its only unsound if you are using standard summation. That is not the only kind, nor is it the only useful kind. The same can be said about zero or negative numbers, or the real number line. Those are also "fanciful fictions"
I mean, do you really think the sum of the positive integers is -1/12? It's very easy to prove that it isn't. (The sum of any two positive integers is guaranteed to be greater than either one of them. Therefore, no sum of positive integers can be negative.)
It's self-contradictory in the definition that you're using for sum. It'd be like saying:
1 apple + 1 orange != 2 fruits because addition only works across the same units.
1. The sum of any two integers is guaranteed to be bounded.
2. Therefore, all sums of integers are bounded.
Many many examples in Math go against natural intuition but they are still useful
for example, you say that the real number isn't "self-contradictory". Look at the banach-tarski paradox, that certainly contradicts intuition. A large part of mathematics is foregoing natural intuitive, and just follow the logical deductions of axioms. That is how we got hyperbolic geometry, which is essential to special relativity.