Buddhism is not exactly a religion or faith. There is no requirement that a Buddhist believe or have faith in anything, as there is in most religions. Through various practices such as meditation, we can train our minds to concentrate single-pointedly on profound questions about the true nature of reality; of how the universe actually works. As in science or mathematics, the “proof” of the Buddhist concept of how the universe works is gained through rigorous logic and experience of the clarity of an un-deluded mind. The Buddha encouraged his students to not accept anything he said on faith, but to examine and test what he told them themselves. Most Buddhist practitioners will at some point in their practice experience “realizations”, which are leaps in understanding, allowing the person to comprehend a little more of the true nature of reality. These realizations serve to reaffirm what has been taught.
Some experienced practitioners have been known to perform feats that do not seem logically possible in our day-to-day experience of reality. This is because with the power of their minds they have transcended day-to-day reality, into a more clear understanding of the true nature of reality. These feats don’t really serve a practicing Buddhist as “concrete proof”, however, since it is understood that our day-to-day minds are deluded. Fantastic physical feats serve more to impress lay people.