The quantum physics textbooks I read were happy to define linear operator-ness in great gory detail, but they never actually came out and said, “This is not something physically happening to the wavefunction. We are just using this math trick to extract an average value.”
I think is is a common problem for many mathematical conventions in physics.
The same thing happened be me in high school physics. I was confused by the torque vector, and I spent an entire year thinking that somehow rotation causes a force perpendicular to the plane of motion. I just could not visualize what the heck was going on.
Finally I realized the direction of the torque vector is an arbitrary convenience. My teacher and textbook both neglected to explain why it works like that.
The “why’s” are important!
Being an artist has nothing to do with the accuracy of this belief.
There are two problems here. First, irrational numbers are the ones that cannot be expressed as a fraction of integers. Transendentals are defined as numbers that are not algebraic. All transcendental numbers are irrational, but the converse does not hold.
Second, pi is defined as the ratio of circumference to diameter, true. This would only be a contradiction if both the circumference and diameter could be integers at the same time, which is impossible.
You are confused about what numbers actually are. Some classes of numbers are useful for certain tasks, but there is no sense in which one class is more ‘real’ than another. I recommend Mathematics, Queen & Servant of Science by Eric Temple Bell for a wonderful overview of mathematics. Chapter 2, “Mathematical Truth”, is relevent to this discussion. Also, see Godel, Escher, Bach, Chapter 11: “Meaning and Form in Mathematics”.