So, the entirety of your argument relies on the 1 + 1 = 1 raindrop analogy. Disproving that invalidates your entire argument. I like a challenge in a debate, but lets get this settled.

Now, lets start with how the analogy starts; two raindrops are falling, side by side. So, we have two separate rain drops. At this point in time, we can say, with certainty, that 1 + 1 = 2. There are two of them, as they are separate, and falling side by side. Now, as they fall, they bump together and, thanks to the properties of water, form together into one rain drop. At this time, if we are to take another observation, there is merely one raindrop. We do not see two separate rain drops, we only see one. Which is as you said.

However, we would not say 1 + 1 = 1. Because, by observation, there is no second 1. There is only the one observed rain drop.

Now, we need to account for where that other rain drop went. We can't simply say it vanished, but the matter of the numerical accuracy still needs to be settled. So, our initial system of measurement must have been poorly chosen, as it doesn't give you a net result of zero (which every equation reaching equilibrium should supply). So, instead of numerically counting the individual drops, we need to quantify them in a different way. Since it's a fluid, we're use milliliters. Now, just to keep numbers simple, lets say the average rain drop is 1 mL. Lets start at the beginning again. We have two rain drops, each consisting of 1 mL of water, falling at the same rate. So it would be 1mL + 1mL = 2mL. Or you can express it 2(1mL) = 2mL. Now, as they fall, they form into 1 rain drop. So, add the fluid volume of the two rain drops, which now equal the single rain drop. We already have done so, twice. The formulas would look identical. These formulas apply for when they are separate and you are determining the total fluid volume of the two individual drops, and they would apply for the moment they are forming together into a single drop. After that moment, observation would only yield that there is a single rain drop, that is 2 mL in volume. It would simply be 2 mL = 2 mL. You can express '2mL' any way you would like, it is the same quantified value.

You should, perhaps, use a better example of Real World Science. It does have some basis. Newtons theory of gravity, for instance, doesn't hold true past the high school level, not taking into account several factors. Yet the current theory of gravity accounts for many more variables than simply distance and volume. This theory is also proving inadequate to explain certain astronomical anomalies (such as a galaxy approximately 8 times the size of our own, which is impossible in our current theories). So, it's being expanded upon again, taking into account even more variables. This is exactly how science works, and how it is meant to work. We explain what we can with what we know, and as we know more, we seek to find ways to explain more.

However, for the mathematical system we are using, the basics hold true. Why do they hold true? Because in order to use the system at all, they have to be true. If you come up with discrepancies such as you had, you simply are using the wrong units, or are neglecting a driving variable. yes, there are mathematical systems where 1 + 1 != 2, but we're talking higher level calculus, which is a completely different mathematical system then what we are using here, and thus, isn't applicable.