I would like to draw a connection with Galois theory. This theory was created to solve the following problem: Given a polynomial, find a formula which solves it. Now, the fundamental theorem of algebra says that each polynomial of degree n has n roots. However, the formula must output a single number. So there is Buridan’s-Ass style problem. The crucial insight behind Galois theory is this: The Buridan’s Ass Problem is the only obstacle in solving a polynomial.
For example, a quadratic equation has two roots, and a solution to the quadratic equation must pick out one. As it happens, each number has two square roots, so the square root operator is similarly ambiguous. To solve the quadratic equation, you must arrange it so that picking a square root picks out a solution. This is why the solution to the quadratic equation has a single square root and no other roots.
My point is this: Solving the Buridan’s Ass Paradox is not easy. It is at least as hard as solving polynomials.
I don’t think the analogy holds water. You’re trying to make a point about breaking symmetries, but in the case of Buridan’s ass you can play a mixed strategy where you choose each option with probability 1⁄2. There’s no need to actually break the symmetry.
I’m not sure where you’re getting the analogy of Buridan’s Ass from “solvability of Galois groups.” Solvability in radicals and picking between 2 (or however many) “equally good” choices feels like stretching. I also don’t know where you’re getting “the formula must output a single number.” The solution to quadratics yields (at most) 2 numbers. The solution to cubics is big and nasty, yielding at most 3. And similarly for quartics, with at most 4. (I’ve only bothered with a few of these. You work in cases, the nastiest of which involves “reducing” the problem to a cubic.)
I see the analogy to “symmetry”, but that’s about it.
Now, solving the Buridan’s Ass paradox is rather easy. Common Law just thinks it’s cheating to point out that starving to death is less optimal than doing something other than starving to death. Or it’s a bias. Or something.
The crucial insight behind Galois theory is this: The Buridan’s Ass Problem is the only obstacle in solving a polynomial.
I could see this being true for polynomials with a closed-form solution. But for polynomials that have to be solved numerically, surely the hard part is just finding a root?
I’m using “only obstacle” in the same way mathematicians use the word “trivial”: It’s the only obstacle to a sufficiently sophisticated mathematician. Now the roots of a polynomial are symmetric algebraically, but not numerically, so Buridan’s Ass is not an obstacle for solving polynomials numerically (although it still shouldn’t be ignored; after all, Newton’s Algorithm starting from a real number never find a complex solution). The only thing that’s left is the “trivial” portion of the problem.
I would like to draw a connection with Galois theory. This theory was created to solve the following problem: Given a polynomial, find a formula which solves it. Now, the fundamental theorem of algebra says that each polynomial of degree n has n roots. However, the formula must output a single number. So there is Buridan’s-Ass style problem. The crucial insight behind Galois theory is this: The Buridan’s Ass Problem is the only obstacle in solving a polynomial.
For example, a quadratic equation has two roots, and a solution to the quadratic equation must pick out one. As it happens, each number has two square roots, so the square root operator is similarly ambiguous. To solve the quadratic equation, you must arrange it so that picking a square root picks out a solution. This is why the solution to the quadratic equation has a single square root and no other roots.
My point is this: Solving the Buridan’s Ass Paradox is not easy. It is at least as hard as solving polynomials.
I don’t think the analogy holds water. You’re trying to make a point about breaking symmetries, but in the case of Buridan’s ass you can play a mixed strategy where you choose each option with probability 1⁄2. There’s no need to actually break the symmetry.
I’m not sure where you’re getting the analogy of Buridan’s Ass from “solvability of Galois groups.” Solvability in radicals and picking between 2 (or however many) “equally good” choices feels like stretching. I also don’t know where you’re getting “the formula must output a single number.” The solution to quadratics yields (at most) 2 numbers. The solution to cubics is big and nasty, yielding at most 3. And similarly for quartics, with at most 4. (I’ve only bothered with a few of these. You work in cases, the nastiest of which involves “reducing” the problem to a cubic.)
I see the analogy to “symmetry”, but that’s about it.
Now, solving the Buridan’s Ass paradox is rather easy. Common Law just thinks it’s cheating to point out that starving to death is less optimal than doing something other than starving to death. Or it’s a bias. Or something.
I could see this being true for polynomials with a closed-form solution. But for polynomials that have to be solved numerically, surely the hard part is just finding a root?
I’m using “only obstacle” in the same way mathematicians use the word “trivial”: It’s the only obstacle to a sufficiently sophisticated mathematician. Now the roots of a polynomial are symmetric algebraically, but not numerically, so Buridan’s Ass is not an obstacle for solving polynomials numerically (although it still shouldn’t be ignored; after all, Newton’s Algorithm starting from a real number never find a complex solution). The only thing that’s left is the “trivial” portion of the problem.