I was musing about the foundations of mathematics the other day, when it occurred to me that you could make a pretty good case that the key foundational idea of mathematics is that of Horn clauses (also known as universal Horn sentences). Horn clauses, despite begin obscure outside certain areas, are ubiquitous. Many (perhaps most?) basic mathematical objects can be described by Horn clauses. Fundamental category theoretic notions have Horn clause interpretations. Even first-order logic, which contains Horn clauses as a special case, can be viewed as having inference rules in the form of Horn clauses applied at the level of proofs.
I thought I’d spend a couple of posts describing Horn clauses, and laying out the case for their ubiquity.
Next post here.