This is one question that I can never get the answer of, because I am too young at this moment. My question is that can a common person like me, not a genius, just a normal person, build mathematics on my own? I mean to say that if I don't refer to any books or resources, can I prove some results on my own?
Of course I am not saying whether I will be able to build all of it, but some of it. I am asking this because there are many things that I have studied in math, which are very difficult to prove. I wonder what kind of brains Newton or Leibniz had to develop everything they did.
I am sorry if this question is stupid or inappropriate.
Yes, and there are a number of levels, but it takes not only time but effort and determination to go from one to the next:
You can start here, working through puzzle books and olympiad questions to get a firm grip on logic, which is the foundation of all mathematics. Without proper grasp of logical deduction, it is difficult to have anything more than a shallow understanding of any area of mathematics. You should learn about truth tables, boolean algebra, first-order logic, and natural deduction. These should put you in good shape.
Here you take any rigorous math textbook, such as Spivak's Calculus, and then systematically go through it, stopping at every theorem and attempting to prove it yourself without reading it yet. If you try enough and feel like it is out of your reach, read the first one or two lines and see if it gives you any new idea. If so, try to continue by yourself. Repeat until you have proven the theorem, with or without some help from the book.
After you can consistently work through systematic introductory expositions of mathematics, you can try two things. One is to try to make new conjectures of your own based on what you know at each point, and prove or disprove them. Often these conjectures will have already been thought of by others, and you might even find it in the next part of the book that you have not come to yet. When this happens, you essentially recreated parts of the mathematics that the book describes. The other is to continue the same way of working through books except that you go for books that seem to be at a right pace for you to do it.
After you've worked through enough books and published articles in some particular topic, you should be well on your way to craft and tackle questions of your own that are not immediately answered by the theorems and techniques that you have learnt, in which case you might well be building mathematical objects and proving properties about them that other people might not have thought of. You would have to search around to know what other people have done, and then perhaps you can build on all that, whether in some similar direction or at a tangent.