This is a mindset, a philosophy, a worldview, that claims that the natural world is all that exists (or can be proven to exist) and may be referred to as naturalism, materialism, scientism, or physicalism. Those all have subtle differences, but essentially all hold the same idea: the supernatural cannot be scientifically proven thus does not exist.
The term "supernatural" is usually used to mean things like, God, angels, demons, ghosts, spirits, miracles, etc. However, the word simply means beyond or above (super) the realm of nature (natural). Anything that is immaterial, not made of matter or energy, can be said to be supernatural. When I grasped this idea, it just opened up doors of understanding for me.
Supernatural is not limited to the fantastical claims of religion and spirituality. The term also encompasses things like logic, mathematics, and induction (assuming that the future will be like the past).
I have always had a hard time discussing, debating, arguing, etc. with naturalists and materialists. For someone to say, "There is no such thing as the supernatural," and demand I give them "evidence" is like standing in the desert at noon and demanding evidence that the sun exists.
Once it finally clicked and I realized that "supernatural" is not spiritual but anything that is not material or physical, then I realized what was going on.
We live in a culture that has become infatuated with science. Science has given us a great many wonderful things. As a way of thinking, we have elevated the scientific method to a place of high honor. Those who practice this method are revered, and the conclusions they reach are sacred.
This way of thinking has so permeated our culture that we don't even realize what is going on when the naturalist asks for "evidence." To them ONLY physical, material, scientific evidence counts. But we know that there is more to life than what our 5 physical senses take in.
How do you account for mathematics in a naturalistic world? What molecule is the number 4 made of?
How do you get a foundation for the Laws of Logic in a naturalistic world? What experiment was done to prove that A = A?
For that matter, what scientific experiment was performed to determine the nature and proper methodology of scientific experiments?
If we are just meat machines guided by the purposeless chemical reactions in our brains, where do you get ethics or morality? Or how do you even determine the need for something like reason or rationality?
How can you be sure the universe is orderly and won't suddenly fall into chaos?
Why do you assume that tomorrow will go just like today?
Why bother arguing or trying to discover any truth at all? If the chemicals in your brain randomly create thoughts about God and the chemicals in my brain randomly create thoughts about atheism, which random chemical reaction to we appeal to for discernment to know which is right?
For that matter, in a materialistic world, what is the basis for even needing to determine what is right?
We have science, yes, but we also have intuition and experience, and feelings and logic, the spiritual, and our volitional consciousness. However, non of this can be explained by blind, unguided material processes.
Here is what the naturalist is doing. They build a nice, cozy little box where everything on the inside is blue. Then they invite you to come into their little box, close the door, and then only using the things they have placed inside the box prove that there are colors other than blue.
However, they say they want proof (material, natural, scientific) that the immaterial exists. Since no such proof can be presented, then the immaterial cannot exist. But that's crazy talk. You can't give evidence of immaterial realities by using material. That's like demanding proof of music using only visual evidence.
The reality is that we are not only physical beings but spiritual beings as well. We are not bodies with a soul, we are souls with a body. We posses a volitional consciousness, that is, we have an awareness and will separate from our physical bodies.
The chemistry in our brains may have some effect on our moods and thoughts, but so too can our thoughts shape the chemical make up of our brain. There may be cases of chemical imbalances that require medical assistance to correct, but there is also a greater frequency of times that we can literally WILL ourselves to refocus, feel better, or think differently. This cannot be simple chemical processes in our brain. This is a separate, immaterial will being imposed upon the brain.
Here are a couple of examples:
My daughter has a very active imagination. She is at that age where she is having trouble determining between dreams, fantasies and memories. As we raise her and teach her, we do not manipulate the chemistry of her brain. We train her mind to think so that she will be able to know the differences. In a purely naturalistic world where brain chemistry is all there is, there is no mechanism to objectively determine the difference between dream and memory.
Several years ago, I spent a few months caring for my great grandfather who had Alzheimer's. During his confused moments, as his brain chemistry was malfunctioning, you could visibly see the frustration as his mind, his consciousness was aware that something was wrong. He was not just a slave to his biology. Like when a limb falls asleep and won't behave like your mind is wanting it to, his brain was not performing like he knew it should. If we are nothing more than material processes in the brain, a malfunctioning chemical process doesn't know it's malfunctioning.
These are two small example in a lifetime of day-after-day experienced that clearly demonstrate that we are more than animated meat sacks being guided by chemical reactions in our head.
For the naturalist to exclude an entire realm of reality and experience from is an act of faith. Despite the evidence to the contrary, the naturalist decides to trust only in what his 5 senses tell him. He places science as the ultimate guide to truth and rejects all else that does not fit into this little box.
Faith is the step of trust that completes the journey, bridges the gap, from evidence to belief. (more on that here: Faith, One Step Further)
Christian faith encompasses the entirety of human knowledge to support the viability of trusting in God. The gap between knowledge and belief is much smaller.
Naturalism denies all sources of knowledge but physical science. By adopting a self-limiting pool of evidence to draw from in the search for truth, the naturalist has a larger gap between his evidence and conclusion than does the Christ.
Hence it takes more faith to be a naturalist than a Christian.