There aren't any contradictions in either classical or quantum physics. There is incompleteness of description in the case of classical physics, for instance in the case of gravity-that describes, inevitably, spacetime singularities, whose resolution requires a theory of quantum gravity. Such a theory won't contradict what is already known, it will fill in the blanks of what isn't, yet, known.

This incompleteness is due to the fact that classical gravity is described by a gauge theory with a noncompact gauge group. What is the structure whose classical limit has this property remains to be understood.

Another source of incompleteness-but not of contradiction-is the indication that there exists matter, that carries charges that are not the charges of the known forces (weak, electromagnetic or strong), beyond that of the gravitational force. The particle content of this form of matter is, currently, unknown.

There is, also, incompleteness-but no contradiction-within quantum physics, which does provide a certain completion, though not fully, beyond classical physics.

The discovery of the Higgs boson showed the first example of a particle, that contributes to known matter without being a fermion and with having zero spin. This focuses attention on an apparent dichotomy: Apart from the Higgs boson, all particles that make up matter carry half-integer spin and all particles that mediate interactions carry integer spin. Why we haven't discovered particles of integer spin, whose interactions are mediated by particles of half-integer spin remains to be understood.

Furthermore, while thermal fluctuations can be described in terms of particles, the objects (whether particles or something else) that can describe quantum fluctuations remain to be discovered. The only case where the objects that resolve quantum fluctuations have been found-at least within a consistent mathematical framework-is for the case of extremal black holes, where they are described by the strings and branes of string theory.

Finally, what is also not understood is why we observe matter quite readily but antimatter with considerable difficulty. The Standard Theory of particle physics provides part of the answer, but not all of the answer.