When you add 1kg of some stuff to 1kg of another stuff and end up with 1.97kg instead of 2kg, have you really falsified 1 + 1 = 2, or have you just confirmed that some portion of your material was vaporized in the reaction?

What does the number 1.97 even mean in a world where 1 + 1 ≠ 2? When you falsify 1 + 1 = 2, you also end up denying the validity of the very experiment that you used in the process. If a hypothesis can only be falsified by self-invalidating experiments, does it count as falsifiable?

Your experiment depends on the way your scale defines "1", "2", and "kg", and can't be used to test this.

That's the point. You can't design an empirical experiment to falsify the definitions of basic concepts.

How do you go about testing the hypothesis that 1 + 1 = 2? Not just showing that a particular physical representation obeys the logical laws of arithmetic, but to test the actual laws themselves.

The point is that it's not a hypothesis. Or, at least there are views about maths where it's not a hypothesis.

So it's something that's not a scientific hypothesis, but is considered knowledge!

(Or, if as in the other comment chain, 1+1=2 is too obvious, you can take any sufficiently non-trivial theorem of your choosing).

Under the view I'm describing any statement like that is a kind of definition, not something that is true or false.

You can investigate whether it is consistent with other definitions. I believe that doing so is empirical work, which I think is consistent with a broad notion of science

I could be wrong but if you wanted to test the hypothesis: given the rules of addition show that 1 + 1 = 2 can be deduced.

The experiment would be to go about the deduction. It is falsifiable.

When we talk about falsification in modern science, we usually mean falsification through empirical experiment. It must involve data from the real world.

If you include thought experiments based on nothing but arbitrary axioms, a lot of crazy claims about supernatural beings and other metaphysical curiosities would also qualify as falsifiable. I'm not sure whether you'd like to go down that rabbit hole. Science is about understanding the actual physical world that we live in -- not some fantasy world no matter how internally consistent it might be.

> 1+1=2 is falsifiable as it relates to an experiment.

1+1=2 does not relate to an experiment; it is a statement about relations of concepts, not material entities.