That's all anybody has ever done. You may as well be claiming that the reals don't exist because no mathematician has conceptualized one non-symbolically and no

That's all anybody has ever done. You may as well be claiming that the reals don't exist because no mathematician has conceptualized one non-symbolically and non-approximately. When you do calculus on the reals, you only ever approximate or encode a finite (really, countable) subset of them.