Is math built on assumptions? I just came across this statement when I was lecturing a student on math and strictly speaking I used: 

Assuming that the value of $x$ equals <something>, ...

One of my students just rose and asked me:

Why do we assume so much in math? Is math really built on assumptions?

I couldn't answer him, for, as far as I know, a lot of things are just assumptions. For example:


*

*$1/\infty$ equals zero,

*$\sqrt{-1}$ is $i$, etc.


So would you guys mind telling whether math is built on assumptions?

Maths is built on assumptions in that the data and logic function correctly.
The utility of mathematics is that it does not reflect some truth of nature as such, but that if certain conditions apply, then these outcomes will happen too.
For example, while we think of Euclidean geometry as applying to straight lines on a plane, it holds also for circles passing through a point.  That is, if we call a circle through a point a 'straight line', and parallel straight lines become circles that are cotangent at that point, then we need only demonstrate the postulates to demonstrate the geometry holds.
Another utility comes that algebra can be extended to non-numerics.  That is, we can let variables be whole equations or matrices, and still do the algebra.
Once one understands a particular case, one can liken it to a known mathematical problem, and convert its solution into something that applies for this case.  I used the various edges of the fano triangle to create a 'perfect game' for eight golfers, such that they play with each member once and against the same member twice (in two pairs).
The general utility is that if one correctly guesses the principles of something as being a known mathematical structure, it is easy to pull answers relative to that something.  
