1+1 = banana ~ What is finite?
1+1 = banana ~ What is finite?
Heh… if 1+1 = banana, then we’ve just redefined “+”.
Definitions matter. 🙂
Finite means “has bounds.” More concretely:
- Everyday sense: Something you could, at least in principle, finish measuring, counting, or doing. (A book with 312 pages, a task list with 7 items, a life with a start and an end.)
- Mathematics: A set with a count n for some natural number (e.g., {a,b,c} has n=3). Quantities are finite if they’re not infinite or unbounded. Finite sequences have a last element; finite graphs have a limited number of nodes/edges; a decimal is “finite” if it terminates (like 0.125).
- Computer science: Things with a bounded description or behavior. E.g., a finite-state machine has only finitely many states; an algorithm “finishes in finite time” if it halts after some fixed number of steps.
- Physics/real world: Finite resources, finite speed, finite size: anything constrained by limits.
Non-examples: The natural numbers {1,2,3,…}, π’s digits, a loop that never halts.
Short version: finite = countable, completable, or bounded
… pick your flavor.
IN QUANTUM LOGIC
In quantum logic, “finite” takes on a subtler flavor than in classical set theory. Let me unpack it:
1. Quantum Logic Basics
- Quantum logic is the logical framework proposed to capture the structure of propositions about quantum systems.
- Instead of Boolean logic (true/false, with distributivity), we use an orthomodular lattice of closed subspaces of a Hilbert space.
- Propositions like “the electron’s spin is up along z” correspond to subspaces, not simple truth values.
2. Finite in This Context
- A finite quantum system → modeled on a finite-dimensional Hilbert space (like spin systems with 2, 3, … dimensions).
- “Finite” means there are only finitely many mutually orthogonal states/propositions possible in the system.
- A finite orthomodular lattice → the logical structure has a finite number of elements (projections, subspaces).
For example:
- A qubit lives in a 2-dimensional Hilbert space → finite-dimensional. Its quantum logic is built from the lattice of its subspaces (0, span{|0⟩}, span{|1⟩}, the whole space). That’s finite.
- But a photon’s position observable (continuous spectrum) → infinite-dimensional Hilbert space → not finite.
3. Contrast With Classical Finite
- In classical logic, “finite” often means “limited number of elements in a set.”
- In quantum logic, “finite” usually points to finite-dimensionality of Hilbert space or finite orthogonality relations between propositions.
So, if we put it poetically:
In classical logic, finite means you can count apples in your basket.
In quantum logic, finite means you can count the dimensions of possibility itself.
