I think, Elliott Sober gives a nice example (cf. Elliott Sober `Quine's Two Dogmas', Proceedings of the Aristotelian Society, Supplementary Volumes, Vol. 74 (2000), pp. 268-269):
For logic and mathematics to be tested empirically, one logical or mathematical statement would have to be pitted against an alternative and a framework of shared background assumptions would have to be supplied that permits the two statements to make different predictions about observations. It is instructive to examine a case in the history of mathematics in which this actually happened. Plateau was a 19th century French mathematician who wanted to figure out what the surface of least area is that fills various closed curves. A simple example of the type of problem that Plateau had in mind is a curve that has the shape of an ellipse. An elliptical curve can be filled by the ellipse it contains; any other surface that fills the curve will bulge into a third dimension and so must have more surface area. Although the answer to Plateau's question is obvious for this example, the answers aren't at all obvious for more complicated three-dimensional curves. What Plateau did was to dip wires bent in different shapes into soapy water and observe the resulting soap bubbles that adhered to the frames when they were removed from the water (Courant, R. and Robbins, H., 1941, What is Mathematics?, New York: Oxford University Press). Given the physical assumption that soap bubbles take on the surface of least area in this experiment, different mathematical hypotheses ('the surface of least area for curve C is s1' versus 'the surface of least area for curve C is s2', for example) make different predictions. This example is interesting in the history of mathematics precisely because it is so atypical. No such test of '2 + 3 = 5' against alternative arithmetic hypotheses has ever been carried out, nor is it remotely clear what this would be like.