saturationalism

Saturationalism is a concept primarily found in the philosophy of mathematics and logic, though it can be extended to other areas of inquiry. It refers to the idea that mathematical or logical systems possess a richness and depth that can be fully explored and understood, even if they are infinite. This means that every true statement about the system, regardless of its complexity or how far it is from the foundational axioms, is in some sense already contained within the system itself.

The term suggests that there is a limit to the novelty or surprisingness that can be discovered within a given system. Once all the fundamental principles and their direct consequences are understood, further exploration will only reveal variations or restatements of what is already implicitly present. It implies a form of completeness, not necessarily in the Gödelian sense of provability, but in the sense of inherent meaning and discoverable truth.

Proponents of saturationalism might argue that mathematicians do not invent new mathematical truths as much as they discover truths that were already latent within established systems. This perspective contrasts with views that emphasize the creative and constructive aspects of mathematical thought, where new axioms or definitions might be seen as genuinely bringing new mathematical entities or truths into existence. The debate around saturationalism touches on the nature of mathematical reality, the limits of human understanding, and the relationship between finite minds and potentially infinite mathematical structures.