What is the mathematical formula to generate a random number?

see "Non-mathematical" is a stupid term, because humans define what is mathematical and what isn't.

Everything in this universe is deterministic. Otherwise the laws of physics wouldn't exist.

Ok, how about "The universe is not governed by a single logically consistent system."

I think you're missing the point. Just because the output is seemingly hard to predict does not mean its random. At the point of observation, the number is now known, and thus when you input it into your function, the output is not random. It is expected based on the input. We are essentially arguing theory vs application. In practice, it will be near impossible to predict an outcome, but it's not, by definition, random.

This is a philosophical question. You asked if it's possible to generate a truly random value with a formula. It's not. Any inference you make based on that is your own.

The laws of physics aren't inherent to the universe. They describe behavior we've observed. Deterministic laws do not describe quantum phenomena we've observed. Your logic is backwards.

"quantum phenomena" is a ridiculously tiny portion of all the physical laws we uncovered that we know are deterministic. We don't have the equipment to test and measure all the particle physics tests with full accuracy, but going full retard like quantum theorists do and claiming that therefore there are particles that do random shit rather than admitting they can't understand or measure particles is just hilarious.

nothing is random.
there are just measurable values which we wouldn't be able to derive because we don't have accurate measurements of all of the relevant inputs

It amazes me that people who can't even guess something so simple as a coinflip since they can't measure all the variables pull randomness theories out of their asses about particles. This is no different than religions habits of filling the knowledge gaps with mystery and unscientific drivel.

>I think you're missing the point.
Right back at you. You are treating the seed variable as seperate from the function, and just restated what I already said - a function/formula is deterministic by definition. So to get a random output you need a random input - and random inputs exist, for example the spin of an electron passing through a detector at a certain point in time.

You can't have a "randomness" function, because all functions are built from deterministic steps. The outcome of a sequence of deterministic steps is invariably deterministic. HOWEVER, just because math is composed of logic and functions - and is therefore deterministic - does not mean that a truly random variable is impossible to produce.