Making wild tigers cower in fear with nothing more than an intense gaze
Walking barefoot on the naked blade of a samurai sword
Carrying boiling hot cauldrons with the bare skin of the forearm
Being able to perceive the opponent's intention of striking before they even move
These are only a very small sampling of some of the claims made by traditional martial arts systems that their practitioners have accomplished, or claim to be able to train you to do. But how can these things be possible? How can this chi mumbo jumbo even have the slightest shred of credibility if it doesn't match up with the scientific method? How can I measure chi, intention, kiais, or anything else? Again we come back to apples and oranges. All of the claims made by Eastern martial arts are based in their world view, not ours. To them, they are not dealing with electrons and protons, but yin and yang. Rather than gravitation and magnetism, they concern themselves with concentration and non-attachment.
First we can address the primary issue most people have with taking the "superhuman powers" martial arts can give you at face value: why don't we see people doing these things on a regular basis? Why aren't people using these abilities to their advantage or to gain money or impress others? The answer is quite simple; only the most advanced and dedicated students can ever hope to manifest these skills, and after investing so much time into their art, two very specific things occur: first and foremost, they have become philanthropic, compassionate, others-centered people due to their training. So any impulse they have to use these capabilities for selfish gain is greatly diminished. Secondly, they now have a vested interest in their art form, and want to make sure it is disseminated to responsible students! Imagine training a skill for 40 years to utter perfection, and then just giving it all away in a 10-minute Youtube clip. These individuals intentionally stay out of the public eye, lest their information and techniques fall into the wrong hands.
And yet how convenient that this answer blankets any need for proof. As it has been said, "Except ye see signs and wonders, ye will not believe." The arts are there simply to better you as a person, not to give you exceptional abilities. If you approach a school demanding proof and evidence, what can the teacher give you? You already have your opinions and viewpoints, and are simply looking for anything to support them. If you cannot empty your cup, the teacher can give you nothing. Ultimately, a dedication to these types of arts come down to one thing: a leap of faith. (Of course, being poked and prodded by your sensei or sifu along the way)!