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Chaos theory states that we can't predict future because we can't measure initial conditions of a system to infinite precision. I get that. That alone doesn't mean that the future is not determined, it only means it is unpredictable (by us).

However, given the fact that every measurement yields an irrational number with infinite decimals, how is it possible that future is in itself deterministic? I think that for some system to be deterministic, it has to have a finite amount of data in itself (in this case, decimals.) I can't wrap my head around the notion that some observable variable has infinitely long number attached to itself, but obviously it has!

If nature has irrational, infinite values for every variable we observe, how is it possible for future to happen in any way? What exactly determines the outcome then?

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    $\begingroup$ Why do you think that a deterministic system has to have a finite number of decimals? You also have to make the distinction between what is existing and what we can measure. In my view, most observables are real numbers and by definition have infinite decimals. Even the exact number 2.1 has an infinite number of decimals: 2.10000..., even though we might only measure a finite number. If you take the crude view that all physical laws are differential equations, then inputting a real number will give one unique outcome, no problem there. $\endgroup$ Commented Nov 12, 2022 at 16:56
  • $\begingroup$ Are you thinking that chaos theory applies to every situation? It doesn't... $\endgroup$
    – D. Halsey
    Commented Nov 13, 2022 at 23:57

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The first thing to clarify is that chaotic systems are mathematical abstractions. Attribute values in chaotic systems are continuous (typically real or complex numbers), so they require an infinite number of decimal places to specify them exactly. That causes no difficulty in mathematics.

The second thing to be clear about is that chaotic systems are deterministic. If we know the exact state of the system at a point in time then its whole future (and past) evolution is completely determined. However, states of the system that have very similar values at one point in time diverge rapidly from one another. So if we have only limited knowledge about the state of the system (say, we only know the values of its attributes to two decimal places) then we quickly lose the ability to predict future (or past) states of the system, even approximately. This is what we mean when we say that a chaotic system is unpredictable.

So a chaotic system is a deterministic mathematical system with a continuous state space which is unpredictable if we only have limited knowledge of its state.

The key question is whether chaotic systems are useful when it comes to constructing a model of the real world.

The limited knowledge aspect that makes chaotic systems unpredictable is a good model of reality - we only ever have limited knowledge about the state of an actual physical system. However, we do not know whether the locations of objects in space in the real world are continuous - we haven't shown that space is not continuous, but we haven't shown that it is either. So the continuity aspect of chaotic system may or may not be a good model of reality. And we are fairly sure that the deterministic nature of chaotic systems is not a good model of reality at the quantum level.

So chaos theory can be a useful model of the real world as long as we restrict our attention to large macroscopic objects e.g. the evolution of the solar system.

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  • $\begingroup$ Not only do we not know that the universe is continuous, we don't know that real numbers "exist" or are the right mathematical tool to model the real world. Most of mathematical physics works well using alternatives to real numbers that don't assume infinite precision. But the implications for determinism are different in the same way they would be if the universe is not continuous. $\endgroup$
    – matt_black
    Commented Nov 13, 2022 at 17:56
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Chaos is not tied to rationality.

Two systems could have irrational but close initial conditions, and remain close for all times. For instance, two small dense spheres dropped from a height of approximately $\sqrt{2}$ (or any other irrational number) meters will always arrive on the ground at approximately the same time. (That’s even been tested on the Moon.) There is no chaos there, even if the initial conditions cannot be expressed using finitely many digits.

This is different from other situations (the famous Lorenz system comes to mind) where, because of non-linearity in the system, small changes in the initial conditions leads to vastly different outcomes, even if you feed rational (but close) initial conditions to the system.

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For starters, nature probably doesn't have to write down the values of its variables.

how is it possible for future to happen

I won't pretend to know the answer to this question, but if your worry is how the universe keeps tabs on infinite variables at infinite precision, the answer is that you can see its very existence as doing this booking. As for measuring, if infinite digits are needed to exactly qualify an apple's weight, all this information can be contained in the needle display of an ideal analogical scale.

In the same vein, if one pictures the universe as "calculating" its next state, if you see the universe as a computer (though I'm not sure this picture brings any useful insight), then it's certainly as analogical computer.

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  • $\begingroup$ Related: Is the universe fundamentally deterministic?, Is throwing dice a stochastic or a deterministic process?. $\endgroup$
    – stafusa
    Commented Nov 12, 2022 at 17:30
  • $\begingroup$ If infinitude is all you get with numerical representation of observables, does that mean that numbers cannot fully "encapsulate" the reality? $\endgroup$
    – tetrametra
    Commented Nov 12, 2022 at 21:37
  • $\begingroup$ @tetrametra There might be ways to represent irrationals that don't demand unending sequences of digits we don't know about; or maybe they're not necessary, maybe not even real numbers a needed and you can describe everything only using natural numbers (think of quanta, or using very small units in classical physics - though I wouldn't bet on that, since there is some evidence that even imaginary numbers are necessary for properly describing quantum phenomena). But ultimately, at least right now, your question might be too philosophical for this site. $\endgroup$
    – stafusa
    Commented Nov 12, 2022 at 21:45
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Measurement does not yield a number with infinite decimals. Instead, a measurement for an observable yields a number precise to the most miniature precision of the instrument used to measure it. For example, say you measure the diameter of a ball to be 12.1 cm with a ruler having a least count of 0.1 cm, the measurement is accurate to 0.1 cm, and you say the diameter of the ball is 12.1 cm $\pm$ 0.1 cm and not 12.1000000cm.

Now imagine a situation where we have 1000 balls, and I am claiming this system of 1000 balls is chaotic. Say we do some calculations and predict that for ball number 42, at t=20s, the ball's velocity should be sqrt(2) cm/s, given some initial conditions.

Now nature is evolving all of these balls at each time step. So how can nature evolve this system from t=20s to t=21s if one of the variables (velocity for ball 42) is irrational?

Answer: It does not matter. Language is critical here, and I intentionally anthropomorphized nature to emphasize this point. Nature is not a computer; it need not measure observables to calculate its future. The "observables" are intrinsic to nature; nature exists and evolves even without us (humans) doing a measurement. Doing a measurement for an observable is one of the many ways a simulacrum for reality is obtained (the reality always remains hidden, though).

Note: The question about a measurement changing the physical outcome is widely debated in quantum mechanics; however, my argument is limited only to classical treatment.

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