To a scientist, existence means "you, or someone, can find it, at least potentially". To a non-constructive mathematician (the common kind) existence means "you can't prove it doesn't exist, and I choose to believe it does, at least in the current game I am playing."
Mathematicians use the term "real numbers" to describe precisely the set of numbers you get when you take all the numbers that could plausibly be computed, and then declare that all the holes between numbers we can individually compute/describe are completely full of numbers. This is justified because you can't prove that any specific one of them is impossible to write.
(But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant
https://en.m.wikipedia.org/wiki/Chaitin%27s_constant but you could write it in infinite time if God told you the digits.)
> But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant https://en.m.wikipedia.org/wiki/
Just because you can create a definition, doesn't mean that definition names a thing. We can define "the set of all sets that do not contain themselves", but such a definition is nonsensical in that trying to apply it creates a contradiction. To me, the incomputability of Chaitin's constant seems very similar, except it's contradiction is buried inside the halting problem.
Computer scientists don't have a choice. And not all mathematicians do accept non-computable objects. Constructive and intuitionistic mathematics are two approaches that bring some questions (doubt?) of the continuum.
Computer scientist don't accept noncomputable objects. Mathematicians do.
To a scientist, existence means "you, or someone, can find it, at least potentially". To a non-constructive mathematician (the common kind) existence means "you can't prove it doesn't exist, and I choose to believe it does, at least in the current game I am playing."
Mathematicians use the term "real numbers" to describe precisely the set of numbers you get when you take all the numbers that could plausibly be computed, and then declare that all the holes between numbers we can individually compute/describe are completely full of numbers. This is justified because you can't prove that any specific one of them is impossible to write.
(But there are some so called "real" numbers that are impossible to calculate, even approximately, like Chaitin's constant https://en.m.wikipedia.org/wiki/Chaitin%27s_constant but you could write it in infinite time if God told you the digits.)