This has all been fascinating, helpful, and frustrating to me all at once. What a great thread!
Most of the unreasonableness has to do with quantum mechanics. For instance, we know that energy and mass are related by one equation, and energy and wavelength by another. So you mash them together and you get an equation relating mass and wavelength. This looks like nonsense, but it turns out that particles actually do exhibit the mass-wavelength relation that was derived.
The only thing I could find to back myself up on this was Wikipedia’s article on rational numbers:
I take this to mean that if I throw a dart at the real number line, I am going to hit an irrational number with 100% probability, even though there are an infinite number of rationals there. There are just so many more irrationals.
I just like to bring stuff like this up occasionally, especially when people say things like “Clearly i, the square root of -1, is bogus.” In many ways, real numbers are far more difficult and counterintuitive than complex numbers.
I like Isaac Asimov’s reply to his philosophy professor, when asked to produce i pieces of chalk: “Well, can you show me 1/2 a piece of chalk?” It just ain’t possible.
I realize on preview that this whole post has been a hijack, and probably makes the OP’s head spin, but I feel it is interesting and I’m posting it anyway. 
“We assumed the horse to be a sphere to make the math easier.”
Stranger
Yes, yes, but there’s also a 100% probability that you don’t hit π, to back up C K’s point. But then, there’s also a 100% probability that you don’t hit 5, (to undermine it?).
The idea that the real world contains rationals but doesn’t contain any irrationals is really odd. [The idea that the real world contains numbers at all is weird speech, it seems to me, but that’s a different issue]
Well yes, given that the Lebesgue measure of the rational numbers is 0, if you have a uniform distribution on the reals, the probability of choosing a number at random and getting a rational is 0. (Doesn’t mean that it’s “impossible”, though; that’s the thing with uncountable probability spaces.)
But once again, we’re in an idealized mathematical world here. The real number line doesn’t exist in the “real world”. I think what Dex meant is that we live in a world that’s really in fact discrete. We could make the argument that only the natural numbers exist in the physical world.
Of course, that depends on whether we can approximate the physical world with mathematics, or rather the physical world is an approximation of the idealized mathematical world. 
For example, the OP said that mathematics is a human construct, and I get his point, but there is a philosophical debate whether mathematics is really “invented” or “discovered”. Some mathematics seems so natural that we can’t but wonder whether it exists independently from us. Would an intelligent alien species have the same math as us? I think we’ve discussed this question before on this board.
I wouldn’t use fakey as a synonym for imaginary. I think i is a great example of the kind of thing the OP was asking about. It is perfectly reasonable to ask what the square root of -1 is, and to assign a symbol for it. It is also perfectly reasonable to manipulate it. The interesting thing is that i, which seems to be purely symbolic and which isn’t something you can point to directly in the real world, actually is important in describing real world things. It was the simplest example I could think of for something that seemed to be just mathematicians playing but which is at the heart of reality.
Of course, the very fact that it is capable of being used to describe real world things means we can point to it in the real world, just as much (and as little) as we can point to any other mathematical object in the real world.
I think people are being way too complicated here.
1. Notice a slight aberration within something physical
Notice how an object that is released from a particular height will fall to the ground.
2. ?
Start your critical thinking and consider possibilities. First, lets name this force… we will call it gravity. Now let us try to figure out what we want to learn about this new force called gravity. How powerful is gravity. How fast does it make objects fall? Does it cause them to fall at a constant speed? Or does it make them accelerate (ie, does the object keep speeding up the longer it is falling?)
Now perform some experiments dropping various objects from various heights. Measure the distances and record the time it takes to hit the ground.
3. loads of maths
Start calculating all the data you collected. Your data tells you that objects–no matter how heavy, all fall at the same speed… interesting, you say.
Your data tells you that objects do not fall at a constant speed. They seem to accelerate… that is, they reach faster speeds if they fall from higher up. You only measured falls from a height of 1000ft, so you haven’t learned about terminal velocity or wind resistance.
4.?
Now you can use your knowledge and your math to make predictions on the REAL WORLD!!
For instance. Even though you never dropped a 500lbs anvil from 2000ft, you can use your math skills and your previous data to calculate how fast the anvil will go, and how long it will take to hit the ground.
5. experiment to prove your theory
You work the math, and come up with your numbers. Now go ahead and drop it from 2000ft. Voila. Your math was correct and you used it to predict something in the real world.
6. profit!
But wait… one of your colleagues just got back from the moon. All of his data is different than yours. Things do not fall as fast on the moon he tells you.
So you guys start to think about this and realize (through math) that the force of gravity on the Earth is 6 times as strong as that on the moon.
After measuring the moon, you realize that the Earth is 6 times more massive than the moon. So you think this is the cause.
So now you can make a hypothesis that the massof the planet effects the gravity. You can now make mathmatical predictions about the strength of gravity on… say Mars. Or Jupiter.
This is it in the simple form. Even though we haven’t been to, say, a black hole. We can use the information we know about mass and gravity to make calculations on how objects will react near black holes.