Can the "real world" provide evidence that mathematical knowledge is legitimate? I think its many peoples' intuition that the successful application of math to science and engineering (e.g., that we can use math to build bridges) shows that math is true.

The question is whether what we find in the physical world could tell us whether math is true. Let's consider two sorts of cases. One is what we might call mathematical laws -- 1+1=2 is a particularly simple example. An algebraic law like x 2 - y 2 = (x+y)(x-y) is another. The second sort of case includes things like Newton's law of gravitation -- F 12 = G(m 1 m 2 )/r 2 -- or some mathematical description of the characteristics of steel beams used for bridges. This may be closer to what you have in mind. Start with the first sort of case. Suppose we have two 1-liter beakers of water. We pour them together, measure the volume and find that it's two liters. Have we confirmed the mathematical claim that 1+1=2? If so, what do we make of the fact that if we put pure alcohol rather than water in one of those beakers, when we put the two together we get about 1.94 liters? Does that count against 1+1=2? It's pretty clear that neither experiment tells us anything about whether 1+1 equals 2;...