The scientific method basically is:
- someone gets an idea - called a theory
- this theory is tested by experiment
- if the result is positive, others repeat the experiment to make sure
- if all agree, the theory is accepted as fact
Now, there is a lot of details between these points! But, that is essentially the flow of things.
Let's take math. We define "1" as the symbol to represent a single item. We define "2" to represent two of them. Someone suggests that adding 1 and 1 (1 + 1) makes 2. We try it out, and it works! So, we can prove that 1+1=2. That is a simple case, but mathematics works by going from a very basic definition, and playing with the numbers to see what happens. Over time, formulas and proofs are built up so that advanced mathematics can do all sorts of neat stuff.
Logic is trickier. But, people studying philosophy often begin with logic, which turns out to be a branch of mathematics. There are similar ways of proving things, like if a is to b, and b is to c, than a is to c. This can be proved using math, and used in logic problems.
Religion is a bit different. Philosophy is a tool, in fact the RC's at the seminary I went to had to have a Philosophy degree before starting their Theology one, and the Anglicans there were required to have a minimum of 3 courses in Philosophy. Reason plays a role in Theology, but so does faith, which is definitely a more nebulous concept. You say you can feel God. I can too, so I am happy with your statement, but our feelings are extremely hard to prove! Still, we can use them, and describe them, and it turns out those of us with these "feelings" have a great deal in common. This acts as evidence for what is going on, but not something that can easily be tested, like a scientific experiment.
Science has it's tools, and Theology/religion does as well. It is just tricky to use one set of tools in the other arena.
University studies in both Science, and Theology.