I hate to break it to you, but if setting two things beside two other things didn't yield four things, then number theory would never have contrived to say so.

At what point are there two plus two things, and at what point are there four things? Would you not agree that a) the distinction itself between things happens in the brain and b) the idea of the four things being two separate groups with two elements each is solely in the mind? If not, I'd very much like to see some empirical evidence for the addition operation being carried out.

Mathematics are so firmly grounded in the physical reality that when observations don't line up with what our math tells us, we must change our understanding of reality, not of math.

English is so firmly grounded in the physical reality that when observations don't line up with what our english tells us, we must change our understanding of reality, not of english.

I hope the absurdity is obvious, and that there are no problems to make models of the world with english alone. So, do you find it more likely that math is connected to the world because we link it up explicitly or because it is an intrinsic property of the world itself?