It's been a while since I did abstract algebra, but I'm pretty sure that once you have the additive and multiplicative identities, the rest of the reals can be generated. Which is still a constructive process.
Regardless, the existence of the real numbers is not a matter of need. Their existence is a consequence of how mathematics is defined. Over-simplified, it's a case of if addition and multiplication work, then the real numbers must exist.
Usually, maths doesn't require us to overthink about anything metaphysical. Things either are or they aren't, the problem-solving approach taken to demonstrate a result one way or the other is the fascinating part.
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It's been a while since I did abstract algebra, but I'm pretty sure that once you have the additive and multiplicative identities, the rest of the reals can be generated. Which is still a constructive process.
Regardless, the existence of the real numbers is not a matter of need. Their existence is a consequence of how mathematics is defined. Over-simplified, it's a case of if addition and multiplication work, then the real numbers must exist.
Usually, maths doesn't require us to overthink about anything metaphysical. Things either are or they aren't, the problem-solving approach taken to demonstrate a result one way or the other is the fascinating part.