In this paper, we will develop a significantly more general notion of classical Ramsey numbers (extending most other graph-theoretic generalizations) and make some preliminary characterizations of these new Ramsey numbers using simple algebraic tools. Throughout, we make a case arguing that, while our access to specific values of Ramsey numbers (or, in general, precise numerical solutions to Ramsey-theoretic problems) may be limited, the interplay between and overall structure of Ramseyian objects is likely tractable. To support the relevancy of this perspective, we conclude by demonstrating that the Green-Tao Theorem, the Twin Prime conjecture, Zhang's bounded prime gap theorem, and Polignac's conjecture can be viewed as statements about Ramsey numbers.