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## Background

I'm defining the **Classical Reality** as everything having a given state therefore with enough knowledge about the state of everything at the smallest particle level we can theoretically predict any seemingly random event.

I'm defining **Quantum Reality** as things being in an unresolved state until observed and therefore the probability of any random event is unknowable even with complete knowledge of state for every particle in the universe due to there being a probability distribution built into the fabric of reality itself.

I've run into a contradiction (In my head so it may just seem like one with my intuition) while considering the idea of constructing minimal models between inputs (facts) and an output (categorization and extrapolated facts).

Simply put in a classical model 1 + 1 always equals 2 and can be abstractly represented in mathematics with the probability(1+1 = 2) = 100% or true.

It seems to me that in a quantum reality the appropriate abstraction should be probability(1+1 = 2) is equal to the limit as observations go to infinity of the number of times we observe that 1 + 1 = 2. This would likely be less then 100% or infinitely close. I think specific results would be dependent on the reality of time and what could be considered an observation.

## Question

If the world is quantum does that imply that numerically based math (which is an abstraction by nature) cannot always hold?

Therefore, in a quantum universe is P(1+1=2) < 100% even though all observations should show P(1+1=2) = 100% since observed results in quantum mechanics are still a discrete set.

Do the implications of math being an abstraction in some way avoid the question entirely.

## Relevant thoughts

It makes sense that since we are abstractly modeling a world based on observations which in both a quantum and classical reality should adhere to laws of state. We should always be able to use integers or categorical sets to represent state with large enough numbers.

Imagine adding the mass of two objects together broken down by component molecules. (With impossibly perfect measurements) We would expect a discrete set of outputs and that can be perfectly modeled with whole integers.

Yet in a world of quantum mechanics and state being decided upon observation is it not likely that adding those same masses would only add up to the expected number most of the time, so the best model we would be able to produce would be a (discrete representation of a continuous number) a number to some decimal point plus or minus the error.

Some things like space between objects are continuous already so are impossible to represent discretely.

## Answer?

Does that imply the answer to my question is that, some things are discrete others are continuous. Math cannot precisely define anything that isn't discrete anyways so our abstraction is broken before we even consider a Quantum vs Classical reality?

Newtonian physics can't be predicted, so your premise is flawed. https://physics.stackexchange.com/questions/403574/what-situations-in-classical-physics-are-non-deterministic

– user4894 – 2018-10-18T18:35:03.760One problem is that even if the world is classical "numerically based math" cannot always hold. If you place two drops of water close enough to each other the "answer" will be 1, not 2. Math does not mimic the world quite so straightforwardly, it is about idealizations, and "quantum math" is the same as classical math, just differently applied. Also, the probabilities of future events

areknowable with the complete knowledge of quantum state, that is exactly what the laws of quantum mechanics predict. They just do not predict which events will actually happen, like classical laws do. – Conifold – 2018-10-18T19:31:38.8071Also, your statement "Math cannot precisely define anything that isn't discrete anyways" isn't actually true. The real number line is a perfectly good mathematical construct, and it isn't discrete. If what you're saying is that you can't necessarily write down a number corresponding to something, that's true, but it doesn't mean it's mathematically undefined. – David Thornley – 2018-10-18T20:41:31.880

@DavidThornley but it's not like you can write out most numbers from the real number line without deciding on a precision and therefore error. The second you decide on the precision of the real number line you make it discrete. True some numbers can be represented as exact fractions or irrational numbers by their relationships (and I don't know enough about that relation to know which is infinitely more infinite). I think it's fair but maybe inaccurate to call 1/3 and pi imprecisely defined, since any use of them must compute to an accuracy and thereby a margin of error. – Sarzorus – 2018-10-19T16:22:35.160

Backwards omega sequences also create nondeterministic cases in Newtonian mechanics, cf the literature around the bomber paradox, Yablo's paradox, etc. – Not_Here – 2018-12-18T04:13:05.590

@Sarzorus There are uncountably many real numbers (and irrational numbers, and transcendental numbers), and countably many integers, ratios, and numbers that can be written out by some form of computer program. 1/3 can be precisely described, although not in decimal notation. Pi can be described in the sense that you can write a program that will print it out (obviously, infinitely many digits requires infinitely long run time, and none of these pesky real-life restrictions on space, time, and energy). – David Thornley – 2018-12-18T17:51:12.827

Sure. The "law" of non-contradiction is just a One mode of thinking. Just one way to get a handle on the world. Albeit non-contradiction is an important mode. But we need more than one approach to get a handle on the universe. – Gordon – 2019-01-18T22:30:38.720