We can begin by trying to define what actually constitutes physicality. By and large, what we mean when we say that something is physical is that it exists a) in time and space or b) in a mind. To discover the immaterial, therefore, is simply a matter of demonstrating the existence of that which is nether a nor b. To explore this I am drawing from the work of my favorite theological philosopher, Edward Feser in his superb book, The Last Superstition. Here are some examples:
- The "one over many" argument. While there may be very many examples of triangles such as scalene, obtuse and isosceles and many shades of blue such as cobalt, turquoise and ultramarine, "triangularity" and "blueness" are not reducible to any particular triangle or blue thing. Indeed, even if there were no physical examples of any triangle or anything blue, they would still be truths that could come to be exemplified in the future. They also can, and many times are, exemplified even when no human mind is aware of them. Hence, triangularity, blueness and many other "universals" are neither material nor dependent on a human mind for their existence.
- The argument from abstract objects. Geometry deals with perfect lines, angles, circles, etc and discovers objective facts about them. As the facts are objective, we haven't invented them and they could not be altered (like material things). Clearly then, they do not depend of our minds. Also, no physical objects have the perfection that geometrical ones do so clearly they do not depend on the material world.
- The argument from mathematics. Like other universals, math is necessary and unchangeable - exactly the opposite of material things. 2+7=9 was true before there was a physical universe or any mind to apprehend it and would be true even after both were long gone. Another interesting twist with this one is though there is an infinite series of numbers, there is only a finite amount of physical things and mental events. Therefore, the series of numbers can't be equated with the physical or the mental. Take 10 minutes to listen to mathematician David Berlinski discussing this here.
- The argument from the nature of propositions. A proposition like "Kurt Cobain is a member of the 27 Club" would remain true even if the entire world and all the minds in it suddenly went out of existence. Interestingly, even if no mind or material world had ever existed, the proposition "there is neither a material world nor any human mind" would still be true - proving that it is neither material nor mental.
- The argument from science. Science is the business of discovering objective mind-independent facts (though that often is not really the case). As such, to accept the results of science is to accept the notion that there are such things as mind-independent universals. Here's an interesting video of MIT physicist Gerald Schroeder discussing the implications of the laws of nature as universals.
- The argument from words and concepts. How is it that I understand what you mean when you say "puppy?" Clearly, there is a universal understanding of a word that is shared over and above our various utterances of it. If one were to argue that there is no standard conception of what various words mean then we would run into some serious problems, such as the fact that it would render all communication impossible since we would never be truly using the same words - even, perhaps, when speaking to ourselves! Would it be possible to suggest that I think about my own Pythagorean theorem and you think about yours (which is different from mine)? Obviously not. Rather, we are sharing one universal notion. One that is neither physical nor mental.
Granted, all of this is a tad heady and it should be acknowledged that there is a good deal of philosophical back and forth about this approach - which I think doesn't at all defeat the premise. I strongly suggest picking up The Last Superstition and giving it a careful read. Each chapter builds on the previous in a deep but accessible presentation that lays out what I consider to be as air tight a case as can be made for something.