Double proton transfer in the formic acid dimer has been investigated with Car-Parrinello ab initio molecular dynamics calculations. The electronic structure of the dimer has been obtained using gradient-corrected density functional theory based on the B-LYP (Becke exchange [Phys. Rev. A 38, 3098 (1988)] and Lee-Yang-Parr correlation [Phys. Rev. B 37, 785 (1988)] functional. The optimized equilibrium and saddle-point geometries, obtained by simulated annealing, are in good agreement with previous ab initio quantum chemical predictions and experiment. Thermal and quantum fluctuations of nuclei along the double proton transfer reaction path have also been investigated at T=300 K. Thermal fluctuations give a broad distribution of nuclei around the minimum energy path on the potential energy surface. Quantum fluctuations, investigated using ab initio path integral molecular dynamics, make the distribution even broader around the equilibrium structure, and cause the distribution to deviate appreciably from the minimum energy path on approaching the reaction barrier. In particular, the system passes through higher energy regions than the geometrical saddle point by tunneling; an observation which is consistent with the conventional understanding of heavy-light-heavy mass combination reactions. While there is asynchronous movement of the two protons around the equilibrium structure, synchronous movement becomes relevant on approaching the reaction barrier.