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If laws and initial conditions are enough to determine everything, how is this reconciled with the idea that quantities have limited precision (real numbers) and hence there will always be errors?

Even in principle, each quantity can be made more precise…infinitely. How would a deterministic world then work?

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  • Can you reference what you claim, else the question is not clear enough. Determinism does not mean everything can be determined (by humans).
    – tkruse
    Commented May 19 at 17:35
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    How is limited precision and errors of human measurements and calculations relevant to ontological determinism? Ontology is not so limited, not even to being describable by real numbers and other human conceptions.
    – Conifold
    Commented May 19 at 22:00
  • @Conifold But don’t the laws of nature work according to mathematics? If the quantities in these equations can have infinite precision, how can anything be determined if there’s multiple different solutions to each equation with varying degrees of precision?
    – user74135
    Commented May 19 at 22:34
  • The laws of nature do not have to work according to mathematics, that is just our limited way of expressing them. But let's say the world is platonic and they do. Mathematics platonically determines unique solutions to all that infinite precision, it does not have to approximate anything to "varying degrees" as humans do.
    – Conifold
    Commented May 19 at 22:53
  • @Conifold Are infinite precisions actually real though? Wouldn’t the precision eventually have to stop at some limit? How can it go forever?
    – user74135
    Commented May 19 at 23:06

3 Answers 3

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AFAIK it's not about numbers and their precision. It's about computational irreducibility. Yes, everything could be predicted given the known initial parameters, but the computing required for that would be beyond anyone's reach.

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  • Even in principle, each quantity can be made more precise…infinitely. How would a deterministic world then work even if we had immense computational power?
    – user74135
    Commented May 19 at 20:14
  • Determinism isn't about us being able to actually simulate anything, it's just postulating that the universe mathematically speaking predictable. Essentially, it postulates that the universe satisfies some autonomous differential equation. Commented May 20 at 3:41
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According to Heisenberg's Uncertainty Principle, it isn't just that we can't know with arbitrary accuracy conjugate physical quantities simultaneously, they simply don't exist to arbitrary accuracy simultaneously, even in principle. In order to predict a trajectory with perfect accuracy, you need to know its current position, its current momentum, and have some deterministic way those change over time. It's fundamental to the nature of reality that there is never perfect precision in the first two. You can only get to those values probabilistically. Then the physical quantities due propagate over time according to a deterministic rule, the Schrodinger Equation. You may start with probabilities and end with probabilities, but its deterministic all along the way.

The concept of error doesn't quite make sense when you talk about real world quantities. They are what they are. You can say they are uncertain and perhaps call that error. That suggests that there is something wrong with the measuring device or there are hidden variables. Bell's Theorem proves there aren't local hidden variables. It can be shown the uncertainties are independent of measurement.

There are errors inherent to calculation, measurement, and such, but that doesn't have much bearing on determinism.

I'm not sure why people ponder over determinism though. So long as there is some preferable course of action and people can be trained to more likely pursue that action over alternatives, concepts of responsibility and morality make sense. A malfunctioning robot can be fixed.

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It is not reconciled at all. A deterministic world would work differently than this real world.

In reality, there is no such thing as absolute (=infinite) precision. That is denied by the Heisenberg Uncertainty Principle, which in some languages is translated as Heisenberg Inaccuracy Principle. We can only deal with approximations and averages.

Infinite precision is also mathematically impossible as each event would then require the transfer of infinite amount of information in a finite time.

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  • Perhaps infinite precision is a failure of our description of things but not of determinism itself
    – user74135
    Commented May 20 at 6:43
  • Determinism is only an assumption of infinite precision and the absence of quantum mechanics. Determinism is only a simplified model of reality, not a description of reality. Commented May 20 at 7:26
  • How would imprecise events transmit any less information than precise ones, albeit wrong information?
    – tkruse
    Commented May 21 at 11:24
  • @tkruse Infinite precision means infinite amount of decimals or bits in every quantity. Imprecise values can do with less. Commented May 21 at 11:52

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