Edit: These ideas are also discussed here and here (thanks to Qiaochu Yuan: I found out about those links by him linking back to this post). Although topology is usually motivated as a study of spatial structures, you can interpret topological spaces as being a particular type of logic, and give a purely logical, non-spatial […]
Let be the set of propositions considered by some rational logician (call her Sue). Further, suppose that is closed under the propositional connectives , , . Here are two related but different preorders on : if logically entails . if Sue considers at least as likely to be true as is. Let be the equivalence […]
The normal square root function can be considered to be multi-valued. Let’s momentarily accept the heresy of saying that the square root of a negative number is , so that our function will be total. How can we represent the situation of this branching “function” topologically?
By “voting”, I mean the following general problem: Suppose there are candidates and voters. Each voter produces a total ordering of all candidates. A voting procedure is a function which takes as input all orderings, and produces an output ranking of all candidates. Arrow’s impossibility theorem states that there is really no satisfactory voting procedure […]
In the book Good and Real, author Gary Drescher, who received his PhD from MIT’s AI lab, defends the view that determinism is a consistent and coherent view of the world. In doing so, he enters many different arenas: ethics, decision theory, and physics. In his chapter on quantum mechanics, he defends the “many-worlds” interpretation […]
A function from to is called even if for all , . We might call it even about the point if, for all , . Conversely, we can call a function strongly non-even if for all , , . Finding strongly non-even functions is easy, as any injective function provides a trivial example. We can […]
You may have heard about the Tarski-Seidenberg theorem, which says that the first-order theory of the reals is decidable, that the first-order theory of the complex numbers is similarly decidable, or that the first order theory of the integers without multiplication is decidable. In the course of John Harrison‘s logic textbook Handbook of Practical Logic […]