You Could Have Discovered Quantum Mechanics

Quantum mechanics is what you would inevitably come up with if you started from probability theory, and then said, let’s try to generalize it so that the numbers we used to call “probabilities” can be negative numbers. As such, the theory could have been invented by mathematicians in the nineteenth century without any input from experiment. It wasn’t, but it could have been… And yet, with all the structures mathematicians studied, none of them came up with quantum mechanics until experiment forced it on them. And that’s a perfect illustration of why experiments are relevant in the first place! More often than not, the only reason we need experiments is that we’re not smart enough.

—Quantum Computing Since Democritus

The chapter itself goes into more details. You should buy a copy. The author blogs.