the concept of determinism.

My question is, is anyone familiar with other forms of determinism which can definitively not be reduced to or be shown to be equivalent to differential equations in any sense, not even in principle?
 
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If we have a discrete-time system, where the time variable jumps forward in constant steps, then it would be a difference equation.

A system with memory effects, like a particle acted on by a force field that depends on where the particle was 1 second ago and not on its current position, is not described by a differential equation (if a DE is defined in the usual way). It can be formally written like a DE that has time derivatives of arbitrarily high order in it.
 
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Auto-Didact said:
My question is, is anyone familiar with other forms of determinism which can definitively not be reduced to or be shown to be equivalent to differential equations in any sense, not even in principle?

(Deterministic) cellular automata?

hilbert2 said:
A system with memory effects, like a particle acted on by a force field that depends on where the particle was 1 second ago and not on its current position, is not described by a differential equation (if a DE is defined in the usual way). It can be formally written like a DE that has time derivatives of arbitrarily high order in it.

It can also be written (rigorously, not just formally) as a DE in a Banach space (of infinite dimension).
 
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  • #4
hilbert2 said:
If we have a discrete-time system, where the time variable jumps forward in constant steps, then it would be a difference equation.

...if a DE is defined in the usual way
I am not defining DEs in the usual way, but instead including all similar concepts such as difference equations, recurrence relations, iterative maps, dynamical systems, and other commonly employed mathematical abstractions used in physics and applied mathematics.
S.G. Janssens said:
(Deterministic) cellular automata?
This is more in line with what I'm envisioning, but as far as I can tell deterministic cellular automata are outputs which function by producing more outputs based on inputs of their local environment. In other words, they are themselves solutions to some functions representative of what I'm arbitrarily calling DEs.
 
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What about random processes whose result at some stage has probability one? Or whose average approaches some other process as a limit (e.g. the Central Limit Theorem)?
 
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  • #6
hilbert2 said:
If we have a discrete-time system. . .
Well, we might . . . but how would you ever know (measure it?) ??

hilbert2 said:
. . .where the time variable jumps forward in constant steps. . .
That makes me think about the guy with the arrow. . . . :olduhh:

.
 
  • #7
Is it relevant to bring in Chaos here?
 
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FactChecker said:
What about random processes whose result at some stage has probability one? Or whose average approaches some other process as a limit (e.g. the Central Limit Theorem)?
Elaborate?
sophiecentaur said:
Is it relevant to bring in Chaos here?
As much as I love chaos, I'm afraid it isn't.

What I'm looking for is some other mathematical description of determinism, which cannot be reduced to being equivalent to DEs, or else a proof that such another mathematical description is impossible, i.e. that the class of deterministic conceptualizations = the class of objects which fall under my non-standard DE moniker.
 
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  • #9
FactChecker said:
What about random processes whose result at some stage has probability one? Or whose average approaches some other process as a limit (e.g. the Central Limit Theorem)?
Auto-Didact said:
Elaborate?
I'm not sure that I can give a formal description. There are things whose state at particular times is determined (or approximately determined in the limit) even though the intermediate states are not deterministic. Do they fit into your original question?
 
  • #10
FactChecker said:
I'm not sure that I can give a formal description. There are things whose state at particular times is determined (or approximately determined in the limit) even though the intermediate states are not deterministic. Do they fit into your original question?
They might. If I understand correctly you are saying that some aspects of outcomes based on probability theory and its theorems are in advance determined i.e. deterministic in some more abstract sense than determinism as usually understood.

Stated in this way I would I agree with this. The next question is, is the underlying reason for this more abstract determinism analogous or reducible to my non-standard DE definition or not?
 
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What is determinism?

Determinism is the philosophical concept that all events, including human actions, are ultimately determined by causes external to the will. In other words, everything that happens in the universe is the result of a chain of cause and effect, and there is no room for free will or random chance.

How do mathematical models relate to determinism?

Mathematical models are used to represent and analyze complex systems, such as the behavior of physical objects or the interactions between individuals in a society. These models are based on mathematical equations and can be used to predict future outcomes, making them useful tools for studying determinism and its effects on the world.

Can mathematical models accurately predict the future?

While mathematical models can provide valuable insights and predictions, they are not infallible. The accuracy of a model depends on the quality of the data and assumptions used, and there will always be some level of uncertainty in any prediction. Additionally, the concept of determinism itself is debated and there may be factors that cannot be accounted for in a mathematical model.

What are some criticisms of determinism and mathematical models?

One common criticism of determinism is that it undermines the concept of free will and personal responsibility. Some argue that if all actions are predetermined, individuals cannot be held accountable for their choices. As for mathematical models, they may oversimplify complex systems and fail to account for all variables, leading to inaccurate predictions. Additionally, reliance on mathematical models can limit creativity and innovation in problem-solving.

How can determinism and mathematical models be applied in real-world situations?

Determinism and mathematical models have a wide range of applications in fields such as physics, economics, and social sciences. For example, they can be used to predict the trajectory of a rocket, analyze market trends, or study the spread of diseases. Understanding determinism and utilizing mathematical models can help us make more informed decisions and better understand the world around us.

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