Mathematics as a tool to ask new questions

One thing that sometimes bugged me quite a bit during education (but still does occasionally) is the people would describe in detail why something happened, but fail to describe why something else didn't happen. This became a bit confusing if two situations seemed similar but the end result was very different. It made me wonder if completely different physics was at work in otherwise similar conditions. Usually the answer turns out to be trivial : all of the same fundamental processes are occuring, it's just that their magnitude differs. Or of course there might be emergent effects from subtle differences that profoundly change the final result, without changing the small-scale processes except in magnitude. This has happened to me quite often, but I'm frustratingly unable to think of any good examples right now.

I don't mean to imply that this is directly analogous to mathematics, because mathematics isn't a direct description of physical reality. But I think there's a broad similarity there.

However, mathematics is not one harmonious entity – its tools fit together reasonably well, but not perfectly well. We have to mind the gap between them. Division is a useful tool, and zero is a useful tool, but dividing by zero is beyond the useful range of division.

Aside from facts and paradoxes, mathematics can also produce unusual models which seem intentionally detached from the world that surrounds us. Let us consider one very simple example. The picture below shows a knotted string. Its ends are glued together to prevent it unknotting when pulled one way or another.

We cannot untie a knot like this just by gently pulling it, we have to cut it. However, an alternative approach asks whether a knot can be unknotted by considering it in some imaginary space instead of the usual space. For example, the knot in the picture above is a so-called slice knot, which can be unknotted easily if we observe it in four spatial dimensions, rather than the three-dimensional space we’re used to.

Most famously, non-Euclidean geometry, which was developed as a thought experiment by mathematicians in the middle of the 19th century, argued that some straight lines may be curved. It became indispensable to the 20th-century discovery of the relativity theory, which argued that light, instead of travelling in a straight line, sometimes travels along a curve, or even around a circle.

http://theconversation.com/how-maths-can-help-us-answer-questions-we-havent-thought-of-yet-102051