How does Mathematics work?

To understand what I mean by the title, let's think of what we usually do in a derivation in science. We first take a situation (say, a bob swinging from a thread), represent it by some numbers (say the mass of the bob, length of thread, etc.), do some mathematical jugglery on them, interpret the result, apply it to the situation and verify it. Now, though we work in math and take it to physics, why does it still work?

If that didn't sound like a sound question to you, let's take an example:
We take,

s = ut + (1/2) .a.t^2 (One of Newton's equations of motion)

Now, we differentiate it with respect with respect to time,

ds/dt = u + (1/2). [a.(2t)]
v = u + at [since ds/dt = v]

Now, the last equation has been mathematically obtained from the original equation. Imagine that we took that original equation from an experiment (i.e., it is a 'model' for our 'observation'), so that we know it to be empirically true. On that model, we did some mathematical operations on the mathematical variables and obtained the last equation. But perhaps surprisingly, that last equation also turns out to be empirically true. It was not obtained directly from physical phenomena, it was obtained from a mathematical model. A physical phenomenon was converted to a model, some operations were done on the model, then when the new model was re-'translated' to the physical world, it is seen to be true! Maths seems to have an amazing power of 'predicting' natural phenomena, even though by itself it has nothing to do with natural phenomena.

I'm not sure how it does that, but I believe this might be the reason: First of all, the physical processes don't themselves correspond to the numbers. There do exist things called numbers which are the basis of mathematics. Maths knows how to manipulate these imaginaryentities called numbers to give other numbers. It is we who associate these things called numbers with physical phenomena. Then we associate mathematical processes like addition with some physical process. For example, when we place one ball in a basket and then place another, we take it to correspond to addition of 1 and 1. We use these 'representative elements' and work with them. Then, after the operations (which take the place of physical 'processes') are over and numbers (which we assume to denote some physical measure) have been obtained, we take it back to the physical reality. Since we know details like what represents what, we can easily retranslate it to physics. What we are doing is, it seems, just simplifying our derivations by using numbers instead of actual physical properties and processes. Since we take all the needed details from the physical model and use them in the math model, both give the same result. This is the view I now have of this thing called Philosophy of Mathematics.

This view seems to be supported by the Fictionalism view of Hartry Field. Here's a quote:
"In 1980, Hartry Field published Science Without Numbers, which turned Quine's indispensability argument on its head. Where Quine suggested that mathematics was indispensable for our best scientific theories, and therefore should be accepted as true, Field suggested that mathematics was dispensable, and therefore should be rejected as false. He did this by giving a complete axiomatization of Newtonian mechanics that didn't reference numbers or functions at all.
[...] Having shown how to do science without using mathematics, he rehabilitates mathematics as a kind of "useful fiction". He proves that mathematical physics is a conservative extension of his non-mathematical physics (that is, every physical fact provable in mathematical physics is already provable from his system), so that the mathematics is a reliable process whose physical applications are all true, even though its own statements are false. Thus, when doing mathematics, we can see ourselves as telling a sort of story, talking "as if" numbers existed.

On this account, there are no metaphysical or epistemological problems special to mathematics. The only worries left are the general worries about non-mathematical physics, and about fiction in general."

That, to a great extent, justifies "the unreasonable effectiveness of mathematics" in science. It's almost comic to think that we are doing all this complicated integrations and transformations to imaginary entities of which we talk "as if" they existed... :-)