Are the fundamental mathematical assumptions on which all mathematical proofs rest unproven or unprovable?
Here’s a stab at answering this question for non-mathematicians (warning — oversimplification ahead): Mathematicians and philosophers are still arguing about the logical basis of what mathematics is. The basic process of mathematics is that you start by assuming a few basic fundamental truths. These are called axioms. These axioms often just amount to a definition of the concepts and symbols you are using. Then you use logic to derive new truths (theorems) from your axioms. Some mathematics consists of deriving new theorems from axioms that are already widely accepted. Other mathematics consists of inventing new systems of axioms and seeing what kinds of theorems you can prove with them. In some ways, this means that all mathematical theorems are just very elaborate tautologies. The statement “1+1=2” depends on your definition of what “1”, “+”, and “=” mean. There are some things that make a system of axioms good. You want to assume as little as possible, so you don’t want to be able
