Does Quantum vs Classical Reality implicate contradictions in mathematical models

Question

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?

Answer 1

Your concept of a quantum reality is not consistent with how Physicists treat quantum mechanics, so it is reasonable that your quantum reality may lead to troublesome issues with mathematical abstractions when physicists have no such issues.

First thing to note: QM is defined using the language of mathematics. Those mathematical abstractions must hold for QM to be able to define the world as it does.

Second, QM will never say 1+1=2 has a probability of being true less than 100% of the time. QM will always say 1+1=2. It doesn't break math. However, what QM may do is pose a question which becomes x+y=2, and ask the probability of it being true. What the uncertainty of QM says is that, for physically observable properties, the probability of x and y both being 1 is not 100%, and you wont know what their real value is until you operate on them.

Now there is a peculiar corner case to consider when the brain doing said mathematical abstractions is governed by QM. This means that there is a less than 100% chance that when the brain is trying to think about "1+1=2", it is actually thinking about 1+1=2 precisely. There's a chance that key electrons in that brain which were essential for capturing the essence of "1+1=2" skip town and go elsewhere. But that's a very different beast entirely.

Indeed, the fact that we do not regularly observe such events in day to day life is why classical mechanics does not govern them. And the desire for our brains to operate in this nice crisp clean way without fear of quantum uncertainty creating funny loops, we have developed the major interpretations of quantum mechanics, offering arguments for how a classical world and a QM world could interact.

Answer 2

Quantum uses waves and probabilities to try to explain beyond what we are able to see with the current technology, so Einstein is famous for creating a formula that describes some behaviour at some scale but that is not the ultimate way, quantum seems to be a better way of describing reality than any other but still we cannot be sure it is the ultimate, however we can say that the path to a unified field equation will come from quantum to macroscopic and no the other way around, but my main msg is:

We don't have the ultimate understanding yet that is why we find useful all kinds of aproaches that can allow us to represent what is going on in certain "system"

Answer 3

You are conflating two ideas of "1 + 1 = 2". There's the mathematical way (if you're using the Peano postulates, you're saying "the sum of the successor of 0 and the successor of 0 is the successor of the successor of 0". This exists independently of any sort of physics.

If you apply arithmetic to the physical world, you find that it doesn't apply consistently. Add a drop of water to a drop of water and get one drop of water. Add a half-liter of ethanol to a half-liter of water and get something less than a liter. Bang two appropriately-sized lumps of pure U-235 together hard enough, collect every bit of matter that comes out of the U-235 (the really tricky part), weigh it, and find you've lost some mass. There are some quantities that appear to be conserved (and, if they aren't, mean that the laws of physics vary in unexpected ways - see Noether's theorem), but not all of them.

What quantum physics does is put limits on what we can know. We know that the operation of the Universe depends on particles that appear and disappear faster than we can notice them. Any intuition developed with classical physics will be misleading when dealing with quantum mechanics or relativity. There is no contradiction in this, just applying concepts incorrectly in different situations.