Reality cannot be turned into mathematics
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The way i think of it is via three concepts viz. "Randomness", "Cause and Effect", "Correlation (and not causation)". Randomness assuming true uncaused nondeterministic, by definition cannot be modeled by deterministic mathematics. Cause-and-Effect in the real world is actually aggregate multiple causes giving rise to multiple effects and so trying to follow a single cause to effect presupposes you can have knowledge of every single parameter (i.e. nothing hidden) in all the causes and the same for all the effects. Correlation problem arises naturally from a combination of the above two.
Ultimately then, in the Universe; does true randomness exist or is everything deterministic and we just cannot know all parameters involved?
The philosophical text "Yoga Vasistha" uses the following example to illustrate the above; a crow alights on a palm/coconut tree and a fruit falls; is it cause-and-effect or a random occurrence? Can we ever know?
This is why all mathematical models are a simplification and we always make simplifying assumptions on what to ignore when constructing a model and agree to accept the model when the error delta of the result is small enough for our needs. Hence assumptions like "fair dice", "fair coin" etc. in probability theory.
Note that this is not to say we cannot get close to Reality; all of Science is proof that we can, but that there will always be that gap/error delta between Reality and our Model of it. For many cases it may not matter eg. all our technology today, but in some very important cases it does eg. Consciousness.
You might find the following books useful in this regard;
1) Ivar Ekeland has a series of popular mathematics books viz. a) Mathematics and the Unexpected b) The Broken Dice, and other Mathematical Tales of Chance c) The Best of all Possible Worlds: Mathematics and Destiny which i have found pretty interesting and thought-provoking in this regard - https://en.wikipedia.org/wiki/Ivar_Ekeland Read the first one in particular.
2) Timothy Gowers' Mathematics: A Very Short Introduction is more focused on Concepts/Abstraction/Models and hence relevant here.
An infinite/unbounded universe can literally be modeled by a single random infinite sequence, though. Even a random binary sequence. Think about it: Every logical structure, every pattern, must recur infinitely many times. A complete description of our universe, also, will be among them.
> Cause-and-Effect in the real world is actually aggregate multiple causes giving rise to multiple effects and so trying to follow a single cause to effect presupposes you can have knowledge of every single parameter (i.e. nothing hidden) in all the causes and the same for all the effects.
This is not a real objection, though.
Consider this: Let's say you have a physical model of the solar system and you want to specify the position of Venus to fantastically high accuracy, absolute or near-absolute precision. First, you'd have to take into account the position of every body in the solar system, then you would need to fold in the influence of nearby stars, then the whole galaxy, then the full stress-energy content of the observable universe -- and beyond that our notions of "position" themselves break down. In our universe, everything is in principle simulatable, but no finite agent can ever measure or compute to arbitrary precision.
So you can't say "arbitrary precision can't be measured, thus the universe is unsimulatable in principle" because the first statement there is trivial and the second doesn't really follow and is still far out of reach.
> For many cases it may not matter eg. all our technology today, but in some very important cases it does eg. Consciousness.
Consciousness is the most common objection, and yet there's absolutely nothing to suggest that it isn't simulatable. Hence the enduring popularity of "the universe isn't real and we're post-singularity ancestor simulation" theories, and similar. They're popular precisely because there's nothing to disprove them, and it's not even clear that they can be disproved.
Your understanding is wrong; That is not how Randomness and Infinity works. I used specific qualifiers "true uncaused nondeterministic" for Randomness which mean that there is no underlying process, no known deterministic causes, no known sequence/list, absolute unpredictability etc. Now remember the difference between "Countable Infinity" and "Uncountable Infinity" and how in the latter case no list can ever capture every element in those sets (see Cantor's diagonalization argument). So when truly nondeterministic randomness operates over an uncountable sample space, your argument does not hold.
For a concrete example, think about the "Copenhagen Interpretation" of Randomness and the Uncertainty Principle in Quantum Mechanics.
> Consider this: Let's say you have a physical model of the solar system and you want to specify the position of Venus to fantastically high accuracy, absolute or near-absolute precision. First, you'd have to take into account the position of every body in the solar system, then you would need to fold in the influence of nearby stars, then the whole galaxy, then the full stress-energy content of the observable universe -- and beyond that our notions of "position" themselves break down. In our universe, everything is in principle simulatable, but no finite agent can ever measure or compute to arbitrary precision ... So you can't say "arbitrary precision can't be measured, thus the universe is unsimulatable in principle" because the first statement there is trivial and the second doesn't really follow and is still far out of reach.
Again No; "In our universe, everything is in principle simulatable" is false since you cannot know every single parameter affecting the object under study. Precision is a consequence of that. There are parameters which we do not know and cannot know since the Universe is not deterministic at atomic scale (QM again). This fundamental probabilistic indeterminacy underpins everything.
In this context you should also look into "Complex Systems Science"(CSS) where components self-organize into whole systems with unpredictable behaviour and emergent properties none of which can be known beforehand from the components.
> Consciousness is the most common objection, and yet there's absolutely nothing to suggest that it isn't simulatable. Hence the enduring popularity of "the universe isn't real and we're post-singularity ancestor simulation" theories, and similar. They're popular precisely because there's nothing to disprove them, and it's not even clear that they can be disproved.
Nope. One model of "Consciousness" is as an emergent property where biological/chemical/electrical/etc. systems come together in a Complex System eg. an Organism i.e."Us". This is the materialistic scientific view. It is unique to that instance of organism and cannot be simulated in-toto since it comes into existence only in the whole organism and nowhere else.
Finally; i want to point out that while a Mathematical Model can approximate Reality, a Computer Model is an approximation of the Mathematical Model and hence twice removed from Reality.