Toy models in theoretical physics are invented to make simpler the modelling of complex physical systems while preserving at least a few key features of the originals. Sometimes toy models get a life of their own and have the chance of emerging as paradigms. Such an upgraded role, on the one hand, makes these models likely to be considered for validation through (possibly new) experimental tests. On the other, the role played by mathematical proof - evoked in Wigner's "unreasonable effectiveness of Mathematics in the Natural Science"-could be so enhanced as to become in a sense more compelling than experiments. "Theoretical Mathematics', a new synthesis of mathematics and theoretical physics proposed by Jaffe and Quinn in the 1990's [12], looks at pure mathematics as a sort of experimental testing ground for certain physical theories (and in particular for associated toy models).