"Up to now, my examples of missed opportunities have been mathematical discoveries which actually occurred, although they could have occurred a long time earlier. In such cases one can be sure that an opportunity existed, but it existed only in the past. I now come to the more difficult task of identifying missed opportunities that are still open. Here one can no longer be sure that the opportunity is real, but if it is real then it has the virtue of existing in the present. The past opportunities which I discussed have one important feature in common. In every case there was an empirical finding that two disparate or incompatible mathematical concepts were juxtaposed in the description of a single situation. Taking the four examples in turn, the pairs of disparate concepts were respectively: modular functions and Lie algebras, field equations and particle dynamics, Lorentz invariance and Galilean invariance, quaternion algebra and Grassmann algebra. In each case the opportunity offered to the pure mathematician was to create a wider conceptual framework within which the pair of disparate elements would find a harmonious coexistence. I take this to be my methodological principle in looking for opportunities that are still open. I look for situations in which the juxtaposition of a pair of incompatible concepts is acknowledged but unexplained." - from Missed Opportunities by Freeman Dyson.
Dyson made a name for himself by showing the compatibility of the Schwinger, Feynman, and Tomonaga approaches to QED. He presents two unresolved juxtapositions of incompatible mathematical elements still under intense investigation:
A) General relativity and quantum mechanics
B) Feynman's sum over histories and existing theories of normed vector spaces.
Can you pose other Dyson pairs of incompatible mathematical structures (perhaps not as grand as Dyson's) that might present some interesting opportunities for integration under a yet-to-be-discovered (or invented) overarching theory? That is, can you extend Dyson's list?