In mathematics, it is common to start from some foundational assumptions, and see what you can prove. The assumptions are called axioms, and the notable generalities you can prove are called theorems.
The classic example is the Pythagorean Theorem. You learned this theorem in high school because it's extremely useful in everyday life. You may even have proved it in class. If you start from the axioms of (Euclidean) geometry, you can prove that the square of the length of the long side of a right triangle is equal to the sum of the squares of the lengths of the other two sides.
Yes, for a lot of people, mathematical proofs put them right to sleep. However, I think they understand the point of the exercise. There are important truths that you can't easily work out in your head. It's important to check the steps, even if you're not the one doing it.
There are some ideas in mathematics we think might be true, but which we have yet to be able to prove. One of the most famous is the Goldbach Conjecture:
Every even integer greater than 2 is a Goldbach number, a number that can be expressed as the sum of two primes.
Boffins have been trying to crack this nut since Goldbach came up with it in 1742. No one has yet succeeded, though several complex proofs have been proposed.
Imagine a mathematician, well call him Oiler, who claims that the Goldbach Conjecture is true. Oiler claims he has thought his way through the proof in his head, and he's confident the Conjecture is true.
Naturally, we are thrilled, and we ask Oiler to write down a detailed proof to support his conclusion. Oiler refuses. Oiler tells us that the Goldbach Theorem is special. It's a special class of theorem that's true, but which you can only see to be true if you DO NOT write down the proof and check your work carefully. After we pick our jaws up off the floor, we would quiz Mr. Oiler about his claim just to make sure we understand him correctly.
We would then proceed to cover Oiler in tar, and feather him. We would ridicule Oiler savagely.
Yes, this is an analogy for superstitious thinking. What is superstitious thinking?
Superstitious thinking is counting the hits but not the misses. It's about looking for ways to confirm your theories, but never looking for ways to prove them wrong. It's a matter of ignoring the possibility that an effect would exist even if the theory was false. It's a matter of succumbing to human cognitive biases.
Superstitious thinking can be overcome by using statistics, random sampling, blind testing, and an array of other techniques for checking our work. Basically, it's a matter of using science to suppress human bias and get to the truth.
There are a lot of people in the world who believe in the paranormal, and when you ask them to prove their case, their favorite answer is that God (or the aliens, or the fairies, or whatever) won't be tested. They tell us you can only see them when you don't use science to check your work. Now, how is this any different from Mr. Oiler in the above?
I'm not seeing a significant difference.