Is quantum logic producing interesting/different mathematics?

Question

Is quantum logic producing interesting/different mathematics?

Is it different from the intuitionist approach to mathematics? How?

Answer 1

There is an approach to quantum logic where you get topoi with quantum logic. An elementary topos is sometimes regarded as a "place" where you can do mathematics, but where classical logic doesn't necessarily apply. Thus you get a different sort of mathematics.

I know very little about these quantum topoi, so I cannot detail in what way their mathematics differ from the classical one. But I think the two articles referenced in the below PlanetMath articles may (or may not - I haven't read them) answer your question.

http://planetmath.org/encyclopedia/QuantumLogicsTopoi2.html

http://planetmath.org/encyclopedia/QuantumStateSpace.html

Answer 2

Quantum logic differs from classical logic because it lacks the distributive property e.g. A * (B + C) = AB + AC. The justification for its removal is based on what is called the "no cloning theorem" in quantum information and computation circles. Note: this does not in any way invalidate the dist. property for purely formal systems, only that it cannot be carried over into the formal descriptions of physical systems that obey the laws of quantum mechanics.

Intuitionistic logic differs from classical logic because it lacks the double negative e.g. ~(~A) = A. The reason for its removal is based on a philosophy of LEJ Brouwer that only constructable objects should be admitted in mathematics, meaning that a proof that denied an objects impossibility was a necessary but insufficient condition for admitting it. That may or may not be true; it depends on where you stand concerning the philosophy of mathematics.

Of course, both of these restrictions to classical logic may be taken together i.e. constructable quantum logics. These are the kind that a quantum topos theory would describe.