They exist in the mathematical sense that they are basically the solution to an equation. We know the equation, we know there is a solution, but we don't know what the solution is, and it has no physical relevance. Many mathematicians (constructivists) would say that this means that they don't exist until you can find a way of finding the exact solution.
For many of these numbers, we know that there cannot be any rigorous procedure by which they can be found, because if there was, there would be a logical contradiction.
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They exist in the mathematical sense that they are basically the solution to an equation. We know the equation, we know there is a solution, but we don't know what the solution is, and it has no physical relevance. Many mathematicians (constructivists) would say that this means that they don't exist until you can find a way of finding the exact solution.
For many of these numbers, we know that there cannot be any rigorous procedure by which they can be found, because if there was, there would be a logical contradiction.