So I watched Fight Club and now I have a social experiment. I am going to go in front of my neighborhood's local market(across from a large high school) while students are having their lunch and just start randomly pummeling punches after punches at myself and beat myself up really bad in front of everyone and see how they react. My only concern is if this is illegal what I'm doing, or if I will need permission from police and the school to do this first?
People in the nearby area would likely contact emergency services. As a matter of policy, emergency services would conclude that you are demonstrating behavior consistent with a person who is severely mentally ill. Because you have become a hazard to yourself, they would pursue emergency psychiatric care. This would involve being taken to a hospital or other facility, where you would undergo screening and a battery of tests. This process is usually no shorter than 72 hours, but can be significantly longer if medical officials believe that you a persistent hazard to yourself.
...in front of everyone and see how they react.
In horror. They just watched a person perform severe self-harm in a public setting. They would believe that you are mentally ill.
My only concern is if this is illegal...
Self-harm is not strictly illegal in all jurisdictions, but you would be taken into protective custody by police in most, if not all, jurisdictions.
I don't fully understand the philosophical nature of your question, since we have lots of instances of this sort of behavior being responded to in a fairly consistent fashion.
are you asking if you could actually be able to recruit Men like in the movie? , if yes , the result will be more likely to fail. without knowing what exactly influence/motivate those Men to start something like the fight club is impossible. more the place and time you picked is far from what you should be using , i think. in doing experiments you should be more careful on what are the variables you are considering . ex. in the movie it was night time, not too many people(Men) around outside the bar. the bar as i remembered was not that empty so you might wanna consider it was a weekend (just for assumptions). i would pick that time because the movie tackled topic against industrialism/commercialism/consumers (most likely against the day job and against routines) weekends would be a good choice to do so because people(Men) should be tired over the week and would want escapes. My point being is your first concern is not if its illegal to punch your own face in public but to start thinking and planning better.